notebooks/data_preparation.ipynb

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  "metadata": {
    "colab": {
      "provenance": [],
      "collapsed_sections": [
        "MI-XcwChd7ku",
        "5kwucgEZigYa",
        "4vNxTg1aivD3",
        "sh_p9O748gZS",
        "d9edpW3r41K4",
        "U9Iq5FId5FkE"
      ],
      "toc_visible": true,
      "mount_file_id": "1mLBWb7ozd3fPSfH36tKKE1cxNeqIkBXK",
      "authorship_tag": "ABX9TyM84nL0Y+QjTnqg4Fdv6m/B"
    },
    "kernelspec": {
      "name": "python3",
      "display_name": "Python 3"
    },
    "language_info": {
      "name": "python"
    }
  },
  "cells": [
    {
      "cell_type": "markdown",
      "source": [
        "# Dataset Collection\n",
        "\n",
        "-\tThe data was collected from Google Cloud Storage (GCS) where it was available for free in buckets for bulk Access.\n",
        "-\tThe command line tool `gsutil` was used to access ArXive’s physics PDF buckets and downloaded into local machine.\n",
        "-\tThe size was about 7.19GB of 22.3K PDFs of different versions  -\tThe dataset was then uploaded into Google Drive to be easily accessed through Google Collab.  "
      ],
      "metadata": {
        "id": "MI-XcwChd7ku"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "from google.colab import drive\n",
        "drive.mount('/content/drive')"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "bYmRswToiIv5",
        "outputId": "2e123419-6770-4175-ee7f-da91ed0db28b"
      },
      "execution_count": null,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "Drive already mounted at /content/drive; to attempt to forcibly remount, call drive.mount(\"/content/drive\", force_remount=True).\n"
          ]
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "## Importing Libraries :"
      ],
      "metadata": {
        "id": "RG1m3_WF6tu-"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "!pip install pypdf\n",
        "!pip install PyMuPDF"
      ],
      "metadata": {
        "id": "xImFW7pA6vkA"
      },
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "code",
      "source": [
        "# PymuPDF library\n",
        "import pymupdf\n",
        "# PyPDF related libraries\n",
        "from pypdf import PdfReader\n",
        "from PyPDF2 import PdfReader\n",
        "# PDFminer.six related libraries\n",
        "from io import StringIO\n",
        "from pdfminer.converter import TextConverter\n",
        "from pdfminer.layout import LAParams\n",
        "from pdfminer.pdfdocument import PDFDocument\n",
        "from pdfminer.pdfinterp import PDFResourceManager, PDFPageInterpreter\n",
        "from pdfminer.pdfpage import PDFPage\n",
        "from pdfminer.pdfparser import PDFParser\n",
        "# Other libraries\n",
        "import os\n",
        "import nltk\n",
        "nltk.download('punkt')"
      ],
      "metadata": {
        "id": "W1TywdCM68zO"
      },
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "source": [
        "# PDF To Text\n",
        "Since the files we have are **native PDFs** (which means text is already digitally encoded ) there is **no need to apply any OCR** (Optical character recognition) techniques.\n",
        "\n",
        "- This means we will use **PyMuPDF** ,  **PyPDF2** or **PDFMiner.six**\n",
        "- We will test all of them on one PDF file and see the results\n",
        "- The evaluation is going to be done manually (human evalution)"
      ],
      "metadata": {
        "id": "Qih2ffqxvZWX"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "# creating sample pdf file path string\n",
        "pdf_file =  \"/content/drive/MyDrive/UH - Final Year Project/Dataset/pdf/9905/9905061v3.pdf\""
      ],
      "metadata": {
        "id": "cBwgXUwprIiJ"
      },
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "markdown",
      "source": [
        "## Testing **PyMuPDF**"
      ],
      "metadata": {
        "id": "5kwucgEZigYa"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "# testing pymupdf on sample pdf file\n",
        "doc = pymupdf.open(pdf_file)\n",
        "pymupdf_text = \"\\n\".join([page.get_text() for page in doc])"
      ],
      "metadata": {
        "id": "WRcRjYEBr6fk"
      },
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "code",
      "source": [
        "print(pymupdf_text)"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "xKXlidp-sJG0",
        "outputId": "2e808438-7a83-4e7e-cbc0-743d1f68e74d",
        "collapsed": true
      },
      "execution_count": null,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "arXiv:physics/9905061v3  [physics.plasm-ph]  8 Jun 1999\n",
            "DPNU-99-14\n",
            "Electron acceleration to ultrarelativistic energies in a collisionless\n",
            "oblique shock wave\n",
            "Naoki Bessho and Yukiharu Ohsawa\n",
            "Department of Physics, Nagoya University, Nagoya 464-8602, Japan\n",
            "(July 14, 2011)\n",
            "Abstract\n",
            "Electron motion in an oblique shock wave is studied by means of a one-\n",
            "dimensional, relativistic, electromagnetic, particle simulation code with full\n",
            "ion and electron dynamics. It is found that an oblique shock can produce\n",
            "electrons with ultra-relativistic energies; Lorentz factors with γ >\n",
            "∼100 have\n",
            "been observed in our simulations. The physical mechanisms for the reﬂection\n",
            "and acceleration are discussed, and the maximum energy is estimated.\n",
            "If\n",
            "the electron reﬂection occurs near the end of a large-amplitude pulse, those\n",
            "particles will then be trapped in the pulse and gain a great deal of energy.\n",
            "The theory predicts that the electron energies can become especially high at\n",
            "certain propagation angles. This is veriﬁed by the simulations.\n",
            "52.65.Cc, 52.35.Tc, 52.35.Mw, 98.70.Sa\n",
            "Typeset using REVT\n",
            "EX\n",
            "1\n",
            "\n",
            "I. INTRODUCTION\n",
            "Electron acceleration has been an important issue in plasma physics and astrophysics. In\n",
            "solar physics, for instance, the acceleration of electrons as well as of ions has received a great\n",
            "deal of attention; in solar ﬂares, gamma rays with energies of several tens of MeV, which are\n",
            "emitted by the bremsstrahlung from the high-energy electrons, are often observed [1], [2]. In\n",
            "astrophysics, much more energetic electrons have been discussed. From the observations of\n",
            "X-rays and gamma rays, it is now believed that electrons with energies up to ∼100 TeV are\n",
            "produced by a shell-type supernova remnant SN 1006 [3], [4]. It is also believed that in the\n",
            "Crab Nebula high-energy electrons accelerated up to ∼100 TeV are produced by the pulsar\n",
            "wind [5]. Theories based on the Fermi acceleration model have been proposed to account for\n",
            "these high-energy phenomena [6], [7]. As for laboratory plasmas, in an attempt to realize\n",
            "plasma-based accelerators, the electron acceleration in a relativistic space charge wave has\n",
            "been intensively studied by simulations and experiments [8].\n",
            "Ion acceleration in a magnetosonic wave has been studied by many authors [9]- [22].\n",
            "In a single-ion-species plasma, the large positive potential formed in the shock region can\n",
            "reﬂect some ions and give great energies to them [9]- [18]. In a multi-ion-species plasma\n",
            "such as space plasmas, some of the majority ions (hydrogen) can also gain energies by\n",
            "the same mechanism.\n",
            "Furthermore, the transverse electric ﬁeld in the shock wave can\n",
            "accelerate all particles of all kinds of (minority) heavy ions to nearly the same speed [19],\n",
            "[20]. In addition, in a turbulent plasma where particles can interact with many diﬀerent\n",
            "large-amplitude magnetosonic pulses, some energetic ions can be further accelerated by\n",
            "a diﬀerent mechanism [21]. These processes have been extensively studied by theory and\n",
            "particle simulation by many authors and have been applied to the production of high-energy\n",
            "particles in solar ﬂares and interplanetary shocks [9]- [20].\n",
            "Strong electron acceleration in shock waves, however, has not been shown by particle\n",
            "simulations. In this paper we will theoretically and numerically study electron motion in\n",
            "a magnetosonic shock wave propagating obliquely to a magnetic ﬁeld. We will show that\n",
            "an oblique shock can produce ultra-relativistic electrons. The preliminary result has been\n",
            "reported in Ref. [23].\n",
            "In Sec. II, we outline some basic properties of oblique shocks. In Sec. III, we theoretically\n",
            "discuss the possibility of electron reﬂection. If electrons are reﬂected near the end of the\n",
            "main pulse, then after the reﬂection they can have great energies in the main pulse.\n",
            "Several authors discussed stationary pulse solutions for the nonlinear magnetosonic waves\n",
            "[24]- [27]. However, if the magnetic ﬁeld is rather strong so that ωce/ωpe\n",
            ">\n",
            "∼1, where ωce\n",
            "is the electron cyclotron frequency and ωpe is the plasma frequency, or if the amplitude is\n",
            "quite large, then the pulse propagation is not perfectly stationary [13], [14]; small-amplitude\n",
            "ﬂuctuations are generated in and around the main pulse.\n",
            "Even though the potential is\n",
            "positive in the main pulse in a shock wave, the electron reﬂection studied in this paper can\n",
            "occur in such a non-stationary pulse.\n",
            "In Sec. IV, we estimate the maximum energy of a reﬂected electron and discuss its\n",
            "dependence on plasma parameters. In Sec. V, we further study the electron acceleration by\n",
            "using a one-dimensional (one space coordinate and three velocity components), relativistic,\n",
            "electromagnetic, particle simulation code with full ion and electron dynamics. It will be\n",
            "shown that some electrons can be reﬂected near the end of the main pulse and are trapped\n",
            "2\n",
            "\n",
            "in the shock region. In the simulations, an appreciable number of electrons have energies\n",
            "γ > 100, where γ is the Lorentz factor. Further, we will examine the dependence of γ on the\n",
            "propagation angle and shock speed. It is veriﬁed that, as the theory predicts, there are special\n",
            "angles and shock speeds at which the electron acceleration is especially strong. Our work is\n",
            "summarized in Sec. VI. In Appendix A, we describe relations among physical quantities in a\n",
            "magnetosonic wave. In Appendix B, we give a rough estimate for the magnitude of parallel\n",
            "electric ﬁeld E∥on the basis of a simple physical picture.\n",
            "II. VELOCITIES AND FIELDS IN AN OBLIQUE WAVE\n",
            "We consider a magnetosonic wave propagating in the x direction with a speed vsh in\n",
            "an external magnetic ﬁeld in the (x, z) plane. We assume that the ﬁeld quantities depend\n",
            "on x only; ∂/∂y = ∂/∂z = 0. Then, from the equation ∇· B = 0, it follows that the\n",
            "x component of the magnetic ﬁeld is constant, Bx = Bx0. The other components By and\n",
            "Bz are functions of x, and in the pulse region By as well as Bz can have ﬁnite values. In\n",
            "the wave frame where the time derivatives are zero (∂/∂t = 0), the electric ﬁeld in the y\n",
            "direction is constant, Ey = Ey0, and Ez is zero. Hence we have B = (Bx0, By, Bz) and\n",
            "E = (Ex, Ey0, 0). In Sections II-IV, theoretical analyses will be made mainly in the wave\n",
            "frame.\n",
            "The velocity of the guiding-center position of an electron, vg, may be written as\n",
            "vg = (B/B)v∥+ vd,\n",
            "(1)\n",
            "with vd being the drift velocity\n",
            "vd = cE × B/B2 −(cµ/e)(B × ∇B)/B2.\n",
            "(2)\n",
            "Here c is the speed of light, −e is the electron charge (e > 0), v∥is the velocity parallel to\n",
            "the magnetic ﬁeld, and µ is the magnetic moment\n",
            "µ = mev2\n",
            "⊥/(2B),\n",
            "(3)\n",
            "with me the electron mass and v⊥the gyration speed perpendicular to the magnetic ﬁeld.\n",
            "In the far upstream region, we have no ∇B-drift. Also, the z component of the velocity\n",
            "averaged over all the electrons in a small volume element must be zero, ⟨vgz0⟩= 0; the\n",
            "subscript 0 refers to the quantities in the far upstream region. Accordingly, from the z\n",
            "component of Eq. (1) we have the average parallel velocity\n",
            "⟨v∥0⟩= cEy0\n",
            "B0\n",
            "Bx0\n",
            "Bz0\n",
            ",\n",
            "(4)\n",
            "in the wave frame (see Fig. 1). Also, because ⟨vgx0⟩= −vsh, the x component of Eq. (1)\n",
            "gives\n",
            "vsh = −Bx0\n",
            "B0\n",
            "⟨v∥0⟩−cEy0Bz0\n",
            "B2\n",
            "0\n",
            ".\n",
            "(5)\n",
            "Thus Ey0 is related to the shock speed vsh through\n",
            "3\n",
            "\n",
            "Ey0 = −vshBz0/c.\n",
            "(6)\n",
            "For the deﬁniteness, we assume that Bx0, Bz0 and vsh are all positive; and thus Ey0 < 0.\n",
            "The nonlinear wave theory tells us the wave structure. A shock wave will have a positive\n",
            "electric potential ϕ(x). The quantities ϕ, Bz, ni, and ne have similar proﬁles [28]- [29] (see\n",
            "Appendix A); here ni, and ne are the ion and electron densities, respectively. On the other\n",
            "hand, Ex and By are proportional to the x derivatives of these quantities; for instance,\n",
            "Ex = −∂ϕ/∂x.\n",
            "These relations among the quantities are obtained for small-amplitude\n",
            "waves. In the following theoretical analysis, we assume that the relations are also valid for\n",
            "large-amplitude waves. As we will see later in Fig. 7, simulation results also support this\n",
            "assumption.\n",
            "We denote by xm the x position at which the potential takes its maximum value; hence,\n",
            "Ex(xm) ≃0 and By(xm) ≃0. For later use, we also note that the quantity Bz/B takes its\n",
            "maximum value at x = xm; that is, if Bz(x) is smaller than Bz(xm), then\n",
            "Bz(xm)2\n",
            "B2\n",
            "x0 + Bz(xm)2 >\n",
            "Bz(x)2\n",
            "B2\n",
            "x0 + By(x)2 + Bz(x)2.\n",
            "(7)\n",
            "This can be proved by the following equation\n",
            "Bz(xm)2\n",
            "B2\n",
            "x0 + Bz(xm)2 −\n",
            "Bz(x)2\n",
            "B2\n",
            "x0 + By(x)2 + Bz(x)2 = [Bz(xm)2 −Bz(x)2]B2\n",
            "x0 + Bz(xm)2By(x)2\n",
            "[B2\n",
            "x0 + Bz(xm)2][B2\n",
            "x0 + By(x)2 + Bz(x)2] . (8)\n",
            "Because Bz(xm) is the maximum value of Bz(x), the term [Bz(xm)2 −Bz(x)2]B2\n",
            "x0 is positive.\n",
            "Hence the right-hand side of the above equation is positive.\n",
            "We know the stationary, ﬁnite-amplitude solutions for perpendicular waves [24]- [27]. The\n",
            "stationary solitary wave solution is valid when the magnetic ﬁeld is weak, ωce/ωpe << 1, and\n",
            "when the wave amplitude is not large, 1 < M < 2, where M is the Alfv´\n",
            "en Mach number;\n",
            "M = vsh/vA with vA the Alfv´\n",
            "en speed, v2\n",
            "A = B2\n",
            "0/(4πnimi). Simulations and experiments\n",
            "show that large-amplitude pulses can propagate nearly steadily, even when the amplitudes\n",
            "are large or the magnetic ﬁelds are strong. Oblique pulses can also propagate nearly steadily.\n",
            "As the magnetic ﬁeld becomes stronger or the wave amplitude is increased, non-stationary\n",
            "eﬀects will become important [13], [14].\n",
            "III. ELECTRON REFLECTION\n",
            "Because the potential is usually positive in the shock region, some of the ions can be\n",
            "reﬂected when they go up the potential. As we will see later, electrons can also be reﬂected;\n",
            "its mechanism is, however, diﬀerent from that of ion reﬂection. We give a physical picture\n",
            "of the electron reﬂection.\n",
            "From the nonrelativistic equation of motion for an electron\n",
            "me\n",
            "dv\n",
            "dt = −e\n",
            " \n",
            "E + v × B\n",
            "c\n",
            "!\n",
            ",\n",
            "(9)\n",
            "we have an equation for the kinetic energy in the wave frame\n",
            "4\n",
            "\n",
            "me(v2 −v2\n",
            "0)/2 = e(ϕ −ϕ0) −eEy0\n",
            "Z\n",
            "vydt.\n",
            "(10)\n",
            "Here we have used the relation Ez = 0. The velocity v includes gyro, parallel, and drift\n",
            "motions.\n",
            "We deﬁne length s as\n",
            "ds = (B/Bx0)vgxdt,\n",
            "(11)\n",
            "where vgx is the x component of the electron velocity\n",
            "vgx = (Bx0/B)v∥+ vdx,\n",
            "(12)\n",
            "(see Fig. 2). Thus ds/dt is given by\n",
            "ds\n",
            "dt = v∥+ B\n",
            "Bx0\n",
            "vdx.\n",
            "(13)\n",
            "We can interpret ds as an inﬁnitesimal length along the ﬁeld line corresponding to the length\n",
            "dx = vgxdt; ds = (B/Bx0)dx . In general the guiding-center velocity vg is not parallel to\n",
            "the magnetic ﬁeld B. Hence the guiding center does not move along the ﬁeld line. For a\n",
            "stationary, one-dimensional problem (∂/∂t = 0 and ∂/∂y = ∂/∂z = 0), all quantities depend\n",
            "only on x. As a result, for any stationary function f(x) we have\n",
            "Z x2\n",
            "x1\n",
            "f(x)(B/Bx0)dx =\n",
            "Z s2\n",
            "s1\n",
            "f(s)ds,\n",
            "(14)\n",
            "where s1 and s2 are the positions along the same ﬁeld line; their x positions are x1 and x2,\n",
            "respectively.\n",
            "If we deﬁne quantity F as\n",
            "E∥= −dF/ds,\n",
            "(15)\n",
            "then with the aid of (11) and the relation E∥= E · B/B, we have\n",
            "F = −\n",
            "Z (ExBx0 + Ey0By)\n",
            "B\n",
            "B\n",
            "Bx0\n",
            "dx,\n",
            "(16)\n",
            "which can be written as\n",
            "F = −\n",
            "Z \u0012\n",
            "Ex −vshBz0By\n",
            "cBx0\n",
            "\u0013\n",
            "dx.\n",
            "(17)\n",
            "Using the electric potential ϕ and the vector potential A (B = ∇× A), we ﬁnd\n",
            "F = ϕ −(vsh/c)(Bz0/Bx0)Az,\n",
            "(18)\n",
            "where Az is the z component of the vector potential,\n",
            "Az = −\n",
            "Z\n",
            "Bydx.\n",
            "(19)\n",
            "5\n",
            "\n",
            "The quantities ϕ and Bz have similar proﬁles, while By is proportional to ∂Bz/∂x [28], [29].\n",
            "Hence Az and F have proﬁles similar to ϕ.\n",
            "If we deﬁne quantity Fl in the laboratory frame (the subscript l denotes the laboratory\n",
            "frame) as\n",
            "Fl = −\n",
            "Z\n",
            "El∥(Bl/Blx0)dxl,\n",
            "(20)\n",
            "then for a stationary, one-dimensional problem, that is, for a case where wave proﬁles can\n",
            "be written as f(xl, tl) = f(xl −vshtl), we ﬁnd\n",
            "Fl = −\n",
            "Z \u0012\n",
            "Elx −vshBlz0Bly\n",
            "cBlx0\n",
            "\u0013\n",
            "dxl.\n",
            "(21)\n",
            "One can readily show that it is related to F in the wave frame as\n",
            "F = γshFl,\n",
            "(22)\n",
            "where γsh is deﬁned as\n",
            "γsh = (1 −v2\n",
            "sh/c2)−1/2.\n",
            "(23)\n",
            "In Appendix B, we give a rough estimate of the maximum value of F, Eq. (B9), found from\n",
            "qualitative physical considerations.\n",
            "Combining Eqs. (10) and (18), we eliminate ϕ\n",
            "mev2/2 −mev2\n",
            "0/2 = e(F −F0) −(eEy0/Bx0)(Az −Az0 +\n",
            "Z\n",
            "vyBx0dt).\n",
            "(24)\n",
            "By virtue of the deﬁnition of Az, (19), and the relation dx = vxdt, this equation can be\n",
            "written as\n",
            "mev2/2 −mev2\n",
            "0/2 = e(F −F0) + e(Ey0/Bx0)\n",
            "Z\n",
            "(vxBy −vyBx0)dt.\n",
            "(25)\n",
            "Then, using the z component of the equation of motion, we ﬁnd\n",
            "mev2/2 −mev2\n",
            "0/2 = e(F −F0) −mec(Ey0/Bx0)(vz −vz0).\n",
            "(26)\n",
            "In the drift approximation, the kinetic energy can be expressed as\n",
            "mev2/2 = me(v2\n",
            "∥+ v2\n",
            "d)/2 + µB.\n",
            "(27)\n",
            "Hence, averaging Eq. (26) over the electron cyclotron period, we have\n",
            "(me/2)(v2\n",
            "∥+ v2\n",
            "d) + µB + mec(Ey0/Bx0)vgz −eF\n",
            "= (me/2)(v2\n",
            "∥0 + v2\n",
            "d0) + µB0 + mec(Ey0/Bx0)vz0 −eF0.\n",
            "(28)\n",
            "With the aid of the z component of Eq. (1), we can eliminate vgz and obtain the following\n",
            "equation\n",
            "6\n",
            "\n",
            "(me/2)(v∥−vrv)2 + K = e(F −F0) −µ(B −B0)\n",
            "+ (me/2)(v2\n",
            "∥0 + v2\n",
            "d0) −mec(Ey0/Bx0)(vdz −vz0) −(me/2)v2\n",
            "d,\n",
            "(29)\n",
            "where vrv and K are deﬁned as\n",
            "vrv = −cEy0Bz/(Bx0B)\n",
            "(> 0),\n",
            "(30)\n",
            "K = −mev2\n",
            "rv/2\n",
            "(< 0).\n",
            "(31)\n",
            "If we eliminate Ey0 by substituting Eq. (6) in Eq. (30), vrv can be expressed as\n",
            "vrv = vshBz0Bz/(Bx0B).\n",
            "(32)\n",
            "We will see below that the velocity vgx is reversed when v∥= vrv. In the pulse region, the\n",
            "quantity e(F −F0) can be much larger in magnitude than the other terms on the right-hand\n",
            "side of Eq. (29) (see Appendix B). We denote the left-hand side of Eq. (29) by Ee, i.e.,\n",
            "Ee = (me/2)(v∥−vrv)2 + K. As shown in Fig. 3, at a ﬁxed point x, Ee has its minimum\n",
            "value K when v∥= vrv. It is negative in the region\n",
            "0 < v∥< 2vrv.\n",
            "(33)\n",
            "Because vdx = cEy0Bz/B2, it follows from Eq. (12) that vgx = 0 if v∥= vrv. As can be\n",
            "seen from Eq. (4), most of the electrons have negative parallel velocities in the far upstream\n",
            "region. Accordingly, we suppose that the initial parallel velocity is negative, v∥< 0. If\n",
            "v∥gradually increases and exceeds vrv at a certain position, then the x component of the\n",
            "guiding center velocity, vgx, changes from negative to positive values there, i.e., it is reversed.\n",
            "As an electron moves, i.e., as x changes, the quantities K, F, B, etc. also vary. At any\n",
            "point x, K is the minimum value of the left-hand side of Eq. (29). If there is a region where\n",
            "the values of the right-hand side of Eq. (29) become smaller than K, then the electron\n",
            "cannot enter there; it will be reﬂected. We recall that ϕ, F, and Bz have similar proﬁles\n",
            "and that at x = xm the quantity (Bz/B) has a maximum value [see Eq. (7)]. Because K is\n",
            "proportional to (Bz/B)2, it will be quite small (|K| is large) around the point x = xm, and\n",
            "F is large. In front of or behind the main pulse, F is relatively small and K is relatively\n",
            "large; e(F −F0) could be smaller than K. Thus, if the electron reﬂection occurs, it will be\n",
            "in such regions.\n",
            "In quasi-perpendicular shocks, where (Bz0/Bx0) >> 1, vrv is large. Hence K has large\n",
            "negative values. Thus the electron reﬂection in this mechanism will be diﬃcult.\n",
            "Figure 4 shows a schematic diagram of the trajectory of guiding center. Here it is assumed\n",
            "that the reﬂection takes place at point D; the dotted line shows the orbit of a passing electron.\n",
            "As an electron moves from point A to C, it moves to the negative y direction because of the\n",
            "Ex × Bz drift. It gains kinetic energy ∆E1 from the electric potential\n",
            "∆E1 = eϕ(xC) −eϕ(xA)\n",
            "(> 0).\n",
            "(34)\n",
            "At the same time, it loses energy ∆E2 because of the electric ﬁeld Ey0\n",
            "∆E2 = −eEy0(yC −yA)\n",
            "(< 0).\n",
            "(35)\n",
            "7\n",
            "\n",
            "The net change in the energy is therefore\n",
            "∆E = ∆E1 + ∆E2.\n",
            "(36)\n",
            "Even though the magnitudes of ∆E1 and ∆E2 are quite large, they almost cancel when an\n",
            "electron moves with drift approximation; in particular, in a perpendicular pulse they have\n",
            "exactly the same magnitude and ∆E = 0 [30]. However, if an electron is reﬂected and moves\n",
            "from D to E, then it would gain energies from both Ex and Ey0. As a result, the increase in\n",
            "energy is\n",
            "∆E = ∆E1 + ∆E3,\n",
            "(37)\n",
            "where ∆E3 is deﬁned as\n",
            "∆E3 = −eEy0(yE −yA)\n",
            "(> 0).\n",
            "(38)\n",
            "IV. ESTIMATE OF ENERGY INCREASE\n",
            "Next, we will obtain the maximum energy of a reﬂected electron. Because it can have\n",
            "quite high energy, we will use the relativistic equation of motion\n",
            "me\n",
            "d(γv)\n",
            "dt\n",
            "= −e\n",
            " \n",
            "E + v × B\n",
            "c\n",
            "!\n",
            ",\n",
            "(39)\n",
            "where γ is the Lorentz factor. For the stationary, one-dimensional system being considered,\n",
            "this can be integrated to give the energy conservation equation\n",
            "meγc2 −meγ0c2 = e(ϕ −ϕ0) −eEy0\n",
            "Z\n",
            "vydt.\n",
            "(40)\n",
            "This is a relativistic form of Eq. (10). Again, using the quantities F, Az, and the equation\n",
            "of motion, we can rewrite Eq. (40) as\n",
            "meγc2 −meγ0c2 = e(F −F0) −mec(Ey0/Bx0)(γvz −γ0vz0).\n",
            "(41)\n",
            "If we deﬁne quantity h as\n",
            "h = mec2 −mevshBz0vz/Bx0,\n",
            "(42)\n",
            "then from Eq. (41) we have\n",
            "γ = [e(F −F0) + h0γ0]/h.\n",
            "(43)\n",
            "Equation (42) indicates that h is positive if Bz0/Bx0 is of order unity. If Bz0/Bx0 is much\n",
            "greater than unity, then h can be negative. In addition, Eq. (43) shows that, if the initial\n",
            "value of h is positive, h0 > 0, then h is always positive in the region where (F −F0) is\n",
            "positive.\n",
            "8\n",
            "\n",
            "We show in Fig. 5 schematic diagram of the function [e(F −F0) + h0γ0]. We suppose\n",
            "that F has a maximum value at x = xm and has a minimum value right behind the main\n",
            "pulse. If the reﬂection occurs, it would be in this dip, as suggested in the previous section.\n",
            "The reﬂection point will be denoted by xr. In the top panel this function e(F −F0)+h0γ0 is\n",
            "always positive. In the second one it becomes negative right behind the main pulse; however\n",
            "its maximum value is positive (h0 can be either positive or negative). In the bottom one\n",
            "it is always negative. For the present acceleration mechanism, the top panel is the most\n",
            "important; the strong acceleration takes place in this case.\n",
            "Such dips would not be present in the perfectly stationary solutions [24]- [29]. In large-\n",
            "amplitude magnetosonic waves, however, non-stationary dips can be generated [13], [14].\n",
            "We here just assume that there is a dip behind the main pulse and discuss its eﬀects on\n",
            "particle orbits.\n",
            "First we consider the top panel in Fig. 5: An electron with positive h0 is reﬂected at a\n",
            "certain point near the end of the main pulse. We will show that, after the reﬂection, h has\n",
            "its minimum (positive) value at x = xm; hence γ has its maximum value there.\n",
            "As mentioned earlier, the parallel velocity v∥is already positive when an electron is\n",
            "reﬂected, i.e., when vgx is reversed, even if its initial value is negative. After the reﬂection,\n",
            "v∥will be increased by the (positive) parallel electric force. Because E∥changes its sign at\n",
            "x = xm, v∥will have a maximum value at x = xm (unless vx changes to negative values again\n",
            "before reaching the point xm ). The parallel velocity will become the dominant component in\n",
            "the electron velocity, v ∼v∥B/B. The quantity Bz/B as well as ϕ and Bz has a maximum\n",
            "value at this point. Thus vz, which can be approximated as vz ∼v∥Bz/B, will also have the\n",
            "maximum value there. Because h0 and (Fm −F0) are both positive, h is positive at x = xm.\n",
            "In addition, Eq. (42) indicates that h decreases (draws closer to zero) as vz increases. Hence,\n",
            "h of a reﬂected electron will have a minimum positive value at x = xm. Its energy γ will\n",
            "therefore have a maximum value there. If h can be close to zero, then γ would become quite\n",
            "large.\n",
            "If Bx0 is small and vz0 has a large positive value, then h0 becomes small (can be even\n",
            "negative). We now consider such particles; the second panel in Fig. 5. Also in this case,\n",
            "vz will have larger values after the reﬂection; thus h will decrease. At the reﬂection point\n",
            "in the dip, the quantity [e(F −F0) + h0γ0] is negative. Consequently, h(xr) < 0. If the\n",
            "reﬂection occurs, the electron would move in the positive x direction. Because vz increases,\n",
            "h will further decrease. Hence, the particle would never be able to penetrate the region\n",
            "again where hγ > 0. The strong acceleration in this mechanism, therefore, will not take\n",
            "place.\n",
            "Next, we consider a case where the quantity [e(F −F0) + h0γ0] is always negative (the\n",
            "bottom panel in Fig. 5). Then, h is always negative. If the reﬂection occurs at x = xr,\n",
            "then |h| would increase while |e(F −F0) + h0γ0| would decrease. Hence, even if the particle\n",
            "reaches the point xm, γ would not have an extremely large value. Therefore, the strong\n",
            "acceleration is not expected.\n",
            "We can estimate the maximum value γm in the wave frame by substituting the maximum\n",
            "value of F, Eq. (B12), in Eq. (43). The maximum γ in the laboratory frame can be obtained\n",
            "by the Lorentz transformation as\n",
            "γlm = γsh(1 + vxvsh/c2)γm ≃γshγm.\n",
            "(44)\n",
            "9\n",
            "\n",
            "Now let us consider a case where h becomes quite small, h ≃0, at x = xm. There the\n",
            "electron speed would be close to the speed of light c. Hence, roughly, we have vz ∼cBz/B.\n",
            "The quantity h can then be expressed as\n",
            "h = mec2\n",
            "\u0012\n",
            "1 −γsh\n",
            "vsh\n",
            "c\n",
            "Bz\n",
            "B tan θ\n",
            "\u0013\n",
            ",\n",
            "(45)\n",
            "where tan θ = Blz0/Blx0. Here we have used the relations Bz0 = γshBlz0 and Bx0 = Blx0.\n",
            "Because Bz becomes quite large in the shock region, we may put as Bz/B ∼1. Equation\n",
            "(45) then suggests that the electron energy can become extremely large when\n",
            "γsh(vsh/c) tan θ = 1,\n",
            "(46)\n",
            "or, equivalently\n",
            "c cos θ = vsh.\n",
            "(47)\n",
            "V. SIMULATION STUDIES\n",
            "A. Simulation Method\n",
            "We further study the electron acceleration by using a one-dimensional (one space co-\n",
            "ordinate and three velocity components), relativistic, electromagnetic, particle simulation\n",
            "code with full ion and electron dynamics [31]. The total system length is Lx = 4096∆g\n",
            "for most of the simulation runs, where ∆g is the grid spacing. All lengths and velocities in\n",
            "the simulations were normalized to ∆g and ωpe∆g, respectively, where ωpe is the spatially\n",
            "averaged plasma frequency.\n",
            "We use a bounded plasma model. The plasma is limited to the region 400 < x < 3696.\n",
            "The particles are specularly reﬂected at x = 400 and at x = 3696. The radiation leaving the\n",
            "plasma region is absorbed in the vacuum regions, 0 < x < 400 and 3696 < x < 4096. Thus\n",
            "the electromagnetic interactions between the two plasma boundaries through the vacuum\n",
            "regions are made negligibly small [12].\n",
            "In most of the plasma region, i.e., in 500 < x < 3696, each particle species initially has\n",
            "a uniform density and has a Maxwellian velocity distribution function. In the small region,\n",
            "400 < x < 480, the plasma density is four times as high as that in the main region. That\n",
            "is, the initial plasma density can be written as\n",
            "ne = 4n0\n",
            "for 400 < x < x1,\n",
            "ne = 4n0 exp[−(x −x1)2/(2d2)]\n",
            "for x1 < x < 500,\n",
            "ne = n0\n",
            "for 500 < x < 3696.\n",
            "Here x1 = 480 and d = 12. In the high-density region, 400 < x < 480, particles have\n",
            "shifted Maxwellian velocity distribution functions with average velocity vps; fj ∼exp[−(v −\n",
            "vps)2/(2v2\n",
            "Tj)], where the subscript j refers to particle species.\n",
            "These particles act as a\n",
            "piston. They push the neighboring particles and excite a shock wave. We can change the\n",
            "10\n",
            "\n",
            "shock strength by changing the magnitude of vps; the velocity vps is perpendicular to the\n",
            "external magnetic ﬁeld so that no particles have extremely large parallel speeds initially.\n",
            "More detailed description about the simulation code can be found in Ref. [12].\n",
            "The simulation parameters are as follows. The number of simulation particles is Ni =\n",
            "Ne = 262, 144.\n",
            "The ion-to-electron mass ratio is mi/me = 100.\n",
            "The ratio ωce/ωpe is\n",
            "ωce/ωpe = 3.0 in the far upstream region.\n",
            "The electron and ion thermal velocities are\n",
            "vTe/c = 0.38 and vTi/c = 0.01, respectively. The Alfv´\n",
            "en speed is vA/c = 0.3. The electron\n",
            "skin depth is c/ωpe = 4∆g. The time step is suﬃciently small, ωpe∆t = 0.02, so that ∆t is\n",
            "much smaller than the plasma and cyclotron periods even in the shock region. As in the\n",
            "theoretical model, the external magnetic ﬁeld is in the (x, z) plane, and waves propagate\n",
            "in the x direction.\n",
            "B. Simulation Results\n",
            "First, we study a shock with the propagation angle θ = 45◦; i.e., Bx0 = Bz0. Figure 6\n",
            "shows proﬁles of Bz in the shock wave at various times. Here, the magnetic ﬁeld is normalized\n",
            "to Bz0. The Mach number is observed to be M = 2.3, i.e., the shock speed is vsh = 2.3vA.\n",
            "The maximum value of Bz is 4∼5 times as large as Bz0. The main pulse propagates nearly\n",
            "steadily. However, the proﬁle is not perfectly stationary.\n",
            "Figure 7 displays ﬁeld proﬁles at ωpet = 680; here the magnetic and electric ﬁelds are\n",
            "both normalized to B0. As the theory predicts [28], [29], Ey and Bz have similar proﬁles,\n",
            "while Ex, Ez, and By are quite small in magnitude at the point where Bz has its maximum\n",
            "value. Figure 8 shows phase space plots of electrons at ωpet = 680. In the shock region,\n",
            "ultra-relativistic electrons are produced. The maximum value of Lorentz factors is γ ≃130.\n",
            "We also see that px and pz have greater values than py, which is consistent with the picture\n",
            "that the parallel velocity is dominant in the motion of high-energy electrons. From Figs. 7\n",
            "and 8, we see that some electrons are reﬂected near the end of the main pulse and that the\n",
            "high-energy electrons are present in the shock region.\n",
            "Figure 9 shows proﬁles of E∥, F, and the electric potential ϕ at ωpet = 650 and at\n",
            "ωpet = 680. The quantities ˜\n",
            "F and ˜\n",
            "ϕ denote eF/(mec2) and eϕ/(mec2), respectively. The\n",
            "quantities F and ϕ were obtained in the simulation by the following equations\n",
            "F(x) = −\n",
            "Z x\n",
            "E∥(B/Bx0)dx,\n",
            "(48)\n",
            "ϕ(x) = −\n",
            "Z x\n",
            "Exdx.\n",
            "(49)\n",
            "As mentioned earlier, roughly speaking, the parallel electric ﬁeld E∥is positive in the region\n",
            "where ∂Bz/∂x and ∂F/∂x are negative. It is negative in the region where they are positive.\n",
            "Comparing the plots at diﬀerent times, we see that the wave proﬁles are not perfectly\n",
            "stationary. They vary with time with small-amplitudes. In particular, as shown in the plot\n",
            "at ωpet = 680, F sometimes becomes negative near the end of the main pulse, where the\n",
            "electron reﬂection can take place.\n",
            "Next, we will study trajectories of electrons; we will ﬁnd that it is the reﬂected electrons\n",
            "that gain great energies. Figure 10 displays time variations of (x −vsht), y, z, and γ of\n",
            "11\n",
            "\n",
            "electrons in a shock with vsh = 1.75vA; (x −vsht) is the x position in the wave frame. The\n",
            "lengths are normalized to the electron skin depth c/ωpe. (Here the system size is a half of\n",
            "that in the previous run; the other parameters such as Ne/Lx are the same.) Two typical\n",
            "examples are presented here. The thick lines show an electron that was accelerated, and\n",
            "the thin lines represent an electron that was not accelerated; even the low-energy electron\n",
            "here has energy γ ≃10 in the shock region and γ ≃5 behind the shock. In all of the four\n",
            "panels, the low- and high-energy electrons both have small-amplitude, cyclotron oscillations\n",
            "with short periods, ωpet <\n",
            "∼20; note that the cyclotron period depends on γ as well as the\n",
            "ﬁeld strength B. At the same time, the accelerated electron has a large-amplitude, long-\n",
            "period oscillation with period ωpet ≃170; it begins when the particle enters the shock,\n",
            "ωpet = 180, and continues until the end of this simulation run. (Simulations show that\n",
            "this period increases with γ. To observe this oscillation as many times as possible, we have\n",
            "chosen here the case where γ is not so large.) For comparison with Fig. 4, we show the\n",
            "times corresponding to points B, C, D, and E. From the long-period oscillation of (x −vsht)\n",
            "in the top panel, we see that the electron that will become high energy is reﬂected and then\n",
            "trapped by the shock wave. On the other hand, the quantity (x −vsht) of the low-energy\n",
            "electron keeps decreasing, which means that this particle passes through the shock region\n",
            "without strong interactions. The second panel shows that the y position of the accelerated\n",
            "electron also oscillates, while the third one shows that z increases rapidly after the reﬂection.\n",
            "The average value of velocity vz over the period from ωpet = 220 to 400 is vz/c ≃0.6, and\n",
            "the value of vz at maximum γ is vz/c ≃0.8. The maximum γ in the bottom panel is γ ≃40.\n",
            "By the Lorentz transformation of these data, we can ﬁnd v and B in the wave frame.\n",
            "At the time and position of maximum γ, they are v = c(0.27, 0.13, 0.95) and B =\n",
            "B0(0.70, 0.16, 2.56). This conﬁrms our statement in the previous section that v∥is the\n",
            "dominant component in v there; i.e., vz ≃v∥Bz/B.\n",
            "According to the theory, (47), electron energies can be extremely high at some angles\n",
            "and shock speeds. To examine this prediction, we have carried out several simulations with\n",
            "diﬀerent values of the propagation angle θ, keeping other parameters unchanged; Lx =\n",
            "4096∆g, ωce/ωpe = 3.0, and vsh ≃2.1vA. Figure 11 shows the maximum value of γ as a\n",
            "function of the propagation angle θ. Here the electron energy takes the highest value at the\n",
            "angle θ ≃52◦. For these simulation parameters, Eq. (47) predicts that γ takes a peak value\n",
            "at θ ≃53◦. Thus the theory and simulation are in good agreement. Further, we carried out\n",
            "simulations with diﬀerent values of the shock speed vsh; the propagation angle is ﬁxed to be\n",
            "θ = 45◦. Figure 12 shows the maximum value of γ as a function of the shock speed vsh. The\n",
            "observed values have a peak at vsh/c ≃0.7. For these parameters, Eq. (47) predicts that γ\n",
            "takes large values for vsh/c ≃0.71. Again, the theory and simulation are in good agreement.\n",
            "VI. SUMMARY\n",
            "We have studied electron motion in a shock wave propagating obliquely to a magnetic\n",
            "ﬁeld.\n",
            "First, we analytically discussed electron motion in an oblique shock wave.\n",
            "It is\n",
            "pointed out that, if an electron is reﬂected near the end of a large-amplitude pulse, it would\n",
            "gain a great deal of energy from the electric ﬁeld formed in the wave. The condition for\n",
            "the reﬂection is then examined. Further, the maximum energy of a reﬂected electron is\n",
            "12\n",
            "\n",
            "estimated, and its dependence on plasma parameters is discussed. Next, we investigated the\n",
            "shock evolution and associated electron acceleration by using a one-dimensional, relativistic,\n",
            "electromagnetic, particle simulation code with full ion and electron dynamics. It was shown\n",
            "that an oblique shock can produce ultra-relativistic electrons; Lorentz factors with γ >\n",
            "∼100\n",
            "have been observed. As the theory predicts, at certain propagation angles electron energies\n",
            "become extremely high. The electron reﬂection and resultant acceleration takes place when\n",
            "the wave amplitude is large and the magnetic ﬁeld is rather strong, ωce\n",
            ">\n",
            "∼ωpe.\n",
            "In the present paper, strong electron acceleration has been demonstrated. As future\n",
            "work, it would be desirable to develop further the quantitative theory for large-amplitude\n",
            "oblique waves. The evaluation of the electric ﬁeld strength along the magnetic ﬁeld will be\n",
            "especially important, which will enable us to estimate the maximum energy more accurately.\n",
            "From the view point of nonlinear wave propagation, motion of trapped electrons and their\n",
            "eﬀects on the wave evolution are also quite interesting.\n",
            "APPENDIX A: RELATIONS AMONG QUANTITIES\n",
            "We here describe relations among the physical variables, using the approximation appro-\n",
            "priate for low-frequency magnetosonic waves with small amplitudes (for the details of the\n",
            "calculations, see Ref. [29]). We assume quasi neutrality, n ≃ni ≃ne and use the following\n",
            "stretched variables\n",
            "τ = ǫ3/2t,\n",
            "(A1)\n",
            "ξ = ǫ1/2(x −vp0t),\n",
            "(A2)\n",
            "(vp0 is the wave propagation speed in the long-wavelength limit in a ﬁnite beta plasma) and\n",
            "expansion\n",
            "n = n0 + ǫn1 + ǫ2n2 + ...,\n",
            "(A3)\n",
            "Ey = ǫEy1 + ǫ2Ey2 + ...,\n",
            "(A4)\n",
            "Bz = Bz0 + ǫBz1 + ǫ2Bz2 + ...,\n",
            "(A5)\n",
            "then, we can express the lowest order perturbations in terms of n1 as\n",
            "Bz1/B0 = cEy1/(B0vp0) = [(v2\n",
            "p0 −c2\n",
            "s)/v2\n",
            "A sin θ](n1/n0),\n",
            "(A6)\n",
            "cEz1\n",
            "B0\n",
            "= −vp0By1\n",
            "B0\n",
            "= (ωce −ωci)\n",
            "ωceωci\n",
            "(v2\n",
            "p0 −c2\n",
            "s)v2\n",
            "p0\n",
            "(v2\n",
            "p0 −v2\n",
            "A cos2 θ)\n",
            "cot θ\n",
            "n0\n",
            "∂n1\n",
            "∂ξ .\n",
            "(A7)\n",
            "Here cs is the sound speed, c2\n",
            "s = (Γepe0 + Γipi0)/(n0mi) with Γj (j = e or i) the speciﬁc heat\n",
            "ratio and pj0 the equilibrium pressure. We can obtain KdV equation by proceeding to the\n",
            "next order, O(ǫ5/2).\n",
            "The relations (A6) and (A7) are valid, even when the wave proﬁle is not soliton-like.\n",
            "That is, perturbations propagating in the same direction with vx ∼vp0 have the relations\n",
            "(A6) and (A7).\n",
            "13\n",
            "\n",
            "APPENDIX B: MAGNITUDES OF E∥AND F\n",
            "For a nonrelativistic, small-amplitude oblique magnetosonic wave, the electric ﬁeld par-\n",
            "allel to the magnetic ﬁeld, E∥, has been analytically obtained [32]. However, for a large-\n",
            "amplitude wave, it will be quite diﬃcult to obtain E∥in a rigorous manner. Here, on the\n",
            "basis of a simple physical picture, we will give a rough estimate of the parallel electric ﬁeld\n",
            "E∥in a relativistic, large-amplitude, oblique shock wave. Further, by using it, we will obtain\n",
            "the maximum value of the quantity F.\n",
            "We consider in the wave frame.\n",
            "In the ﬂuid model, the equation of motion may be\n",
            "written as\n",
            "mj\n",
            "d(γjvj)\n",
            "dt\n",
            "= qjE + qj\n",
            "vj × B\n",
            "c\n",
            ".\n",
            "(B1)\n",
            "Here the pressure term is neglected. The subscript j refers to ions j = i or electrons j = e.\n",
            "The time derivative d/dt is deﬁned as\n",
            "d\n",
            "dt = ∂\n",
            "∂t + vj · ∇\n",
            ".\n",
            "(B2)\n",
            "We denote by b the unit vector along the magnetic ﬁeld. Then multiplying Eq. (B1) by b,\n",
            "we have\n",
            "mj\n",
            "d(γjb · vj)\n",
            "dt\n",
            "−mjγjvj · db\n",
            "dt = qjE∥.\n",
            "(B3)\n",
            "Let t1 be the time when a ﬂuid element is at the leading edge of the shock (the location\n",
            "where the shock proﬁle begins to sharply rise) and t2 be the time when it is at the point\n",
            "xm; xm is the location where the electric potential and magnetic ﬁeld take their maximum\n",
            "values. Then, integrating from t1 to t2, we ﬁnd for the ions\n",
            "mi(γi2b2 · vi2 −γi1b1 · vi1) −mi⟨γivi⟩· (b2 −b1) ≃eE∥\n",
            "∆\n",
            "|⟨vix⟩|.\n",
            "(B4)\n",
            "Here the bracket indicates the mean value over the time period (t2 −t1), and ∆is the\n",
            "shock width. We have a relation (t2 −t1) ≃∆/|⟨vix⟩|. For oblique shocks, ∆will be of\n",
            "the order of the ion inertial length, ∆≃c/ωpi. The velocity at time t1 is vi1 = −vshex,\n",
            "where ex is the unit vector in the x direction. The velocity vix will be slowed down by\n",
            "the longitudinal electric ﬁeld. Thus vix2 will be smaller in magnitude than vix1. Because\n",
            "Bz is large at the point xm and By(xm) ∼0, the unit vector b2 is nearly parallel to the z\n",
            "direction. Consequently, the inner product (b2 · vi2) will be quite small; here we neglect the\n",
            "z component of vi2. Since the x component of b1 is given by Bx0/B0, we have\n",
            "mi(γi1vsh + ⟨γivix⟩)Bx0/B0 ≃eE∥∆/|⟨vix⟩|.\n",
            "(B5)\n",
            "This gives the strength of the parallel electric ﬁeld as\n",
            "eE∥≃miv2\n",
            "sh\n",
            "∆\n",
            "|⟨vix⟩|\n",
            "vsh\n",
            " \n",
            "γi1 + ⟨γivix⟩\n",
            "vsh\n",
            "! Bx0\n",
            "B0\n",
            ".\n",
            "(B6)\n",
            "14\n",
            "\n",
            "Because |vix| is smaller than vsh, the range of |⟨vix⟩| is expected to be 1/2 <\n",
            "∼|⟨vix⟩|/vsh <\n",
            "1, except maybe for extremely strong shock waves. In fact, if we assume a simple time\n",
            "dependence of vix\n",
            "vix = −vsh[1 −a1(t −t1)/(t2 −t1)]\n",
            "for\n",
            "t1 ≤t ≤t2,\n",
            "(B7)\n",
            "with a1 a constant (0 ≤a1 ≤1), then, after integrating over time, we have ⟨vix⟩= −vsh(1 −\n",
            "a1/2).\n",
            "The Lorentz factor for the ﬂuid ion will be close to unity. Accordingly, for instance, if\n",
            "the ion ﬂuid velocity vix is decreased to a value ∼−vsh/2, then the value of the content in\n",
            "the parentheses would be about 1/4.\n",
            "The electron speed will be ∼vsh when the electrons encounter the shock and will be close\n",
            "to the speed of light when they are at the point xm. If we assume that the electron velocity\n",
            "is nearly parallel to the ﬁeld line in the shock region, then (v · db/dt) can be neglected.\n",
            "Hence we have\n",
            "me(−γe2c + γe1vshBx0/B0) ≃−eE∥(t2 −t1).\n",
            "(B8)\n",
            "The second term on the left-hand side, γe1vshBx0/B0, can be neglected compared with the\n",
            "ﬁrst term γe2c. The ions and electrons will have nearly the same time period (t2 −t1) to\n",
            "keep charge neutrality. Substituting Eq. (B6) in (B8) yields\n",
            "γe2 ≃mivsh\n",
            "mec\n",
            " \n",
            "γi1 + ⟨γivix⟩\n",
            "vsh\n",
            "! Bx0\n",
            "B0\n",
            ".\n",
            "(B9)\n",
            "At x = xm, vz can be estimated as vz ≃−cBz/B (for reﬂected electrons the sign is reversed).\n",
            "For a large-amplitude shock, the ratio Bz/B is an order-unity quantity there. The quantity\n",
            "h deﬁned by Eq. (42) is obtained as\n",
            "h2 ≃mec2[1 + vshBz0/(cBx0)].\n",
            "(B10)\n",
            "If we average Eq. (43) over electrons in a small volume element, we have\n",
            "⟨hγ⟩= e(F −F0) + mec2⟨γ0⟩.\n",
            "(B11)\n",
            "Here we used the relation ⟨vz0⟩= 0. If electrons have the velocity vz ≃−cBz/B at x = xm,\n",
            "then we ﬁnd the maximum value of F from Eq. (43) as\n",
            "e(Fm −F0) ≃h2γe2 −mec2⟨γ0⟩.\n",
            "(B12)\n",
            "In many practical cases, we can take ⟨γ0⟩to be order unity.\n",
            "Roughly speaking, the quantity e(Fm −F0) is of the order of micvsh. It is, therefore,\n",
            "much greater than electron thermal energy mev2\n",
            "Te (∼µB) and drift kinetic energy mev2\n",
            "d\n",
            "(∼mev2\n",
            "sh).\n",
            "15\n",
            "\n",
            "REFERENCES\n",
            "[1] S. R. Kane, K. Kai, T. Kosugi, S. Enome, P. B. Landecker, and D. L. McKenzie,\n",
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            "[2] M. Yoshimori, Y. Takai, K. Morimoto, and K. Suga, Publ. Astron. Soc. Japan 44, L107\n",
            "(1992).\n",
            "[3] K. Koyama, R. Petre, E. V. Gotthelf, et al., Nature 378, 255 (1995).\n",
            "[4] T. Tanimori, Y. Hayami, S. Kamei, et al., Astrophys. J. 497, L25 (1998).\n",
            "[5] T. Tanimori, K. Sakurazawa, S. A. Dazeley, et al., Astrophys. J. 492, L33 (1998).\n",
            "[6] R. D. Blandford and D. Eichler, Phys. Rep. 154, 1 (1987).\n",
            "[7] S. P. Reynolds and D. C. Ellison, Astrophys. J. 399, L75 (1992).\n",
            "[8] C. Joshi, C. E. Clayton, W. B. Mori, J. M. Dawson, and T. Katsouleas, Comments\n",
            "Plasma Phys. Controlled Fusion 16, 65 (1994).\n",
            "[9] D. Biskamp and H. Welter, Nuclear Fusion 12, 663 (1972).\n",
            "[10] M. M. Leory, D. K. Winske, C. C. Goodrich, C. S. Wu, and K. Papadopoulos, J.\n",
            "Geophys. Res. 87, 5081 (1982).\n",
            "[11] D. W. Forslund, K. B. Quest, J. U. Brackbill, and K. Lee, J. Geophys. Res. 89, 2142\n",
            "(1984).\n",
            "[12] Y. Ohsawa, Phys. Fluids 28, 2130 (1985).\n",
            "[13] Y. Ohsawa, J. Phys. Soc. Jpn. 55, 1047 (1986).\n",
            "[14] B. Lembege, and J. M. Dawson, Phys. Fluids B1, 1001 (1989).\n",
            "[15] R. L. Tokar, S. P. Gary and K. B. Quest, Phys. Fluids 30, 2569 (1987).\n",
            "[16] Y. Ohsawa, J. Phys. Soc. Jpn. 59, 2782 (1990).\n",
            "[17] R. Z. Sagdeev and V. D. Shapiro, Pis’ma Zh. Exps. Theor. Fiz. 17, 387 (1973) [JETP\n",
            "Lett. 17, 279 (1973)].\n",
            "[18] M. A. Lee, V. D. Shapiro, and R. Z. Sagdeev, J. Geophys. Res. 101, 4777 (1996).\n",
            "[19] M. Toida and Y. Ohsawa, J. Phys. Soc. Jpn. 64, 2038 (1995).\n",
            "[20] M. Toida and Y. Ohsawa, Solar Physics 171, 161 (1997).\n",
            "[21] K. Maruyama, N. Bessho, and Y. Ohsawa, Phys. Plasmas 5, 3257 (1998).\n",
            "[22] B. Rau and T. Tajima, Phys. Plasmas 5, 3575 (1998).\n",
            "[23] N. Bessho, K. Maruyama, and Y. Ohsawa, J. Phys. Soc. Jpn. 68, 1 (1999).\n",
            "[24] J. H. Adlam and J. E. Allen, Philos. Mag. 3, 448 (1958).\n",
            "[25] L. Davis, R. L¨\n",
            "ust, and A. Schl¨\n",
            "uter, Z. Naturforsch. Teil A13, 916 (1958).\n",
            "[26] C. S. Gardner and G. K. Morikawa, Commun. Pure Appl. Math. 18, 35 (1965).\n",
            "[27] Y. Ohsawa, Phys. Fluids 29, 2474 (1986).\n",
            "[28] T. Kakutani, H. Ono, T. Taniuti, and C. C. Wei, J. Phys. Soc. Jpn. 24, 1159 (1968).\n",
            "[29] Y. Ohsawa, Phys. Fluids 29, 1844 (1986).\n",
            "[30] Y. Ohsawa, J. Phys. Soc. Jpn. 58, 4445 (1989).\n",
            "[31] P. C. Liewer, A. T. Lin, J. M. Dawson, and M. Z. Caponi, Phys. Fluids 24, 1364 (1981).\n",
            "[32] Y. Ohsawa, J. Phys. Soc. Jpn. 57, 929 (1988).\n",
            "16\n",
            "\n",
            "FIGURES\n",
            "FIG. 1. Velocities and ﬁelds in the far upstream region in the wave frame. The velocity vd is\n",
            "Ey0 × B0 drift, and −vsh is equal to average guiding-center velocity in the x direction ⟨vgx0⟩.\n",
            "FIG. 2. Schematic diagram of magnetic ﬁeld, electron velocity, dx, and length ds.\n",
            "FIG. 3. Plot of function Ee(v∥) = (me/2)(v∥−vrv)2 + K at a ﬁxed x position.\n",
            "FIG. 4. Schematic diagram of electron orbit in (x, y) plane.\n",
            "FIG. 5. Schematic diagram of quantity [e(F −F0) + h0γ0]. In the top panel, this quantity\n",
            "is always positive. In the second one, it is negative in the dip. In the bottom one, it is always\n",
            "negative.\n",
            "FIG. 6. Magnetic ﬁeld proﬁles of an oblique shock at various times.\n",
            "FIG. 7. Snapshots of ﬁeld proﬁles. Electric and magnetic ﬁeld proﬁles at ωpet = 680 are plotted.\n",
            "They are normalized to B0.\n",
            "FIG. 8. Phase space plots of electrons.\n",
            "FIG. 9. Proﬁles of E∥, F, and ϕ at two diﬀerent times. Here, ˜\n",
            "F and ˜\n",
            "ϕ denote eF/(mec2) and\n",
            "eϕ/(mec2), respectively.\n",
            "FIG. 10. Time variations of (x −vsht ), y, z, and γ of electrons. The lengths are normalized to\n",
            "the electron skin depth c/ωpe.\n",
            "FIG. 11. Maximum γ versus propagation angle θ. The propagation speeds for these shocks are\n",
            "vsh ≃2.1vA.\n",
            "FIG. 12.\n",
            "Maximum γ versus shock speed vsh. The propagation angle θ is ﬁxed to be θ = 45◦\n",
            "17\n",
            "\n",
            "z\n",
            "x\n",
            "B\n",
            "-vsh v\n",
            "d\n",
            "<v  >\n",
            "//0\n",
            "y\n",
            "Ey0\n",
            "0\n",
            "\n",
            "v dt\n",
            "g\n",
            "dx\n",
            "ds\n",
            "B\n",
            "\n",
            "vrv\n",
            "v\n",
            "//\n",
            "E  (   )\n",
            "v\n",
            "//\n",
            "e\n",
            "0\n",
            "K\n",
            "\n",
            "y\n",
            "x\n",
            "A\n",
            "B\n",
            "C\n",
            "D\n",
            "E\n",
            "\n",
            "γ\n",
            "e(F-F )+h\n",
            "0\n",
            "0\n",
            "0\n",
            "x\n",
            "x\n",
            "xr\n",
            "xm\n",
            "x\n",
            "xr xm\n",
            "xm\n",
            "xr\n",
            "\n",
            "250\n",
            "375\n",
            "500\n",
            "625\n",
            "ωpet=400\n",
            "2\n",
            "0\n",
            "4\n",
            "500\n",
            "600\n",
            "700\n",
            "800\n",
            "Bz\n",
            "Bz0\n",
            "x/(c/ωpe)\n",
            "\n",
            "-1\n",
            "0\n",
            "1\n",
            "Ex\n",
            "0\n",
            "1\n",
            "2\n",
            "Ey\n",
            "-0.5\n",
            "0\n",
            "0.5\n",
            "Ez\n",
            "-0.5\n",
            "0\n",
            "0.5\n",
            "By\n",
            "0\n",
            "2\n",
            "4\n",
            "525\n",
            "550\n",
            "575\n",
            "600\n",
            "Bz\n",
            "x/(c/ωpe)\n",
            "\n",
            "0\n",
            "50\n",
            "100\n",
            "px\n",
            "0\n",
            "50\n",
            "100\n",
            "py\n",
            "0\n",
            "50\n",
            "100\n",
            "pz\n",
            "0\n",
            "40\n",
            "80\n",
            "120\n",
            "525\n",
            "550\n",
            "575\n",
            "600\n",
            "γ\n",
            "x/(c/ωpe)\n",
            "\n",
            "-0.3\n",
            "0\n",
            "0.3\n",
            "E//\n",
            "ωpet=650\n",
            "B0\n",
            "0\n",
            "10\n",
            "540 555 570\n",
            "x/(c/ωpe)\n",
            "F\n",
            "φ\n",
            "-0.3\n",
            "0\n",
            "0.3\n",
            "ωpet=680\n",
            "525 540 555\n",
            "0\n",
            "10\n",
            "x/(c/ωpe)\n",
            "F\n",
            "φ\n",
            "\n",
            "0\n",
            "50\n",
            "10\n",
            "0\n",
            "-10\n",
            "200\n",
            "100\n",
            "0\n",
            "40\n",
            "20\n",
            "0\n",
            "x-v  t\n",
            "sh\n",
            "y\n",
            "z\n",
            "γ\n",
            "200\n",
            "400\n",
            "600\n",
            "ωpet\n",
            "B CD\n",
            "E\n",
            "Fig.3\n",
            "BCD E\n",
            "\n",
            "20\n",
            "40\n",
            "60\n",
            "80\n",
            "100\n",
            "40\n",
            "45\n",
            "50\n",
            "55\n",
            "60\n",
            "Angle\n",
            "γ\n",
            "\n",
            "50\n",
            "100\n",
            "150\n",
            "0.5\n",
            "0.6\n",
            "0.7\n",
            "0.8\n",
            "vsh/c\n",
            "γ\n",
            "\n"
          ]
        }
      ]
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      "cell_type": "markdown",
      "source": [
        "## Testing **PyPDF2**"
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      "cell_type": "code",
      "source": [
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        "pdf_file"
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      "outputs": []
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        "# testing PyPDF2\n",
        "reader = PdfReader(pdf_file)\n",
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        "    text += page.extract_text() + \"\\n\""
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          "text": [
            "arXiv:physics/0001001v1  [physics.flu-dyn]  1 Jan 2000Under consideration for publication in J. Fluid Mech. 1\n",
            "Capillary-gravity wave transport over\n",
            "spatially random drift\n",
            "By GUILLAUME BAL∗and TOM CHOU†\n",
            "∗Department of Mathematics, University of Chicago, Chicago , IL 60637\n",
            "†Department of Mathematics, Stanford University, Stanford , CA 94305\n",
            "(Received 2 February 2008)\n",
            "We derive transport equations for the propagation of water w ave action in the pres-\n",
            "ence of a static, spatially random surface drift. Using the W igner distribution W(x,k,t)\n",
            "to represent the envelope of the wave amplitude at position xcontained in waves with\n",
            "wavevector k, we describe surface wave transport over static ﬂows consis ting of two length\n",
            "scales; one varying smoothly on the wavelength scale, the ot her varying on a scale com-\n",
            "parable to the wavelength. The spatially rapidly varying bu t weak surface ﬂows augment\n",
            "the characteristic equations with scattering terms that ar e explicit functions of the cor-\n",
            "relations of the random surface currents. These scattering terms depend parametrically\n",
            "on the magnitudes and directions of the smoothly varying dri ft and are shown to give\n",
            "rise to a Doppler coupled scattering mechanism. The Doppler interaction in the presence\n",
            "of slowly varying drift modiﬁes the scattering processes an d provides a mechanism for\n",
            "coupling long wavelengths with short wavelengths. Conserv ation of wave action (CWA),\n",
            "typically derived for slowly varying drift, is extended to s ystems with rapidly varying\n",
            "ﬂow. At yet larger propagation distances, we derive from the transport equations, an\n",
            "equation for wave energy diﬀusion. The associated diﬀusion constant is also expressed\n",
            "in terms of the surface ﬂow correlations. Our results provid e a formal set of equations\n",
            "to analyse transport of surface wave action, intensity, ene rgy, and wave scattering as a\n",
            "function of the slowly varying drifts and the correlation fu nctions of the random, highly\n",
            "oscillatory surface ﬂows.\n",
            "1. Introduction\n",
            "Water wave dynamics are altered by interactions with spatia lly varying surface ﬂows.\n",
            "The surface ﬂows modify the free surface boundary condition s that determine the dis-\n",
            "persion for propagating water waves. The eﬀect of smoothly v arying (compared to the\n",
            "wavelength) currents have been analysed using ray theory (P eregrine (1976), Jonsson\n",
            "(1990)) and the principle of conservation of wave action (CW A) (cf. Longuet-Higgins\n",
            "& Stewart (1961), Mei (1979), White (1999), Whitham (1974) a nd references within).\n",
            "These studies have largely focussed on long ocean gravity wa ves propagating over even\n",
            "larger scale spatially varying drifts. Water waves can also scatter from regions of un-\n",
            "derlying vorticity regions smaller than the wavelength Fab rikant & Raevsky (1994) and\n",
            "Cerda & Lund (1993). Boundary conditions that vary on capill ary length scales, as well\n",
            "as wave interactions with structures comparable to or small er than the wavelength can\n",
            "also lead to wave scattering (Chou, Lucas & Stone (1995), Gou , Messiter & Schultz\n",
            "(1993)), attenuation (Chou & Nelson (1994), Lee et al. (1993)), and Bragg reﬂections\n",
            "(Chou (1998), Naciri & Mei (1988)). Nonetheless, water wave propagation over random\n",
            "2 Bal & Chou\n",
            "static underlying currents that vary on both large and small length scales, and their\n",
            "interactions, have received relatively less attention.\n",
            "In this paper, we will only consider static irrotational cur rents, but derive the transport\n",
            "equations for surface waves in the presence of underlying ﬂo ws that vary on bothlong\n",
            "and short (on the order of the wavelength) length scales. Rat her than computing wave\n",
            "scattering from speciﬁc static ﬂow conﬁgurations (Gerber ( 1993), Trulson & Mei (1993),\n",
            "Fabrikant & Raevsky (1994)), we take a statistical approach by considering ensemble\n",
            "averages over realisations of the static randomness. Diﬀer ent statistical approaches have\n",
            "been applied to wave propagation over a random depth (Elter & Molyneux (1972)), third\n",
            "sound localization in superﬂuid Helium ﬁlms (Kleinert (199 0)), and wave diﬀusion in the\n",
            "presence of turbulent ﬂows (Howe (1973), Rayevskiy (1983), Fannjiang & Ryzhik (1999)).\n",
            "In the next section we derive the linearised capillary-grav ity wave equations to low-\n",
            "est order in the irrotational surface ﬂow. The ﬂuid mechanic al boundary conditions are\n",
            "reduced to two partial diﬀerential equations that couple th e surface height to velocity\n",
            "potential at the free surface. We treat only the “high freque ncy” limit (Ryzhik, Papani-\n",
            "colaou, & Keller (1996)) where wavelengths are much smaller than wave propagation dis-\n",
            "tances under consideration. In Section 3, we introduce the W igner distribution W(x,k,t)\n",
            "which represents the wave energy density and allows us to tre at surface currents that\n",
            "vary simultaneously on two separated length scales. The dyn amical equations developed\n",
            "in section 2 are then written in terms of an evolution equatio n forW. Upon expanding\n",
            "Win powers of wavelength/propagation distance, we obtain tr ansport equations.\n",
            "In Section 4, we present our main mathematical result, equat ion (4.1), an equation\n",
            "describing the transport of surface wave action. Appendix A gives details of some of\n",
            "the derivation. The transport equation includes advection by the slowly varying drift,\n",
            "plus scattering terms that are functions of the correlation s of the rapidly varying drift,\n",
            "representing water wave scattering. Upon simultaneously t reating both smoothly varying\n",
            "and rapidly varying ﬂows using a two-scale expansion, we ﬁnd that scattering from rapidly\n",
            "varying ﬂows depends parametrically on the smoothly varyin g ﬂows. In the Results and\n",
            "Discussion, we discuss the regimes of validity, consider sp eciﬁc forms for the correlation\n",
            "functions, and detail the conditions for doppler coupling. CWA is extended to include\n",
            "rapidly varying drift provided that the correlations of the drift satisfy certain constraints.\n",
            "We also physically motivate the reason for considering two s cales for the underlying\n",
            "drift. In the limit of still larger propagation distances, a fter multiple wave scattering,\n",
            "wave propagation leaves the transport regime and becomes di ﬀusive (Sheng (1995)). A\n",
            "diﬀusion equation for water wave energy is also given, with a n outline of its derivation\n",
            "given in Appendix B.\n",
            "2. Surface wave equations\n",
            "Assume an underlying ﬂow V(x,z)≡(U1(x,z),U2(x,z),Uz(x,z))≡(U(x,z),Uz(x,z)),\n",
            "where the 1,2 components denote the two-dimensional in-pla ne directions. This static\n",
            "ﬂow may be generated by external, time independent sources s uch as wind or inter-\n",
            "nal ﬂows beneath the water surface. The surface deformation due to V(x,z) is denoted\n",
            "¯η(x) where x≡(x,y) is the two-dimensional in-plane position vector. An addit ional\n",
            "variation in height due to the velocity v(x,z) associated with surface waves is denoted\n",
            "η(x,t). When all ﬂows are irrotational, we can deﬁne their associa ted velocity poten-\n",
            "tialsV(x,z)≡(∇x+ˆz∂z)Φ(x,z) andv(x,z,t)≡(∇x+ˆz∂z)ϕ(x,z,t). Incompressibility\n",
            "requires\n",
            "∆ϕ(x,z,t) +∂2\n",
            "zϕ(x,z,t) = ∆Φ( x,z) +∂2\n",
            "zΦ(x,z) = 0, (2.1)\n",
            "Water wave transport 3\n",
            "+ε−y\n",
            "|y|~1/k\n",
            "x1x2\n",
            "O(1)X\n",
            "-+y\n",
            "O(L)x\n",
            "x\n",
            "Figure 1. The relevant scales in water wave transport. Initially, the system size, observation\n",
            "point, and length scale of the slowly varying drift is O(L), with surface wave wavelength and\n",
            "scale of the random surface current of O(1). Upon rescaling, the system size becomes O(1), while\n",
            "the wavelength and random ﬂow variations are O(ε).\n",
            "where ∆ = ∇2\n",
            "xis the two-dimensional Laplacian. The kinematic condition (Whitham\n",
            "(1974)) applied at z= ¯η(x) +η(x,t)≡ζ(x,t) is\n",
            "∂tη(x,t) +U(x,ζ)· ∇xζ(x,t) =Uz(x,z=ζ) +∂zϕ(x,z=ζ,t). (2.2)\n",
            "Upon expanding (2.2) to linear order in ηandϕabout the static free surface, the right\n",
            "hand side becomes\n",
            "Uz(x,ζ) +∂zϕ(x,ζ,t) =Uz(x,¯η) +η(x,t)∂zUz(x,¯η) +∂zϕ(x,¯η,t) +O(η2).(2.3)\n",
            "At the static surface ¯ η,U(x,¯η)·∇x¯η(x) =Uz(x,¯η). Now assume that the underlying ﬂow\n",
            "is weak enough such that Uz(x,z≈0) and ¯ηare both small. A rigid surface approximation\n",
            "is appropriate for small Froude numbers U2/c2\n",
            "φ∼ |∇x¯η|2∼Uz(x,0)/|U(x,0)| ≪1 (cφ\n",
            "is the surface wave phase velocity) when the free surface bou ndary conditions can be\n",
            "approximately evaluated at z= 0 (Fabrikant & Raevsky (1994)). Although we have\n",
            "assumedUz(x,z≈0) =∂zΦ(x,z≈0)≈0 and a vanishing static surface deformation\n",
            "¯η(x)≈0,∇x·U(x,0) =−∂zUz(x,0)̸= 0.\n",
            "Combining the above approximations with the dynamic bounda ry conditions (derived\n",
            "from balance of normal surface stresses at z= 0 (Whitham (1974))), we have the pair of\n",
            "coupled equations\n",
            "∂tη(x,t) +∇x·(U(x,z= 0)η(x,t)) = lim\n",
            "z→0−∂zϕ(x,z,t)\n",
            "lim\n",
            "z→0−[ρ∂tϕ(x,z,t) +ρU(x,z)· ∇xϕ(x,z,t)] =σ∆η(x,t)−ρgη(x,t)(2.4)\n",
            "whereσandgare the air-water surface tension and gravitational accele ration, respec-\n",
            "tively. Although it is straightforward to expand to higher o rders in ¯η(x) andη(x,t), or to\n",
            "include underlying vorticity, we will limit our study to equ ations (2.4) in order to make\n",
            "the development of the transport equations more transparen t.\n",
            "The typical system size, or distance of wave propagation sho wn in Fig. 1 is of O(L) with\n",
            "4 Bal & Chou\n",
            "L≫1. Wavelengths however, are of O(1). To implement our high frequency (Ryzhik,\n",
            "Papanicolaou, & Keller (1996)) asymptotic analyses, we res cale the system such that all\n",
            "distances are measured in units of L≡ε−1. We eventually take the limit ε→0 as an\n",
            "approximation for small, ﬁnite ε. Surface velocities, potentials, and height displacement s\n",
            "are now functions of the new variables x→x/ε,z→z/εandt→t/ε. We shall further\n",
            "nondimensionalise all distances in terms of the capillary l engthℓc=√\n",
            "σ/gρ. Time,\n",
            "velocity potentials, and velocities are dimensionalised i n units of√\n",
            "ℓc/g,√\n",
            "gℓ3c, and√gℓc\n",
            "respectively, e.g.for water,U= 1 corresponds to a surface drift velocity of ∼16.3cm/s.\n",
            "SinceUz(x,z≈0)≈0, we deﬁne the ﬂow at the surface by\n",
            "U(x,z= 0)≡U(x) +√εδU(x/ε). (2.5)\n",
            "In these rescaled coordinates, U(x) denotes surface ﬂows varying on length scales of\n",
            "O(1) much greater than a typical wavelength, while δU(x/ε) varies over lengths of O(ε)\n",
            "comparable to a typical wavelength. The amplitude of the slowly varying ﬂow U(x) is\n",
            "O(ε0), while that of the rapidly varying ﬂow δU(x/ε), is assumed to be of O(√ε). A more\n",
            "detailed discussion of the physical motivation for conside ring the√εscaling is deferred\n",
            "to the Results and Discussion. After rescaling, the boundar y conditions (2.4) evaluated\n",
            "atz= 0 become\n",
            "∂tη(x,t) +∇x·[(\n",
            "U(x) +√εδU(x/ε))\n",
            "η(x)]\n",
            "= lim\n",
            "z→0−∂zϕ(x,0)\n",
            "∂tϕ(x,t) +U(x)· ∇xϕ(x,t) +√εδU(x/ε)· ∇xϕ(x,t) =ε∆η(x,t)−ε−1η(x,t).\n",
            "(2.6)\n",
            "Although drift that varies slowly along one wavelength can b e treated with characteristics\n",
            "and WKB theory, random ﬂows varying on the wavelength scale r equire a statistical\n",
            "approach. Without loss of generality, we choose δUto have zero mean and an isotropic\n",
            "two-point correlation function ⟨δUi(x)δUj(x′)⟩ ≡Rij(|x−x′|), where (i,j) = (1,2) and\n",
            "⟨...⟩denotes an ensemble average over realisations of δU(x).\n",
            "We now deﬁne the spatial Fourier decompositions for the dyna mical wave variables\n",
            "ϕ(x,−h⩽z⩽ζ,t) =∫\n",
            "qϕ(q,t)e−iq·xcoshq(h+z)\n",
            "coshqh, η(x,t) =∫\n",
            "qη(q,t)e−iq·x,\n",
            "(2.7)\n",
            "the static surface ﬂows\n",
            "U(x) =∫\n",
            "qU(q)e−iq·x, δU(x\n",
            "ε)\n",
            "=∫\n",
            "qδU(q)e−iq·x/ε, (2.8)\n",
            "and the correlations\n",
            "Rij(x) =∫\n",
            "qRij(q)e−iq·x, (2.9)\n",
            "where q= (q1,q2) is an in-plane two dimensional wavevector, q≡ |q|=√\n",
            "q2\n",
            "1+q2\n",
            "2, and∫\n",
            "q≡(2π)−2∫\n",
            "dq1dq2. The Fourier integrals for ηexclude q= 0 due to the incompress-\n",
            "ibility constraint∫\n",
            "xη(x,t) = 0, while the q= 0 mode for ϕgives an irrelevant constant\n",
            "shift to the velocity potential. Note that ϕin (2.7) manifestly satisﬁes (2.1). Substituting\n",
            "Water wave transport 5\n",
            "(2.8) into the boundary conditions (2.4), we obtain,\n",
            "∂tη(k,t)−i∫\n",
            "qη(k−q)U(q)·k−i√ε∫\n",
            "qη(k−q/ε)δU(q)·k=ϕ(k,t)ktanhεkh\n",
            "∂tϕ(k,t)−i∫\n",
            "qU(q)·(k−q)ϕ(k−q)−i√ε∫\n",
            "qδU(q)·(k−q/ε)ϕ(k−q/ε)\n",
            "=−(εk2+ε−1)η(k).\n",
            "(2.10)\n",
            "where theδU(q) are correlated according to\n",
            "⟨δUi(p)δUj(q)⟩=Rij(|p|)δ(p+q). (2.11)\n",
            "Since the correlation Rij(x) is symmetric in i↔j, and depends only upon the magnitude\n",
            "|x|,Rij(|p−q|) is real.\n",
            "In the case where δU= 0 and U(x)≡U0is strictly uniform, equations (2.10) can\n",
            "be simpliﬁed by assuming a e−iωtdependence for all dynamical variables. Uniform drift\n",
            "yields the familiar capillary-gravity wave dispersion rel ation\n",
            "ω(k) =√\n",
            "(k3+k)tanhkh+U0·k≡Ω(k) +U0·k. (2.12)\n",
            "However, for what follows, we wish to derive transport equat ions for surface waves (action,\n",
            "energy, intensity) in the presence of a spatially varying dr ift containing two length scales:\n",
            "U=U(x) +√εδU(x/ε).\n",
            "3. The Wigner distribution and asymptotic analyses\n",
            "The intensity of the dynamical wave variables can be represe nted by the product of two\n",
            "Green functions evaluated at positions x±εy/2. The diﬀerence in their evaluation points,\n",
            "εy, resolves the waves of wavevector |k| ∼2π/(εy). Elter & Molyneux (1972) used this\n",
            "representation to study shallow water wave propagation ove r a random bottom. However,\n",
            "for the arbitrary depth surface wave problem, where the Gree n function is not simple,\n",
            "and where two length scales are treated, it is convenient to u se the Fourier representation\n",
            "of the Wigner distribution (Wigner (1932), G´ erard et al. (1 997), Ryzhik, Papanicolaou,\n",
            "& Keller (1996)).\n",
            "Deﬁneψ= (ψ1,ψ2)≡(η(x),ϕ(x,z= 0)) and the Wigner distribution:\n",
            "Wij(x,k,t)≡(2π)−2∫\n",
            "eik·yψi(\n",
            "x−εy\n",
            "2,t)\n",
            "ψ∗\n",
            "j(\n",
            "x+εy\n",
            "2,t)\n",
            "dy (3.1)\n",
            "where xis a central ﬁeld point from which we consider two neighbouri ng points x±εy\n",
            "2,\n",
            "and their intervening wave ﬁeld. Fourier transforming the xvariable using the deﬁnition\n",
            "(2.7) we ﬁnd,\n",
            "Wij(p,k,t) = (2πε)−2ψi(p\n",
            "2−k\n",
            "ε,t)\n",
            "ψ∗\n",
            "j(\n",
            "−p\n",
            "2−k\n",
            "ε,t)\n",
            ". (3.2)\n",
            "The total wave energy, comprising gravitational, kinetic, and surface tension contribu-\n",
            "6 Bal & Chou\n",
            "tions is\n",
            "E=1\n",
            "2∫\n",
            "x[\n",
            "|∇xη(x)|2+|η(x)|2]\n",
            "+1\n",
            "2∫\n",
            "x∫0\n",
            "−hdz|U(x,z) +ˆ zUz(x,z) +v(x,z)|2\n",
            "−1\n",
            "2∫\n",
            "x∫0\n",
            "−hdz|U(x,z) +ˆ zUz(x,z)|2\n",
            "=1\n",
            "2∫\n",
            "k(k2+ 1)|η(k)|2+ktanhkh|ϕ(k,z= 0)|2.(3.3)\n",
            "The energy above has been expanded to an order in η(x,t) andϕ(x,z,t) consistent with\n",
            "the approximations used to derive (2.4). In arriving at the l ast equality in (3.3), we have\n",
            "integrated by parts, used the Fourier decompositions (2.7) and imposed an impenetrable\n",
            "bottom condition at z=−h. The wave energy density carried by wavevector kis (G´ erard\n",
            "et al. (1997))\n",
            "E(k,t) =1\n",
            "2Tr [A(k)W(k,t)], (3.4)\n",
            "whereA11(k) =k2+1,A22(k) =ktanhkh,A 12=A21= 0. Thus, the Wigner distribution\n",
            "epitomises the local surface wave energy density.\n",
            "In the presence of slowly varying drift, we identify W(x,k,t) as the localWigner\n",
            "distribution at position xrepresenting waves of wavevector k. The time evolution of its\n",
            "Fourier transform W(p,k,t), can be derived by considering time evolution of the vector\n",
            "ﬁeldψimplied by the boundary conditions (2.4):\n",
            "˙ψj(k,t) +iLjℓ(k)ψℓ(k,t) =i∫\n",
            "qU(q)·(k−qδj2)ψj(k−q,t)\n",
            "+i√ε∫\n",
            "qδU(q)·(k−qδj2/ε)ψj(k−q/ε,t),(3.5)\n",
            "where the operator L(k) is deﬁned by\n",
            "L(k) =\n",
            "0i|k|tanhε|k|h\n",
            "−i(εk2+ε−1) 0\n",
            ". (3.6)\n",
            "We have redeﬁned the physical wavenumber to be k/εso thatk∼O(1). Upon using\n",
            "(3.5) and the deﬁnition (3.2), (see Appendix A)\n",
            "˙Wij(p,k,t) =iWiℓ(p,k,t)L†\n",
            "ℓj(k\n",
            "ε+p\n",
            "2)\n",
            "−iLiℓ(k\n",
            "ε−p\n",
            "2)\n",
            "Wℓj(p,k,t)\n",
            "+i∫\n",
            "qU(q)·(\n",
            "−k\n",
            "ε+p\n",
            "2−qδi2)\n",
            "Wij(p−q,k+εq/2,t)\n",
            "−i∫\n",
            "qU(q)·(\n",
            "−k\n",
            "ε−p\n",
            "2+qδj2)\n",
            "Wij(p−q,k−εq/2,t)\n",
            "+i√ε∫\n",
            "qδU(q)·(\n",
            "−k\n",
            "ε+p\n",
            "2−q\n",
            "εδi2)\n",
            "Wij(p−q/ε,k+q/2,t)\n",
            "−i√ε∫\n",
            "qδU(q)·(\n",
            "−k\n",
            "ε−p\n",
            "2+q\n",
            "εδj2)\n",
            "Wij(p−q/ε,k−q/2,t),(3.7)\n",
            "where only the index ℓ= 1,2 has been summed over. If we now assume that W(x,k,t)\n",
            "can be expanded in functions that vary independently at the t wo relevant length scales,\n",
            "Water wave transport 7\n",
            "functions of the ﬁeld p(dual to x) can be replaced by functions of a slow variation in p\n",
            "and a fast oscillation ξ/ε;p→p+ξ/ε.\n",
            "This amounts to the Fourier equivalent of a two-scale expans ion in which xis replaced\n",
            "byxandy=x/ε(Ryzhik, Papanicolaou, & Keller (1996)). The two new indepe ndent\n",
            "wavevectors pandξare both of O(1). Expanding the Wigner distribution in powers of√εand using p→p+ξ/ε,\n",
            "W(p,k,t)→W0(p,ξ,k,t) +√εW1/2(p,ξ,k,t) +εW1(p,ξ,k,t) +O(ε3/2),(3.8)\n",
            "we expand each quantity appearing in (3.7) in powers of√εand equate like powers.\n",
            "Upon expanding the oﬀ-diagonal operator L(−k/ε+p/2) =ε−1L0(k)+L1(k,p)+O(ε),\n",
            "where\n",
            "L0(k) =\n",
            "0iktanhkh\n",
            "−i(k2+ 1) 0\n",
            ",L1(k,p)≡\n",
            "0ip·kf(k)\n",
            "ip·k 0\n",
            "(3.9)\n",
            "and\n",
            "f(k)≡ −hk+ sinhkhcoshkh\n",
            "2kcosh2kh. (3.10)\n",
            "3.1.Orderε−1terms\n",
            "The terms of O(ε−1) in (3.7) are\n",
            "W0(p,ξ,k,t)L†\n",
            "0(k+)−L0(k−)W0(p,ξ,k,t) = 0,k±≡k±ξ\n",
            "2(3.11)\n",
            "To solve (3.11), we use the eigenvalues and normalised eigen vectors for L0and its complex\n",
            "adjoint L†\n",
            "0,\n",
            "τΩ(k)−iγ,bτ=\n",
            "iτ√\n",
            "α(k)/2\n",
            "1√\n",
            "2α(k)\n",
            ";τΩ(k) +iγ,cτ=\n",
            "iτ√\n",
            "2α(k)\n",
            "√\n",
            "α(k)/2\n",
            ",(3.12)\n",
            "whereα(k)≡Ω(k)\n",
            "k2+ 1,τ=±1, andiγ→0 is a small imaginary term. A W0(p,ξ,k,t)\n",
            "that manifestly satisﬁes (3.12) can be constructed by expan ding in the basis of 2 ×2\n",
            "matrices composed from the eigenvectors:\n",
            "W0(p,ξ,k,t) =∑\n",
            "τ,τ′=±aττ′(p,k,t)bτ(k−)b†\n",
            "τ′(k+).(3.13)\n",
            "Right[left] multiplying (3.11) (using (3.13)) by the eigen vectors of the adjoint prob-\n",
            "lem,cτ(k−)[\n",
            "c†\n",
            "τ(k+)]\n",
            ", we ﬁnd that a+−=a−+= 0, anda−−(x,k,t)≡a−(x,k,t) =\n",
            "a++(x,−k,t)≡a+(x,−k,t). Furthermore, a+,a−̸= 0 only ifξ= 0. Thus W0has the\n",
            "form\n",
            "W0(p,ξ,k,t) =W0(p,k,t)δ(ξ). (3.14)\n",
            "From the deﬁnition of W0, we see that the (1,1) component of W0is the local envelop\n",
            "of the ensemble averaged wave intensity |η(x,k,t)|2≃a+(x,k,t)α(k). Similarly, from the\n",
            "energy (Eq. (3.4)), we see immediately that the local ensemb le averaged energy density\n",
            "⟨E(x,k,t)⟩=A11(k)α(k)⟨a(x,k,t)⟩+A22(k)⟨a(x,k,t)⟩\n",
            "= Ω(k)⟨a(x,k,t)⟩.(3.15)\n",
            "8 Bal & Chou\n",
            "Therefore, since the starting dynamical equations are line ar, we can identify ⟨a(x,k,t)⟩as\n",
            "the ensemble averaged local wave action associated with wav es of wavevector k(Henyey\n",
            "et al. (1988)). The wave action ⟨a(x,k,t)⟩, rather than the energy density ⟨E(x,k,t)⟩\n",
            "is the conserved quantity (Longuet-Higgins & Stewart (1961 ), Mei (1979), Whitham\n",
            "(1974)).\n",
            "The physical origin of γarises from causality, but can also be explicitly derived fr om\n",
            "considerations of an inﬁnitesimally small viscous dissipa tion (Chou, Lucas & Stone\n",
            "(1995)). Although we have assumed γ→0, for our model to be valid, the viscosity\n",
            "need only be small enough such that surface waves are not atte nuated before they have\n",
            "a chance to multiply scatter and enter the transport or diﬀus ion regimes under consid-\n",
            "eration. This constraint can be quantiﬁed by noting that in t he frequency domain, wave\n",
            "dissipation is given by γ= 2νk2(Landau (1985)) where νis the kinematic viscosity and\n",
            "cg(k)≡ |∇kΩ(k)| (3.16)\n",
            "is the group velocity. The corresponding decay length k−1\n",
            "d∼cg(k)/(νk2) must be greater\n",
            "than the relevant wave propagation distance. Therefore, we require\n",
            "ε2cg(k/ε)\n",
            "νk2≫(1,ε−1) (3.17)\n",
            "for (transport, diﬀusion) theories to be valid. The inequal ity (3.17) gives an upper bound\n",
            "for the viscosity\n",
            "νk2≪(εcg(k/ε),ε2cg(k/ε)) (3.18)\n",
            "which is most easily satisﬁed in the shallow water wave regim e for transport. Otherwise\n",
            "we must at least require ν <o(√ε). The upper bounds for ν(and hence γ) given above\n",
            "provide one criterion for the validity of transport theory.\n",
            "3.2.Orderε−1/2terms\n",
            "Collecting terms in (3.7) of order ε−1/2, we obtain\n",
            "W1/2(p,ξ,k,t)L†\n",
            "0(k+)−L0(k−)W1/2(p,ξ,k,t) +∫\n",
            "qU(q)·ξW1/2(p−q,ξ−q,k,t)\n",
            "−∫\n",
            "qδU(q)·k−W0(p,ξ−q,k+q/2,t) +∫\n",
            "qδU(q)·k+W0(p,ξ−q,k−q/2,t)\n",
            "−∫\n",
            "qδU(q)·q[W0(p,ξ−q,k+q/2,t)S+SW0(p,ξ−q,k−q/2,t)] = 0\n",
            "(3.19)\n",
            "where S=[\n",
            "0 0\n",
            "0 1]\n",
            ".\n",
            "Similarly decomposing W1/2in the basis matrices composed of bτ(k−)b†\n",
            "τ′(k+) (as in\n",
            "3.13), substituting W0(p,0,k,t)δ(ξ) from (3.13 into (3.19), and inverse Fourier trans-\n",
            "forming in the slow variable p, we obtain\n",
            "W1/2(x,k,ξ,t) =∑\n",
            "τ,τ′=±δU(ξ)·Γτ,τ′(x,ξ,k,t)bτ(k−)b†\n",
            "τ′(k+)\n",
            "τ′Ω(k+)−τΩ(k−) +U(x)·ξ+ 2iγ, (3.20)\n",
            "Water wave transport 9\n",
            "where\n",
            "Γτ,τ′(x,ξ,k,t)≡k−aτ′(x,k+,t)c†\n",
            "τ(k−)bτ′(k+)−k+aτ(x,k−,t)b†\n",
            "τ(k−)cτ′(k+)\n",
            "+ξ\n",
            "2∑\n",
            "µ=±[\n",
            "aµ(x,k+,t)c†\n",
            "τ(k−)bµ(k+) +aµ(x,k−,t)b†\n",
            "µ(k−)cτ′(k+)]\n",
            ".\n",
            "(3.21)\n",
            "3.3.Orderε0terms\n",
            "The terms of order ε0in (3.7) read\n",
            "˙W0(p,k,t) =iW0(p,k,t)L†\n",
            "1(−p)−iL1(p)W0(p,k,t)−i∫\n",
            "qk·U(q)q· ∇kW0(p−q,ξ,k,t)\n",
            "+i∫\n",
            "qU(q)·pW0(p−q,ξ,k,t)−i∫\n",
            "qU(q)·q[SW0(p−q,ξ,k,t) +W0(p−q,ξ,k,t)S]\n",
            "+i∫\n",
            "qδU(q)·k+W1/2(p,ξ−q,k−q/2,t)−i∫\n",
            "qδU(q)·k−W1/2(p,ξ−q,k+q/2,t)\n",
            "−∫\n",
            "qδU(q)·q[\n",
            "SW1/2(p,ξ−q,k+q/2,t) +W1/2(p,ξ−q,k−q/2,t)S]\n",
            "+iW1L†\n",
            "0−iL0W1+∫\n",
            "qU(q)·ξW1(p−q,ξ,k,t).\n",
            "(3.22)\n",
            "To obtain an equation for the statistical ensemble average ⟨a+(x,k,t)⟩, we multiply\n",
            "(3.22) by c†\n",
            "+(k) on the left and by c+(k) on the right and substitute W1/2from equation\n",
            "(3.20). We obtain a closed equation for a(x,k,t)≡ ⟨a+(x,k,t)⟩(we henceforth suppress\n",
            "the⟨...⟩notation for a(x,k,t) andE(x,k,t)) by truncating terms containing W1. Clearly,\n",
            "from (3.12), c†\n",
            "+(k)(iW1L†\n",
            "0−iL0W1)c+(k) = 0. Furthermore, we assume ⟨ξW1(p−\n",
            "q,ξ,k,t)⟩ ≈0 which follows from ergodicity of dynamical systems, and ha s been used in\n",
            "the propagation of waves in random media (see Ryzhik, Papani colaou, & Keller (1996),\n",
            "Bal et al. (1999)). The transport equations resulting from t his truncation are rigorously\n",
            "justiﬁed in the scalar case (Spohn (1977), Erd¨ os & Yau (1998 )).\n",
            "4. The surface wave transport equation\n",
            "The main mathematical result of this paper, an evolution equ ation for the ensemble\n",
            "averaged wave action a(x,k,t) follows from equation (3.22) above (cf. Appendix A) and\n",
            "reads,\n",
            "˙a(x,k,t) +∇kω(x,k)· ∇xa(x,k,t)− ∇xω(x,k)· ∇ka(x,k,t)\n",
            "=−Σ(k)a(x,k,t) +∫\n",
            "qσ(q,k)a(x,q,t),(4.1)\n",
            "where\n",
            "ω(x,k) =√\n",
            "(k3+k)tanhkh+U(x)·k≡Ω(k) +U(x)·k. (4.2)\n",
            "The left hand side in (4.1) corresponds to wave action propag ation in the absence of\n",
            "random ﬂuctuations. It is equivalent to the equations obtai ned by the ray theory, or\n",
            "a WKB expansion (see section 5.1). The two terms on the right h and side of (4.1)\n",
            "represent refraction, or “scattering” of wave action out of and into waves with wavevector\n",
            "10 Bal & Chou\n",
            "krespectively. In deriving (4.1) we have inverse Fourier tra nsformed back to the slow ﬁeld\n",
            "point variable x, and used the relation ( α(k)−f(k)α−1(k))k≡ ∇kΩ(k). To obtain (4.1),\n",
            "we assumed Rij(q)qi=Rij(q)qj= 0, which would always be valid for divergence-free\n",
            "ﬂows in two dimensions. Although the perturbation δUis not divergence-free in general,\n",
            "∇·δU(x,z= 0) = −∂zδUz(x,0)̸= 0, using symmetry considerations, we will show in\n",
            "section 5.2 that Rij(q)qi=Rij(q)qj= 0.\n",
            "Explicitly, the scattering rates are\n",
            "Σ(k)≡2π∫\n",
            "qqiRij(q−k)kj∑\n",
            "τ=±b†\n",
            "+(k)cτ(q)b†\n",
            "τ(q)c+(k)δ(τω(x,τq)−ω(x,k))\n",
            "σ(q,k)≡2π∑\n",
            "τ=±τqiRij(τq−k)kj|b†\n",
            "τ(τq)c+(k)|2δ(τω(x,q)−ω(x,k))\n",
            "(4.3)\n",
            "where\n",
            "b†\n",
            "+(k)cτ(q)b†\n",
            "τ(q)c+(k) =(τα(k) +α(q))(τα(q) +α(k))\n",
            "4α(k)α(q)\n",
            "|b†\n",
            "τ(k)cτ′(q)|2=(τα(q) +α(k))2\n",
            "4α(k)α(q).(4.4)\n",
            "Physically, Σ( k) is a decay rate arising from scattering of action out of wave vector k.\n",
            "The kernel σ(q,k) represents scattering of action from wavevector qintoaction with\n",
            "wavevector k. Note that the slowly varying drift U(x) enters parametrically in the scat-\n",
            "tering viaω(x,k) in theδ−function supports. The arguments ω(x,k) in theδ−functions\n",
            "mean that we can consider the transport of waves of each ﬁxed f requencyω0≡ω(x,k)\n",
            "independently.\n",
            "The typical distance travelled by a wave before it is signiﬁc antly redirected is deﬁned\n",
            "by the mean free path\n",
            "ℓmfp=cg(k)\n",
            "Σ(k)∼O(1). (4.5)\n",
            "The mean free path described here carries a diﬀerent interpr etation from that considered\n",
            "in weakly nonlinear, or multiple scattering theories (Zakh arov, L’vov & Falkovich (1992))\n",
            "where one treats a low density of scatterers. Rather than str ong, rare scatterings over\n",
            "every distance ℓmfp∼O(1), we have considered constant, but weak interaction with\n",
            "an extended, random ﬂow ﬁeld. Although here, each scatterin g isO(ε) and weak, over\n",
            "a distance of O(1), approximately ε−1interactions arise, ultimately producing ℓmfp∼\n",
            "O(1).\n",
            "5. Results and Discussion\n",
            "We have derived transport equations for water wave propagat ion interacting with\n",
            "static, random surface ﬂows containing two explicit length scales. We have further as-\n",
            "sumed that the amplitude of δUscales asεβwithβ= 1/2: The random ﬂows are\n",
            "correspondingly weakened as the high frequency limit is tak en. Since scattering strength\n",
            "is proportional to the power spectrum of the random ﬂows and i s quadratic in δU, the\n",
            "mean free path can be estimated heuristically by ℓmfp∼cg(k)/Σ(k)ε1−2β. Forβ >1/2,\n",
            "the scattering is too weak and the mean free path diverges. In this limit, waves are nearly\n",
            "freely propagating and can be described by the slowly varyin g ﬂows alone, or WKB the-\n",
            "ory. Ifβ <1/2,ℓmfp→0 and the scattering becomes so frequent that over a propagat ion\n",
            "distance of O(1), the large number of scatterings lead to diﬀusive (cf. Se ction 5.4) be-\n",
            "Water wave transport 11\n",
            "haviour (Sheng (1995)). Therefore, only random ﬂows that ha ve the scaling β= 1/2\n",
            "contribute to the wave transport regime.\n",
            "We also note that β > 0 precludes any wave localisation phenomena. In a two-\n",
            "dimensional random environment, the localisation length o ver which wave diﬀusion is\n",
            "inhibited is approximately (Sheng (1995))\n",
            "ℓloc∼ℓmfpexp(\n",
            "ε−1kℓmfp)\n",
            "∼ε1−2βexp(\n",
            "ε−2β)\n",
            ". (5.1)\n",
            "As long as the random potential is scaled weaker ( β >0),ℓloc→ ∞, and strong lo-\n",
            "calisation will not take hold. In the following subsections , we systematically discuss the\n",
            "salient features of water wave transport contained in Eq. (4 .1) and derive wave diﬀusion\n",
            "for propagation distances ≳O(1).\n",
            "5.1.Slowly varying drift: U(x)̸= 0,δU= 0\n",
            "First consider the case where surface ﬂows vary only on scale s much larger than the\n",
            "longest wavelength 2 π/kconsidered, i.e.,δU= 0. The left-hand side in (4.1) represents\n",
            "wave action transport over slowly varying drift and may desc ribe short wavelength modes\n",
            "propagating over ﬂows generated by underlying long ocean wa ves.\n",
            "We ﬁrst demonstrate that the nonscattering terms of the tran sport equation (4.1) is\n",
            "equivalent to the results obtained by ray theory (WKB expans ion) and conservation of\n",
            "wave action (CWA) (Longuet-Higgins & Stewart (1961), Mei (1 979), Peregrine (1976),\n",
            "White (1999), Whitham (1974)). Assume the WKB expansion (Ke ller (1958), Bender &\n",
            "Orszag (1978))\n",
            "ηε=Aη(x,t)eiS(x,t)/εandϕε=Aϕ(x,t)eiS(x,t)/ε, (5.2)\n",
            "with smoothly varying AηandAϕ. Upon using the above ansatz in (3.1) and setting\n",
            "ε→0, we have a(x,k,t) =|A|2(x,t)δ(k− ∇xS(x,t)) where |A|2= 2α(k)|Aϕ|2=\n",
            "2α−1(k)|Aη|2. Substitution of this expression for a(x,k,t) into (4.1), we obtain the fol-\n",
            "lowing possible equations for S(x,t) and|A|2(x,t)\n",
            "∂tS+ω(x,∇xS) = 0, (5.3)\n",
            "∂t|A|2(x,t) +∇x·(\n",
            "|A|2∇kω(x,∇xS))\n",
            "= 0. (5.4)\n",
            "The ﬁrst equation is the eikonal equation, while the second e quation is the wave action\n",
            "amplitude equation. Recalling that |Aη|2=α(k)|A|2/2, we obtain the following transport\n",
            "equation for the height amplitude:\n",
            "∂t(|Aη|2\n",
            "α(∇xS))\n",
            "+∇x·(|Aη|2\n",
            "α(∇xS)∇kω(x,∇xS))\n",
            "= 0. (5.5)\n",
            "Equation (5.5) is the same as Eq. (8) of White (1999), except t hat his ¯Ω is replaced here\n",
            "withαdue to our inclusion of surface tension.\n",
            "Wave action conservation can be understood by noting that\n",
            "d\n",
            "dta(X(t),K(t),t) = 0, (5.6)\n",
            "where the characteristics ( X(t),K(t)) satisfy the Hamilton equations\n",
            "dX(t)\n",
            "dt=∇kω(X(t),K(t)), anddK(t)\n",
            "dt=−∇xω(X(t),K(t)). (5.7)\n",
            "The solutions to the ordinary diﬀerential equations (5.7) a re the characteristic curves\n",
            "used to solve (5.3) and (5.4) (Courant & Hilbert (1962)).\n",
            "12 Bal & Chou\n",
            "5.2.Correlation functions and conservation laws\n",
            "We now consider the case where δU̸= 0. The scattering rates deﬁned by (4.3) depend\n",
            "upon the precise form of the random ﬂow correlation Rij. There are actually six additional\n",
            "terms in (4.3) in the calculation of σand Σ, which vanish because\n",
            "2∑\n",
            "j=1Rij(q)qj= 0 for i= 1,2. (5.8)\n",
            "We prove relation (5.8) provided that δUz(k,kz) andδUz(k,−kz) have the same proba-\n",
            "bility distribution. Thus,\n",
            "⟨δUi(p,pz)δUz(k,kz)⟩=⟨δUi(p,pz)δUz(k,−kz)⟩ (5.9)\n",
            "This symmetry condition is reasonable, and is compatible wi th the divergence-free con-\n",
            "dition forδUin three dimensions. We show that Hypothesis (5.9) implies ( 5.8) by ﬁrst\n",
            "using incompressibility∑2\n",
            "j=1δUj(k,kz)kj+δUz(k,kz)kz= 0:\n",
            "2∑\n",
            "j=1δ(p+k)Rij(k)kj=2∑\n",
            "j=1⟨δUi(p,0)δUj(k,0)kj⟩\n",
            "=2∑\n",
            "j=1⟨δUi(p,0)∫\n",
            "kzδUj(k,kz)kj⟩\n",
            "=−∫∞\n",
            "−∞⟨δUi(p,0)δUz(k,kz)⟩kzdkz\n",
            "= 0,\n",
            "where the last equality follows from (5.9). Thus, (5.8) is ve riﬁed, and (4.3) derived.\n",
            "The formRij(|q|)qi=Rij(|q|)qj= 0, requires the correlation function to be transverse:\n",
            "Rij(|q|) =R(q)[\n",
            "δij−qiqj\n",
            "q2]\n",
            ", (5.10)\n",
            "whereR(q) is a scalar function of q. The correlation kernels in the scattering integrals\n",
            "can now be written as\n",
            "qiRij(|τq−k|)kj=R(|τq−k|)[\n",
            "q·k−q·(τq−k)k·(τq−k)\n",
            "|τq−k|2]\n",
            "=τR(|τq−k|)\n",
            "|τq−k|2q2k2sin2θ(5.11)\n",
            "whereθdenotes the angle between qandk. The scattering must also satisfy the support\n",
            "of theδ-functions; for U(x) = 0 only |q|=|k|satisfy the the δ−function constraints. In\n",
            "the presence of slowly varying drift, the evolution of a(x,|k| ̸=|q|) can “doppler” couple\n",
            "toa(x,q,t).\n",
            "It is straightforward to show from the explicit expressions (4.3) that\n",
            "Σ(k) =∫\n",
            "qσ(k,q). (5.12)\n",
            "This relation indicates that the scattering operator on the right hand side of (4.1) is\n",
            "conservative: Integrating (4.1) over the whole phase space yields\n",
            "d\n",
            "dt∫\n",
            "x∫\n",
            "ka(x,k,t) = 0. (5.13)\n",
            "Water wave transport 13\n",
            "Figure 2. (a). Contour plot of ω(q). Each grayscale corresponds to a diﬀerent constant value\n",
            "ofω(q) =ω(k)≡ω0. (b). The band of qthat satisﬁes 0 .625< ω0<0.6625. Wavevectors qand\n",
            "kthat lie in this band can couple a(x,k, t) toa(x,q, t) via wave scattering.\n",
            "Equation (5.13) is the generalization of CWA to include scat tering of action from rapidly\n",
            "varying random ﬂows δU(x/ε). Although a(x,k,t) is conserved, the total water wave\n",
            "energy E(x,k,t) = Ω( k)a(x,k,t) will not be conserved. For example, if U(x) is small\n",
            "enough such that the δ−function in the σ(q,k) integral is triggered only when τ= +1,\n",
            "d\n",
            "dtE=d\n",
            "dt∫\n",
            "x∫\n",
            "k[ω(x,k)−k·U(x)]a(x,k,t) =−d\n",
            "dt∫\n",
            "x∫\n",
            "kk·U(x)a(x,k,t)̸= 0.(5.14)\n",
            "This nonconservation results from the energy that must be su pplied in order to sustain\n",
            "the stationary underlying ﬂow. For small U(x), the quantity ω(x,k)a(x,k,t) is conserved.\n",
            "In that case, the evolution of ω(x,k)a(x,k,t) obeys an equation identical to (4.1). When\n",
            "there is doppler coupling with τ=−1, an additional term arises and ω(x,k)a(x,k,t) is\n",
            "no longer conserved under scattering.\n",
            "5.3.Doppler coupled scattering\n",
            "In addition to the correlation functions, the wave action sc attering terms involving Σ( k)\n",
            "and integrals over σ(q,k) depend also on the support of the δ−function. Consider action\n",
            "contained in water waves of ﬁxed wavevector k. When U(x) = 0, only τ= +1 terms\n",
            "contribute to the the integration over qas long as |q|=|k|. In this case, we can deﬁne\n",
            "the angle q·k=k2cosθand reduce the cross-sections to single angular integrals o ver\n",
            "qiRij(|q−k|)qj=R(⏐⏐2ksinθ\n",
            "2⏐⏐)k2\n",
            "4sin2θ\n",
            "sin2θ\n",
            "2, τ= +1. (5.15)\n",
            "In this case ( U(x) = 0), assuming R(|q|) is monotonically decreasing, the most important\n",
            "contribution to the scattering occurs when qandkare collinear.\n",
            "When U(x)̸= 0, andτ= +1, the sets of qwhich satisfy Ω( q) +U(x)·q= Ω(k) +\n",
            "U(x)·k≡ω0trace out closed ellipse-like curves and are shown in the con tour plots\n",
            "ofω(q) in Figure 2(a). The parameters used are U(x)·k1=−0.5k1andh=∞(the\n",
            "−k1,−q1directions are deﬁned by the direction of U(x)).\n",
            "Each grayscale corresponds to a curve deﬁned by ﬁxed ω(k) =ω0. All wavevectors q\n",
            "in each contour contribute to the integration in the express ions for Σ( k) andω(q,k).\n",
            "Thus, slowly varying drift can induce an indirect doppler co upling between waves with\n",
            "14 Bal & Chou\n",
            "diﬀerent wavenumbers, with the most drastic coupling occur ring at the two far ends of\n",
            "a particular oval curve. For example, in Figure 2(b), the dar k band denotes qsuch that\n",
            "ω(q) =ω0when 0.625< ω0<0.6625. The wavevectors q≈(−0.3,0) and q≈(0.8,0)\n",
            "are two of many that contribute to the scattering terms. Ther efore, the evolution of\n",
            "a(x,k≈(−0.3,0),t) also depends on a(x,q≈(0.8,0),t) via the second term on the\n",
            "right side of (4.1).\n",
            "Provided U(x) is suﬃciently large, the τ=−1 terms can also contribute to scatter-\n",
            "ing. The dissipative scattering rate Σ( k)a(x,k,t) will change quantitatively since addi-\n",
            "tional q’s will contribute to Σ( k). However, this decay process depends only on kand\n",
            "is not coupled to a(x,|q| ̸=|k|,t). Wavevectors qthat satisfy the δ−function in the\n",
            "σ(q,k)a(x,q,t) term will, as when τ= +1, lead to indirect doppler coupling. This oc-\n",
            "curs whenω(q) =−ω0and, as we shall see, allows doppler coupling of waves with mo re\n",
            "widely varying wavelengths than compared to the τ= +1 case. Observe that if τ=−1\n",
            "terms arise, the drift frame energy a(x,k,t)ω(x,k) is no longer conserved. Figure 3(a)\n",
            "plotsω(q1,q2= 0) forU(x) = 1<√\n",
            "2,U(x) =√\n",
            "2, andU(x) = 1.6>√\n",
            "2. Sinceω0\n",
            "andω(q) are identical functions, −ω0can take on values below the upper dotted line\n",
            "(ω0≲0.22 forU= 1.6). Therefore, coupling for τ=−1 andq2= 0 occurs for values of\n",
            "−ω0between the dotted lines. Note that depending upon the value ofω0, coupling can\n",
            "occur at two or four diﬀerent points q= (q1,0). Figure 3(b) shows a contour plot of |ω(q)|\n",
            "as a function of ( q1,q2). A level set lying between the dotted lines in ( a) will slice out two\n",
            "bands; one band corresponds to all values of kthat couple to qlying in the associated\n",
            "second band. The two bands determined by the interval 0 .414<−ω0<0.468 are shown\n",
            "in Fig. 3(c). For any klying in the inner band of Fig. 3(c), all qlying in the outer band\n",
            "will contribute to doppler coupling for τ=−1, and vice versa . As−ω0is increased, the\n",
            "inner(outer) band decreases(increases) in size, with the c entral band vanishing when −ω0\n",
            "approaches the upper dotted line in ( a) where the τ=−1 coupling evaporates. If −ω0\n",
            "is decreased, the two bands merge, then disappear as −ω0reaches the lower limit. Fig.\n",
            "3(d) is an expanded view of the two bands for small 0 .0756<−ω0<0.1368. Note that\n",
            "a small island of qorkappears for very small wavevectors. The water wave scatteri ng\n",
            "represented by σ(q,k) can therefore couple very long wavelength modes with very s hort\n",
            "wavelength modes (the two larger bands to the right in Fig. 3( d)). However, the strength\n",
            "of this coupling is still determined by the magnitude of qiRij(|q−k|)kj, which may be\n",
            "small for |q−k|large.\n",
            "The depth dependence of doppler coupling will be relevant wh enhq,hk ≲1 whereq\n",
            "andkare the magnitudes of the wavevectors of two doppler-couple d waves. For τ= +1,\n",
            "ﬁnite depth reduces the ellipticity of the coupling bands, r esulting in weaker doppler\n",
            "eﬀects. Since the water wave phase velocity decreases with h, a ﬁnite depth will also\n",
            "reduce the critical U(x) required for τ=−1 doppler coupling. For small U(x), it is clear\n",
            "that theδ−functions associated with the τ=−1 terms inσ(q,k) are ﬁrst triggered when\n",
            "theqandkare antiparallel, U·k=−k|U|,U·q= +q|U|.\n",
            "Figure 4(a) shows the phase velocity for various depths h. In order for τ=−1 to\n",
            "contribute to scattering, U⩾cφ(k;h). ForU≈1.6, this condition holds in the h=∞case\n",
            "for 0.5≲k≲2 (the dashed region of cφ(k,∞)). Recall that our starting equations (2.4)\n",
            "are valid only in the small Froude number limit. However, for water waves propagating\n",
            "over inﬁnite depth, τ=−1 coupling requires U >U min= min k{cφ(k)}, withcφ(kmin)≃\n",
            "22cm/s. Therefore, in such “supersonic” cases, where τ=−1 is relevant, our treatment\n",
            "is accurate only at wavevectors k∗such thatU≪cφ(k∗;h),e.g.,the thick solid portion\n",
            "ofcφ(k;∞). ForU≳Umin, theτ=−1 term can couple wavevectors q≈0≪kmin\n",
            "withk≈2−3≫kmin. The rich τ=−1 doppler coupling displayed in Figures 3 is\n",
            "Water wave transport 15\n",
            "Figure 3. Conditions for doppler coupling when τ=−1. (a). Plot of ω(q1, q2= 0;h=∞) for\n",
            "U= 1,U=√\n",
            "2, and U= 1.6. Only for U >√\n",
            "2 does ω(q1, q2= 0;h=∞)<0. (b). Contour\n",
            "plot of |ω(q)|. Each grayscale corresponds to a diﬀerent constant value of ω(q) =ω(k)≡ −ω0.\n",
            "(c). The bands of qsatisfying 0 .414<−ω0<0.468. (d). An expanded view of the coupling\n",
            "bands for 0 .0756<−ω0<0.1368. Note that wavenumbers of very small modulus can couple\n",
            "with wavenumbers of signiﬁcantly larger modulus.\n",
            "particular to water waves with a dispersion relation ω(q) that behaves as q3/2,U·q, or\n",
            "q1/2depending on the wavelength. Doppler coupling in water wave propagation is very\n",
            "diﬀerent from that arising in acoustic wave propagation in a n incompressible, randomly\n",
            "ﬂowing ﬂuid (Howe (1973), Fannjiang & Ryzhik (1999), Vedant ham & Hunter (1997))\n",
            "whereω(q) =cs|q|. An additional doppler coupling analogous to the τ=−1 coupling\n",
            "for water waves arises only for supersonic random ﬂows when U(x)⩾cs, independent of\n",
            "q. In such instances, compressibility eﬀects must also be con sidered.\n",
            "Figure 4(b) plots the minimum drift velocity Umin(h) whereτ=−1 doppler coupling\n",
            "ﬁrst occurs at any wavevector. The wavevector at which coupl ing ﬁrst occurs is also\n",
            "shown by the dashed curve. For shallow water, h≪√\n",
            "3,Umin(h)∝√\n",
            "hand very long\n",
            "wavelengths couple ﬁrst (small k(Umin)). For depths h >√\n",
            "3 (∼3cm for water), the\n",
            "minimum drift required quickly increases to U∗(∞) =√\n",
            "2, while the initial coupling\n",
            "occurs at increasing wavevectors until at inﬁnite depth, wh ere the ﬁrst wavevector to\n",
            "doppler couple approaches k→1 (in water, this corresponds to wavelengths of ∼6.3cm).\n",
            "The conditions for τ=−1 doppler coupling outlined in Figures 2 and 3 apply to both\n",
            "16 Bal & Chou\n",
            "0 1 2 3 4 5\n",
            " k0123 phase velocity cφ(k;h)\n",
            "h=∞\n",
            "h=2\n",
            "h=1\n",
            "h=0.5\n",
            "0 1 2 3 4\n",
            "h00.511.5Umin for τ=−1 coupling\n",
            "k(Umin)\n",
            "Umin1.021/2\n",
            "(a).(b).\n",
            "Figure 4. U > c φ(k) is required for τ=−1 coupling. (a). The phase velocity cφ(k) for various\n",
            "depths h. The velocity shown by the solid horizontal line U≈1.6> cφ(k;h=∞) for 0 .5≲k≲2.\n",
            "(b). The minimum Umin(h) required for existence of τ=−1 coupling at any wavevector k, and\n",
            "the wavevector k(Umin) at which this ﬁrst happens.\n",
            "Σ(k) andσ(q,k), with the proviso that qandkare parallel for Σ( k) and antiparallel\n",
            "forσ(q,k). However, even when U <U minsuch that only τ= +1 applies, the set of q\n",
            "corresponding to a constant value of ω(k) =ω0, traces out a noncircular curve. There is\n",
            "doppler coupling between wavenumbers q̸=kas long asU̸= 0.\n",
            "5.4.Surface wave diﬀusion\n",
            "We now consider the radiative transfer equation (4.1) over p ropagation distances long\n",
            "compared to the mean free path ℓmfp. Imposing an additional rescaling and measuring all\n",
            "distances in terms of the mean free path, we introduce anothe r scalingǫ−1, proportional\n",
            "to the number of mean free paths travelled. Since β= 1/2, transport of wave action\n",
            "prevails when O(ε)<|x| ∼O(1), while diﬀusion holds when O(ǫ−1)∼ |x|<ℓloc.\n",
            "Since waves of each frequency satisfy (4.1) independently, we consider the diﬀusion of\n",
            "waves of constant frequency ω0. To derive the diﬀusion equation, we assume for simplicity\n",
            "thatUis constant and small such that ω0+ω(x,q)̸= 0 (theτ=−1 terms are never\n",
            "triggered by the δ−functions). Expanding all quantities in the transport equa tion (4.1)\n",
            "in powers of ǫ, we ﬁnd\n",
            "˙a0+¯U· ∇xa0− ∇x·D· ∇xa0= 0. (5.16)\n",
            "The derivation of this equation is given in Appendix B. The di ﬀusion tensor Dis given\n",
            "in (B 12) and is a function of the power spectrum Rij. The eﬀective drift ¯Uis given by\n",
            "(B 7):\n",
            "¯U=∫\n",
            "k∇kω(k)δ(k·U+ Ω(k)−ω0)∫\n",
            "kδ(k·U+ Ω(k)−ω0). (5.17)\n",
            "Up to a change of basis, we can assume that U=Ue1, whereU >0. Then the set of\n",
            "points k·U+ Ω(k)−ω0= 0 is symmetric with respect to the x1−axis and ¯Uis parallel\n",
            "toU. Also notice that the total energy given in (3.15) is asympto tically conserved in the\n",
            "Water wave transport 17\n",
            "diﬀusive regime. Indeed, the total energy variations are gi ven by (5.14). Assuming that\n",
            "all water waves have frequency ω0, we have in the diﬀusive regime\n",
            "d\n",
            "dtE=−d\n",
            "dt∫\n",
            "x∫\n",
            "kk·U(x)a(x,k,t)\n",
            "≈ −(∫\n",
            "x˙a0(x,t))∫\n",
            "kk·Uδ(ω0−ω(k)),\n",
            "sinceUis constant. Recasting the diﬀusion equation as ˙ a0=−∇x·(¯Ua0+D· ∇xa0),\n",
            "we deduce that∫\n",
            "x˙a0(x,t) = 0,\n",
            "which conserves the total energy E.\n",
            "Now consider the simpliﬁed case U≡0,h=∞and Ω ∞(k) =√\n",
            "k3+k. Since U=\n",
            "¯U= 0, (5.15) holds and we have for all k,\n",
            "∫\n",
            "qqiRij(|q−k|)qjq=0. (5.18)\n",
            "We deduce that the corrector χin (B 8) is given by\n",
            "χ(k) =−∇kΩ∞(k)\n",
            "Σ(k)=−|∇kΩ∞(k)|\n",
            "Σ(k)ˆk=−cg\n",
            "Σ(k)ˆk,\n",
            "where k=kˆk. The isotropic diﬀusion tensor Dis thus given by\n",
            "D=1\n",
            "Σ(k)Vω0∫\n",
            "q|∇qΩ∞(q)|2ˆqˆqTδ(Ω∞(q)−ω0) =c2\n",
            "g(k)\n",
            "2Σ(k)I, (5.19)\n",
            "whereIis the 2 ×2 identity matrix. Thus, the diﬀusion equation for a0(x,t) assumes the\n",
            "standard form (Sheng (1995))\n",
            "˙a0−c2\n",
            "g(k)\n",
            "2Σ(k)∆a0= 0. (5.20)\n",
            "6. Summary and Conclusions\n",
            "In this paper, we have used the Wigner distribution to derive the transport equations\n",
            "for water wave propagation over a spatially random drift com posed of a slowly varying\n",
            "partU(x), and a rapidly varying part√εδU(x/ε). The slowly varying part determines\n",
            "the characteristics on which the waves propagate. We recove r the standard result obtained\n",
            "from WKB theory: conservation of wave action. Provided Rij(q)qj= 0, we extend CWA\n",
            "to include wave scattering from correlations Rijof the rapidly varying random ﬂow.\n",
            "Evolution equations for the nonconserved wave intensity an d energy density can be read-\n",
            "ily obtained from (4.1). Moreover, conservation of drift fr ame energy a(x,k,t)ω(x,k)\n",
            "requires small U <U minand absence of τ=−1 contributions to scattering.\n",
            "Explicit expressions for the scattering rates Σ( k) andσ(q,k) are given in Eqs. (4.3).\n",
            "For ﬁxedω(k), we ﬁnd the set of qsuch that the δ−functions in (4.3) are supported.\n",
            "This set of qindicates the wavevectors of the background surface ﬂow tha t can me-\n",
            "diate doppler coupling of the water waves. Although widely v arying wavenumbers can\n",
            "doppler couple, supported by the δ−function constraints, particularly for τ=−1, the\n",
            "correlation Rij(|q−k|) also decreases for large |q−k|. For long times, multiple weak scat-\n",
            "tering nonetheless exchanges action among disparate waven umbers within the transport\n",
            "18 Bal & Chou\n",
            "regime. Our collective results, including water wave actio n diﬀusion, provide a model for\n",
            "describing linear ocean wave propagation over random ﬂows o f diﬀerent length scales.\n",
            "The scattering terms in (4.1) also provide a means to correla te sea surface wave spectra\n",
            "to statistics Rijof ﬁner scale random ﬂows.\n",
            "Although many situations arise where the underlying ﬂow is r otational (White (1999)),\n",
            "the irrotational approximation used simpliﬁes the treatme nt and allows a relatively simple\n",
            "derivation of the transport and diﬀusive regimes of water wa ve propagation. The recent\n",
            "extension by White (1999) of CWA to include rotational ﬂows a lso suggests that an\n",
            "explicit consideration of velocity and pressure can be used to generalise the present\n",
            "study to include rotational random ﬂows. Other feasible ext ensions include the analysis\n",
            "of a time varying random ﬂow, as well as separating the underl ying ﬂows into static and\n",
            "wave dynamic components.\n",
            "The authors thank A. Balk, M. Moscoso, G. Papanicolaou, L. Ry zhik, and I. Smol-\n",
            "yarenko for helpful comments and discussion. GB was support ed by AFOSR grant 49620-\n",
            "98-1-0211 and NSF grant DMS-9709320. TC was supported by NSF grant DMS-9804780.\n",
            "Appendix A. Derivation of the transport equation\n",
            "Some of the steps in the derivation of (4.1) are outlined here . By taking the time\n",
            "derivative of Wijin (3.2) and using the deﬁnition (3.5) for ˙ψ, we obtain\n",
            "(2πε)2˙Wij(p,k,t) = (2πε)2iWiℓ(p,k)L∗\n",
            "ℓj(k\n",
            "ε−p\n",
            "2)\n",
            "−(2πε)2iLiℓ(k\n",
            "ε+p\n",
            "2)\n",
            "Wℓj(p,k)\n",
            "+i∫\n",
            "qU(q)·(\n",
            "−k\n",
            "ε+p\n",
            "2−qδi2)\n",
            "ψi(\n",
            "−k\n",
            "ε+p\n",
            "2−q)\n",
            "ψ∗\n",
            "j(\n",
            "−k\n",
            "ε−p\n",
            "2)\n",
            "−i∫\n",
            "qU∗(q)·(\n",
            "−k\n",
            "ε−p\n",
            "2−qδj2)\n",
            "ψi(\n",
            "−k\n",
            "ε+p\n",
            "2)\n",
            "ψ∗\n",
            "j(\n",
            "−k\n",
            "ε−p\n",
            "2−q)\n",
            "+i√ε∫\n",
            "qδU(q)·(\n",
            "−k\n",
            "ε+p\n",
            "2−q\n",
            "εδi2)\n",
            "ψi(\n",
            "−k\n",
            "ε+p\n",
            "2−q\n",
            "ε)\n",
            "ψ∗\n",
            "j(\n",
            "−k\n",
            "ε−p\n",
            "2)\n",
            "−i√ε∫\n",
            "qδU∗(q)·(\n",
            "−k\n",
            "ε−p\n",
            "2−q\n",
            "εδj2)\n",
            "ψi(\n",
            "−k\n",
            "ε+p\n",
            "2)\n",
            "ψ∗\n",
            "j(\n",
            "−k\n",
            "ε−p\n",
            "2−q\n",
            "ε)\n",
            "(A 1)\n",
            "To rewrite the above expression as a function of Wijonly, we relabel appropriately,\n",
            "e.g.,\n",
            "−k\n",
            "ε−p\n",
            "2=−k′\n",
            "ε−p′\n",
            "2\n",
            "−k\n",
            "ε+p\n",
            "2−q=−k′\n",
            "ε+p′\n",
            "2(A 2)\n",
            "for the third term on the right hand side of (A1). Similarly re labelling for all relevant\n",
            "terms yields the integral equation (3.7).\n",
            "TheO(ε−1/2) terms of (3.7) determine W1/2. Decomposing\n",
            "W1/2(p,ξ,k)≡∑\n",
            "τ,τ′=±a(1/2)\n",
            "τ,τ′(p,ξ,k)bτ(k−)b†\n",
            "τ′(k+) (A 3)\n",
            "and substituting into (3.19) we ﬁnd the coeﬃcients a(1/2)\n",
            "τ,τ′, where in this case a(1/2)\n",
            "+−,a(1/2)\n",
            "−+̸=\n",
            "Water wave transport 19\n",
            "0. Due to the nonlocal nature of the third term on the right of ( 3.19), we must ﬁrst inverse\n",
            "Fourier transform the slow wavevector variable back to x.\n",
            "To extract the O(ε0) terms from (3.7) we need to expand Lto orderε0, theL1term.\n",
            "Similarly, the terms W(p−q,ξ,k±εq/2) must be expanded:\n",
            "W(p−q,ξ,k±εq/2) =W(p−q,ξ,k)±ε\n",
            "2q· ∇kW(p−q,ξ,k) +O(ε2).(A 4)\n",
            "Theεq·∇kW(p−q,ξ,k) terms combine with the −ε−1U(q)·k−+ε−1U(q)·k+terms\n",
            "from the third and fourth terms in (3.7) to give the third term on the right of Eq. (3.22).\n",
            "TheδU-dependent, order ε0terms (the sixth, seventh, and eighth terms on the right side\n",
            "of (3.22)) come from collecting\n",
            "±√εδU(q)·(\n",
            "−k\n",
            "ε±ξ\n",
            "2ε)√εW1/2(p,ξ−q,k±q/2) (A 5)\n",
            "from the last two terms in (3.7). The ensemble averaged time e volution of the Wigner\n",
            "amplitudeaσ(x,k) can be succinctly written in the form:\n",
            "˙a+(x,k,t)− ∇xω(x,k)· ∇ka+(x,k,t) +∇kω(x,k)· ∇xa+(x,k,t)\n",
            "= Σ+,µ(k)aµ(x,k,t) +∫\n",
            "qσ+,µ(q,k)aµ(x,k,t).\n",
            "(A 6)\n",
            "Using the form for W0found from (3.11) in (3.19) to ﬁnd W1/2, we substitute into\n",
            "(3.22) to ﬁnd (4.1), the transport equation for one of the dia gonal intensities of the Wigner\n",
            "distribution. We have explicitly used eigenbasis orthonor mality b†\n",
            "τ(k)·cτ′(k) =δτ,τ′and\n",
            "the fact that a−(x,k,t) =a+(x,−k,t).\n",
            "Appendix B. Derivation of the diﬀusion equation\n",
            "The derivation of diﬀusion of water wave action is outlined b elow and follows the\n",
            "established mathematical treatment of Larsen & Keller (197 4) and Dautray & Lions\n",
            "(1993). For simplicity we assume that the ﬂow Uis constant and small enough so that\n",
            "for a considered range of frequencies, the relation ω(q) +ω(k) = 0 is never satisﬁed\n",
            "for any kandq̸=0. The diﬀusion approximation is valid after long times and la rge\n",
            "distances of propagation X(see Fig. 1) such that the wave has multiply scattered and its\n",
            "dynamics are determined by a random walk. We therefore resca le time and space as\n",
            "˜t=t\n",
            "ǫ2, ˜x=x\n",
            "ǫ. (B 1)\n",
            "The small parameter ǫin this further rescaling represents the transport mean fre e path\n",
            "ℓmfpand not the wavelength as in the initial rescaling used to der ive the transport\n",
            "equation. We drop the tilde symbol for convenience and rewri te the transport equation\n",
            "in the new variables:\n",
            "˙aǫ(x,k,t)+1\n",
            "ǫ∇kω(k)·∇xaǫ(x.k,t) =1\n",
            "ǫ2∫\n",
            "qQ(q,k)(aǫ(x,q,t)−aǫ(x,k,t))δ(ω(q)−ω(k)),\n",
            "(B 2)\n",
            "with obvious notation for Q(q,k). Since the frequency is ﬁxed, the equation is posed\n",
            "forksatisfyingω(k) =ω0. The transport equation assumes the form (B 2) because the\n",
            "scattering operator is conservative. Since U̸= 0, wave action is transported by the ﬂow,\n",
            "and diﬀusion takes place on top of advection. Therefore, we i ntroduce the main drift ¯U,\n",
            "20 Bal & Chou\n",
            "which will be computed explicitly later, and deﬁne the drift -free unknown ˜ aǫ(x,k,t) as\n",
            "˜aǫ(x,k,t) =aǫ(x+¯U\n",
            "ǫt,k,t). (B 3)\n",
            "It is easy to check that ˜ aǫsatisﬁes the same transport equation as aǫwhere the drift\n",
            "term∇kωhas been replaced by ∇kω−¯U.\n",
            "We now derive the limit of ˜ aǫasǫ→0. Consider the classical asymptotic expansion\n",
            "˜aǫ= ˜a0+ǫ˜a1+ǫ2˜a2+.... (B 4)\n",
            "Upon substitution into (B 2) and equating like powers of ǫ, we obtain at order ǫ−2, for\n",
            "ﬁxed frequency ω0,\n",
            "∫\n",
            "qQ(q,k)(˜a0(x,q,t)−˜a0(x,k,t))δ(ω(q)−ω0) = 0. (B 5)\n",
            "It follows from the Krein-Rutman theory (Dautray & Lions (19 93)) that ˜a0is independent\n",
            "ofq. At orderǫ−1, we obtain\n",
            "(∇kω(k)−¯U)· ∇x˜a0=∫\n",
            "qQ(q,k)(˜a1(x,q,t)−˜a1(x,k,t))δ(ω(q)−ω0). (B 6)\n",
            "The compatibility condition for this equation to admit a sol ution requires both sides to\n",
            "vanish upon integration over δ(ω(k)−ω0)dk. Therefore, ¯Usatisﬁes\n",
            "¯U=1\n",
            "Vω0∫\n",
            "k∇kω(k)δ(ω(k)−ω0), where Vω0=∫\n",
            "kδ(ω(k)−ω0).(B 7)\n",
            "Once the constraint is satisﬁed, we deduce from Krein-Rutma n theory the existence of a\n",
            "vector-valued mean zero corrector χsolving\n",
            "(∇kω(k)−¯U) =∫\n",
            "qQ(q,k)(χ(q)−χ(k))δ(ω(q)−ω0)≡ Lχ. (B 8)\n",
            "There is no general analytic expression for χ, which must in practice be solved numer-\n",
            "ically. This is typical of problems where the domain of integ ration in qdoes not have\n",
            "enough symmetries (cf. Allaire & Bal (1999), Bal (1999)). We now have\n",
            "˜a1(x,k,t) =χ(k)· ∇x˜a0(x,t). (B 9)\n",
            "It remains to consider O(ǫ0) in the asymptotic expansion. This yields\n",
            "˙˜a0+ (∇kω(k)−¯U)· ∇x˜a1=L(˜a2). (B 10)\n",
            "The compatibility condition, obtained by integrating both sides over k, yields the wave\n",
            "action diﬀusion equation\n",
            "˙˜a0− ∇x·D· ∇x˜a0= 0, (B 11)\n",
            "where the diﬀusion tensor is given by\n",
            "D=−1\n",
            "Vω0∫\n",
            "k(∇kω(k)−¯U)χT(k) =−1\n",
            "Vω0∫\n",
            "kL(χ)χT(k). (B 12)\n",
            "The second form shows that Dis positive deﬁnite since Lis a nonpositive operator. The\n",
            "formal asymptotic expansion can be justiﬁed rigorously usi ng the techniques in Dautray\n",
            "& Lions (1993). As ǫ→0, we obtain that the error between ˜ aǫand ˜a0is at most of\n",
            "orderǫ. Therefore, we have that aǫconverges to a0satisfying the following drift-diﬀusion\n",
            "equation\n",
            "˙a0+¯U\n",
            "ǫ· ∇xa0− ∇x·D· ∇xa0= 0, (B 13)\n",
            "Water wave transport 21\n",
            "with suitable initial conditions. Equation (B 13) is the coo rdinate-scaled version of (5.16).\n",
            "REFERENCES\n",
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            "Chou, T., & Nelson, D. R. 1994 Surface Wave Scattering from Nonuniform Interfaces J.\n",
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            "White, B. S. 1999 Wave action on currents with vorticity. J. Fluid Mech. 386, 329-344.\n",
            "Whitham, G. B. 1974 Linear & Nonlinear waves. Wiley Interscience.\n",
            "Wigner, E. 1932 On the quantum correction for thermodynamic equilibri um.Phys. Rev. 40,\n",
            "742-749.\n",
            "Zakharov, V. E., V. S. L’vov & G. Falkovich 1992 Kolgomorov Spectra of Turbulence I:\n",
            "Wave Turbulence. Springer-Verlag, Berlin.\n",
            "\n"
          ]
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "# Testing on Math Notations\n",
        "In this step we will try to test our tools on a PDF of brute mathematical notations to see which one will do better"
      ],
      "metadata": {
        "id": "d9edpW3r41K4"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "## Using **PyPDF2**"
      ],
      "metadata": {
        "id": "_tSlAClw6OmA"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "# defining math notations pdf\n",
        "math_pdf = \"/content/drive/MyDrive/UH - Final Year Project/Math Notations List - Cambridge -.pdf\"\n",
        "\n",
        "# testing math notations on PyPDF2\n",
        "reader = PdfReader(math_pdf)\n",
        "pypdf_math_text = \"\\n\".join([page.extract_text()for page in reader.pages])"
      ],
      "metadata": {
        "id": "l9zrN9au43dL"
      },
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "code",
      "source": [
        "print(pypdf_math_text)"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "collapsed": true,
        "id": "eIaH9G8x5Mx1",
        "outputId": "ec7e6272-c14a-4e20-95da-6050ffcabe2b"
      },
      "execution_count": null,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            " \n",
            "  \n",
            "  \n",
            " \n",
            "     \n",
            " \n",
            "    \n",
            " \n",
            "Notation List  \n",
            " \n",
            "  \n",
            "For Cambridge International Mathematics \n",
            "Qualifications \n",
            " \n",
            " \n",
            "   \n",
            "For use from 2020 \n",
            " \n",
            " \n",
            "Notation List for Cambridge International Ma thematics Qualifications (For use from 2020) \n",
            " \n",
            "2 Mathematical notation \n",
            "Examinations for CIE syllabuses may use relevant notation from the following list. \n",
            "1 Set notation \n",
            "∈ is an element of \n",
            "∉ is not an element of \n",
            "{x1, x2, …} the set with elements x1, x2, … \n",
            "{x : …} the set of all x such that … \n",
            "n(A) the number of elements in set A \n",
            "∅ the empty set \n",
            " the universal set \n",
            ", the universal set (for 0607 IGCSE International Mathematics) \n",
            "A′ the complement of the set A \n",
            "ℕ the set of natural numbers, {1, 2, 3, …} \n",
            "ℤ the set of integers, {0, ±1, ±2, ±3, …} \n",
            "ℚ the set of rational numbers, :,  ppqq∈ \n",
            "\tℤ, 0q≠\n",
            " \n",
            "ℝ the set of real numbers \n",
            "ℂ the set of complex numbers \n",
            "(x, y) the ordered pair x, y \n",
            "⊆ is a subset of \n",
            "⊂ is a proper subset of \n",
            "⋃ union \n",
            "⋂ intersection \n",
            "[a, b] the closed interval { x ∈ ℝ : a ⩽ x ⩽ b} \n",
            "[a, b) the interval { x ∈ ℝ : a ⩽ x < b} \n",
            "(a, b] the interval { x ∈ ℝ : a < x ⩽ b} \n",
            "(a, b) the open interval { x ∈ ℝ : a < x < b} \n",
            "(S, ∘) the group consisting of the set S with binary operation ∘ \n",
            "2 Miscellaneous symbols \n",
            "= is equal to \n",
            "≠ is not equal to \n",
            "≡ is identical to or is congruent to \n",
            "≈ is approximately equal to \n",
            "~ is distributed as ≅ is isomorphic to \n",
            "∝ is proportional to \n",
            "< is less than ⩽ is less than or equal to \n",
            "> is greater than ⩾ is greater than or equal to \n",
            "∞ infinity \n",
            "⇒ implies \n",
            "⇐ is implied by \n",
            "⇔ implies and is implied by (is equivalent to) \n",
            "  \n",
            "Notation List for Cambridge International Ma thematics Qualifications (For use from 2020) \n",
            " \n",
            "3 3 Operations \n",
            "a + b a plus b \n",
            "a – b a minus b \n",
            "a × b, ab a multiplied by b \n",
            "a ÷ b, a\n",
            "b a divided by b \n",
            "1n\n",
            "i\n",
            "ia\n",
            "=∑  a1 + a2 + … + an \n",
            "a the non-negative square root of a, for a ∈ ℝ, a ⩾ 0 \n",
            "na the (real) nth root of a, for a ∈ ℝ, where 0na.  for a ⩾ 0 \n",
            "| a | the modulus of a \n",
            "n! n factorial \n",
            "()n\n",
            "r the binomial coefficient !\n",
            "!( )!n\n",
            "rn r− for n, r ∈ ℤ and 0 ⩽ r ⩽ n \n",
            "4 Functions \n",
            "f(x) the value of the function f at x \n",
            "f : A → B f is a function under which each element of set A has an image in set B \n",
            "f : x ↦ y the function f maps the element x to the element y \n",
            "f –1 the inverse function of the one-one function f \n",
            "gf the composite function of f and g which is defined by gf( x) = g(f( x)) \n",
            "lim f( )\n",
            "xax\n",
            "→ the limit of f( x) as x tends to a \n",
            "∆x, δx an increment of x \n",
            "d\n",
            "dy\n",
            "x the derivative of y with respect to x \n",
            "d\n",
            "dn\n",
            "ny\n",
            "x the nth derivative of y with respect to x \n",
            "f ′(x), f ″(x), …, f (n)(x) the first, second, …, nth derivatives of f( x) with respect to x \n",
            "dyx∫ the indefinite integral of y with respect to x \n",
            "db\n",
            "ayx∫ the definite integral of y with respect to x between the limits x = a and x = b \n",
            ", , xx&& & K the first, second, … derivatives of x with respect to t \n",
            "5 Exponential and logarithmic functions \n",
            "e base of natural logarithms \n",
            "ex, exp( x) exponential function of x \n",
            "log a x logarithm to the base a of x \n",
            "ln x natural logarithm of x \n",
            "lg x, log 10 x logarithm of x to base 10 \n",
            "  \n",
            "Notation List for Cambridge International Ma thematics Qualifications (For use from 2020) \n",
            " \n",
            "4 6 Circular and hyperbolic functions \n",
            "}sin, cos, tan\n",
            "cosec, sec,  cot the circular functions \n",
            "111\n",
            "111sin , cos , tan\n",
            "cosec , sec , cot−−−\n",
            "−−−\n",
            "\n",
            " the inverse circular functions \n",
            "}sinh, cosh, tanh\n",
            "cosech,  sech,  coth the hyperbolic functions \n",
            "111\n",
            "111sinh , cosh , tanh\n",
            "cosech ,  sech ,  coth−−−\n",
            "−−−\n",
            "\n",
            " the inverse hyperbolic functions \n",
            "7 Complex numbers \n",
            "i the imaginary unit, i2 = –1 \n",
            "z a complex number, z = x + iy = r(cos θ + i sin θ ) \n",
            "Re z the real part of z, Re z = x \n",
            "Im z the imaginary part of z, Im z = y \n",
            "| z | the modulus of z, 22||zx y=+  \n",
            "arg z the argument of z, arg z = θ where –π < θ ⩽ π \n",
            "z* the complex conjugate of z, x – iy \n",
            "8 Matrices \n",
            "Μ a matrix Μ \n",
            "Μ−1 the inverse of the non-singular square matrix Μ \n",
            "det M, | M | the determinant of the square matrix Μ \n",
            "I an identity (or unit) matrix \n",
            "9 Vectors \n",
            "a the vector a \n",
            "ABuuur\n",
            " the vector represented in magnitude and direction by the directed line segment \n",
            "AB \n",
            "â a unit vector in the direction of a \n",
            "i, j, k unit vectors in the directions of the Cartesian coordinate axes \n",
            "(), xxyyz\n",
            "\n",
            " the vectors xi + yj (in 2 dimensions) and xi + yj + zk (in 3 dimensions) \n",
            "| a |, a the magnitude of a \n",
            ", ABA Buuur\n",
            " the magnitude of ABuuur\n",
            " \n",
            "a.b the scalar product of a and b \n",
            "a × b the vector product of a and b \n",
            "  \n",
            "Notation List for Cambridge International Ma thematics Qualifications (For use from 2020) \n",
            " \n",
            "5 10 Probability and statistics \n",
            "A, B, C, … events \n",
            "A ⋃ B union of the events A and B \n",
            "A ⋂ B intersection of the events A and B \n",
            "P(A) probability of the event A \n",
            "A′ complement of the event A \n",
            "P(A | B) probability of the event A conditional on the event B \n",
            "nCr the number of combinations of r objects from n, Cnr=()n\n",
            "r=!\n",
            "!( )!n\n",
            "rn r− \n",
            "nPr the number of permutations of r objects from n, !P\n",
            "() !n nrnr=\n",
            "− \n",
            "X, Y, R, … random variables \n",
            "x, y, r, … values of the random variables X, Y, R, … \n",
            "x1, x2, … observations \n",
            "f1, f2, … frequencies with which the observations x1, x2, … occur \n",
            "p(x) probability function P( X = x) of the discrete random variable X \n",
            "p1, p2, … probabilities of the values x1, x2, … of the discrete random variable X \n",
            "f(x) value of the probability density function of a continuous random variable X \n",
            "F(x) value of the cumulative distribution function of a continuous random variable X \n",
            "E(X) expectation of the random variable X \n",
            "E(g(X)) expectation of g( X) \n",
            "Var(X) variance of the random variable X \n",
            "GX(t) probability generating function fo r the discrete random variable X \n",
            "MX(t) moment generating function for the random variable X \n",
            "B(n, p) binomial distribution with parameters n and p \n",
            "Geo(p) geometric distribution with parameter p \n",
            "Po(λ) Poisson distribution with parameter λ \n",
            "N(µ, σ2) normal distribution with mean µ and variance σ2 \n",
            "µ population mean \n",
            "σ2 population variance \n",
            "σ population standard deviation \n",
            "x sample mean, \n",
            "11n\n",
            "i\n",
            "ix xn==∑  \n",
            "s2 unbiased estimate of population variance from a sample, 22\n",
            "11()1n\n",
            "i\n",
            "is xxn== −−∑  \n",
            "ρ product moment correlation coefficient for a population \n",
            "r product moment correlation coefficient for a sample \n",
            "φ probability density function of th e standardised normal variable Z ~ N(0, 1) \n",
            "Φ cumulative distribution function of the standardised normal variable Z ~ N(0, 1) \n",
            "H0, H 1 null and alternative hypotheses for a hypothesis test \n",
            " \n"
          ]
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "## Using **PDFMiner.six**\n",
        "\n",
        "\n"
      ],
      "metadata": {
        "id": "YvqLozqM6V2Y"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "# testing PDFMiner.six on math notations\n",
        "output_string = StringIO()\n",
        "with open(math_pdf, 'rb') as in_file:\n",
        "    parser = PDFParser(in_file)\n",
        "    doc = PDFDocument(parser)\n",
        "    rsrcmgr = PDFResourceManager()\n",
        "    device = TextConverter(rsrcmgr, output_string, laparams=LAParams())\n",
        "    interpreter = PDFPageInterpreter(rsrcmgr, device)\n",
        "    for page in PDFPage.create_pages(doc):\n",
        "        interpreter.process_page(page)\n",
        "\n",
        "pdfminer_math_text = output_string.getvalue()"
      ],
      "metadata": {
        "id": "jo1HTYqB6jkG"
      },
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "code",
      "source": [
        "print(pdfminer_math_text)"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "collapsed": true,
        "id": "LlDgGsxE6tnl",
        "outputId": "371a4362-9ccd-4686-b937-1b29dc5841c3"
      },
      "execution_count": null,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "Notation List  \n",
            "\n",
            "For Cambridge International Mathematics \n",
            "Qualifications \n",
            "\n",
            "For use from 2020 \n",
            "\n",
            " \n",
            " \n",
            " \n",
            " \n",
            " \n",
            " \n",
            " \n",
            " \n",
            " \n",
            " \n",
            " \n",
            " \n",
            " \n",
            " \n",
            " \n",
            " \n",
            " \n",
            " \n",
            " \n",
            " \n",
            " \n",
            " \n",
            " \n",
            " \n",
            " \n",
            " \n",
            " \n",
            "\fNotation List for Cambridge International Mathematics Qualifications (For use from 2020) \n",
            "\n",
            "Mathematical notation \n",
            "\n",
            "Examinations for CIE syllabuses may use relevant notation from the following list. \n",
            "\n",
            "1  Set notation \n",
            "\n",
            "∈ \n",
            "∉ \n",
            "{x1, x2, …} \n",
            "{x : …} \n",
            "n(A) \n",
            "∅ \n",
            "\n",
            ", \n",
            "A′ \n",
            "ℕ \n",
            "ℤ \n",
            "\n",
            "ℚ \n",
            "\n",
            "ℝ \n",
            "ℂ \n",
            "(x, y) \n",
            "⊆ \n",
            "⊂ \n",
            "⋃ \n",
            "⋂ \n",
            "[a, b] \n",
            "[a, b) \n",
            "(a, b] \n",
            "(a, b) \n",
            "(S, ∘) \n",
            "\n",
            "2  Miscellaneous symbols \n",
            "\n",
            "is an element of \n",
            "is not an element of \n",
            "the set with elements x1, x2, … \n",
            "the set of all x such that … \n",
            "the number of elements in set A \n",
            "the empty set \n",
            "the universal set \n",
            "the universal set (for 0607 IGCSE International Mathematics) \n",
            "the complement of the set A \n",
            "the set of natural numbers, {1, 2, 3, …} \n",
            "the set of integers, {0, ±1, ±2, ±3, …} \n",
            "\n",
            "the set of rational numbers, \n",
            "\n",
            "\n",
            "\n",
            "\n",
            "\n",
            "p\n",
            "q\n",
            "\n",
            ":\n",
            "\n",
            "p q\n",
            ",  \n",
            "\n",
            "∈\n",
            "\n",
            "\tℤ,\n",
            "\n",
            "q\n",
            "\n",
            "\n",
            "0\n",
            "≠ \n",
            "\n",
            "\n",
            "the set of real numbers \n",
            "the set of complex numbers \n",
            "the ordered pair x, y \n",
            "is a subset of \n",
            "is a proper subset of \n",
            "union \n",
            "intersection \n",
            "the closed interval {x ∈ ℝ : a ⩽ x ⩽ b} \n",
            "the interval {x ∈ ℝ : a ⩽ x < b} \n",
            "the interval {x ∈ ℝ : a < x ⩽ b} \n",
            "the open interval {x ∈ ℝ : a < x < b} \n",
            "the group consisting of the set S with binary operation ∘ \n",
            "\n",
            "= \n",
            "≠ \n",
            "≡ \n",
            "≈ \n",
            "~ \n",
            "≅ \n",
            "∝ \n",
            "< \n",
            "⩽ \n",
            "> \n",
            "⩾ \n",
            "∞ \n",
            "⇒ \n",
            "⇐ \n",
            "⇔ \n",
            "\n",
            "is equal to \n",
            "is not equal to \n",
            "is identical to or is congruent to \n",
            "is approximately equal to \n",
            "is distributed as \n",
            "is isomorphic to \n",
            "is proportional to \n",
            "is less than \n",
            "is less than or equal to \n",
            "is greater than \n",
            "is greater than or equal to \n",
            "infinity \n",
            "implies \n",
            "is implied by \n",
            "implies and is implied by (is equivalent to) \n",
            "\n",
            "2 \n",
            "\n",
            " \n",
            " \n",
            " \n",
            " \n",
            " \n",
            "\fNotation List for Cambridge International Mathematics Qualifications (For use from 2020) \n",
            "\n",
            "3  Operations \n",
            "\n",
            "a + b \n",
            "a – b \n",
            "a × b, ab \n",
            "\n",
            "a ÷ b, \n",
            "\n",
            "a\n",
            "b\n",
            "\n",
            "n\n",
            "\n",
            "∑  \n",
            "a\n",
            "i\n",
            "\n",
            "i\n",
            "\n",
            "1\n",
            "=\n",
            "a  \n",
            "n a  \n",
            "| a | \n",
            "n! \n",
            "\n",
            "a plus b \n",
            "a minus b \n",
            "a multiplied by b \n",
            "\n",
            "a divided by b \n",
            "\n",
            "a1 + a2 + … + an \n",
            "\n",
            "the non-negative square root of a, for a ∈ ℝ, a ⩾ 0 \n",
            "the (real) nth root of a, for a ∈ ℝ, where \n",
            "the modulus of a \n",
            "n factorial \n",
            "\n",
            "n a (cid:46) \n",
            "0\n",
            "\n",
            " for a ⩾ 0 \n",
            "\n",
            "( )n\n",
            "\n",
            "r\n",
            "\n",
            "4  Functions \n",
            "\n",
            "the binomial coefficient \n",
            "\n",
            "!\n",
            "\n",
            "n\n",
            "r n r−\n",
            "!(\n",
            "\n",
            ")!\n",
            "\n",
            " for n, r ∈ ℤ and 0 ⩽ r ⩽ n \n",
            "\n",
            "f(x) \n",
            "f : A → B \n",
            "f : x ↦ y \n",
            "f –1 \n",
            "gf \n",
            "lim f( )\n",
            "x\n",
            "x\n",
            "a\n",
            "→\n",
            "∆x, δx \n",
            "y\n",
            "d\n",
            "x\n",
            "d\n",
            "n\n",
            "y\n",
            "d\n",
            "n\n",
            "d\n",
            "x\n",
            "f ′(x), f ″(x), …, f (n)(x) \n",
            "∫\n",
            "∫\n",
            "a\n",
            "x x& && K  \n",
            ",   ,  \n",
            "\n",
            "y x\n",
            "d\n",
            "\n",
            "dy x\n",
            "\n",
            "b\n",
            "\n",
            "the value of the function f at x \n",
            "f is a function under which each element of set A has an image in set B \n",
            "the function f maps the element x to the element y \n",
            "the inverse function of the one-one function f \n",
            "the composite function of f and g which is defined by gf(x) = g(f(x)) \n",
            "the limit of f(x) as x tends to a \n",
            "\n",
            "an increment of x \n",
            "\n",
            "the derivative of y with respect to x \n",
            "\n",
            "the nth derivative of y with respect to x \n",
            "\n",
            "the first, second, …, nth derivatives of f(x) with respect to x \n",
            "the indefinite integral of y with respect to x \n",
            "\n",
            "the definite integral of y with respect to x between the limits x = a and x = b \n",
            "\n",
            "the first, second, … derivatives of x with respect to t \n",
            "\n",
            "5  Exponential and logarithmic functions \n",
            "\n",
            "e \n",
            "ex, exp(x) \n",
            "loga x \n",
            "ln x \n",
            "lg x, log10 x \n",
            "\n",
            "base of natural logarithms \n",
            "exponential function of x \n",
            "logarithm to the base a of x \n",
            "natural logarithm of x \n",
            "logarithm of x to base 10 \n",
            "\n",
            "3 \n",
            "\n",
            " \n",
            " \n",
            " \n",
            " \n",
            " \n",
            " \n",
            " \n",
            " \n",
            " \n",
            " \n",
            "\fNotation List for Cambridge International Mathematics Qualifications (For use from 2020) \n",
            "\n",
            "6  Circular and hyperbolic functions \n",
            "\n",
            "sin,   cos,   tan\n",
            "cosec,   sec,   cot\n",
            "\n",
            "}\n",
            "\n",
            "the circular functions \n",
            "\n",
            "\n",
            "\n",
            "\n",
            "\n",
            "−\n",
            "\n",
            "1\n",
            "\n",
            "the inverse circular functions \n",
            "\n",
            "1\n",
            "\n",
            "−\n",
            "\n",
            "1\n",
            "−\n",
            "\n",
            "1\n",
            "−\n",
            "\n",
            ",   tan\n",
            "1\n",
            "−\n",
            "\n",
            "1\n",
            "sin ,   cos\n",
            "−\n",
            "cosec ,   sec ,   cot\n",
            "sinh,   cosh,   tanh\n",
            "cosech,   sech,   coth\n",
            "\n",
            "}\n",
            "\n",
            "1\n",
            "−\n",
            "\n",
            "−\n",
            "\n",
            "1\n",
            "\n",
            "sinh ,   cosh ,   tanh\n",
            "cosech ,   sech ,   coth\n",
            "\n",
            "1\n",
            "−\n",
            "\n",
            "−\n",
            "\n",
            "1\n",
            "\n",
            "−\n",
            "\n",
            "1\n",
            "\n",
            "7  Complex numbers \n",
            "\n",
            "i \n",
            "z \n",
            "Re z \n",
            "Im z \n",
            "| z | \n",
            "arg z \n",
            "z* \n",
            "\n",
            "8  Matrices \n",
            "\n",
            "Μ \n",
            "Μ−1 \n",
            "det M, | M | \n",
            "I \n",
            "\n",
            "9  Vectors \n",
            "\n",
            "a \n",
            "uuur\n",
            "AB\n",
            "\n",
            "(\n",
            "\n",
            "x\n",
            "y\n",
            "\n",
            "),  \n",
            "\n",
            "â \n",
            "i, j, k \n",
            "\n",
            "\n",
            "\n",
            "\n",
            "| a |, a \n",
            "uuur\n",
            "AB AB\n",
            ",  \n",
            "\n",
            "x\n",
            "y\n",
            "z\n",
            "\n",
            "\n",
            "\n",
            "\n",
            "\n",
            "\n",
            "a.b \n",
            "a × b \n",
            "\n",
            "the hyperbolic functions \n",
            "\n",
            "\n",
            "\n",
            "\n",
            "\n",
            "1\n",
            "−\n",
            "\n",
            " the inverse hyperbolic functions \n",
            "\n",
            "the imaginary unit, i2 = –1 \n",
            "a complex number, z = x + iy = r(cos θ + i sin θ ) \n",
            "the real part of z, Re z = x \n",
            "the imaginary part of z, Im z = y \n",
            "the modulus of z, \n",
            "the argument of z, arg z = θ where –π < θ ⩽ π \n",
            "the complex conjugate of z, x – iy \n",
            "\n",
            "|z\n",
            "\n",
            "=\n",
            "\n",
            "+\n",
            "\n",
            "y\n",
            "\n",
            "x\n",
            "\n",
            "2\n",
            "\n",
            "2\n",
            "\n",
            "|\n",
            "\n",
            "a matrix Μ \n",
            "the inverse of the non-singular square matrix Μ \n",
            "the determinant of the square matrix Μ \n",
            "an identity (or unit) matrix \n",
            "\n",
            "the vector a \n",
            "the vector represented in magnitude and direction by the directed line segment \n",
            "AB \n",
            "a unit vector in the direction of a \n",
            "unit vectors in the directions of the Cartesian coordinate axes \n",
            "\n",
            "the vectors xi + yj (in 2 dimensions) and xi + yj + zk (in 3 dimensions) \n",
            "\n",
            "the magnitude of a \n",
            "uuur\n",
            "the magnitude of  AB\n",
            "\n",
            "the scalar product of a and b \n",
            "the vector product of a and b \n",
            "\n",
            "4 \n",
            "\n",
            " \n",
            " \n",
            " \n",
            " \n",
            " \n",
            " \n",
            " \n",
            " \n",
            " \n",
            " \n",
            " \n",
            "\fNotation List for Cambridge International Mathematics Qualifications (For use from 2020) \n",
            "\n",
            "10 Probability and statistics \n",
            "\n",
            "A, B, C, … \n",
            "A ⋃ B \n",
            "A ⋂ B \n",
            "P(A) \n",
            "A′ \n",
            "P(A | B) \n",
            "nCr \n",
            "\n",
            "nPr \n",
            "\n",
            "X, Y, R, … \n",
            "x, y, r, … \n",
            "x1, x2, … \n",
            "f1, f2, … \n",
            "p(x) \n",
            "p1, p2, … \n",
            "f(x) \n",
            "F(x) \n",
            "E(X) \n",
            "E(g(X)) \n",
            "Var(X) \n",
            "GX(t) \n",
            "MX(t) \n",
            "B(n, p) \n",
            "Geo(p) \n",
            "Po(λ) \n",
            "N(µ, σ2) \n",
            "µ \n",
            "σ2 \n",
            "σ \n",
            "\n",
            "x  \n",
            "\n",
            "s2 \n",
            "\n",
            "ρ \n",
            "r \n",
            "φ \n",
            "Φ \n",
            "H0, H1 \n",
            "\n",
            "events \n",
            "union of the events A and B \n",
            "intersection of the events A and B \n",
            "probability of the event A \n",
            "complement of the event A \n",
            "probability of the event A conditional on the event B \n",
            "the number of combinations of r objects from n,  Cn\n",
            "\n",
            "r =( )n\n",
            "\n",
            "r\n",
            "\n",
            "=\n",
            "\n",
            "!\n",
            "\n",
            "n\n",
            "r n r−\n",
            "!(\n",
            "\n",
            ")!\n",
            "\n",
            "the number of permutations of r objects from n, \n",
            "\n",
            "n\n",
            "\n",
            "P\n",
            "r\n",
            "\n",
            "=\n",
            "\n",
            "n\n",
            "\n",
            "!\n",
            "\n",
            "(\n",
            "\n",
            "n r\n",
            "−\n",
            "\n",
            ")!\n",
            "\n",
            "random variables \n",
            "values of the random variables X, Y, R, … \n",
            "observations \n",
            "frequencies with which the observations x1, x2, … occur \n",
            "probability function P(X = x) of the discrete random variable X \n",
            "probabilities of the values x1, x2, … of the discrete random variable X \n",
            "value of the probability density function of a continuous random variable X \n",
            "value of the cumulative distribution function of a continuous random variable X \n",
            "expectation of the random variable X \n",
            "expectation of g(X) \n",
            "variance of the random variable X \n",
            "probability generating function for the discrete random variable X \n",
            "moment generating function for the random variable X \n",
            "binomial distribution with parameters n and p \n",
            "geometric distribution with parameter p \n",
            "Poisson distribution with parameter λ \n",
            "normal distribution with mean µ and variance σ2 \n",
            "population mean \n",
            "population variance \n",
            "population standard deviation \n",
            "\n",
            "sample mean, \n",
            "\n",
            "x\n",
            "\n",
            "1 n\n",
            "= ∑  \n",
            "x\n",
            "i\n",
            "n =\n",
            "i\n",
            "1\n",
            "\n",
            "unbiased estimate of population variance from a sample,  2\n",
            "s\n",
            "\n",
            "=\n",
            "\n",
            "n\n",
            "\n",
            "n\n",
            "\n",
            "1\n",
            "\n",
            "− ∑\n",
            "\n",
            "1\n",
            "\n",
            "i\n",
            "\n",
            "1\n",
            "=\n",
            "\n",
            "(\n",
            "\n",
            "x\n",
            "i\n",
            "\n",
            "2\n",
            "\n",
            "−\n",
            "\n",
            "x\n",
            "\n",
            ")\n",
            "\n",
            "product moment correlation coefficient for a population \n",
            "product moment correlation coefficient for a sample \n",
            "probability density function of the standardised normal variable Z ~ N(0, 1) \n",
            "cumulative distribution function of the standardised normal variable Z ~ N(0, 1) \n",
            "null and alternative hypotheses for a hypothesis test \n",
            "\n",
            "5 \n",
            "\n",
            " \n",
            " \n",
            " \n",
            " \n",
            " \n",
            "\f\n"
          ]
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "## **Conclusion :**\n",
        "> **PyPDF2** had better formatting while **PDFMiner.Six** had better symbole identification"
      ],
      "metadata": {
        "id": "QUnfcZyV6jpf"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "# PDF Text Cleanup Attempt\n",
        "In this step we will try to fix the problems found with both tools to see if we can overcome one of them and use it as a tool to create textual database\n",
        "- We'll try to fix **PyPDF2** by :\n",
        "  - Fixing its symbole recognition problem (some symboles are given latex like notations instead of the symbole itself)\n",
        "  - Fixing its spacing problem (some variables are joined into other words which makes them harder to recognize)"
      ],
      "metadata": {
        "id": "2-8WDVSxHiPU"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "## Substitution-based Symbole correction"
      ],
      "metadata": {
        "id": "U9Iq5FId5FkE"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "def replace_symbols(text, replacements):\n",
        "  \"\"\"Replaces custom symbols in text with their LaTeX equivalents from a dictionary.\"\"\"\n",
        "  new_text = text\n",
        "  for symbol, replacement in replacements.items():\n",
        "    new_text = new_text.replace(symbol, replacement)\n",
        "  return new_text"
      ],
      "metadata": {
        "id": "Gr6rEvMtOyun"
      },
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "code",
      "source": [
        "replacements = {\n",
        "\t\"/bardbl\" : \"║\" ,\n",
        "\t\"/angb∇acket∇ight\" : \">\" ,\n",
        "\t\"/angb∇acketleft\" : \"<\" ,\n",
        "\t\"/parenleftBigg\"  : \"(\" ,\n",
        "\t\"/parenrightBigg\" : \")\" ,\n",
        "\t\"/integraldisplay\" : \"∫\"\n",
        "}\n",
        "\n",
        "# testing symbole replacement function\n",
        "replaced_text = replace_symbols(pypdf_text, replacements)\n",
        "print(replaced_text)"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "iYNwdMi1vlJp",
        "outputId": "4a281f9f-ff1a-42a9-9bdb-928aecdbf041",
        "collapsed": true
      },
      "execution_count": null,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "arXiv:physics/9905061v3  [physics.plasm-ph]  8 Jun 1999DPNU-99-14\n",
            "Electron acceleration to ultrarelativistic energies in a c ollisionless\n",
            "oblique shock wave\n",
            "Naoki Bessho and Yukiharu Ohsawa\n",
            "Department of Physics, Nagoya University, Nagoya 464-8602 , Japan\n",
            "(July 14, 2011)\n",
            "Abstract\n",
            "Electron motion in an oblique shock wave is studied by means o f a one-\n",
            "dimensional, relativistic, electromagnetic, particle si mulation code with full\n",
            "ion and electron dynamics. It is found that an oblique shock c an produce\n",
            "electrons with ultra-relativistic energies; Lorentz fact ors with γ>∼100 have\n",
            "been observed in our simulations. The physical mechanisms f or the reﬂection\n",
            "and acceleration are discussed, and the maximum energy is es timated. If\n",
            "the electron reﬂection occurs near the end of a large-amplit ude pulse, those\n",
            "particles will then be trapped in the pulse and gain a great de al of energy.\n",
            "The theory predicts that the electron energies can become es pecially high at\n",
            "certain propagation angles. This is veriﬁed by the simulati ons.\n",
            "52.65.Cc, 52.35.Tc, 52.35.Mw, 98.70.Sa\n",
            "Typeset using REVT EX\n",
            "1\n",
            "I. INTRODUCTION\n",
            "Electron acceleration has been an important issue in plasma physics and astrophysics. In\n",
            "solar physics, for instance, the acceleration of electrons as well as of ions has received a great\n",
            "deal of attention; in solar ﬂares, gamma rays with energies o f several tens of MeV, which are\n",
            "emitted by the bremsstrahlung from the high-energy electro ns, are often observed [1], [2]. In\n",
            "astrophysics, much more energetic electrons have been disc ussed. From the observations of\n",
            "X-rays and gamma rays, it is now believed that electrons with energies up to ∼100 TeV are\n",
            "produced by a shell-type supernova remnant SN 1006 [3], [4]. It is also believed that in the\n",
            "Crab Nebula high-energy electrons accelerated up to ∼100 TeV are produced by the pulsar\n",
            "wind [5]. Theories based on the Fermi acceleration model hav e been proposed to account for\n",
            "these high-energy phenomena [6], [7]. As for laboratory pla smas, in an attempt to realize\n",
            "plasma-based accelerators, the electron acceleration in a relativistic space charge wave has\n",
            "been intensively studied by simulations and experiments [8 ].\n",
            "Ion acceleration in a magnetosonic wave has been studied by m any authors [9]- [22].\n",
            "In a single-ion-species plasma, the large positive potenti al formed in the shock region can\n",
            "reﬂect some ions and give great energies to them [9]- [18]. In a multi-ion-species plasma\n",
            "such as space plasmas, some of the majority ions (hydrogen) c an also gain energies by\n",
            "the same mechanism. Furthermore, the transverse electric ﬁ eld in the shock wave can\n",
            "accelerate all particles of all kinds of (minority) heavy io ns to nearly the same speed [19],\n",
            "[20]. In addition, in a turbulent plasma where particles can interact with many diﬀerent\n",
            "large-amplitude magnetosonic pulses, some energetic ions can be further accelerated by\n",
            "a diﬀerent mechanism [21]. These processes have been extens ively studied by theory and\n",
            "particle simulation by many authors and have been applied to the production of high-energy\n",
            "particles in solar ﬂares and interplanetary shocks [9]- [20 ].\n",
            "Strong electron acceleration in shock waves, however, has n ot been shown by particle\n",
            "simulations. In this paper we will theoretically and numeri cally study electron motion in\n",
            "a magnetosonic shock wave propagating obliquely to a magnet ic ﬁeld. We will show that\n",
            "an oblique shock can produce ultra-relativistic electrons . The preliminary result has been\n",
            "reported in Ref. [23].\n",
            "In Sec. II, we outline some basic properties of oblique shock s. In Sec. III, we theoretically\n",
            "discuss the possibility of electron reﬂection. If electron s are reﬂected near the end of the\n",
            "main pulse, then after the reﬂection they can have great ener gies in the main pulse.\n",
            "Several authors discussed stationary pulse solutions for t he nonlinear magnetosonic waves\n",
            "[24]- [27]. However, if the magnetic ﬁeld is rather strong so thatωce/ωpe>∼1, where ωce\n",
            "is the electron cyclotron frequency and ωpeis the plasma frequency, or if the amplitude is\n",
            "quite large, then the pulse propagation is not perfectly sta tionary [13], [14]; small-amplitude\n",
            "ﬂuctuations are generated in and around the main pulse. Even though the potential is\n",
            "positive in the main pulse in a shock wave, the electron reﬂec tion studied in this paper can\n",
            "occur in such a non-stationary pulse.\n",
            "In Sec. IV, we estimate the maximum energy of a reﬂected elect ron and discuss its\n",
            "dependence on plasma parameters. In Sec. V, we further study the electron acceleration by\n",
            "using a one-dimensional (one space coordinate and three vel ocity components), relativistic,\n",
            "electromagnetic, particle simulation code with full ion an d electron dynamics. It will be\n",
            "shown that some electrons can be reﬂected near the end of the m ain pulse and are trapped\n",
            "2\n",
            "in the shock region. In the simulations, an appreciable numb er of electrons have energies\n",
            "γ >100, where γis the Lorentz factor. Further, we will examine the dependen ce ofγon the\n",
            "propagation angle and shock speed. It is veriﬁed that, as the theory predicts, there are special\n",
            "angles and shock speeds at which the electron acceleration i s especially strong. Our work is\n",
            "summarized in Sec. VI. In Appendix A, we describe relations a mong physical quantities in a\n",
            "magnetosonic wave. In Appendix B, we give a rough estimate fo r the magnitude of parallel\n",
            "electric ﬁeld E║on the basis of a simple physical picture.\n",
            "II. VELOCITIES AND FIELDS IN AN OBLIQUE WAVE\n",
            "We consider a magnetosonic wave propagating in the xdirection with a speed vshin\n",
            "an external magnetic ﬁeld in the ( x, z) plane. We assume that the ﬁeld quantities depend\n",
            "onxonly; ∂/∂y =∂/∂z = 0. Then, from the equation ∇ ·B= 0, it follows that the\n",
            "xcomponent of the magnetic ﬁeld is constant, Bx=Bx0. The other components Byand\n",
            "Bzare functions of x, and in the pulse region Byas well as Bzcan have ﬁnite values. In\n",
            "the wave frame where the time derivatives are zero ( ∂/∂t= 0), the electric ﬁeld in the y\n",
            "direction is constant, Ey=Ey0, and Ezis zero. Hence we have B= (Bx0, By, Bz) and\n",
            "E= (Ex, Ey0,0). In Sections II-IV, theoretical analyses will be made mai nly in the wave\n",
            "frame.\n",
            "The velocity of the guiding-center position of an electron, vg, may be written as\n",
            "vg= (B/B)v║+vd, (1)\n",
            "withvdbeing the drift velocity\n",
            "vd=cE×B/B2−(cµ/e)(B× ∇B)/B2. (2)\n",
            "Herecis the speed of light, −eis the electron charge ( e >0),v║is the velocity parallel to\n",
            "the magnetic ﬁeld, and µis the magnetic moment\n",
            "µ=mev2\n",
            "⊥/(2B), (3)\n",
            "withmethe electron mass and v⊥the gyration speed perpendicular to the magnetic ﬁeld.\n",
            "In the far upstream region, we have no ∇B-drift. Also, the zcomponent of the velocity\n",
            "averaged over all the electrons in a small volume element mus t be zero, <vgz0>= 0; the\n",
            "subscript 0 refers to the quantities in the far upstream regi on. Accordingly, from the z\n",
            "component of Eq. (1) we have the average parallel velocity\n",
            "<v║0>=cEy0\n",
            "B0Bx0\n",
            "Bz0, (4)\n",
            "in the wave frame (see Fig. 1). Also, because <vgx0>=−vsh, thexcomponent of Eq. (1)\n",
            "gives\n",
            "vsh=−Bx0\n",
            "B0<v║0> −cEy0Bz0\n",
            "B2\n",
            "0. (5)\n",
            "ThusEy0is related to the shock speed vshthrough\n",
            "3\n",
            "Ey0=−vshBz0/c. (6)\n",
            "For the deﬁniteness, we assume that Bx0,Bz0andvshare all positive; and thus Ey0<0.\n",
            "The nonlinear wave theory tells us the wave structure. A shoc k wave will have a positive\n",
            "electric potential ϕ(x). The quantities ϕ,Bz,ni, and nehave similar proﬁles [28]- [29] (see\n",
            "Appendix A); here ni, and neare the ion and electron densities, respectively. On the oth er\n",
            "hand, ExandByare proportional to the xderivatives of these quantities; for instance,\n",
            "Ex=−∂ϕ/∂x . These relations among the quantities are obtained for smal l-amplitude\n",
            "waves. In the following theoretical analysis, we assume tha t the relations are also valid for\n",
            "large-amplitude waves. As we will see later in Fig. 7, simula tion results also support this\n",
            "assumption.\n",
            "We denote by xmthexposition at which the potential takes its maximum value; hen ce,\n",
            "Ex(xm)≃0 and By(xm)≃0. For later use, we also note that the quantity Bz/Btakes its\n",
            "maximum value at x=xm; that is, if Bz(x) is smaller than Bz(xm), then\n",
            "Bz(xm)2\n",
            "B2\n",
            "x0+Bz(xm)2>Bz(x)2\n",
            "B2\n",
            "x0+By(x)2+Bz(x)2. (7)\n",
            "This can be proved by the following equation\n",
            "Bz(xm)2\n",
            "B2\n",
            "x0+Bz(xm)2−Bz(x)2\n",
            "B2\n",
            "x0+By(x)2+Bz(x)2=[Bz(xm)2−Bz(x)2]B2\n",
            "x0+Bz(xm)2By(x)2\n",
            "[B2\n",
            "x0+Bz(xm)2][B2\n",
            "x0+By(x)2+Bz(x)2].(8)\n",
            "Because Bz(xm) is the maximum value of Bz(x), the term [ Bz(xm)2−Bz(x)2]B2\n",
            "x0is positive.\n",
            "Hence the right-hand side of the above equation is positive.\n",
            "We know the stationary, ﬁnite-amplitude solutions for perp endicular waves [24]- [27]. The\n",
            "stationary solitary wave solution is valid when the magneti c ﬁeld is weak, ωce/ωpe<<1, and\n",
            "when the wave amplitude is not large, 1 < M < 2, where Mis the Alfv´ en Mach number;\n",
            "M=vsh/vAwithvAthe Alfv´ en speed, v2\n",
            "A=B2\n",
            "0/(4πnimi). Simulations and experiments\n",
            "show that large-amplitude pulses can propagate nearly stea dily, even when the amplitudes\n",
            "are large or the magnetic ﬁelds are strong. Oblique pulses ca n also propagate nearly steadily.\n",
            "As the magnetic ﬁeld becomes stronger or the wave amplitude i s increased, non-stationary\n",
            "eﬀects will become important [13], [14].\n",
            "III. ELECTRON REFLECTION\n",
            "Because the potential is usually positive in the shock regio n, some of the ions can be\n",
            "reﬂected when they go up the potential. As we will see later, e lectrons can also be reﬂected;\n",
            "its mechanism is, however, diﬀerent from that of ion reﬂecti on. We give a physical picture\n",
            "of the electron reﬂection.\n",
            "From the nonrelativistic equation of motion for an electron\n",
            "medv\n",
            "dt=−e(\n",
            "E+v×B\n",
            "c)\n",
            ", (9)\n",
            "we have an equation for the kinetic energy in the wave frame\n",
            "4\n",
            "me(v2−v2\n",
            "0)/2 =e(ϕ−ϕ0)−eEy0∫\n",
            "vydt. (10)\n",
            "Here we have used the relation Ez= 0. The velocity vincludes gyro, parallel, and drift\n",
            "motions.\n",
            "We deﬁne length sas\n",
            "ds= (B/B x0)vgxdt, (11)\n",
            "where vgxis the xcomponent of the electron velocity\n",
            "vgx= (Bx0/B)v║+vdx, (12)\n",
            "(see Fig. 2). Thus d s/dtis given by\n",
            "ds\n",
            "dt=v║+B\n",
            "Bx0vdx. (13)\n",
            "We can interpret d sas an inﬁnitesimal length along the ﬁeld line corresponding to the length\n",
            "dx=vgxdt; ds= (B/B x0)dx. In general the guiding-center velocity vgis not parallel to\n",
            "the magnetic ﬁeld B. Hence the guiding center does not move along the ﬁeld line. F or a\n",
            "stationary, one-dimensional problem ( ∂/∂t= 0 and ∂/∂y=∂/∂z= 0), all quantities depend\n",
            "only on x. As a result, for any stationary function f(x) we have\n",
            "∫x2\n",
            "x1f(x)(B/B x0)dx=∫s2\n",
            "s1f(s)ds, (14)\n",
            "where s1ands2are the positions along the same ﬁeld line; their xpositions are x1andx2,\n",
            "respectively.\n",
            "If we deﬁne quantity Fas\n",
            "E║=−dF/ds, (15)\n",
            "then with the aid of (11) and the relation E║=E·B/B, we have\n",
            "F=−∫(ExBx0+Ey0By)\n",
            "BB\n",
            "Bx0dx, (16)\n",
            "which can be written as\n",
            "F=−∫/parenleftbigg\n",
            "Ex−vshBz0By\n",
            "cBx0/parenrightbigg\n",
            "dx. (17)\n",
            "Using the electric potential ϕand the vector potential A(B=∇ ×A), we ﬁnd\n",
            "F=ϕ−(vsh/c)(Bz0/Bx0)Az, (18)\n",
            "where Azis the zcomponent of the vector potential,\n",
            "Az=−∫\n",
            "Bydx. (19)\n",
            "5\n",
            "The quantities ϕandBzhave similar proﬁles, while Byis proportional to ∂Bz/∂x[28], [29].\n",
            "Hence AzandFhave proﬁles similar to ϕ.\n",
            "If we deﬁne quantity Flin the laboratory frame (the subscript ldenotes the laboratory\n",
            "frame) as\n",
            "Fl=−∫\n",
            "El║(Bl/Blx0)dxl, (20)\n",
            "then for a stationary, one-dimensional problem, that is, fo r a case where wave proﬁles can\n",
            "be written as f(xl, tl) =f(xl−vshtl), we ﬁnd\n",
            "Fl=−∫/parenleftbigg\n",
            "Elx−vshBlz0Bly\n",
            "cBlx0/parenrightbigg\n",
            "dxl. (21)\n",
            "One can readily show that it is related to Fin the wave frame as\n",
            "F=γshFl, (22)\n",
            "where γshis deﬁned as\n",
            "γsh= (1−v2\n",
            "sh/c2)−1/2. (23)\n",
            "In Appendix B, we give a rough estimate of the maximum value of F, Eq. (B9), found from\n",
            "qualitative physical considerations.\n",
            "Combining Eqs. (10) and (18), we eliminate ϕ\n",
            "mev2/2−mev2\n",
            "0/2 =e(F−F0)−(eEy0/Bx0)(Az−Az0+∫\n",
            "vyBx0dt). (24)\n",
            "By virtue of the deﬁnition of Az, (19), and the relation d x=vxdt, this equation can be\n",
            "written as\n",
            "mev2/2−mev2\n",
            "0/2 =e(F−F0) +e(Ey0/Bx0)∫\n",
            "(vxBy−vyBx0)dt. (25)\n",
            "Then, using the zcomponent of the equation of motion, we ﬁnd\n",
            "mev2/2−mev2\n",
            "0/2 =e(F−F0)−mec(Ey0/Bx0)(vz−vz0). (26)\n",
            "In the drift approximation, the kinetic energy can be expres sed as\n",
            "mev2/2 =me(v2\n",
            "║+v2\n",
            "d)/2 +µB. (27)\n",
            "Hence, averaging Eq. (26) over the electron cyclotron perio d, we have\n",
            "(me/2)(v2\n",
            "║+v2\n",
            "d) +µB+mec(Ey0/Bx0)vgz−eF\n",
            "= (me/2)(v2\n",
            "║0+v2\n",
            "d0) +µB0+mec(Ey0/Bx0)vz0−eF0. (28)\n",
            "With the aid of the zcomponent of Eq. (1), we can eliminate vgzand obtain the following\n",
            "equation\n",
            "6\n",
            "(me/2)(v║−vrv)2+K=e(F−F0)−µ(B−B0)\n",
            "+ (me/2)(v2\n",
            "║0+v2\n",
            "d0)−mec(Ey0/Bx0)(vdz−vz0)−(me/2)v2\n",
            "d, (29)\n",
            "where vrvandKare deﬁned as\n",
            "vrv=−cEy0Bz/(Bx0B) (>0), (30)\n",
            "K=−mev2\n",
            "rv/2 ( <0). (31)\n",
            "If we eliminate Ey0by substituting Eq. (6) in Eq. (30), vrvcan be expressed as\n",
            "vrv=vshBz0Bz/(Bx0B). (32)\n",
            "We will see below that the velocity vgxis reversed when v║=vrv. In the pulse region, the\n",
            "quantity e(F−F0) can be much larger in magnitude than the other terms on the ri ght-hand\n",
            "side of Eq. (29) (see Appendix B). We denote the left-hand sid e of Eq. (29) by Ee, i.e.,\n",
            "Ee= (me/2)(v║−vrv)2+K. As shown in Fig. 3, at a ﬁxed point x,Eehas its minimum\n",
            "value Kwhen v║=vrv. It is negative in the region\n",
            "0< v║<2vrv. (33)\n",
            "Because vdx=cEy0Bz/B2, it follows from Eq. (12) that vgx= 0 if v║=vrv. As can be\n",
            "seen from Eq. (4), most of the electrons have negative parall el velocities in the far upstream\n",
            "region. Accordingly, we suppose that the initial parallel v elocity is negative, v║<0. If\n",
            "v║gradually increases and exceeds vrvat a certain position, then the xcomponent of the\n",
            "guiding center velocity, vgx, changes from negative to positive values there, i.e., it is reversed.\n",
            "As an electron moves, i.e., as xchanges, the quantities K,F,B, etc. also vary. At any\n",
            "point x,Kis the minimum value of the left-hand side of Eq. (29). If ther e is a region where\n",
            "the values of the right-hand side of Eq. (29) become smaller t hanK, then the electron\n",
            "cannot enter there; it will be reﬂected. We recall that ϕ,F, and Bzhave similar proﬁles\n",
            "and that at x=xmthe quantity ( Bz/B) has a maximum value [see Eq. (7)]. Because Kis\n",
            "proportional to ( Bz/B)2, it will be quite small ( |K|is large) around the point x=xm, and\n",
            "Fis large. In front of or behind the main pulse, Fis relatively small and Kis relatively\n",
            "large; e(F−F0) could be smaller than K. Thus, if the electron reﬂection occurs, it will be\n",
            "in such regions.\n",
            "In quasi-perpendicular shocks, where ( Bz0/Bx0)>>1,vrvis large. Hence Khas large\n",
            "negative values. Thus the electron reﬂection in this mechan ism will be diﬃcult.\n",
            "Figure 4 shows a schematic diagram of the trajectory of guidi ng center. Here it is assumed\n",
            "that the reﬂection takes place at point D; the dotted line sho ws the orbit of a passing electron.\n",
            "As an electron moves from point A to C, it moves to the negative ydirection because of the\n",
            "Ex×Bzdrift. It gains kinetic energy ∆ E1from the electric potential\n",
            "∆E1=eϕ(xC)−eϕ(xA) (>0). (34)\n",
            "At the same time, it loses energy ∆ E2because of the electric ﬁeld Ey0\n",
            "∆E2=−eEy0(yC−yA) (<0). (35)\n",
            "7\n",
            "The net change in the energy is therefore\n",
            "∆E= ∆E1+ ∆E2. (36)\n",
            "Even though the magnitudes of ∆ E1and ∆ E2are quite large, they almost cancel when an\n",
            "electron moves with drift approximation; in particular, in a perpendicular pulse they have\n",
            "exactly the same magnitude and ∆ E= 0 [30]. However, if an electron is reﬂected and moves\n",
            "from D to E, then it would gain energies from both ExandEy0. As a result, the increase in\n",
            "energy is\n",
            "∆E= ∆E1+ ∆E3, (37)\n",
            "where ∆ E3is deﬁned as\n",
            "∆E3=−eEy0(yE−yA) (>0). (38)\n",
            "IV. ESTIMATE OF ENERGY INCREASE\n",
            "Next, we will obtain the maximum energy of a reﬂected electro n. Because it can have\n",
            "quite high energy, we will use the relativistic equation of m otion\n",
            "med(γv)\n",
            "dt=−e(\n",
            "E+v×B\n",
            "c)\n",
            ", (39)\n",
            "where γis the Lorentz factor. For the stationary, one-dimensional system being considered,\n",
            "this can be integrated to give the energy conservation equat ion\n",
            "meγc2−meγ0c2=e(ϕ−ϕ0)−eEy0∫\n",
            "vydt. (40)\n",
            "This is a relativistic form of Eq. (10). Again, using the quan titiesF,Az, and the equation\n",
            "of motion, we can rewrite Eq. (40) as\n",
            "meγc2−meγ0c2=e(F−F0)−mec(Ey0/Bx0)(γvz−γ0vz0). (41)\n",
            "If we deﬁne quantity has\n",
            "h=mec2−mevshBz0vz/Bx0, (42)\n",
            "then from Eq. (41) we have\n",
            "γ= [e(F−F0) +h0γ0]/h. (43)\n",
            "Equation (42) indicates that his positive if Bz0/Bx0is of order unity. If Bz0/Bx0is much\n",
            "greater than unity, then hcan be negative. In addition, Eq. (43) shows that, if the init ial\n",
            "value of his positive, h0>0, then his always positive in the region where ( F−F0) is\n",
            "positive.\n",
            "8\n",
            "We show in Fig. 5 schematic diagram of the function [ e(F−F0) +h0γ0]. We suppose\n",
            "thatFhas a maximum value at x=xmand has a minimum value right behind the main\n",
            "pulse. If the reﬂection occurs, it would be in this dip, as sug gested in the previous section.\n",
            "The reﬂection point will be denoted by xr. In the top panel this function e(F−F0)+h0γ0is\n",
            "always positive. In the second one it becomes negative right behind the main pulse; however\n",
            "its maximum value is positive ( h0can be either positive or negative). In the bottom one\n",
            "it is always negative. For the present acceleration mechani sm, the top panel is the most\n",
            "important; the strong acceleration takes place in this case .\n",
            "Such dips would not be present in the perfectly stationary so lutions [24]- [29]. In large-\n",
            "amplitude magnetosonic waves, however, non-stationary di ps can be generated [13], [14].\n",
            "We here just assume that there is a dip behind the main pulse an d discuss its eﬀects on\n",
            "particle orbits.\n",
            "First we consider the top panel in Fig. 5: An electron with pos itiveh0is reﬂected at a\n",
            "certain point near the end of the main pulse. We will show that , after the reﬂection, hhas\n",
            "its minimum (positive) value at x=xm; hence γhas its maximum value there.\n",
            "As mentioned earlier, the parallel velocity v║is already positive when an electron is\n",
            "reﬂected, i.e., when vgxis reversed, even if its initial value is negative. After the reﬂection,\n",
            "v║will be increased by the (positive) parallel electric force . Because E║changes its sign at\n",
            "x=xm,v║will have a maximum value at x=xm(unless vxchanges to negative values again\n",
            "before reaching the point xm). The parallel velocity will become the dominant component in\n",
            "the electron velocity, v∼v║B/B. The quantity Bz/Bas well as ϕandBzhas a maximum\n",
            "value at this point. Thus vz, which can be approximated as vz∼v║Bz/B, will also have the\n",
            "maximum value there. Because h0and (Fm−F0) are both positive, his positive at x=xm.\n",
            "In addition, Eq. (42) indicates that hdecreases (draws closer to zero) as vzincreases. Hence,\n",
            "hof a reﬂected electron will have a minimum positive value at x=xm. Its energy γwill\n",
            "therefore have a maximum value there. If hcan be close to zero, then γwould become quite\n",
            "large.\n",
            "IfBx0is small and vz0has a large positive value, then h0becomes small (can be even\n",
            "negative). We now consider such particles; the second panel in Fig. 5. Also in this case,\n",
            "vzwill have larger values after the reﬂection; thus hwill decrease. At the reﬂection point\n",
            "in the dip, the quantity [ e(F−F0) +h0γ0] is negative. Consequently, h(xr)<0. If the\n",
            "reﬂection occurs, the electron would move in the positive xdirection. Because vzincreases,\n",
            "hwill further decrease. Hence, the particle would never be ab le to penetrate the region\n",
            "again where hγ > 0. The strong acceleration in this mechanism, therefore, wi ll not take\n",
            "place.\n",
            "Next, we consider a case where the quantity [ e(F−F0) +h0γ0] is always negative (the\n",
            "bottom panel in Fig. 5). Then, his always negative. If the reﬂection occurs at x=xr,\n",
            "then|h|would increase while |e(F−F0) +h0γ0|would decrease. Hence, even if the particle\n",
            "reaches the point xm,γwould not have an extremely large value. Therefore, the stro ng\n",
            "acceleration is not expected.\n",
            "We can estimate the maximum value γmin the wave frame by substituting the maximum\n",
            "value of F, Eq. (B12), in Eq. (43). The maximum γin the laboratory frame can be obtained\n",
            "by the Lorentz transformation as\n",
            "γlm=γsh(1 +vxvsh/c2)γm≃γshγm. (44)\n",
            "9\n",
            "Now let us consider a case where hbecomes quite small, h≃0, atx=xm. There the\n",
            "electron speed would be close to the speed of light c. Hence, roughly, we have vz∼cBz/B.\n",
            "The quantity hcan then be expressed as\n",
            "h=mec2/parenleftbigg\n",
            "1−γshvsh\n",
            "cBz\n",
            "Btanθ/parenrightbigg\n",
            ", (45)\n",
            "where tan θ=Blz0/Blx0. Here we have used the relations Bz0=γshBlz0andBx0=Blx0.\n",
            "Because Bzbecomes quite large in the shock region, we may put as Bz/B∼1. Equation\n",
            "(45) then suggests that the electron energy can become extre mely large when\n",
            "γsh(vsh/c) tanθ= 1, (46)\n",
            "or, equivalently\n",
            "ccosθ=vsh. (47)\n",
            "V. SIMULATION STUDIES\n",
            "A. Simulation Method\n",
            "We further study the electron acceleration by using a one-di mensional (one space co-\n",
            "ordinate and three velocity components), relativistic, el ectromagnetic, particle simulation\n",
            "code with full ion and electron dynamics [31]. The total syst em length is Lx= 4096∆ g\n",
            "for most of the simulation runs, where ∆ gis the grid spacing. All lengths and velocities in\n",
            "the simulations were normalized to ∆ gandωpe∆g, respectively, where ωpeis the spatially\n",
            "averaged plasma frequency.\n",
            "We use a bounded plasma model. The plasma is limited to the reg ion 400 < x < 3696.\n",
            "The particles are specularly reﬂected at x= 400 and at x= 3696. The radiation leaving the\n",
            "plasma region is absorbed in the vacuum regions, 0 < x < 400 and 3696 < x < 4096. Thus\n",
            "the electromagnetic interactions between the two plasma bo undaries through the vacuum\n",
            "regions are made negligibly small [12].\n",
            "In most of the plasma region, i.e., in 500 < x < 3696, each particle species initially has\n",
            "a uniform density and has a Maxwellian velocity distributio n function. In the small region,\n",
            "400< x < 480, the plasma density is four times as high as that in the mai n region. That\n",
            "is, the initial plasma density can be written as\n",
            "ne= 4n0for 400 < x < x 1,\n",
            "ne= 4n0exp[−(x−x1)2/(2d2)] for x1< x < 500,\n",
            "ne=n0for 500 < x < 3696.\n",
            "Herex1= 480 and d= 12. In the high-density region, 400 < x < 480, particles have\n",
            "shifted Maxwellian velocity distribution functions with a verage velocity vps;fj∼exp[−(v−\n",
            "vps)2/(2v2\n",
            "Tj)], where the subscript jrefers to particle species. These particles act as a\n",
            "piston. They push the neighboring particles and excite a sho ck wave. We can change the\n",
            "10\n",
            "shock strength by changing the magnitude of vps; the velocity vpsis perpendicular to the\n",
            "external magnetic ﬁeld so that no particles have extremely l arge parallel speeds initially.\n",
            "More detailed description about the simulation code can be f ound in Ref. [12].\n",
            "The simulation parameters are as follows. The number of simu lation particles is Ni=\n",
            "Ne= 262 ,144. The ion-to-electron mass ratio is mi/me= 100. The ratio ωce/ωpeis\n",
            "ωce/ωpe= 3.0 in the far upstream region. The electron and ion thermal vel ocities are\n",
            "vTe/c= 0.38 and vTi/c= 0.01, respectively. The Alfv´ en speed is vA/c= 0.3. The electron\n",
            "skin depth is c/ωpe= 4∆ g. The time step is suﬃciently small, ωpe∆t= 0.02, so that ∆ tis\n",
            "much smaller than the plasma and cyclotron periods even in th e shock region. As in the\n",
            "theoretical model, the external magnetic ﬁeld is in the ( x, z) plane, and waves propagate\n",
            "in the xdirection.\n",
            "B. Simulation Results\n",
            "First, we study a shock with the propagation angle θ= 45◦; i.e., Bx0=Bz0. Figure 6\n",
            "shows proﬁles of Bzin the shock wave at various times. Here, the magnetic ﬁeld is normalized\n",
            "toBz0. The Mach number is observed to be M= 2.3, i.e., the shock speed is vsh= 2.3vA.\n",
            "The maximum value of Bzis 4∼5 times as large as Bz0. The main pulse propagates nearly\n",
            "steadily. However, the proﬁle is not perfectly stationary.\n",
            "Figure 7 displays ﬁeld proﬁles at ωpet= 680; here the magnetic and electric ﬁelds are\n",
            "both normalized to B0. As the theory predicts [28], [29], EyandBzhave similar proﬁles,\n",
            "while Ex,Ez, andByare quite small in magnitude at the point where Bzhas its maximum\n",
            "value. Figure 8 shows phase space plots of electrons at ωpet= 680. In the shock region,\n",
            "ultra-relativistic electrons are produced. The maximum va lue of Lorentz factors is γ≃130.\n",
            "We also see that pxandpzhave greater values than py, which is consistent with the picture\n",
            "that the parallel velocity is dominant in the motion of high- energy electrons. From Figs. 7\n",
            "and 8, we see that some electrons are reﬂected near the end of t he main pulse and that the\n",
            "high-energy electrons are present in the shock region.\n",
            "Figure 9 shows proﬁles of E║,F, and the electric potential ϕatωpet= 650 and at\n",
            "ωpet= 680. The quantities ˜Fand ˜ϕdenote eF/(mec2) and eϕ/(mec2), respectively. The\n",
            "quantities Fandϕwere obtained in the simulation by the following equations\n",
            "F(x) =−∫x\n",
            "E║(B/B x0)dx, (48)\n",
            "ϕ(x) =−∫x\n",
            "Exdx. (49)\n",
            "As mentioned earlier, roughly speaking, the parallel elect ric ﬁeld E║is positive in the region\n",
            "where ∂Bz/∂xand∂F/∂x are negative. It is negative in the region where they are posi tive.\n",
            "Comparing the plots at diﬀerent times, we see that the wave pr oﬁles are not perfectly\n",
            "stationary. They vary with time with small-amplitudes. In p articular, as shown in the plot\n",
            "atωpet= 680, Fsometimes becomes negative near the end of the main pulse, wh ere the\n",
            "electron reﬂection can take place.\n",
            "Next, we will study trajectories of electrons; we will ﬁnd th at it is the reﬂected electrons\n",
            "that gain great energies. Figure 10 displays time variation s of (x−vsht),y,z, and γof\n",
            "11\n",
            "electrons in a shock with vsh= 1.75vA; (x−vsht) is the xposition in the wave frame. The\n",
            "lengths are normalized to the electron skin depth c/ωpe. (Here the system size is a half of\n",
            "that in the previous run; the other parameters such as Ne/Lxare the same.) Two typical\n",
            "examples are presented here. The thick lines show an electro n that was accelerated, and\n",
            "the thin lines represent an electron that was not accelerate d; even the low-energy electron\n",
            "here has energy γ≃10 in the shock region and γ≃5 behind the shock. In all of the four\n",
            "panels, the low- and high-energy electrons both have small- amplitude, cyclotron oscillations\n",
            "with short periods, ωpet<∼20; note that the cyclotron period depends on γas well as the\n",
            "ﬁeld strength B. At the same time, the accelerated electron has a large-ampl itude, long-\n",
            "period oscillation with period ωpet≃170; it begins when the particle enters the shock,\n",
            "ωpet= 180, and continues until the end of this simulation run. (Si mulations show that\n",
            "this period increases with γ. To observe this oscillation as many times as possible, we ha ve\n",
            "chosen here the case where γis not so large.) For comparison with Fig. 4, we show the\n",
            "times corresponding to points B, C, D, and E. From the long-pe riod oscillation of ( x−vsht)\n",
            "in the top panel, we see that the electron that will become hig h energy is reﬂected and then\n",
            "trapped by the shock wave. On the other hand, the quantity ( x−vsht) of the low-energy\n",
            "electron keeps decreasing, which means that this particle p asses through the shock region\n",
            "without strong interactions. The second panel shows that th eyposition of the accelerated\n",
            "electron also oscillates, while the third one shows that zincreases rapidly after the reﬂection.\n",
            "The average value of velocity vzover the period from ωpet= 220 to 400 is vz/c≃0.6, and\n",
            "the value of vzat maximum γisvz/c≃0.8. The maximum γin the bottom panel is γ≃40.\n",
            "By the Lorentz transformation of these data, we can ﬁnd vandBin the wave frame.\n",
            "At the time and position of maximum γ, they are v=c(0.27,0.13,0.95) and B=\n",
            "B0(0.70,0.16,2.56). This conﬁrms our statement in the previous section that v║is the\n",
            "dominant component in vthere; i.e., vz≃v║Bz/B.\n",
            "According to the theory, (47), electron energies can be extr emely high at some angles\n",
            "and shock speeds. To examine this prediction, we have carrie d out several simulations with\n",
            "diﬀerent values of the propagation angle θ, keeping other parameters unchanged; Lx=\n",
            "4096∆ g,ωce/ωpe= 3.0, and vsh≃2.1vA. Figure 11 shows the maximum value of γas a\n",
            "function of the propagation angle θ. Here the electron energy takes the highest value at the\n",
            "angle θ≃52◦. For these simulation parameters, Eq. (47) predicts that γtakes a peak value\n",
            "atθ≃53◦. Thus the theory and simulation are in good agreement. Furth er, we carried out\n",
            "simulations with diﬀerent values of the shock speed vsh; the propagation angle is ﬁxed to be\n",
            "θ= 45◦. Figure 12 shows the maximum value of γas a function of the shock speed vsh. The\n",
            "observed values have a peak at vsh/c≃0.7. For these parameters, Eq. (47) predicts that γ\n",
            "takes large values for vsh/c≃0.71. Again, the theory and simulation are in good agreement.\n",
            "VI. SUMMARY\n",
            "We have studied electron motion in a shock wave propagating o bliquely to a magnetic\n",
            "ﬁeld. First, we analytically discussed electron motion in a n oblique shock wave. It is\n",
            "pointed out that, if an electron is reﬂected near the end of a l arge-amplitude pulse, it would\n",
            "gain a great deal of energy from the electric ﬁeld formed in th e wave. The condition for\n",
            "the reﬂection is then examined. Further, the maximum energy of a reﬂected electron is\n",
            "12\n",
            "estimated, and its dependence on plasma parameters is discu ssed. Next, we investigated the\n",
            "shock evolution and associated electron acceleration by us ing a one-dimensional, relativistic,\n",
            "electromagnetic, particle simulation code with full ion an d electron dynamics. It was shown\n",
            "that an oblique shock can produce ultra-relativistic elect rons; Lorentz factors with γ>∼100\n",
            "have been observed. As the theory predicts, at certain propa gation angles electron energies\n",
            "become extremely high. The electron reﬂection and resultan t acceleration takes place when\n",
            "the wave amplitude is large and the magnetic ﬁeld is rather st rong, ωce>∼ωpe.\n",
            "In the present paper, strong electron acceleration has been demonstrated. As future\n",
            "work, it would be desirable to develop further the quantitat ive theory for large-amplitude\n",
            "oblique waves. The evaluation of the electric ﬁeld strength along the magnetic ﬁeld will be\n",
            "especially important, which will enable us to estimate the m aximum energy more accurately.\n",
            "From the view point of nonlinear wave propagation, motion of trapped electrons and their\n",
            "eﬀects on the wave evolution are also quite interesting.\n",
            "APPENDIX A: RELATIONS AMONG QUANTITIES\n",
            "We here describe relations among the physical variables, us ing the approximation appro-\n",
            "priate for low-frequency magnetosonic waves with small amp litudes (for the details of the\n",
            "calculations, see Ref. [29]). We assume quasi neutrality, n≃ni≃neand use the following\n",
            "stretched variables\n",
            "τ=ǫ3/2t, (A1)\n",
            "ξ=ǫ1/2(x−vp0t), (A2)\n",
            "(vp0is the wave propagation speed in the long-wavelength limit i n a ﬁnite beta plasma) and\n",
            "expansion\n",
            "n=n0+ǫn1+ǫ2n2+..., (A3)\n",
            "Ey=ǫEy1+ǫ2Ey2+..., (A4)\n",
            "Bz=Bz0+ǫBz1+ǫ2Bz2+..., (A5)\n",
            "then, we can express the lowest order perturbations in terms ofn1as\n",
            "Bz1/B0=cEy1/(B0vp0) = [(v2\n",
            "p0−c2\n",
            "s)/v2\n",
            "Asinθ](n1/n0), (A6)\n",
            "cEz1\n",
            "B0=−vp0By1\n",
            "B0=(ωce−ωci)\n",
            "ωceωci(v2\n",
            "p0−c2\n",
            "s)v2\n",
            "p0\n",
            "(v2\n",
            "p0−v2\n",
            "Acos2θ)cotθ\n",
            "n0∂n1\n",
            "∂ξ. (A7)\n",
            "Herecsis the sound speed, c2\n",
            "s= (Γ epe0+Γipi0)/(n0mi) with Γ j(j=eori) the speciﬁc heat\n",
            "ratio and pj0the equilibrium pressure. We can obtain KdV equation by proc eeding to the\n",
            "next order, O(ǫ5/2).\n",
            "The relations (A6) and (A7) are valid, even when the wave proﬁ le is not soliton-like.\n",
            "That is, perturbations propagating in the same direction wi thvx∼vp0have the relations\n",
            "(A6) and (A7).\n",
            "13\n",
            "APPENDIX B: MAGNITUDES OF E║AND F\n",
            "For a nonrelativistic, small-amplitude oblique magnetoso nic wave, the electric ﬁeld par-\n",
            "allel to the magnetic ﬁeld, E║, has been analytically obtained [32]. However, for a large-\n",
            "amplitude wave, it will be quite diﬃcult to obtain E║in a rigorous manner. Here, on the\n",
            "basis of a simple physical picture, we will give a rough estim ate of the parallel electric ﬁeld\n",
            "E║in a relativistic, large-amplitude, oblique shock wave. Fu rther, by using it, we will obtain\n",
            "the maximum value of the quantity F.\n",
            "We consider in the wave frame. In the ﬂuid model, the equation of motion may be\n",
            "written as\n",
            "mjd(γjvj)\n",
            "dt=qjE+qjvj×B\n",
            "c. (B1)\n",
            "Here the pressure term is neglected. The subscript jrefers to ions j=ior electrons j=e.\n",
            "The time derivative d /dtis deﬁned as\n",
            "d\n",
            "dt=∂\n",
            "∂t+vj· ∇. (B2)\n",
            "We denote by bthe unit vector along the magnetic ﬁeld. Then multiplying Eq . (B1) by b,\n",
            "we have\n",
            "mjd(γjb·vj)\n",
            "dt−mjγjvj·db\n",
            "dt=qjE║. (B3)\n",
            "Lett1be the time when a ﬂuid element is at the leading edge of the sho ck (the location\n",
            "where the shock proﬁle begins to sharply rise) and t2be the time when it is at the point\n",
            "xm;xmis the location where the electric potential and magnetic ﬁe ld take their maximum\n",
            "values. Then, integrating from t1tot2, we ﬁnd for the ions\n",
            "mi(γi2b2·vi2−γi1b1·vi1)−mi<γivi> ·(b2−b1)≃eE║∆\n",
            "|<vix>|. (B4)\n",
            "Here the bracket indicates the mean value over the time perio d (t2−t1), and ∆ is the\n",
            "shock width. We have a relation ( t2−t1)≃∆/|<vix>|. For oblique shocks, ∆ will be of\n",
            "the order of the ion inertial length, ∆ ≃c/ωpi. The velocity at time t1isvi1=−vshex,\n",
            "where exis the unit vector in the xdirection. The velocity vixwill be slowed down by\n",
            "the longitudinal electric ﬁeld. Thus vix2will be smaller in magnitude than vix1. Because\n",
            "Bzis large at the point xmandBy(xm)∼0, the unit vector b2is nearly parallel to the z\n",
            "direction. Consequently, the inner product ( b2·vi2) will be quite small; here we neglect the\n",
            "zcomponent of vi2. Since the xcomponent of b1is given by Bx0/B0, we have\n",
            "mi(γi1vsh+<γivix>)Bx0/B0≃eE║∆/|<vix>|. (B5)\n",
            "This gives the strength of the parallel electric ﬁeld as\n",
            "eE║≃miv2\n",
            "sh\n",
            "∆|<vix>|\n",
            "vsh(\n",
            "γi1+<γivix>\n",
            "vsh)Bx0\n",
            "B0. (B6)\n",
            "14\n",
            "Because |vix|is smaller than vsh, the range of |<vix>|is expected to be 1 /2<∼ |<vix>|/vsh<\n",
            "1, except maybe for extremely strong shock waves. In fact, if we assume a simple time\n",
            "dependence of vix\n",
            "vix=−vsh[1−a1(t−t1)/(t2−t1)] for t1≤t≤t2, (B7)\n",
            "witha1a constant (0 ≤a1≤1), then, after integrating over time, we have <vix>=−vsh(1−\n",
            "a1/2).\n",
            "The Lorentz factor for the ﬂuid ion will be close to unity. Acc ordingly, for instance, if\n",
            "the ion ﬂuid velocity vixis decreased to a value ∼ −vsh/2, then the value of the content in\n",
            "the parentheses would be about 1/4.\n",
            "The electron speed will be ∼vshwhen the electrons encounter the shock and will be close\n",
            "to the speed of light when they are at the point xm. If we assume that the electron velocity\n",
            "is nearly parallel to the ﬁeld line in the shock region, then ( v·db/dt) can be neglected.\n",
            "Hence we have\n",
            "me(−γe2c+γe1vshBx0/B0)≃ −eE║(t2−t1). (B8)\n",
            "The second term on the left-hand side, γe1vshBx0/B0, can be neglected compared with the\n",
            "ﬁrst term γe2c. The ions and electrons will have nearly the same time period (t2−t1) to\n",
            "keep charge neutrality. Substituting Eq. (B6) in (B8) yield s\n",
            "γe2≃mivsh\n",
            "mec(\n",
            "γi1+<γivix>\n",
            "vsh)Bx0\n",
            "B0. (B9)\n",
            "Atx=xm,vzcan be estimated as vz≃ −cBz/B(for reﬂected electrons the sign is reversed).\n",
            "For a large-amplitude shock, the ratio Bz/Bis an order-unity quantity there. The quantity\n",
            "hdeﬁned by Eq. (42) is obtained as\n",
            "h2≃mec2[1 +vshBz0/(cBx0)]. (B10)\n",
            "If we average Eq. (43) over electrons in a small volume elemen t, we have\n",
            "<hγ>=e(F−F0) +mec2<γ0>. (B11)\n",
            "Here we used the relation <vz0>= 0. If electrons have the velocity vz≃ −cBz/Batx=xm,\n",
            "then we ﬁnd the maximum value of Ffrom Eq. (43) as\n",
            "e(Fm−F0)≃h2γe2−mec2<γ0>. (B12)\n",
            "In many practical cases, we can take <γ0>to be order unity.\n",
            "Roughly speaking, the quantity e(Fm−F0) is of the order of micvsh. It is, therefore,\n",
            "much greater than electron thermal energy mev2\n",
            "Te(∼µB) and drift kinetic energy mev2\n",
            "d\n",
            "(∼mev2\n",
            "sh).\n",
            "15\n",
            "REFERENCES\n",
            "[1] S. R. Kane, K. Kai, T. Kosugi, S. Enome, P. B. Landecker, an d D. L. McKenzie,\n",
            "Astrophys. J. 271, 376 (1983).\n",
            "[2] M. Yoshimori, Y. Takai, K. Morimoto, and K. Suga, Publ. As tron. Soc. Japan 44, L107\n",
            "(1992).\n",
            "[3] K. Koyama, R. Petre, E. V. Gotthelf, et al., Nature 378, 255 (1995).\n",
            "[4] T. Tanimori, Y. Hayami, S. Kamei, et al., Astrophys. J. 497, L25 (1998).\n",
            "[5] T. Tanimori, K. Sakurazawa, S. A. Dazeley, et al., Astrophys. J. 492, L33 (1998).\n",
            "[6] R. D. Blandford and D. Eichler, Phys. Rep. 154, 1 (1987).\n",
            "[7] S. P. Reynolds and D. C. Ellison, Astrophys. J. 399, L75 (1992).\n",
            "[8] C. Joshi, C. E. Clayton, W. B. Mori, J. M. Dawson, and T. Kat souleas, Comments\n",
            "Plasma Phys. Controlled Fusion 16, 65 (1994).\n",
            "[9] D. Biskamp and H. Welter, Nuclear Fusion 12, 663 (1972).\n",
            "[10] M. M. Leory, D. K. Winske, C. C. Goodrich, C. S. Wu, and K. P apadopoulos, J.\n",
            "Geophys. Res. 87, 5081 (1982).\n",
            "[11] D. W. Forslund, K. B. Quest, J. U. Brackbill, and K. Lee, J . Geophys. Res. 89, 2142\n",
            "(1984).\n",
            "[12] Y. Ohsawa, Phys. Fluids 28, 2130 (1985).\n",
            "[13] Y. Ohsawa, J. Phys. Soc. Jpn. 55, 1047 (1986).\n",
            "[14] B. Lembege, and J. M. Dawson, Phys. Fluids B1, 1001 (1989).\n",
            "[15] R. L. Tokar, S. P. Gary and K. B. Quest, Phys. Fluids 30, 2569 (1987).\n",
            "[16] Y. Ohsawa, J. Phys. Soc. Jpn. 59, 2782 (1990).\n",
            "[17] R. Z. Sagdeev and V. D. Shapiro, Pis’ma Zh. Exps. Theor. F iz.17, 387 (1973) [JETP\n",
            "Lett.17, 279 (1973)].\n",
            "[18] M. A. Lee, V. D. Shapiro, and R. Z. Sagdeev, J. Geophys. Re s.101, 4777 (1996).\n",
            "[19] M. Toida and Y. Ohsawa, J. Phys. Soc. Jpn. 64, 2038 (1995).\n",
            "[20] M. Toida and Y. Ohsawa, Solar Physics 171, 161 (1997).\n",
            "[21] K. Maruyama, N. Bessho, and Y. Ohsawa, Phys. Plasmas 5, 3257 (1998).\n",
            "[22] B. Rau and T. Tajima, Phys. Plasmas 5, 3575 (1998).\n",
            "[23] N. Bessho, K. Maruyama, and Y. Ohsawa, J. Phys. Soc. Jpn. 68, 1 (1999).\n",
            "[24] J. H. Adlam and J. E. Allen, Philos. Mag. 3, 448 (1958).\n",
            "[25] L. Davis, R. L¨ ust, and A. Schl¨ uter, Z. Naturforsch. Te ilA13, 916 (1958).\n",
            "[26] C. S. Gardner and G. K. Morikawa, Commun. Pure Appl. Math .18, 35 (1965).\n",
            "[27] Y. Ohsawa, Phys. Fluids 29, 2474 (1986).\n",
            "[28] T. Kakutani, H. Ono, T. Taniuti, and C. C. Wei, J. Phys. So c. Jpn. 24, 1159 (1968).\n",
            "[29] Y. Ohsawa, Phys. Fluids 29, 1844 (1986).\n",
            "[30] Y. Ohsawa, J. Phys. Soc. Jpn. 58, 4445 (1989).\n",
            "[31] P. C. Liewer, A. T. Lin, J. M. Dawson, and M. Z. Caponi, Phy s. Fluids 24, 1364 (1981).\n",
            "[32] Y. Ohsawa, J. Phys. Soc. Jpn. 57, 929 (1988).\n",
            "16\n",
            "FIGURES\n",
            "FIG. 1. Velocities and ﬁelds in the far upstream region in the wave frame. The velocity vdis\n",
            "Ey0×B0drift, and −vshis equal to average guiding-center velocity in the xdirection <vgx0>.\n",
            "FIG. 2. Schematic diagram of magnetic ﬁeld, electron veloci ty, dx, and length d s.\n",
            "FIG. 3. Plot of function Ee(v║) = (me/2)(v║−vrv)2+Kat a ﬁxed xposition.\n",
            "FIG. 4. Schematic diagram of electron orbit in ( x, y) plane.\n",
            "FIG. 5. Schematic diagram of quantity [ e(F−F0) +h0γ0]. In the top panel, this quantity\n",
            "is always positive. In the second one, it is negative in the di p. In the bottom one, it is always\n",
            "negative.\n",
            "FIG. 6. Magnetic ﬁeld proﬁles of an oblique shock at various t imes.\n",
            "FIG. 7. Snapshots of ﬁeld proﬁles. Electric and magnetic ﬁel d proﬁles at ωpet= 680 are plotted.\n",
            "They are normalized to B0.\n",
            "FIG. 8. Phase space plots of electrons.\n",
            "FIG. 9. Proﬁles of E║, F, and ϕat two diﬀerent times. Here, ˜Fand ˜ϕdenote eF/(mec2) and\n",
            "eϕ/(mec2), respectively.\n",
            "FIG. 10. Time variations of ( x−vsht),y,z, and γof electrons. The lengths are normalized to\n",
            "the electron skin depth c/ωpe.\n",
            "FIG. 11. Maximum γversus propagation angle θ. The propagation speeds for these shocks are\n",
            "vsh≃2.1vA.\n",
            "FIG. 12. Maximum γversus shock speed vsh. The propagation angle θis ﬁxed to be θ= 45◦\n",
            "17\n",
            "z\n",
            "xB -vshvd\n",
            "<v  >//0yEy0\n",
            "0\n",
            "vdtgdx\n",
            "ds\n",
            "B\n",
            "vrvv//E  (   )v//e\n",
            "0\n",
            "K\n",
            "y\n",
            "xAB\n",
            "CDE\n",
            "γ e(F-F )+h000\n",
            "x\n",
            "x\n",
            "xrxm\n",
            "xxrxmxmxr\n",
            "250 375 500 625ωpet=400 2\n",
            "04500600700800\n",
            "Bz\n",
            "Bz0\n",
            "x/(c/ωpe)\n",
            "-101\n",
            "Ex\n",
            "012Ey\n",
            "-0.500.5\n",
            "Ez\n",
            "-0.500.5By\n",
            "024\n",
            "525 550 575 600Bz\n",
            "x/(c/ωpe)\n",
            "050100\n",
            "px\n",
            "050100\n",
            "py\n",
            "050100\n",
            "pzz\n",
            "04080120\n",
            "525 550 575 600γ\n",
            "x/(c/ωpe)\n",
            "-0.300.3\n",
            "E//ωpet=650\n",
            "B0\n",
            "010\n",
            "540 555 570x/(c/ωpe)Fφ-0.300.3ωpet=680\n",
            "525 540 555010\n",
            "x/(c/ωpe)Fφ\n",
            "050\n",
            "10\n",
            "0\n",
            "-10\n",
            "200\n",
            "100\n",
            "0\n",
            "40\n",
            "20\n",
            "0x-v  tsh\n",
            "y\n",
            "z\n",
            "γ\n",
            "200 400 600ωpetBCD E\n",
            "Fig.3BCD E\n",
            "20406080100\n",
            "40 45 50 55 60Angleγ\n",
            "50100150\n",
            "0.5 0.6 0.7 0.8vsh/cγ\n"
          ]
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "## NLTK-based text spacing correction"
      ],
      "metadata": {
        "id": "uPlsPnb65Mrv"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "def improve_spacing(text):\n",
        "  \"\"\"Attempts to improve spacing in a sentence using NLTK tokenization.\"\"\"\n",
        "  tokens = nltk.word_tokenize(text)  # Split text into words (tokens)\n",
        "  return \" \".join(tokens)  # Join tokens with spaces"
      ],
      "metadata": {
        "id": "3Jis1KZUOioO"
      },
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "code",
      "source": [
        "# testing NLTK spacing solution\n",
        "improved_text = improve_spacing(replaced_text)\n",
        "improved_text"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 143
        },
        "id": "lYBzr-gmEGF9",
        "outputId": "47fdbb3d-463f-440c-f2c2-ddc48e56ea99"
      },
      "execution_count": null,
      "outputs": [
        {
          "output_type": "execute_result",
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            "text/plain": [
              "'arXiv : physics/9905061v3 [ physics.plasm-ph ] 8 Jun 1999DPNU-99-14 Electron acceleration to ultrarelativistic energies in a c ollisionless oblique shock wave Naoki Bessho and Yukiharu Ohsawa Department of Physics , Nagoya University , Nagoya 464-8602 , Japan ( July 14 , 2011 ) Abstract Electron motion in an oblique shock wave is studied by means o f a one- dimensional , relativistic , electromagnetic , particle si mulation code with full ion and electron dynamics . It is found that an oblique shock c an produce electrons with ultra-relativistic energies ; Lorentz fact ors with γ > ∼100 have been observed in our simulations . The physical mechanisms f or the reﬂection and acceleration are discussed , and the maximum energy is es timated . If the electron reﬂection occurs near the end of a large-amplit ude pulse , those particles will then be trapped in the pulse and gain a great de al of energy . The theory predicts that the electron energies can become es pecially high at certain propagation angles . This is veriﬁed by the simulati ons . 52.65.Cc , 52.35.Tc , 52.35.Mw , 98.70.Sa Typeset using REVT EX 1 I . INTRODUCTION Electron acceleration has been an important issue in plasma physics and astrophysics . In solar physics , for instance , the acceleration of electrons as well as of ions has received a great deal of attention ; in solar ﬂares , gamma rays with energies o f several tens of MeV , which are emitted by the bremsstrahlung from the high-energy electro ns , are often observed [ 1 ] , [ 2 ] . In astrophysics , much more energetic electrons have been disc ussed . From the observations of X-rays and gamma rays , it is now believed that electrons with energies up to ∼100 TeV are produced by a shell-type supernova remnant SN 1006 [ 3 ] , [ 4 ] . It is also believed that in the Crab Nebula high-energy electrons accelerated up to ∼100 TeV are produced by the pulsar wind [ 5 ] . Theories based on the Fermi acceleration model hav e been proposed to account for these high-energy phenomena [ 6 ] , [ 7 ] . As for laboratory pla smas , in an attempt to realize plasma-based accelerators , the electron acceleration in a relativistic space charge wave has been intensively studied by simulations and experiments [ 8 ] . Ion acceleration in a magnetosonic wave has been studied by m any authors [ 9 ] - [ 22 ] . In a single-ion-species plasma , the large positive potenti al formed in the shock region can reﬂect some ions and give great energies to them [ 9 ] - [ 18 ] . In a multi-ion-species plasma such as space plasmas , some of the majority ions ( hydrogen ) c an also gain energies by the same mechanism . Furthermore , the transverse electric ﬁ eld in the shock wave can accelerate all particles of all kinds of ( minority ) heavy io ns to nearly the same speed [ 19 ] , [ 20 ] . In addition , in a turbulent plasma where particles can interact with many diﬀerent large-amplitude magnetosonic pulses , some energetic ions can be further accelerated by a diﬀerent mechanism [ 21 ] . These processes have been extens ively studied by theory and particle simulation by many authors and have been applied to the production of high-energy particles in solar ﬂares and interplanetary shocks [ 9 ] - [ 20 ] . Strong electron acceleration in shock waves , however , has n ot been shown by particle simulations . In this paper we will theoretically and numeri cally study electron motion in a magnetosonic shock wave propagating obliquely to a magnet ic ﬁeld . We will show that an oblique shock can produce ultra-relativistic electrons . The preliminary result has been reported in Ref . [ 23 ] . In Sec . II , we outline some basic properties of oblique shock s. In Sec . III , we theoretically discuss the possibility of electron reﬂection . If electron s are reﬂected near the end of the main pulse , then after the reﬂection they can have great ener gies in the main pulse . Several authors discussed stationary pulse solutions for t he nonlinear magnetosonic waves [ 24 ] - [ 27 ] . However , if the magnetic ﬁeld is rather strong so thatωce/ωpe > ∼1 , where ωce is the electron cyclotron frequency and ωpeis the plasma frequency , or if the amplitude is quite large , then the pulse propagation is not perfectly sta tionary [ 13 ] , [ 14 ] ; small-amplitude ﬂuctuations are generated in and around the main pulse . Even though the potential is positive in the main pulse in a shock wave , the electron reﬂec tion studied in this paper can occur in such a non-stationary pulse . In Sec . IV , we estimate the maximum energy of a reﬂected elect ron and discuss its dependence on plasma parameters . In Sec . V , we further study the electron acceleration by using a one-dimensional ( one space coordinate and three vel ocity components ) , relativistic , electromagnetic , particle simulation code with full ion an d electron dynamics . It will be shown that some electrons can be reﬂected near the end of the m ain pulse and are trapped 2 in the shock region . In the simulations , an appreciable numb er of electrons have energies γ > 100 , where γis the Lorentz factor . Further , we will examine the dependen ce ofγon the propagation angle and shock speed . It is veriﬁed that , as the theory predicts , there are special angles and shock speeds at which the electron acceleration i s especially strong . Our work is summarized in Sec . VI . In Appendix A , we describe relations a mong physical quantities in a magnetosonic wave . In Appendix B , we give a rough estimate fo r the magnitude of parallel electric ﬁeld E║on the basis of a simple physical picture . II . VELOCITIES AND FIELDS IN AN OBLIQUE WAVE We consider a magnetosonic wave propagating in the xdirection with a speed vshin an external magnetic ﬁeld in the ( x , z ) plane . We assume that the ﬁeld quantities depend onxonly ; ∂/∂y =∂/∂z = 0 . Then , from the equation ∇ ·B= 0 , it follows that the xcomponent of the magnetic ﬁeld is constant , Bx=Bx0 . The other components Byand Bzare functions of x , and in the pulse region Byas well as Bzcan have ﬁnite values . In the wave frame where the time derivatives are zero ( ∂/∂t= 0 ) , the electric ﬁeld in the y direction is constant , Ey=Ey0 , and Ezis zero . Hence we have B= ( Bx0 , By , Bz ) and E= ( Ex , Ey0,0 ) . In Sections II-IV , theoretical analyses will be made mai nly in the wave frame . The velocity of the guiding-center position of an electron , vg , may be written as vg= ( B/B ) v║+vd , ( 1 ) withvdbeing the drift velocity vd=cE×B/B2− ( cµ/e ) ( B× ∇B ) /B2 . ( 2 ) Herecis the speed of light , −eis the electron charge ( e > 0 ) , v║is the velocity parallel to the magnetic ﬁeld , and µis the magnetic moment µ=mev2 ⊥/ ( 2B ) , ( 3 ) withmethe electron mass and v⊥the gyration speed perpendicular to the magnetic ﬁeld . In the far upstream region , we have no ∇B-drift . Also , the zcomponent of the velocity averaged over all the electrons in a small volume element mus t be zero , < vgz0 > = 0 ; the subscript 0 refers to the quantities in the far upstream regi on . Accordingly , from the z component of Eq . ( 1 ) we have the average parallel velocity < v║0 > =cEy0 B0Bx0 Bz0 , ( 4 ) in the wave frame ( see Fig . 1 ) . Also , because < vgx0 > =−vsh , thexcomponent of Eq . ( 1 ) gives vsh=−Bx0 B0 < v║0 > −cEy0Bz0 B2 0 . ( 5 ) ThusEy0is related to the shock speed vshthrough 3 Ey0=−vshBz0/c . ( 6 ) For the deﬁniteness , we assume that Bx0 , Bz0andvshare all positive ; and thus Ey0 < 0 . The nonlinear wave theory tells us the wave structure . A shoc k wave will have a positive electric potential ϕ ( x ) . The quantities ϕ , Bz , ni , and nehave similar proﬁles [ 28 ] - [ 29 ] ( see Appendix A ) ; here ni , and neare the ion and electron densities , respectively . On the oth er hand , ExandByare proportional to the xderivatives of these quantities ; for instance , Ex=−∂ϕ/∂x . These relations among the quantities are obtained for smal l-amplitude waves . In the following theoretical analysis , we assume tha t the relations are also valid for large-amplitude waves . As we will see later in Fig . 7 , simula tion results also support this assumption . We denote by xmthexposition at which the potential takes its maximum value ; hen ce , Ex ( xm ) ≃0 and By ( xm ) ≃0 . For later use , we also note that the quantity Bz/Btakes its maximum value at x=xm ; that is , if Bz ( x ) is smaller than Bz ( xm ) , then Bz ( xm ) 2 B2 x0+Bz ( xm ) 2 > Bz ( x ) 2 B2 x0+By ( x ) 2+Bz ( x ) 2 . ( 7 ) This can be proved by the following equation Bz ( xm ) 2 B2 x0+Bz ( xm ) 2−Bz ( x ) 2 B2 x0+By ( x ) 2+Bz ( x ) 2= [ Bz ( xm ) 2−Bz ( x ) 2 ] B2 x0+Bz ( xm ) 2By ( x ) 2 [ B2 x0+Bz ( xm ) 2 ] [ B2 x0+By ( x ) 2+Bz ( x ) 2 ] . ( 8 ) Because Bz ( xm ) is the maximum value of Bz ( x ) , the term [ Bz ( xm ) 2−Bz ( x ) 2 ] B2 x0is positive . Hence the right-hand side of the above equation is positive . We know the stationary , ﬁnite-amplitude solutions for perp endicular waves [ 24 ] - [ 27 ] . The stationary solitary wave solution is valid when the magneti c ﬁeld is weak , ωce/ωpe < < 1 , and when the wave amplitude is not large , 1 < M < 2 , where Mis the Alfv´ en Mach number ; M=vsh/vAwithvAthe Alfv´ en speed , v2 A=B2 0/ ( 4πnimi ) . Simulations and experiments show that large-amplitude pulses can propagate nearly stea dily , even when the amplitudes are large or the magnetic ﬁelds are strong . Oblique pulses ca n also propagate nearly steadily . As the magnetic ﬁeld becomes stronger or the wave amplitude i s increased , non-stationary eﬀects will become important [ 13 ] , [ 14 ] . III . ELECTRON REFLECTION Because the potential is usually positive in the shock regio n , some of the ions can be reﬂected when they go up the potential . As we will see later , e lectrons can also be reﬂected ; its mechanism is , however , diﬀerent from that of ion reﬂecti on . We give a physical picture of the electron reﬂection . From the nonrelativistic equation of motion for an electron medv dt=−e ( E+v×B c ) , ( 9 ) we have an equation for the kinetic energy in the wave frame 4 me ( v2−v2 0 ) /2 =e ( ϕ−ϕ0 ) −eEy0∫ vydt . ( 10 ) Here we have used the relation Ez= 0 . The velocity vincludes gyro , parallel , and drift motions . We deﬁne length sas ds= ( B/B x0 ) vgxdt , ( 11 ) where vgxis the xcomponent of the electron velocity vgx= ( Bx0/B ) v║+vdx , ( 12 ) ( see Fig . 2 ) . Thus d s/dtis given by ds dt=v║+B Bx0vdx . ( 13 ) We can interpret d sas an inﬁnitesimal length along the ﬁeld line corresponding to the length dx=vgxdt ; ds= ( B/B x0 ) dx . In general the guiding-center velocity vgis not parallel to the magnetic ﬁeld B . Hence the guiding center does not move along the ﬁeld line . F or a stationary , one-dimensional problem ( ∂/∂t= 0 and ∂/∂y=∂/∂z= 0 ) , all quantities depend only on x . As a result , for any stationary function f ( x ) we have ∫x2 x1f ( x ) ( B/B x0 ) dx=∫s2 s1f ( s ) ds , ( 14 ) where s1ands2are the positions along the same ﬁeld line ; their xpositions are x1andx2 , respectively . If we deﬁne quantity Fas E║=−dF/ds , ( 15 ) then with the aid of ( 11 ) and the relation E║=E·B/B , we have F=−∫ ( ExBx0+Ey0By ) BB Bx0dx , ( 16 ) which can be written as F=−∫/parenleftbigg Ex−vshBz0By cBx0/parenrightbigg dx . ( 17 ) Using the electric potential ϕand the vector potential A ( B=∇ ×A ) , we ﬁnd F=ϕ− ( vsh/c ) ( Bz0/Bx0 ) Az , ( 18 ) where Azis the zcomponent of the vector potential , Az=−∫ Bydx . ( 19 ) 5 The quantities ϕandBzhave similar proﬁles , while Byis proportional to ∂Bz/∂x [ 28 ] , [ 29 ] . Hence AzandFhave proﬁles similar to ϕ . If we deﬁne quantity Flin the laboratory frame ( the subscript ldenotes the laboratory frame ) as Fl=−∫ El║ ( Bl/Blx0 ) dxl , ( 20 ) then for a stationary , one-dimensional problem , that is , fo r a case where wave proﬁles can be written as f ( xl , tl ) =f ( xl−vshtl ) , we ﬁnd Fl=−∫/parenleftbigg Elx−vshBlz0Bly cBlx0/parenrightbigg dxl . ( 21 ) One can readily show that it is related to Fin the wave frame as F=γshFl , ( 22 ) where γshis deﬁned as γsh= ( 1−v2 sh/c2 ) −1/2 . ( 23 ) In Appendix B , we give a rough estimate of the maximum value of F , Eq . ( B9 ) , found from qualitative physical considerations . Combining Eqs . ( 10 ) and ( 18 ) , we eliminate ϕ mev2/2−mev2 0/2 =e ( F−F0 ) − ( eEy0/Bx0 ) ( Az−Az0+∫ vyBx0dt ) . ( 24 ) By virtue of the deﬁnition of Az , ( 19 ) , and the relation d x=vxdt , this equation can be written as mev2/2−mev2 0/2 =e ( F−F0 ) +e ( Ey0/Bx0 ) ∫ ( vxBy−vyBx0 ) dt . ( 25 ) Then , using the zcomponent of the equation of motion , we ﬁnd mev2/2−mev2 0/2 =e ( F−F0 ) −mec ( Ey0/Bx0 ) ( vz−vz0 ) . ( 26 ) In the drift approximation , the kinetic energy can be expres sed as mev2/2 =me ( v2 ║+v2 d ) /2 +µB . ( 27 ) Hence , averaging Eq . ( 26 ) over the electron cyclotron perio d , we have ( me/2 ) ( v2 ║+v2 d ) +µB+mec ( Ey0/Bx0 ) vgz−eF = ( me/2 ) ( v2 ║0+v2 d0 ) +µB0+mec ( Ey0/Bx0 ) vz0−eF0 . ( 28 ) With the aid of the zcomponent of Eq . ( 1 ) , we can eliminate vgzand obtain the following equation 6 ( me/2 ) ( v║−vrv ) 2+K=e ( F−F0 ) −µ ( B−B0 ) + ( me/2 ) ( v2 ║0+v2 d0 ) −mec ( Ey0/Bx0 ) ( vdz−vz0 ) − ( me/2 ) v2 d , ( 29 ) where vrvandKare deﬁned as vrv=−cEy0Bz/ ( Bx0B ) ( > 0 ) , ( 30 ) K=−mev2 rv/2 ( < 0 ) . ( 31 ) If we eliminate Ey0by substituting Eq . ( 6 ) in Eq . ( 30 ) , vrvcan be expressed as vrv=vshBz0Bz/ ( Bx0B ) . ( 32 ) We will see below that the velocity vgxis reversed when v║=vrv . In the pulse region , the quantity e ( F−F0 ) can be much larger in magnitude than the other terms on the ri ght-hand side of Eq . ( 29 ) ( see Appendix B ) . We denote the left-hand sid e of Eq . ( 29 ) by Ee , i.e. , Ee= ( me/2 ) ( v║−vrv ) 2+K . As shown in Fig . 3 , at a ﬁxed point x , Eehas its minimum value Kwhen v║=vrv . It is negative in the region 0 < v║ < 2vrv . ( 33 ) Because vdx=cEy0Bz/B2 , it follows from Eq . ( 12 ) that vgx= 0 if v║=vrv . As can be seen from Eq . ( 4 ) , most of the electrons have negative parall el velocities in the far upstream region . Accordingly , we suppose that the initial parallel v elocity is negative , v║ < 0 . If v║gradually increases and exceeds vrvat a certain position , then the xcomponent of the guiding center velocity , vgx , changes from negative to positive values there , i.e. , it is reversed . As an electron moves , i.e. , as xchanges , the quantities K , F , B , etc . also vary . At any point x , Kis the minimum value of the left-hand side of Eq . ( 29 ) . If ther e is a region where the values of the right-hand side of Eq . ( 29 ) become smaller t hanK , then the electron can not enter there ; it will be reﬂected . We recall that ϕ , F , and Bzhave similar proﬁles and that at x=xmthe quantity ( Bz/B ) has a maximum value [ see Eq . ( 7 ) ] . Because Kis proportional to ( Bz/B ) 2 , it will be quite small ( |K|is large ) around the point x=xm , and Fis large . In front of or behind the main pulse , Fis relatively small and Kis relatively large ; e ( F−F0 ) could be smaller than K. Thus , if the electron reﬂection occurs , it will be in such regions . In quasi-perpendicular shocks , where ( Bz0/Bx0 ) > > 1 , vrvis large . Hence Khas large negative values . Thus the electron reﬂection in this mechan ism will be diﬃcult . Figure 4 shows a schematic diagram of the trajectory of guidi ng center . Here it is assumed that the reﬂection takes place at point D ; the dotted line sho ws the orbit of a passing electron . As an electron moves from point A to C , it moves to the negative ydirection because of the Ex×Bzdrift . It gains kinetic energy ∆ E1from the electric potential ∆E1=eϕ ( xC ) −eϕ ( xA ) ( > 0 ) . ( 34 ) At the same time , it loses energy ∆ E2because of the electric ﬁeld Ey0 ∆E2=−eEy0 ( yC−yA ) ( < 0 ) . ( 35 ) 7 The net change in the energy is therefore ∆E= ∆E1+ ∆E2 . ( 36 ) Even though the magnitudes of ∆ E1and ∆ E2are quite large , they almost cancel when an electron moves with drift approximation ; in particular , in a perpendicular pulse they have exactly the same magnitude and ∆ E= 0 [ 30 ] . However , if an electron is reﬂected and moves from D to E , then it would gain energies from both ExandEy0 . As a result , the increase in energy is ∆E= ∆E1+ ∆E3 , ( 37 ) where ∆ E3is deﬁned as ∆E3=−eEy0 ( yE−yA ) ( > 0 ) . ( 38 ) IV . ESTIMATE OF ENERGY INCREASE Next , we will obtain the maximum energy of a reﬂected electro n. Because it can have quite high energy , we will use the relativistic equation of m otion med ( γv ) dt=−e ( E+v×B c ) , ( 39 ) where γis the Lorentz factor . For the stationary , one-dimensional system being considered , this can be integrated to give the energy conservation equat ion meγc2−meγ0c2=e ( ϕ−ϕ0 ) −eEy0∫ vydt . ( 40 ) This is a relativistic form of Eq . ( 10 ) . Again , using the quan titiesF , Az , and the equation of motion , we can rewrite Eq . ( 40 ) as meγc2−meγ0c2=e ( F−F0 ) −mec ( Ey0/Bx0 ) ( γvz−γ0vz0 ) . ( 41 ) If we deﬁne quantity has h=mec2−mevshBz0vz/Bx0 , ( 42 ) then from Eq . ( 41 ) we have γ= [ e ( F−F0 ) +h0γ0 ] /h . ( 43 ) Equation ( 42 ) indicates that his positive if Bz0/Bx0is of order unity . If Bz0/Bx0is much greater than unity , then hcan be negative . In addition , Eq . ( 43 ) shows that , if the init ial value of his positive , h0 > 0 , then his always positive in the region where ( F−F0 ) is positive . 8 We show in Fig . 5 schematic diagram of the function [ e ( F−F0 ) +h0γ0 ] . We suppose thatFhas a maximum value at x=xmand has a minimum value right behind the main pulse . If the reﬂection occurs , it would be in this dip , as sug gested in the previous section . The reﬂection point will be denoted by xr . In the top panel this function e ( F−F0 ) +h0γ0is always positive . In the second one it becomes negative right behind the main pulse ; however its maximum value is positive ( h0can be either positive or negative ) . In the bottom one it is always negative . For the present acceleration mechani sm , the top panel is the most important ; the strong acceleration takes place in this case . Such dips would not be present in the perfectly stationary so lutions [ 24 ] - [ 29 ] . In large- amplitude magnetosonic waves , however , non-stationary di ps can be generated [ 13 ] , [ 14 ] . We here just assume that there is a dip behind the main pulse an d discuss its eﬀects on particle orbits . First we consider the top panel in Fig . 5 : An electron with pos itiveh0is reﬂected at a certain point near the end of the main pulse . We will show that , after the reﬂection , hhas its minimum ( positive ) value at x=xm ; hence γhas its maximum value there . As mentioned earlier , the parallel velocity v║is already positive when an electron is reﬂected , i.e. , when vgxis reversed , even if its initial value is negative . After the reﬂection , v║will be increased by the ( positive ) parallel electric force . Because E║changes its sign at x=xm , v║will have a maximum value at x=xm ( unless vxchanges to negative values again before reaching the point xm ) . The parallel velocity will become the dominant component in the electron velocity , v∼v║B/B . The quantity Bz/Bas well as ϕandBzhas a maximum value at this point . Thus vz , which can be approximated as vz∼v║Bz/B , will also have the maximum value there . Because h0and ( Fm−F0 ) are both positive , his positive at x=xm . In addition , Eq . ( 42 ) indicates that hdecreases ( draws closer to zero ) as vzincreases . Hence , hof a reﬂected electron will have a minimum positive value at x=xm . Its energy γwill therefore have a maximum value there . If hcan be close to zero , then γwould become quite large . IfBx0is small and vz0has a large positive value , then h0becomes small ( can be even negative ) . We now consider such particles ; the second panel in Fig . 5 . Also in this case , vzwill have larger values after the reﬂection ; thus hwill decrease . At the reﬂection point in the dip , the quantity [ e ( F−F0 ) +h0γ0 ] is negative . Consequently , h ( xr ) < 0 . If the reﬂection occurs , the electron would move in the positive xdirection . Because vzincreases , hwill further decrease . Hence , the particle would never be ab le to penetrate the region again where hγ > 0 . The strong acceleration in this mechanism , therefore , wi ll not take place . Next , we consider a case where the quantity [ e ( F−F0 ) +h0γ0 ] is always negative ( the bottom panel in Fig . 5 ) . Then , his always negative . If the reﬂection occurs at x=xr , then|h|would increase while |e ( F−F0 ) +h0γ0|would decrease . Hence , even if the particle reaches the point xm , γwould not have an extremely large value . Therefore , the stro ng acceleration is not expected . We can estimate the maximum value γmin the wave frame by substituting the maximum value of F , Eq . ( B12 ) , in Eq . ( 43 ) . The maximum γin the laboratory frame can be obtained by the Lorentz transformation as γlm=γsh ( 1 +vxvsh/c2 ) γm≃γshγm . ( 44 ) 9 Now let us consider a case where hbecomes quite small , h≃0 , atx=xm . There the electron speed would be close to the speed of light c. Hence , roughly , we have vz∼cBz/B . The quantity hcan then be expressed as h=mec2/parenleftbigg 1−γshvsh cBz Btanθ/parenrightbigg , ( 45 ) where tan θ=Blz0/Blx0 . Here we have used the relations Bz0=γshBlz0andBx0=Blx0 . Because Bzbecomes quite large in the shock region , we may put as Bz/B∼1 . Equation ( 45 ) then suggests that the electron energy can become extre mely large when γsh ( vsh/c ) tanθ= 1 , ( 46 ) or , equivalently ccosθ=vsh . ( 47 ) V. SIMULATION STUDIES A. Simulation Method We further study the electron acceleration by using a one-di mensional ( one space co- ordinate and three velocity components ) , relativistic , el ectromagnetic , particle simulation code with full ion and electron dynamics [ 31 ] . The total syst em length is Lx= 4096∆ g for most of the simulation runs , where ∆ gis the grid spacing . All lengths and velocities in the simulations were normalized to ∆ gandωpe∆g , respectively , where ωpeis the spatially averaged plasma frequency . We use a bounded plasma model . The plasma is limited to the reg ion 400 < x < 3696 . The particles are specularly reﬂected at x= 400 and at x= 3696 . The radiation leaving the plasma region is absorbed in the vacuum regions , 0 < x < 400 and 3696 < x < 4096 . Thus the electromagnetic interactions between the two plasma bo undaries through the vacuum regions are made negligibly small [ 12 ] . In most of the plasma region , i.e. , in 500 < x < 3696 , each particle species initially has a uniform density and has a Maxwellian velocity distributio n function . In the small region , 400 < x < 480 , the plasma density is four times as high as that in the mai n region . That is , the initial plasma density can be written as ne= 4n0for 400 < x < x 1 , ne= 4n0exp [ − ( x−x1 ) 2/ ( 2d2 ) ] for x1 < x < 500 , ne=n0for 500 < x < 3696 . Herex1= 480 and d= 12 . In the high-density region , 400 < x < 480 , particles have shifted Maxwellian velocity distribution functions with a verage velocity vps ; fj∼exp [ − ( v− vps ) 2/ ( 2v2 Tj ) ] , where the subscript jrefers to particle species . These particles act as a piston . They push the neighboring particles and excite a sho ck wave . We can change the 10 shock strength by changing the magnitude of vps ; the velocity vpsis perpendicular to the external magnetic ﬁeld so that no particles have extremely l arge parallel speeds initially . More detailed description about the simulation code can be f ound in Ref . [ 12 ] . The simulation parameters are as follows . The number of simu lation particles is Ni= Ne= 262 ,144 . The ion-to-electron mass ratio is mi/me= 100 . The ratio ωce/ωpeis ωce/ωpe= 3.0 in the far upstream region . The electron and ion thermal vel ocities are vTe/c= 0.38 and vTi/c= 0.01 , respectively . The Alfv´ en speed is vA/c= 0.3 . The electron skin depth is c/ωpe= 4∆ g. The time step is suﬃciently small , ωpe∆t= 0.02 , so that ∆ tis much smaller than the plasma and cyclotron periods even in th e shock region . As in the theoretical model , the external magnetic ﬁeld is in the ( x , z ) plane , and waves propagate in the xdirection . B. Simulation Results First , we study a shock with the propagation angle θ= 45◦ ; i.e. , Bx0=Bz0 . Figure 6 shows proﬁles of Bzin the shock wave at various times . Here , the magnetic ﬁeld is normalized toBz0 . The Mach number is observed to be M= 2.3 , i.e. , the shock speed is vsh= 2.3vA . The maximum value of Bzis 4∼5 times as large as Bz0 . The main pulse propagates nearly steadily . However , the proﬁle is not perfectly stationary . Figure 7 displays ﬁeld proﬁles at ωpet= 680 ; here the magnetic and electric ﬁelds are both normalized to B0 . As the theory predicts [ 28 ] , [ 29 ] , EyandBzhave similar proﬁles , while Ex , Ez , andByare quite small in magnitude at the point where Bzhas its maximum value . Figure 8 shows phase space plots of electrons at ωpet= 680 . In the shock region , ultra-relativistic electrons are produced . The maximum va lue of Lorentz factors is γ≃130 . We also see that pxandpzhave greater values than py , which is consistent with the picture that the parallel velocity is dominant in the motion of high- energy electrons . From Figs . 7 and 8 , we see that some electrons are reﬂected near the end of t he main pulse and that the high-energy electrons are present in the shock region . Figure 9 shows proﬁles of E║ , F , and the electric potential ϕatωpet= 650 and at ωpet= 680 . The quantities ˜Fand ˜ϕdenote eF/ ( mec2 ) and eϕ/ ( mec2 ) , respectively . The quantities Fandϕwere obtained in the simulation by the following equations F ( x ) =−∫x E║ ( B/B x0 ) dx , ( 48 ) ϕ ( x ) =−∫x Exdx . ( 49 ) As mentioned earlier , roughly speaking , the parallel elect ric ﬁeld E║is positive in the region where ∂Bz/∂xand∂F/∂x are negative . It is negative in the region where they are posi tive . Comparing the plots at diﬀerent times , we see that the wave pr oﬁles are not perfectly stationary . They vary with time with small-amplitudes . In p articular , as shown in the plot atωpet= 680 , Fsometimes becomes negative near the end of the main pulse , wh ere the electron reﬂection can take place . Next , we will study trajectories of electrons ; we will ﬁnd th at it is the reﬂected electrons that gain great energies . Figure 10 displays time variation s of ( x−vsht ) , y , z , and γof 11 electrons in a shock with vsh= 1.75vA ; ( x−vsht ) is the xposition in the wave frame . The lengths are normalized to the electron skin depth c/ωpe . ( Here the system size is a half of that in the previous run ; the other parameters such as Ne/Lxare the same . ) Two typical examples are presented here . The thick lines show an electro n that was accelerated , and the thin lines represent an electron that was not accelerate d ; even the low-energy electron here has energy γ≃10 in the shock region and γ≃5 behind the shock . In all of the four panels , the low- and high-energy electrons both have small- amplitude , cyclotron oscillations with short periods , ωpet < ∼20 ; note that the cyclotron period depends on γas well as the ﬁeld strength B . At the same time , the accelerated electron has a large-ampl itude , long- period oscillation with period ωpet≃170 ; it begins when the particle enters the shock , ωpet= 180 , and continues until the end of this simulation run . ( Si mulations show that this period increases with γ . To observe this oscillation as many times as possible , we ha ve chosen here the case where γis not so large . ) For comparison with Fig . 4 , we show the times corresponding to points B , C , D , and E. From the long-pe riod oscillation of ( x−vsht ) in the top panel , we see that the electron that will become hig h energy is reﬂected and then trapped by the shock wave . On the other hand , the quantity ( x−vsht ) of the low-energy electron keeps decreasing , which means that this particle p asses through the shock region without strong interactions . The second panel shows that th eyposition of the accelerated electron also oscillates , while the third one shows that zincreases rapidly after the reﬂection . The average value of velocity vzover the period from ωpet= 220 to 400 is vz/c≃0.6 , and the value of vzat maximum γisvz/c≃0.8 . The maximum γin the bottom panel is γ≃40 . By the Lorentz transformation of these data , we can ﬁnd vandBin the wave frame . At the time and position of maximum γ , they are v=c ( 0.27,0.13,0.95 ) and B= B0 ( 0.70,0.16,2.56 ) . This conﬁrms our statement in the previous section that v║is the dominant component in vthere ; i.e. , vz≃v║Bz/B . According to the theory , ( 47 ) , electron energies can be extr emely high at some angles and shock speeds . To examine this prediction , we have carrie d out several simulations with diﬀerent values of the propagation angle θ , keeping other parameters unchanged ; Lx= 4096∆ g , ωce/ωpe= 3.0 , and vsh≃2.1vA . Figure 11 shows the maximum value of γas a function of the propagation angle θ . Here the electron energy takes the highest value at the angle θ≃52◦ . For these simulation parameters , Eq . ( 47 ) predicts that γtakes a peak value atθ≃53◦ . Thus the theory and simulation are in good agreement . Furth er , we carried out simulations with diﬀerent values of the shock speed vsh ; the propagation angle is ﬁxed to be θ= 45◦ . Figure 12 shows the maximum value of γas a function of the shock speed vsh . The observed values have a peak at vsh/c≃0.7 . For these parameters , Eq . ( 47 ) predicts that γ takes large values for vsh/c≃0.71 . Again , the theory and simulation are in good agreement . VI . SUMMARY We have studied electron motion in a shock wave propagating o bliquely to a magnetic ﬁeld . First , we analytically discussed electron motion in a n oblique shock wave . It is pointed out that , if an electron is reﬂected near the end of a l arge-amplitude pulse , it would gain a great deal of energy from the electric ﬁeld formed in th e wave . The condition for the reﬂection is then examined . Further , the maximum energy of a reﬂected electron is 12 estimated , and its dependence on plasma parameters is discu ssed . Next , we investigated the shock evolution and associated electron acceleration by us ing a one-dimensional , relativistic , electromagnetic , particle simulation code with full ion an d electron dynamics . It was shown that an oblique shock can produce ultra-relativistic elect rons ; Lorentz factors with γ > ∼100 have been observed . As the theory predicts , at certain propa gation angles electron energies become extremely high . The electron reﬂection and resultan t acceleration takes place when the wave amplitude is large and the magnetic ﬁeld is rather st rong , ωce > ∼ωpe . In the present paper , strong electron acceleration has been demonstrated . As future work , it would be desirable to develop further the quantitat ive theory for large-amplitude oblique waves . The evaluation of the electric ﬁeld strength along the magnetic ﬁeld will be especially important , which will enable us to estimate the m aximum energy more accurately . From the view point of nonlinear wave propagation , motion of trapped electrons and their eﬀects on the wave evolution are also quite interesting . APPENDIX A : RELATIONS AMONG QUANTITIES We here describe relations among the physical variables , us ing the approximation appro- priate for low-frequency magnetosonic waves with small amp litudes ( for the details of the calculations , see Ref . [ 29 ] ) . We assume quasi neutrality , n≃ni≃neand use the following stretched variables τ=ǫ3/2t , ( A1 ) ξ=ǫ1/2 ( x−vp0t ) , ( A2 ) ( vp0is the wave propagation speed in the long-wavelength limit i n a ﬁnite beta plasma ) and expansion n=n0+ǫn1+ǫ2n2+ ... , ( A3 ) Ey=ǫEy1+ǫ2Ey2+ ... , ( A4 ) Bz=Bz0+ǫBz1+ǫ2Bz2+ ... , ( A5 ) then , we can express the lowest order perturbations in terms ofn1as Bz1/B0=cEy1/ ( B0vp0 ) = [ ( v2 p0−c2 s ) /v2 Asinθ ] ( n1/n0 ) , ( A6 ) cEz1 B0=−vp0By1 B0= ( ωce−ωci ) ωceωci ( v2 p0−c2 s ) v2 p0 ( v2 p0−v2 Acos2θ ) cotθ n0∂n1 ∂ξ . ( A7 ) Herecsis the sound speed , c2 s= ( Γ epe0+Γipi0 ) / ( n0mi ) with Γ j ( j=eori ) the speciﬁc heat ratio and pj0the equilibrium pressure . We can obtain KdV equation by proc eeding to the next order , O ( ǫ5/2 ) . The relations ( A6 ) and ( A7 ) are valid , even when the wave proﬁ le is not soliton-like . That is , perturbations propagating in the same direction wi thvx∼vp0have the relations ( A6 ) and ( A7 ) . 13 APPENDIX B : MAGNITUDES OF E║AND F For a nonrelativistic , small-amplitude oblique magnetoso nic wave , the electric ﬁeld par- allel to the magnetic ﬁeld , E║ , has been analytically obtained [ 32 ] . However , for a large- amplitude wave , it will be quite diﬃcult to obtain E║in a rigorous manner . Here , on the basis of a simple physical picture , we will give a rough estim ate of the parallel electric ﬁeld E║in a relativistic , large-amplitude , oblique shock wave . Fu rther , by using it , we will obtain the maximum value of the quantity F. We consider in the wave frame . In the ﬂuid model , the equation of motion may be written as mjd ( γjvj ) dt=qjE+qjvj×B c. ( B1 ) Here the pressure term is neglected . The subscript jrefers to ions j=ior electrons j=e . The time derivative d /dtis deﬁned as d dt=∂ ∂t+vj· ∇ . ( B2 ) We denote by bthe unit vector along the magnetic ﬁeld . Then multiplying Eq . ( B1 ) by b , we have mjd ( γjb·vj ) dt−mjγjvj·db dt=qjE║ . ( B3 ) Lett1be the time when a ﬂuid element is at the leading edge of the sho ck ( the location where the shock proﬁle begins to sharply rise ) and t2be the time when it is at the point xm ; xmis the location where the electric potential and magnetic ﬁe ld take their maximum values . Then , integrating from t1tot2 , we ﬁnd for the ions mi ( γi2b2·vi2−γi1b1·vi1 ) −mi < γivi > · ( b2−b1 ) ≃eE║∆ | < vix > | . ( B4 ) Here the bracket indicates the mean value over the time perio d ( t2−t1 ) , and ∆ is the shock width . We have a relation ( t2−t1 ) ≃∆/| < vix > | . For oblique shocks , ∆ will be of the order of the ion inertial length , ∆ ≃c/ωpi . The velocity at time t1isvi1=−vshex , where exis the unit vector in the xdirection . The velocity vixwill be slowed down by the longitudinal electric ﬁeld . Thus vix2will be smaller in magnitude than vix1 . Because Bzis large at the point xmandBy ( xm ) ∼0 , the unit vector b2is nearly parallel to the z direction . Consequently , the inner product ( b2·vi2 ) will be quite small ; here we neglect the zcomponent of vi2 . Since the xcomponent of b1is given by Bx0/B0 , we have mi ( γi1vsh+ < γivix > ) Bx0/B0≃eE║∆/| < vix > | . ( B5 ) This gives the strength of the parallel electric ﬁeld as eE║≃miv2 sh ∆| < vix > | vsh ( γi1+ < γivix > vsh ) Bx0 B0 . ( B6 ) 14 Because |vix|is smaller than vsh , the range of | < vix > |is expected to be 1 /2 < ∼ | < vix > |/vsh < 1 , except maybe for extremely strong shock waves . In fact , if we assume a simple time dependence of vix vix=−vsh [ 1−a1 ( t−t1 ) / ( t2−t1 ) ] for t1≤t≤t2 , ( B7 ) witha1a constant ( 0 ≤a1≤1 ) , then , after integrating over time , we have < vix > =−vsh ( 1− a1/2 ) . The Lorentz factor for the ﬂuid ion will be close to unity . Acc ordingly , for instance , if the ion ﬂuid velocity vixis decreased to a value ∼ −vsh/2 , then the value of the content in the parentheses would be about 1/4 . The electron speed will be ∼vshwhen the electrons encounter the shock and will be close to the speed of light when they are at the point xm . If we assume that the electron velocity is nearly parallel to the ﬁeld line in the shock region , then ( v·db/dt ) can be neglected . Hence we have me ( −γe2c+γe1vshBx0/B0 ) ≃ −eE║ ( t2−t1 ) . ( B8 ) The second term on the left-hand side , γe1vshBx0/B0 , can be neglected compared with the ﬁrst term γe2c . The ions and electrons will have nearly the same time period ( t2−t1 ) to keep charge neutrality . Substituting Eq . ( B6 ) in ( B8 ) yield s γe2≃mivsh mec ( γi1+ < γivix > vsh ) Bx0 B0 . ( B9 ) Atx=xm , vzcan be estimated as vz≃ −cBz/B ( for reﬂected electrons the sign is reversed ) . For a large-amplitude shock , the ratio Bz/Bis an order-unity quantity there . The quantity hdeﬁned by Eq . ( 42 ) is obtained as h2≃mec2 [ 1 +vshBz0/ ( cBx0 ) ] . ( B10 ) If we average Eq . ( 43 ) over electrons in a small volume elemen t , we have < hγ > =e ( F−F0 ) +mec2 < γ0 > . ( B11 ) Here we used the relation < vz0 > = 0 . If electrons have the velocity vz≃ −cBz/Batx=xm , then we ﬁnd the maximum value of Ffrom Eq . ( 43 ) as e ( Fm−F0 ) ≃h2γe2−mec2 < γ0 > . ( B12 ) In many practical cases , we can take < γ0 > to be order unity . Roughly speaking , the quantity e ( Fm−F0 ) is of the order of micvsh . It is , therefore , much greater than electron thermal energy mev2 Te ( ∼µB ) and drift kinetic energy mev2 d ( ∼mev2 sh ) . 15 REFERENCES [ 1 ] S. R. Kane , K. Kai , T. Kosugi , S. Enome , P. B. Landecker , an d D. L. McKenzie , Astrophys . J . 271 , 376 ( 1983 ) . [ 2 ] M. Yoshimori , Y. Takai , K. Morimoto , and K. Suga , Publ . As tron . Soc . Japan 44 , L107 ( 1992 ) . [ 3 ] K. Koyama , R. Petre , E. V. Gotthelf , et al. , Nature 378 , 255 ( 1995 ) . [ 4 ] T. Tanimori , Y. Hayami , S. Kamei , et al. , Astrophys . J . 497 , L25 ( 1998 ) . [ 5 ] T. Tanimori , K. Sakurazawa , S. A. Dazeley , et al. , Astrophys . J . 492 , L33 ( 1998 ) . [ 6 ] R. D. Blandford and D. Eichler , Phys . Rep. 154 , 1 ( 1987 ) . [ 7 ] S. P. Reynolds and D. C. Ellison , Astrophys . J . 399 , L75 ( 1992 ) . [ 8 ] C. Joshi , C. E. Clayton , W. B. Mori , J. M. Dawson , and T. Kat souleas , Comments Plasma Phys . Controlled Fusion 16 , 65 ( 1994 ) . [ 9 ] D. Biskamp and H. Welter , Nuclear Fusion 12 , 663 ( 1972 ) . [ 10 ] M. M. Leory , D. K. Winske , C. C. Goodrich , C. S. Wu , and K. P apadopoulos , J. Geophys . Res . 87 , 5081 ( 1982 ) . [ 11 ] D. W. Forslund , K. B . Quest , J. U. Brackbill , and K. Lee , J . Geophys . Res . 89 , 2142 ( 1984 ) . [ 12 ] Y. Ohsawa , Phys . Fluids 28 , 2130 ( 1985 ) . [ 13 ] Y. Ohsawa , J. Phys . Soc . Jpn . 55 , 1047 ( 1986 ) . [ 14 ] B. Lembege , and J. M. Dawson , Phys . Fluids B1 , 1001 ( 1989 ) . [ 15 ] R. L. Tokar , S. P. Gary and K. B . Quest , Phys . Fluids 30 , 2569 ( 1987 ) . [ 16 ] Y. Ohsawa , J. Phys . Soc . Jpn . 59 , 2782 ( 1990 ) . [ 17 ] R. Z. Sagdeev and V. D. Shapiro , Pis ’ ma Zh . Exps . Theor . F iz.17 , 387 ( 1973 ) [ JETP Lett.17 , 279 ( 1973 ) ] . [ 18 ] M. A. Lee , V. D. Shapiro , and R. Z. Sagdeev , J. Geophys . Re s.101 , 4777 ( 1996 ) . [ 19 ] M. Toida and Y. Ohsawa , J. Phys . Soc . Jpn . 64 , 2038 ( 1995 ) . [ 20 ] M. Toida and Y. Ohsawa , Solar Physics 171 , 161 ( 1997 ) . [ 21 ] K. Maruyama , N. Bessho , and Y. Ohsawa , Phys . Plasmas 5 , 3257 ( 1998 ) . [ 22 ] B. Rau and T. Tajima , Phys . Plasmas 5 , 3575 ( 1998 ) . [ 23 ] N. Bessho , K. Maruyama , and Y. Ohsawa , J. Phys . Soc . Jpn . 68 , 1 ( 1999 ) . [ 24 ] J. H. Adlam and J. E. Allen , Philos . Mag . 3 , 448 ( 1958 ) . [ 25 ] L. Davis , R. L¨ ust , and A. Schl¨ uter , Z. Naturforsch . Te ilA13 , 916 ( 1958 ) . [ 26 ] C. S. Gardner and G. K. Morikawa , Commun . Pure Appl . Math .18 , 35 ( 1965 ) . [ 27 ] Y. Ohsawa , Phys . Fluids 29 , 2474 ( 1986 ) . [ 28 ] T. Kakutani , H. Ono , T. Taniuti , and C. C. Wei , J. Phys . So c. Jpn . 24 , 1159 ( 1968 ) . [ 29 ] Y. Ohsawa , Phys . Fluids 29 , 1844 ( 1986 ) . [ 30 ] Y. Ohsawa , J. Phys . Soc . Jpn . 58 , 4445 ( 1989 ) . [ 31 ] P. C. Liewer , A. T. Lin , J. M. Dawson , and M. Z. Caponi , Phy s. Fluids 24 , 1364 ( 1981 ) . [ 32 ] Y. Ohsawa , J. Phys . Soc . Jpn . 57 , 929 ( 1988 ) . 16 FIGURES FIG . 1 . Velocities and ﬁelds in the far upstream region in the wave frame . The velocity vdis Ey0×B0drift , and −vshis equal to average guiding-center velocity in the xdirection < vgx0 > . FIG . 2 . Schematic diagram of magnetic ﬁeld , electron veloci ty , dx , and length d s. FIG . 3 . Plot of function Ee ( v║ ) = ( me/2 ) ( v║−vrv ) 2+Kat a ﬁxed xposition . FIG . 4 . Schematic diagram of electron orbit in ( x , y ) plane . FIG . 5 . Schematic diagram of quantity [ e ( F−F0 ) +h0γ0 ] . In the top panel , this quantity is always positive . In the second one , it is negative in the di p. In the bottom one , it is always negative . FIG . 6 . Magnetic ﬁeld proﬁles of an oblique shock at various t imes . FIG . 7 . Snapshots of ﬁeld proﬁles . Electric and magnetic ﬁel d proﬁles at ωpet= 680 are plotted . They are normalized to B0 . FIG . 8 . Phase space plots of electrons . FIG . 9 . Proﬁles of E║ , F , and ϕat two diﬀerent times . Here , ˜Fand ˜ϕdenote eF/ ( mec2 ) and eϕ/ ( mec2 ) , respectively . FIG . 10 . Time variations of ( x−vsht ) , y , z , and γof electrons . The lengths are normalized to the electron skin depth c/ωpe . FIG . 11 . Maximum γversus propagation angle θ . The propagation speeds for these shocks are vsh≃2.1vA . FIG . 12 . Maximum γversus shock speed vsh . The propagation angle θis ﬁxed to be θ= 45◦ 17 z xB -vshvd < v > //0yEy0 0 vdtgdx ds B vrvv//E ( ) v//e 0 K y xAB CDE γ e ( F-F ) +h000 x x xrxm xxrxmxmxr 250 375 500 625ωpet=400 2 04500600700800 Bz Bz0 x/ ( c/ωpe ) -101 Ex 012Ey -0.500.5 Ez -0.500.5By 024 525 550 575 600Bz x/ ( c/ωpe ) 050100 px 050100 py 050100 pzz 04080120 525 550 575 600γ x/ ( c/ωpe ) -0.300.3 E//ωpet=650 B0 010 540 555 570x/ ( c/ωpe ) Fφ-0.300.3ωpet=680 525 540 555010 x/ ( c/ωpe ) Fφ 050 10 0 -10 200 100 0 40 20 0x-v tsh y z γ 200 400 600ωpetBCD E Fig.3BCD E 20406080100 40 45 50 55 60Angleγ 50100150 0.5 0.6 0.7 0.8vsh/cγ'"
            ],
            "application/vnd.google.colaboratory.intrinsic+json": {
              "type": "string"
            }
          },
          "metadata": {},
          "execution_count": 33
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "## **Conclusion :**\n",
        "> We are able to fix some symbole issues for **PyPDF2** but still couldn't fix the spacing problem\n",
        "\n",
        "- Since **PDFminer.six** showed less issues with variable detection which is our main concern , we will use it instead for creating the textual data\n",
        "\n"
      ],
      "metadata": {
        "id": "dO8MFJgrGvMa"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "# Making Textual Data\n",
        "In this part we will try to go from PDF data to text data to complete our text preparation phase where :\n",
        "\n",
        "\n",
        "1.  we test our pipeline on 1 PDF file\n",
        "2.  we create our pipeline's function\n",
        "3.  we apply it to our PDF data\n"
      ],
      "metadata": {
        "id": "IPxnzZMT7xQh"
      }
    },
    {
      "cell_type": "markdown",
      "source": [
        "## Testing on sample PDF"
      ],
      "metadata": {
        "id": "oOVNAy94Je4E"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "# getting filename + adding it to text extention\n",
        "file_name = path.split(\"/\")[-1].split(\".pdf\")[0]+\".txt\"\n",
        "file_name"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 35
        },
        "id": "lsvGG5fNQCkq",
        "outputId": "7f97e035-2522-4000-ed9b-9a88e89f832c"
      },
      "execution_count": null,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "'9905061v3.txt'"
            ],
            "application/vnd.google.colaboratory.intrinsic+json": {
              "type": "string"
            }
          },
          "metadata": {},
          "execution_count": 35
        }
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "path =  \"/content/drive/MyDrive/UH - Final Year Project/Dataset/pdf/9905/9905061v3.pdf\"\n",
        "# getting file name\n",
        "file_name = path.split(\"/\")[-1].split(\".pdf\")[0]+\".txt\"\n",
        "output_string = StringIO()\n",
        "with open(path, 'rb') as in_file:\n",
        "    parser = PDFParser(in_file)\n",
        "    doc = PDFDocument(parser)\n",
        "    rsrcmgr = PDFResourceManager()\n",
        "    device = TextConverter(rsrcmgr, output_string, laparams=LAParams())\n",
        "    interpreter = PDFPageInterpreter(rsrcmgr, device)\n",
        "    for page in PDFPage.create_pages(doc):\n",
        "      interpreter.process_page(page)\n",
        "\n",
        "path2 = \"/content/drive/MyDrive/UH - Final Year Project/Text Files/\"\n",
        "text = output_string.getvalue()\n",
        "\n",
        "f = open(path2+file_name, \"w\")\n",
        "f.write(text)\n",
        "f.close()"
      ],
      "metadata": {
        "id": "r9VkWkIKHljH"
      },
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "code",
      "source": [
        "f = open(path2+file_name, \"r\")\n",
        "print(f.read())"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "collapsed": true,
        "id": "8EhgB5YeM8LL",
        "outputId": "19075c8f-035f-43de-dee1-96448dcd6bbf"
      },
      "execution_count": null,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "Electron acceleration to ultrarelativistic energies in a collisionless\n",
            "oblique shock wave\n",
            "\n",
            "DPNU-99-14\n",
            "\n",
            "Naoki Bessho and Yukiharu Ohsawa\n",
            "Department of Physics, Nagoya University, Nagoya 464-8602, Japan\n",
            "(July 14, 2011)\n",
            "\n",
            "Abstract\n",
            "\n",
            "Electron motion in an oblique shock wave is studied by means of a one-\n",
            "dimensional, relativistic, electromagnetic, particle simulation code with full\n",
            "ion and electron dynamics.\n",
            "It is found that an oblique shock can produce\n",
            "electrons with ultra-relativistic energies; Lorentz factors with γ >∼ 100 have\n",
            "been observed in our simulations. The physical mechanisms for the reﬂection\n",
            "and acceleration are discussed, and the maximum energy is estimated.\n",
            "If\n",
            "the electron reﬂection occurs near the end of a large-amplitude pulse, those\n",
            "particles will then be trapped in the pulse and gain a great deal of energy.\n",
            "The theory predicts that the electron energies can become especially high at\n",
            "certain propagation angles. This is veriﬁed by the simulations.\n",
            "\n",
            "52.65.Cc, 52.35.Tc, 52.35.Mw, 98.70.Sa\n",
            "\n",
            "9\n",
            "9\n",
            "9\n",
            "1\n",
            "\n",
            "n\n",
            "u\n",
            "J\n",
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            "]\n",
            "h\n",
            "p\n",
            "-\n",
            "m\n",
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            "l\n",
            "p\n",
            ".\n",
            "s\n",
            "c\n",
            "i\n",
            "s\n",
            "y\n",
            "h\n",
            "p\n",
            "[\n",
            "\n",
            "3\n",
            "v\n",
            "1\n",
            "6\n",
            "0\n",
            "5\n",
            "0\n",
            "9\n",
            "9\n",
            "/\n",
            "s\n",
            "c\n",
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            "s\n",
            "y\n",
            "h\n",
            "p\n",
            ":\n",
            "v\n",
            "i\n",
            "X\n",
            "r\n",
            "a\n",
            "\n",
            "Typeset using REVTEX\n",
            "\n",
            "1\n",
            "\n",
            " \n",
            " \n",
            " \n",
            " \n",
            " \n",
            " \n",
            "\fI. INTRODUCTION\n",
            "\n",
            "Electron acceleration has been an important issue in plasma physics and astrophysics. In\n",
            "solar physics, for instance, the acceleration of electrons as well as of ions has received a great\n",
            "deal of attention; in solar ﬂares, gamma rays with energies of several tens of MeV, which are\n",
            "emitted by the bremsstrahlung from the high-energy electrons, are often observed [1], [2]. In\n",
            "astrophysics, much more energetic electrons have been discussed. From the observations of\n",
            "X-rays and gamma rays, it is now believed that electrons with energies up to ∼100 TeV are\n",
            "produced by a shell-type supernova remnant SN 1006 [3], [4]. It is also believed that in the\n",
            "Crab Nebula high-energy electrons accelerated up to ∼100 TeV are produced by the pulsar\n",
            "wind [5]. Theories based on the Fermi acceleration model have been proposed to account for\n",
            "these high-energy phenomena [6], [7]. As for laboratory plasmas, in an attempt to realize\n",
            "plasma-based accelerators, the electron acceleration in a relativistic space charge wave has\n",
            "been intensively studied by simulations and experiments [8].\n",
            "\n",
            "Ion acceleration in a magnetosonic wave has been studied by many authors [9]- [22].\n",
            "In a single-ion-species plasma, the large positive potential formed in the shock region can\n",
            "reﬂect some ions and give great energies to them [9]- [18]. In a multi-ion-species plasma\n",
            "such as space plasmas, some of the majority ions (hydrogen) can also gain energies by\n",
            "the same mechanism. Furthermore, the transverse electric ﬁeld in the shock wave can\n",
            "accelerate all particles of all kinds of (minority) heavy ions to nearly the same speed [19],\n",
            "[20]. In addition, in a turbulent plasma where particles can interact with many diﬀerent\n",
            "large-amplitude magnetosonic pulses, some energetic ions can be further accelerated by\n",
            "a diﬀerent mechanism [21]. These processes have been extensively studied by theory and\n",
            "particle simulation by many authors and have been applied to the production of high-energy\n",
            "particles in solar ﬂares and interplanetary shocks [9]- [20].\n",
            "\n",
            "Strong electron acceleration in shock waves, however, has not been shown by particle\n",
            "simulations. In this paper we will theoretically and numerically study electron motion in\n",
            "a magnetosonic shock wave propagating obliquely to a magnetic ﬁeld. We will show that\n",
            "an oblique shock can produce ultra-relativistic electrons. The preliminary result has been\n",
            "reported in Ref. [23].\n",
            "\n",
            "In Sec. II, we outline some basic properties of oblique shocks. In Sec. III, we theoretically\n",
            "discuss the possibility of electron reﬂection. If electrons are reﬂected near the end of the\n",
            "main pulse, then after the reﬂection they can have great energies in the main pulse.\n",
            "\n",
            "Several authors discussed stationary pulse solutions for the nonlinear magnetosonic waves\n",
            ">∼ 1, where ωce\n",
            "[24]- [27]. However, if the magnetic ﬁeld is rather strong so that ωce/ωpe\n",
            "is the electron cyclotron frequency and ωpe is the plasma frequency, or if the amplitude is\n",
            "quite large, then the pulse propagation is not perfectly stationary [13], [14]; small-amplitude\n",
            "ﬂuctuations are generated in and around the main pulse. Even though the potential is\n",
            "positive in the main pulse in a shock wave, the electron reﬂection studied in this paper can\n",
            "occur in such a non-stationary pulse.\n",
            "\n",
            "In Sec. IV, we estimate the maximum energy of a reﬂected electron and discuss its\n",
            "dependence on plasma parameters. In Sec. V, we further study the electron acceleration by\n",
            "using a one-dimensional (one space coordinate and three velocity components), relativistic,\n",
            "electromagnetic, particle simulation code with full ion and electron dynamics.\n",
            "It will be\n",
            "shown that some electrons can be reﬂected near the end of the main pulse and are trapped\n",
            "\n",
            "2\n",
            "\n",
            "\fin the shock region. In the simulations, an appreciable number of electrons have energies\n",
            "γ > 100, where γ is the Lorentz factor. Further, we will examine the dependence of γ on the\n",
            "propagation angle and shock speed. It is veriﬁed that, as the theory predicts, there are special\n",
            "angles and shock speeds at which the electron acceleration is especially strong. Our work is\n",
            "summarized in Sec. VI. In Appendix A, we describe relations among physical quantities in a\n",
            "magnetosonic wave. In Appendix B, we give a rough estimate for the magnitude of parallel\n",
            "electric ﬁeld Ek on the basis of a simple physical picture.\n",
            "\n",
            "II. VELOCITIES AND FIELDS IN AN OBLIQUE WAVE\n",
            "\n",
            "We consider a magnetosonic wave propagating in the x direction with a speed vsh in\n",
            "an external magnetic ﬁeld in the (x, z) plane. We assume that the ﬁeld quantities depend\n",
            "on x only; ∂/∂y = ∂/∂z = 0. Then, from the equation ∇ · B = 0, it follows that the\n",
            "x component of the magnetic ﬁeld is constant, Bx = Bx0. The other components By and\n",
            "Bz are functions of x, and in the pulse region By as well as Bz can have ﬁnite values. In\n",
            "the wave frame where the time derivatives are zero (∂/∂t = 0), the electric ﬁeld in the y\n",
            "direction is constant, Ey = Ey0, and Ez is zero. Hence we have B = (Bx0, By, Bz) and\n",
            "E = (Ex, Ey0, 0). In Sections II-IV, theoretical analyses will be made mainly in the wave\n",
            "frame.\n",
            "\n",
            "The velocity of the guiding-center position of an electron, vg, may be written as\n",
            "\n",
            "with vd being the drift velocity\n",
            "\n",
            "vg = (B/B)vk + vd,\n",
            "\n",
            "vd = cE × B/B2 − (cµ/e)(B × ∇B)/B2.\n",
            "\n",
            "(1)\n",
            "\n",
            "(2)\n",
            "\n",
            "Here c is the speed of light, −e is the electron charge (e > 0), vk is the velocity parallel to\n",
            "the magnetic ﬁeld, and µ is the magnetic moment\n",
            "\n",
            "µ = mev2\n",
            "\n",
            "⊥/(2B),\n",
            "\n",
            "(3)\n",
            "\n",
            "with me the electron mass and v⊥ the gyration speed perpendicular to the magnetic ﬁeld.\n",
            "In the far upstream region, we have no ∇B-drift. Also, the z component of the velocity\n",
            "averaged over all the electrons in a small volume element must be zero, hvgz0i = 0; the\n",
            "subscript 0 refers to the quantities in the far upstream region. Accordingly, from the z\n",
            "component of Eq. (1) we have the average parallel velocity\n",
            "\n",
            "hvk0i =\n",
            "\n",
            "cEy0\n",
            "B0\n",
            "\n",
            "Bx0\n",
            "Bz0\n",
            "\n",
            ",\n",
            "\n",
            "(4)\n",
            "\n",
            "in the wave frame (see Fig. 1). Also, because hvgx0i = −vsh, the x component of Eq. (1)\n",
            "gives\n",
            "\n",
            "vsh = −\n",
            "\n",
            "Bx0\n",
            "B0\n",
            "\n",
            "hvk0i −\n",
            "\n",
            "cEy0Bz0\n",
            "B2\n",
            "0\n",
            "\n",
            ".\n",
            "\n",
            "(5)\n",
            "\n",
            "Thus Ey0 is related to the shock speed vsh through\n",
            "\n",
            "3\n",
            "\n",
            "\fEy0 = −vshBz0/c.\n",
            "\n",
            "(6)\n",
            "\n",
            "For the deﬁniteness, we assume that Bx0, Bz0 and vsh are all positive; and thus Ey0 < 0.\n",
            "\n",
            "The nonlinear wave theory tells us the wave structure. A shock wave will have a positive\n",
            "electric potential ϕ(x). The quantities ϕ, Bz, ni, and ne have similar proﬁles [28]- [29] (see\n",
            "Appendix A); here ni, and ne are the ion and electron densities, respectively. On the other\n",
            "hand, Ex and By are proportional to the x derivatives of these quantities; for instance,\n",
            "Ex = −∂ϕ/∂x. These relations among the quantities are obtained for small-amplitude\n",
            "waves. In the following theoretical analysis, we assume that the relations are also valid for\n",
            "large-amplitude waves. As we will see later in Fig. 7, simulation results also support this\n",
            "assumption.\n",
            "\n",
            "We denote by xm the x position at which the potential takes its maximum value; hence,\n",
            "Ex(xm) ≃ 0 and By(xm) ≃ 0. For later use, we also note that the quantity Bz/B takes its\n",
            "maximum value at x = xm; that is, if Bz(x) is smaller than Bz(xm), then\n",
            "\n",
            "Bz(xm)2\n",
            "x0 + Bz(xm)2 >\n",
            "B2\n",
            "\n",
            "Bz(x)2\n",
            "x0 + By(x)2 + Bz(x)2 .\n",
            "B2\n",
            "\n",
            "(7)\n",
            "\n",
            "This can be proved by the following equation\n",
            "\n",
            "Bz(xm)2\n",
            "x0 + Bz(xm)2 −\n",
            "B2\n",
            "\n",
            "Bz(x)2\n",
            "\n",
            "x0 + By(x)2 + Bz(x)2 =\n",
            "B2\n",
            "\n",
            "[Bz(xm)2 − Bz(x)2]B2\n",
            "x0 + Bz(xm)2][B2\n",
            "[B2\n",
            "\n",
            "x0 + Bz(xm)2By(x)2\n",
            "x0 + By(x)2 + Bz(x)2]\n",
            "\n",
            ". (8)\n",
            "\n",
            "Because Bz(xm) is the maximum value of Bz(x), the term [Bz(xm)2 − Bz(x)2]B2\n",
            "Hence the right-hand side of the above equation is positive.\n",
            "\n",
            "x0 is positive.\n",
            "\n",
            "We know the stationary, ﬁnite-amplitude solutions for perpendicular waves [24]- [27]. The\n",
            "stationary solitary wave solution is valid when the magnetic ﬁeld is weak, ωce/ωpe << 1, and\n",
            "when the wave amplitude is not large, 1 < M < 2, where M is the Alfv´en Mach number;\n",
            "M = vsh/vA with vA the Alfv´en speed, v2\n",
            "A = B2\n",
            "0/(4πnimi). Simulations and experiments\n",
            "show that large-amplitude pulses can propagate nearly steadily, even when the amplitudes\n",
            "are large or the magnetic ﬁelds are strong. Oblique pulses can also propagate nearly steadily.\n",
            "As the magnetic ﬁeld becomes stronger or the wave amplitude is increased, non-stationary\n",
            "eﬀects will become important [13], [14].\n",
            "\n",
            "III. ELECTRON REFLECTION\n",
            "\n",
            "Because the potential is usually positive in the shock region, some of the ions can be\n",
            "reﬂected when they go up the potential. As we will see later, electrons can also be reﬂected;\n",
            "its mechanism is, however, diﬀerent from that of ion reﬂection. We give a physical picture\n",
            "of the electron reﬂection.\n",
            "\n",
            "From the nonrelativistic equation of motion for an electron\n",
            "\n",
            "me\n",
            "\n",
            "dv\n",
            "dt\n",
            "\n",
            "= −e\n",
            "\n",
            "E +\n",
            "\n",
            "v × B\n",
            "\n",
            "c !\n",
            "\n",
            ",\n",
            "\n",
            "(9)\n",
            "\n",
            "we have an equation for the kinetic energy in the wave frame\n",
            "\n",
            "4\n",
            "\n",
            " \n",
            "\fme(v2 − v2\n",
            "\n",
            "0)/2 = e(ϕ − ϕ0) − eEy0\n",
            "\n",
            "vydt.\n",
            "\n",
            "(10)\n",
            "\n",
            "Z\n",
            "\n",
            "Here we have used the relation Ez = 0. The velocity v includes gyro, parallel, and drift\n",
            "motions.\n",
            "\n",
            "We deﬁne length s as\n",
            "\n",
            "where vgx is the x component of the electron velocity\n",
            "\n",
            "ds = (B/Bx0)vgxdt,\n",
            "\n",
            "vgx = (Bx0/B)vk + vdx,\n",
            "\n",
            "(see Fig. 2). Thus ds/dt is given by\n",
            "\n",
            "ds\n",
            "dt\n",
            "\n",
            "= vk +\n",
            "\n",
            "B\n",
            "Bx0\n",
            "\n",
            "vdx.\n",
            "\n",
            "(11)\n",
            "\n",
            "(12)\n",
            "\n",
            "(13)\n",
            "\n",
            "We can interpret ds as an inﬁnitesimal length along the ﬁeld line corresponding to the length\n",
            "dx = vgxdt; ds = (B/Bx0)dx . In general the guiding-center velocity vg is not parallel to\n",
            "the magnetic ﬁeld B. Hence the guiding center does not move along the ﬁeld line. For a\n",
            "stationary, one-dimensional problem (∂/∂t = 0 and ∂/∂y = ∂/∂z = 0), all quantities depend\n",
            "only on x. As a result, for any stationary function f (x) we have\n",
            "\n",
            "x2\n",
            "\n",
            "x1\n",
            "\n",
            "Z\n",
            "\n",
            "f (x)(B/Bx0)dx =\n",
            "\n",
            "s2\n",
            "\n",
            "s1\n",
            "\n",
            "Z\n",
            "\n",
            "f (s)ds,\n",
            "\n",
            "(14)\n",
            "\n",
            "where s1 and s2 are the positions along the same ﬁeld line; their x positions are x1 and x2,\n",
            "respectively.\n",
            "\n",
            "If we deﬁne quantity F as\n",
            "\n",
            "then with the aid of (11) and the relation Ek = E · B/B, we have\n",
            "\n",
            "Ek = −dF/ds,\n",
            "\n",
            "which can be written as\n",
            "\n",
            "F = −\n",
            "\n",
            "Z\n",
            "\n",
            "(ExBx0 + Ey0By)\n",
            "B\n",
            "\n",
            "B\n",
            "Bx0\n",
            "\n",
            "dx,\n",
            "\n",
            "F = −\n",
            "\n",
            "Ex −\n",
            "\n",
            "Z (cid:18)\n",
            "\n",
            "vshBz0By\n",
            "\n",
            "cBx0 (cid:19)\n",
            "\n",
            "dx.\n",
            "\n",
            "Using the electric potential ϕ and the vector potential A (B = ∇ × A), we ﬁnd\n",
            "\n",
            "F = ϕ − (vsh/c)(Bz0/Bx0)Az,\n",
            "\n",
            "where Az is the z component of the vector potential,\n",
            "\n",
            "Az = −\n",
            "\n",
            "Bydx.\n",
            "\n",
            "Z\n",
            "\n",
            "5\n",
            "\n",
            "(15)\n",
            "\n",
            "(16)\n",
            "\n",
            "(17)\n",
            "\n",
            "(18)\n",
            "\n",
            "(19)\n",
            "\n",
            "\fThe quantities ϕ and Bz have similar proﬁles, while By is proportional to ∂Bz/∂x [28], [29].\n",
            "Hence Az and F have proﬁles similar to ϕ.\n",
            "\n",
            "If we deﬁne quantity Fl in the laboratory frame (the subscript l denotes the laboratory\n",
            "\n",
            "frame) as\n",
            "\n",
            "Fl = −\n",
            "\n",
            "Elk(Bl/Blx0)dxl,\n",
            "\n",
            "Z\n",
            "\n",
            "(20)\n",
            "\n",
            "then for a stationary, one-dimensional problem, that is, for a case where wave proﬁles can\n",
            "be written as f (xl, tl) = f (xl − vshtl), we ﬁnd\n",
            "\n",
            "Fl = −\n",
            "\n",
            "Elx −\n",
            "\n",
            "Z (cid:18)\n",
            "\n",
            "vshBlz0Bly\n",
            "\n",
            "cBlx0 (cid:19)\n",
            "\n",
            "dxl.\n",
            "\n",
            "One can readily show that it is related to F in the wave frame as\n",
            "\n",
            "where γsh is deﬁned as\n",
            "\n",
            "F = γshFl,\n",
            "\n",
            "γsh = (1 − v2\n",
            "\n",
            "sh/c2)−1/2.\n",
            "\n",
            "(21)\n",
            "\n",
            "(22)\n",
            "\n",
            "(23)\n",
            "\n",
            "In Appendix B, we give a rough estimate of the maximum value of F , Eq. (B9), found from\n",
            "qualitative physical considerations.\n",
            "\n",
            "Combining Eqs. (10) and (18), we eliminate ϕ\n",
            "\n",
            "mev2/2 − mev2\n",
            "\n",
            "0/2 = e(F − F0) − (eEy0/Bx0)(Az − Az0 +\n",
            "\n",
            "vyBx0dt).\n",
            "\n",
            "(24)\n",
            "\n",
            "Z\n",
            "\n",
            "By virtue of the deﬁnition of Az, (19), and the relation dx = vxdt, this equation can be\n",
            "written as\n",
            "\n",
            "mev2/2 − mev2\n",
            "\n",
            "0/2 = e(F − F0) + e(Ey0/Bx0)\n",
            "\n",
            "(vxBy − vyBx0)dt.\n",
            "\n",
            "Z\n",
            "\n",
            "Then, using the z component of the equation of motion, we ﬁnd\n",
            "\n",
            "mev2/2 − mev2\n",
            "\n",
            "0/2 = e(F − F0) − mec(Ey0/Bx0)(vz − vz0).\n",
            "\n",
            "In the drift approximation, the kinetic energy can be expressed as\n",
            "\n",
            "mev2/2 = me(v2\n",
            "\n",
            "k + v2\n",
            "\n",
            "d)/2 + µB.\n",
            "\n",
            "Hence, averaging Eq. (26) over the electron cyclotron period, we have\n",
            "\n",
            "(me/2)(v2\n",
            "= (me/2)(v2\n",
            "\n",
            "k + v2\n",
            "k0 + v2\n",
            "\n",
            "d) + µB + mec(Ey0/Bx0)vgz − eF\n",
            "d0) + µB0 + mec(Ey0/Bx0)vz0 − eF0.\n",
            "\n",
            "(25)\n",
            "\n",
            "(26)\n",
            "\n",
            "(27)\n",
            "\n",
            "(28)\n",
            "\n",
            "With the aid of the z component of Eq. (1), we can eliminate vgz and obtain the following\n",
            "equation\n",
            "\n",
            "6\n",
            "\n",
            "\f(me/2)(vk − vrv)2 + K = e(F − F0) − µ(B − B0)\n",
            "\n",
            "+ (me/2)(v2\n",
            "\n",
            "k0 + v2\n",
            "\n",
            "d0) − mec(Ey0/Bx0)(vdz − vz0) − (me/2)v2\n",
            "d,\n",
            "\n",
            "where vrv and K are deﬁned as\n",
            "\n",
            "vrv = −cEy0Bz/(Bx0B)\n",
            "\n",
            "(> 0),\n",
            "\n",
            "K = −mev2\n",
            "\n",
            "rv/2\n",
            "\n",
            "(< 0).\n",
            "\n",
            "If we eliminate Ey0 by substituting Eq. (6) in Eq. (30), vrv can be expressed as\n",
            "\n",
            "vrv = vshBz0Bz/(Bx0B).\n",
            "\n",
            "(29)\n",
            "\n",
            "(30)\n",
            "\n",
            "(31)\n",
            "\n",
            "(32)\n",
            "\n",
            "We will see below that the velocity vgx is reversed when vk = vrv. In the pulse region, the\n",
            "quantity e(F − F0) can be much larger in magnitude than the other terms on the right-hand\n",
            "side of Eq. (29) (see Appendix B). We denote the left-hand side of Eq. (29) by Ee, i.e.,\n",
            "Ee = (me/2)(vk − vrv)2 + K. As shown in Fig. 3, at a ﬁxed point x, Ee has its minimum\n",
            "value K when vk = vrv. It is negative in the region\n",
            "\n",
            "0 < vk < 2vrv.\n",
            "\n",
            "(33)\n",
            "\n",
            "Because vdx = cEy0Bz/B2, it follows from Eq. (12) that vgx = 0 if vk = vrv. As can be\n",
            "seen from Eq. (4), most of the electrons have negative parallel velocities in the far upstream\n",
            "region. Accordingly, we suppose that the initial parallel velocity is negative, vk < 0.\n",
            "If\n",
            "vk gradually increases and exceeds vrv at a certain position, then the x component of the\n",
            "guiding center velocity, vgx, changes from negative to positive values there, i.e., it is reversed.\n",
            "As an electron moves, i.e., as x changes, the quantities K, F , B, etc. also vary. At any\n",
            "point x, K is the minimum value of the left-hand side of Eq. (29). If there is a region where\n",
            "the values of the right-hand side of Eq. (29) become smaller than K, then the electron\n",
            "cannot enter there; it will be reﬂected. We recall that ϕ, F , and Bz have similar proﬁles\n",
            "and that at x = xm the quantity (Bz/B) has a maximum value [see Eq. (7)]. Because K is\n",
            "proportional to (Bz/B)2, it will be quite small (|K| is large) around the point x = xm, and\n",
            "F is large. In front of or behind the main pulse, F is relatively small and K is relatively\n",
            "large; e(F − F0) could be smaller than K. Thus, if the electron reﬂection occurs, it will be\n",
            "in such regions.\n",
            "\n",
            "In quasi-perpendicular shocks, where (Bz0/Bx0) >> 1, vrv is large. Hence K has large\n",
            "\n",
            "negative values. Thus the electron reﬂection in this mechanism will be diﬃcult.\n",
            "\n",
            "Figure 4 shows a schematic diagram of the trajectory of guiding center. Here it is assumed\n",
            "that the reﬂection takes place at point D; the dotted line shows the orbit of a passing electron.\n",
            "As an electron moves from point A to C, it moves to the negative y direction because of the\n",
            "Ex × Bz drift. It gains kinetic energy ∆E1 from the electric potential\n",
            "\n",
            "∆E1 = eϕ(xC) − eϕ(xA)\n",
            "\n",
            "(> 0).\n",
            "\n",
            "At the same time, it loses energy ∆E2 because of the electric ﬁeld Ey0\n",
            "\n",
            "∆E2 = −eEy0(yC − yA)\n",
            "\n",
            "(< 0).\n",
            "\n",
            "7\n",
            "\n",
            "(34)\n",
            "\n",
            "(35)\n",
            "\n",
            "\fThe net change in the energy is therefore\n",
            "\n",
            "∆E = ∆E1 + ∆E2.\n",
            "\n",
            "(36)\n",
            "\n",
            "Even though the magnitudes of ∆E1 and ∆E2 are quite large, they almost cancel when an\n",
            "electron moves with drift approximation; in particular, in a perpendicular pulse they have\n",
            "exactly the same magnitude and ∆E = 0 [30]. However, if an electron is reﬂected and moves\n",
            "from D to E, then it would gain energies from both Ex and Ey0. As a result, the increase in\n",
            "energy is\n",
            "\n",
            "where ∆E3 is deﬁned as\n",
            "\n",
            "∆E = ∆E1 + ∆E3,\n",
            "\n",
            "∆E3 = −eEy0(yE − yA)\n",
            "\n",
            "(> 0).\n",
            "\n",
            "(37)\n",
            "\n",
            "(38)\n",
            "\n",
            "IV. ESTIMATE OF ENERGY INCREASE\n",
            "\n",
            "Next, we will obtain the maximum energy of a reﬂected electron. Because it can have\n",
            "\n",
            "quite high energy, we will use the relativistic equation of motion\n",
            "\n",
            "me\n",
            "\n",
            "d(γv)\n",
            "dt\n",
            "\n",
            "= −e\n",
            "\n",
            "E +\n",
            "\n",
            "v × B\n",
            "\n",
            "c !\n",
            "\n",
            ",\n",
            "\n",
            "(39)\n",
            "\n",
            "where γ is the Lorentz factor. For the stationary, one-dimensional system being considered,\n",
            "this can be integrated to give the energy conservation equation\n",
            "\n",
            "meγc2 − meγ0c2 = e(ϕ − ϕ0) − eEy0\n",
            "\n",
            "vydt.\n",
            "\n",
            "(40)\n",
            "\n",
            "Z\n",
            "\n",
            "This is a relativistic form of Eq. (10). Again, using the quantities F , Az, and the equation\n",
            "of motion, we can rewrite Eq. (40) as\n",
            "\n",
            "meγc2 − meγ0c2 = e(F − F0) − mec(Ey0/Bx0)(γvz − γ0vz0).\n",
            "\n",
            "If we deﬁne quantity h as\n",
            "\n",
            "then from Eq. (41) we have\n",
            "\n",
            "h = mec2 − mevshBz0vz/Bx0,\n",
            "\n",
            "γ = [e(F − F0) + h0γ0]/h.\n",
            "\n",
            "(41)\n",
            "\n",
            "(42)\n",
            "\n",
            "(43)\n",
            "\n",
            "Equation (42) indicates that h is positive if Bz0/Bx0 is of order unity. If Bz0/Bx0 is much\n",
            "greater than unity, then h can be negative. In addition, Eq. (43) shows that, if the initial\n",
            "value of h is positive, h0 > 0, then h is always positive in the region where (F − F0) is\n",
            "positive.\n",
            "\n",
            "8\n",
            "\n",
            " \n",
            "\fWe show in Fig. 5 schematic diagram of the function [e(F − F0) + h0γ0]. We suppose\n",
            "that F has a maximum value at x = xm and has a minimum value right behind the main\n",
            "pulse. If the reﬂection occurs, it would be in this dip, as suggested in the previous section.\n",
            "The reﬂection point will be denoted by xr. In the top panel this function e(F − F0) + h0γ0 is\n",
            "always positive. In the second one it becomes negative right behind the main pulse; however\n",
            "its maximum value is positive (h0 can be either positive or negative). In the bottom one\n",
            "it is always negative. For the present acceleration mechanism, the top panel is the most\n",
            "important; the strong acceleration takes place in this case.\n",
            "\n",
            "Such dips would not be present in the perfectly stationary solutions [24]- [29]. In large-\n",
            "amplitude magnetosonic waves, however, non-stationary dips can be generated [13], [14].\n",
            "We here just assume that there is a dip behind the main pulse and discuss its eﬀects on\n",
            "particle orbits.\n",
            "\n",
            "First we consider the top panel in Fig. 5: An electron with positive h0 is reﬂected at a\n",
            "certain point near the end of the main pulse. We will show that, after the reﬂection, h has\n",
            "its minimum (positive) value at x = xm; hence γ has its maximum value there.\n",
            "\n",
            "As mentioned earlier, the parallel velocity vk is already positive when an electron is\n",
            "reﬂected, i.e., when vgx is reversed, even if its initial value is negative. After the reﬂection,\n",
            "vk will be increased by the (positive) parallel electric force. Because Ek changes its sign at\n",
            "x = xm, vk will have a maximum value at x = xm (unless vx changes to negative values again\n",
            "before reaching the point xm ). The parallel velocity will become the dominant component in\n",
            "the electron velocity, v ∼ vkB/B. The quantity Bz/B as well as ϕ and Bz has a maximum\n",
            "value at this point. Thus vz, which can be approximated as vz ∼ vkBz/B, will also have the\n",
            "maximum value there. Because h0 and (Fm − F0) are both positive, h is positive at x = xm.\n",
            "In addition, Eq. (42) indicates that h decreases (draws closer to zero) as vz increases. Hence,\n",
            "h of a reﬂected electron will have a minimum positive value at x = xm. Its energy γ will\n",
            "therefore have a maximum value there. If h can be close to zero, then γ would become quite\n",
            "large.\n",
            "\n",
            "If Bx0 is small and vz0 has a large positive value, then h0 becomes small (can be even\n",
            "negative). We now consider such particles; the second panel in Fig. 5. Also in this case,\n",
            "vz will have larger values after the reﬂection; thus h will decrease. At the reﬂection point\n",
            "in the dip, the quantity [e(F − F0) + h0γ0] is negative. Consequently, h(xr) < 0. If the\n",
            "reﬂection occurs, the electron would move in the positive x direction. Because vz increases,\n",
            "h will further decrease. Hence, the particle would never be able to penetrate the region\n",
            "again where hγ > 0. The strong acceleration in this mechanism, therefore, will not take\n",
            "place.\n",
            "\n",
            "Next, we consider a case where the quantity [e(F − F0) + h0γ0] is always negative (the\n",
            "bottom panel in Fig. 5). Then, h is always negative. If the reﬂection occurs at x = xr,\n",
            "then |h| would increase while |e(F − F0) + h0γ0| would decrease. Hence, even if the particle\n",
            "reaches the point xm, γ would not have an extremely large value. Therefore, the strong\n",
            "acceleration is not expected.\n",
            "\n",
            "We can estimate the maximum value γm in the wave frame by substituting the maximum\n",
            "value of F , Eq. (B12), in Eq. (43). The maximum γ in the laboratory frame can be obtained\n",
            "by the Lorentz transformation as\n",
            "\n",
            "γlm = γsh(1 + vxvsh/c2)γm ≃ γshγm.\n",
            "\n",
            "(44)\n",
            "\n",
            "9\n",
            "\n",
            "\fNow let us consider a case where h becomes quite small, h ≃ 0, at x = xm. There the\n",
            "electron speed would be close to the speed of light c. Hence, roughly, we have vz ∼ cBz/B.\n",
            "The quantity h can then be expressed as\n",
            "\n",
            "h = mec2\n",
            "\n",
            "1 − γsh\n",
            "\n",
            "(cid:18)\n",
            "\n",
            "vsh\n",
            "c\n",
            "\n",
            "Bz\n",
            "B\n",
            "\n",
            "tan θ\n",
            "\n",
            ",\n",
            "\n",
            "(cid:19)\n",
            "\n",
            "(45)\n",
            "\n",
            "where tan θ = Blz0/Blx0. Here we have used the relations Bz0 = γshBlz0 and Bx0 = Blx0.\n",
            "Because Bz becomes quite large in the shock region, we may put as Bz/B ∼ 1. Equation\n",
            "(45) then suggests that the electron energy can become extremely large when\n",
            "\n",
            "or, equivalently\n",
            "\n",
            "γsh(vsh/c) tan θ = 1,\n",
            "\n",
            "c cos θ = vsh.\n",
            "\n",
            "V. SIMULATION STUDIES\n",
            "\n",
            "A. Simulation Method\n",
            "\n",
            "(46)\n",
            "\n",
            "(47)\n",
            "\n",
            "We further study the electron acceleration by using a one-dimensional (one space co-\n",
            "ordinate and three velocity components), relativistic, electromagnetic, particle simulation\n",
            "code with full ion and electron dynamics [31]. The total system length is Lx = 4096∆g\n",
            "for most of the simulation runs, where ∆g is the grid spacing. All lengths and velocities in\n",
            "the simulations were normalized to ∆g and ωpe∆g, respectively, where ωpe is the spatially\n",
            "averaged plasma frequency.\n",
            "\n",
            "We use a bounded plasma model. The plasma is limited to the region 400 < x < 3696.\n",
            "The particles are specularly reﬂected at x = 400 and at x = 3696. The radiation leaving the\n",
            "plasma region is absorbed in the vacuum regions, 0 < x < 400 and 3696 < x < 4096. Thus\n",
            "the electromagnetic interactions between the two plasma boundaries through the vacuum\n",
            "regions are made negligibly small [12].\n",
            "\n",
            "In most of the plasma region, i.e., in 500 < x < 3696, each particle species initially has\n",
            "a uniform density and has a Maxwellian velocity distribution function. In the small region,\n",
            "400 < x < 480, the plasma density is four times as high as that in the main region. That\n",
            "is, the initial plasma density can be written as\n",
            "\n",
            "ne = 4n0\n",
            "\n",
            "for 400 < x < x1,\n",
            "\n",
            "ne = 4n0 exp[−(x − x1)2/(2d2)]\n",
            "ne = n0\n",
            "\n",
            "for 500 < x < 3696.\n",
            "\n",
            "for x1 < x < 500,\n",
            "\n",
            "In the high-density region, 400 < x < 480, particles have\n",
            "Here x1 = 480 and d = 12.\n",
            "shifted Maxwellian velocity distribution functions with average velocity vps; fj ∼ exp[−(v −\n",
            "vps)2/(2v2\n",
            "T j)], where the subscript j refers to particle species. These particles act as a\n",
            "piston. They push the neighboring particles and excite a shock wave. We can change the\n",
            "\n",
            "10\n",
            "\n",
            "\fshock strength by changing the magnitude of vps; the velocity vps is perpendicular to the\n",
            "external magnetic ﬁeld so that no particles have extremely large parallel speeds initially.\n",
            "More detailed description about the simulation code can be found in Ref. [12].\n",
            "\n",
            "The simulation parameters are as follows. The number of simulation particles is Ni =\n",
            "Ne = 262, 144. The ion-to-electron mass ratio is mi/me = 100. The ratio ωce/ωpe is\n",
            "ωce/ωpe = 3.0 in the far upstream region. The electron and ion thermal velocities are\n",
            "vT e/c = 0.38 and vT i/c = 0.01, respectively. The Alfv´en speed is vA/c = 0.3. The electron\n",
            "skin depth is c/ωpe = 4∆g. The time step is suﬃciently small, ωpe∆t = 0.02, so that ∆t is\n",
            "much smaller than the plasma and cyclotron periods even in the shock region. As in the\n",
            "theoretical model, the external magnetic ﬁeld is in the (x, z) plane, and waves propagate\n",
            "in the x direction.\n",
            "\n",
            "B. Simulation Results\n",
            "\n",
            "First, we study a shock with the propagation angle θ = 45◦; i.e., Bx0 = Bz0. Figure 6\n",
            "shows proﬁles of Bz in the shock wave at various times. Here, the magnetic ﬁeld is normalized\n",
            "to Bz0. The Mach number is observed to be M = 2.3, i.e., the shock speed is vsh = 2.3vA.\n",
            "The maximum value of Bz is 4∼5 times as large as Bz0. The main pulse propagates nearly\n",
            "steadily. However, the proﬁle is not perfectly stationary.\n",
            "\n",
            "Figure 7 displays ﬁeld proﬁles at ωpet = 680; here the magnetic and electric ﬁelds are\n",
            "both normalized to B0. As the theory predicts [28], [29], Ey and Bz have similar proﬁles,\n",
            "while Ex, Ez, and By are quite small in magnitude at the point where Bz has its maximum\n",
            "value. Figure 8 shows phase space plots of electrons at ωpet = 680. In the shock region,\n",
            "ultra-relativistic electrons are produced. The maximum value of Lorentz factors is γ ≃ 130.\n",
            "We also see that px and pz have greater values than py, which is consistent with the picture\n",
            "that the parallel velocity is dominant in the motion of high-energy electrons. From Figs. 7\n",
            "and 8, we see that some electrons are reﬂected near the end of the main pulse and that the\n",
            "high-energy electrons are present in the shock region.\n",
            "\n",
            "Figure 9 shows proﬁles of Ek, F , and the electric potential ϕ at ωpet = 650 and at\n",
            "ωpet = 680. The quantities ˜F and ˜ϕ denote eF/(mec2) and eϕ/(mec2), respectively. The\n",
            "quantities F and ϕ were obtained in the simulation by the following equations\n",
            "\n",
            "F (x) = −\n",
            "\n",
            "x\n",
            "\n",
            "Z\n",
            "\n",
            "Ek(B/Bx0)dx,\n",
            "\n",
            "ϕ(x) = −\n",
            "\n",
            "x\n",
            "\n",
            "Exdx.\n",
            "\n",
            "(48)\n",
            "\n",
            "(49)\n",
            "\n",
            "Z\n",
            "As mentioned earlier, roughly speaking, the parallel electric ﬁeld Ek is positive in the region\n",
            "where ∂Bz/∂x and ∂F/∂x are negative. It is negative in the region where they are positive.\n",
            "Comparing the plots at diﬀerent times, we see that the wave proﬁles are not perfectly\n",
            "stationary. They vary with time with small-amplitudes. In particular, as shown in the plot\n",
            "at ωpet = 680, F sometimes becomes negative near the end of the main pulse, where the\n",
            "electron reﬂection can take place.\n",
            "\n",
            "Next, we will study trajectories of electrons; we will ﬁnd that it is the reﬂected electrons\n",
            "that gain great energies. Figure 10 displays time variations of (x − vsht), y, z, and γ of\n",
            "\n",
            "11\n",
            "\n",
            "\felectrons in a shock with vsh = 1.75vA; (x − vsht) is the x position in the wave frame. The\n",
            "lengths are normalized to the electron skin depth c/ωpe. (Here the system size is a half of\n",
            "that in the previous run; the other parameters such as Ne/Lx are the same.) Two typical\n",
            "examples are presented here. The thick lines show an electron that was accelerated, and\n",
            "the thin lines represent an electron that was not accelerated; even the low-energy electron\n",
            "here has energy γ ≃ 10 in the shock region and γ ≃ 5 behind the shock. In all of the four\n",
            "panels, the low- and high-energy electrons both have small-amplitude, cyclotron oscillations\n",
            "with short periods, ωpet <∼ 20; note that the cyclotron period depends on γ as well as the\n",
            "ﬁeld strength B. At the same time, the accelerated electron has a large-amplitude, long-\n",
            "period oscillation with period ωpet ≃ 170; it begins when the particle enters the shock,\n",
            "ωpet = 180, and continues until the end of this simulation run.\n",
            "(Simulations show that\n",
            "this period increases with γ. To observe this oscillation as many times as possible, we have\n",
            "chosen here the case where γ is not so large.) For comparison with Fig. 4, we show the\n",
            "times corresponding to points B, C, D, and E. From the long-period oscillation of (x − vsht)\n",
            "in the top panel, we see that the electron that will become high energy is reﬂected and then\n",
            "trapped by the shock wave. On the other hand, the quantity (x − vsht) of the low-energy\n",
            "electron keeps decreasing, which means that this particle passes through the shock region\n",
            "without strong interactions. The second panel shows that the y position of the accelerated\n",
            "electron also oscillates, while the third one shows that z increases rapidly after the reﬂection.\n",
            "The average value of velocity vz over the period from ωpet = 220 to 400 is vz/c ≃ 0.6, and\n",
            "the value of vz at maximum γ is vz/c ≃ 0.8. The maximum γ in the bottom panel is γ ≃ 40.\n",
            "By the Lorentz transformation of these data, we can ﬁnd v and B in the wave frame.\n",
            "At the time and position of maximum γ, they are v = c(0.27, 0.13, 0.95) and B =\n",
            "B0(0.70, 0.16, 2.56). This conﬁrms our statement in the previous section that vk is the\n",
            "dominant component in v there; i.e., vz ≃ vkBz/B.\n",
            "\n",
            "According to the theory, (47), electron energies can be extremely high at some angles\n",
            "and shock speeds. To examine this prediction, we have carried out several simulations with\n",
            "diﬀerent values of the propagation angle θ, keeping other parameters unchanged; Lx =\n",
            "4096∆g, ωce/ωpe = 3.0, and vsh ≃ 2.1vA. Figure 11 shows the maximum value of γ as a\n",
            "function of the propagation angle θ. Here the electron energy takes the highest value at the\n",
            "angle θ ≃ 52◦. For these simulation parameters, Eq. (47) predicts that γ takes a peak value\n",
            "at θ ≃ 53◦. Thus the theory and simulation are in good agreement. Further, we carried out\n",
            "simulations with diﬀerent values of the shock speed vsh; the propagation angle is ﬁxed to be\n",
            "θ = 45◦. Figure 12 shows the maximum value of γ as a function of the shock speed vsh. The\n",
            "observed values have a peak at vsh/c ≃ 0.7. For these parameters, Eq. (47) predicts that γ\n",
            "takes large values for vsh/c ≃ 0.71. Again, the theory and simulation are in good agreement.\n",
            "\n",
            "VI. SUMMARY\n",
            "\n",
            "We have studied electron motion in a shock wave propagating obliquely to a magnetic\n",
            "It is\n",
            "ﬁeld. First, we analytically discussed electron motion in an oblique shock wave.\n",
            "pointed out that, if an electron is reﬂected near the end of a large-amplitude pulse, it would\n",
            "gain a great deal of energy from the electric ﬁeld formed in the wave. The condition for\n",
            "the reﬂection is then examined. Further, the maximum energy of a reﬂected electron is\n",
            "\n",
            "12\n",
            "\n",
            "\festimated, and its dependence on plasma parameters is discussed. Next, we investigated the\n",
            "shock evolution and associated electron acceleration by using a one-dimensional, relativistic,\n",
            "electromagnetic, particle simulation code with full ion and electron dynamics. It was shown\n",
            "that an oblique shock can produce ultra-relativistic electrons; Lorentz factors with γ >∼ 100\n",
            "have been observed. As the theory predicts, at certain propagation angles electron energies\n",
            "become extremely high. The electron reﬂection and resultant acceleration takes place when\n",
            "the wave amplitude is large and the magnetic ﬁeld is rather strong, ωce\n",
            "\n",
            ">∼ ωpe.\n",
            "\n",
            "In the present paper, strong electron acceleration has been demonstrated. As future\n",
            "work, it would be desirable to develop further the quantitative theory for large-amplitude\n",
            "oblique waves. The evaluation of the electric ﬁeld strength along the magnetic ﬁeld will be\n",
            "especially important, which will enable us to estimate the maximum energy more accurately.\n",
            "From the view point of nonlinear wave propagation, motion of trapped electrons and their\n",
            "eﬀects on the wave evolution are also quite interesting.\n",
            "\n",
            "APPENDIX A: RELATIONS AMONG QUANTITIES\n",
            "\n",
            "We here describe relations among the physical variables, using the approximation appro-\n",
            "priate for low-frequency magnetosonic waves with small amplitudes (for the details of the\n",
            "calculations, see Ref. [29]). We assume quasi neutrality, n ≃ ni ≃ ne and use the following\n",
            "stretched variables\n",
            "\n",
            "τ = ǫ3/2t,\n",
            "\n",
            "ξ = ǫ1/2(x − vp0t),\n",
            "\n",
            "(A1)\n",
            "\n",
            "(A2)\n",
            "\n",
            "(vp0 is the wave propagation speed in the long-wavelength limit in a ﬁnite beta plasma) and\n",
            "expansion\n",
            "\n",
            "n = n0 + ǫn1 + ǫ2n2 + ...,\n",
            "\n",
            "Ey = ǫEy1 + ǫ2Ey2 + ...,\n",
            "\n",
            "Bz = Bz0 + ǫBz1 + ǫ2Bz2 + ...,\n",
            "\n",
            "then, we can express the lowest order perturbations in terms of n1 as\n",
            "\n",
            "Bz1/B0 = cEy1/(B0vp0) = [(v2\n",
            "\n",
            "p0 − c2\n",
            "\n",
            "s)/v2\n",
            "\n",
            "A sin θ](n1/n0),\n",
            "\n",
            "(A3)\n",
            "\n",
            "(A4)\n",
            "\n",
            "(A5)\n",
            "\n",
            "(A6)\n",
            "\n",
            "= −\n",
            "\n",
            "cEz1\n",
            "B0\n",
            "\n",
            "vp0By1\n",
            "B0\n",
            "Here cs is the sound speed, c2\n",
            "s = (Γepe0 + Γipi0)/(n0mi) with Γj (j = e or i) the speciﬁc heat\n",
            "ratio and pj0 the equilibrium pressure. We can obtain KdV equation by proceeding to the\n",
            "next order, O(ǫ5/2).\n",
            "\n",
            "(ωce − ωci)\n",
            "ωceωci\n",
            "\n",
            "cot θ\n",
            "n0\n",
            "\n",
            "∂n1\n",
            "∂ξ\n",
            "\n",
            "(v2\n",
            "p0 − c2\n",
            "p0 − v2\n",
            "\n",
            "s)v2\n",
            "p0\n",
            "A cos2 θ)\n",
            "\n",
            "(A7)\n",
            "\n",
            "(v2\n",
            "\n",
            "=\n",
            "\n",
            ".\n",
            "\n",
            "The relations (A6) and (A7) are valid, even when the wave proﬁle is not soliton-like.\n",
            "That is, perturbations propagating in the same direction with vx ∼ vp0 have the relations\n",
            "(A6) and (A7).\n",
            "\n",
            "13\n",
            "\n",
            "\fAPPENDIX B: MAGNITUDES OF Ek AND F\n",
            "\n",
            "For a nonrelativistic, small-amplitude oblique magnetosonic wave, the electric ﬁeld par-\n",
            "allel to the magnetic ﬁeld, Ek, has been analytically obtained [32]. However, for a large-\n",
            "amplitude wave, it will be quite diﬃcult to obtain Ek in a rigorous manner. Here, on the\n",
            "basis of a simple physical picture, we will give a rough estimate of the parallel electric ﬁeld\n",
            "Ek in a relativistic, large-amplitude, oblique shock wave. Further, by using it, we will obtain\n",
            "the maximum value of the quantity F .\n",
            "We consider in the wave frame.\n",
            "\n",
            "In the ﬂuid model, the equation of motion may be\n",
            "\n",
            "written as\n",
            "\n",
            "mj\n",
            "\n",
            "d(γjvj)\n",
            "dt\n",
            "\n",
            "= qjE + qj\n",
            "\n",
            "vj × B\n",
            "c\n",
            "\n",
            ".\n",
            "\n",
            "(B1)\n",
            "\n",
            "Here the pressure term is neglected. The subscript j refers to ions j = i or electrons j = e.\n",
            "The time derivative d/dt is deﬁned as\n",
            "\n",
            "d\n",
            "dt\n",
            "\n",
            "=\n",
            "\n",
            "∂\n",
            "∂t\n",
            "\n",
            "+ vj · ∇ .\n",
            "\n",
            "(B2)\n",
            "\n",
            "We denote by b the unit vector along the magnetic ﬁeld. Then multiplying Eq. (B1) by b,\n",
            "we have\n",
            "\n",
            "mj\n",
            "\n",
            "d(γjb · vj)\n",
            "dt\n",
            "\n",
            "− mjγjvj ·\n",
            "\n",
            "db\n",
            "dt\n",
            "\n",
            "= qjEk.\n",
            "\n",
            "(B3)\n",
            "\n",
            "Let t1 be the time when a ﬂuid element is at the leading edge of the shock (the location\n",
            "where the shock proﬁle begins to sharply rise) and t2 be the time when it is at the point\n",
            "xm; xm is the location where the electric potential and magnetic ﬁeld take their maximum\n",
            "values. Then, integrating from t1 to t2, we ﬁnd for the ions\n",
            "\n",
            "mi(γi2b2 · vi2 − γi1b1 · vi1) − mihγivii · (b2 − b1) ≃ eEk\n",
            "\n",
            "∆\n",
            "|hvixi|\n",
            "\n",
            ".\n",
            "\n",
            "(B4)\n",
            "\n",
            "Here the bracket indicates the mean value over the time period (t2 − t1), and ∆ is the\n",
            "shock width. We have a relation (t2 − t1) ≃ ∆/|hvixi|. For oblique shocks, ∆ will be of\n",
            "the order of the ion inertial length, ∆ ≃ c/ωpi. The velocity at time t1 is vi1 = −vshex,\n",
            "where ex is the unit vector in the x direction. The velocity vix will be slowed down by\n",
            "the longitudinal electric ﬁeld. Thus vix2 will be smaller in magnitude than vix1. Because\n",
            "Bz is large at the point xm and By(xm) ∼ 0, the unit vector b2 is nearly parallel to the z\n",
            "direction. Consequently, the inner product (b2 · vi2) will be quite small; here we neglect the\n",
            "z component of vi2. Since the x component of b1 is given by Bx0/B0, we have\n",
            "\n",
            "mi(γi1vsh + hγivixi)Bx0/B0 ≃ eEk∆/|hvixi|.\n",
            "\n",
            "This gives the strength of the parallel electric ﬁeld as\n",
            "\n",
            "eEk ≃\n",
            "\n",
            "miv2\n",
            "sh\n",
            "∆\n",
            "\n",
            "|hvixi|\n",
            "\n",
            "vsh  \n",
            "\n",
            "γi1 +\n",
            "\n",
            "hγivixi\n",
            "\n",
            "vsh !\n",
            "\n",
            "Bx0\n",
            "B0\n",
            "\n",
            ".\n",
            "\n",
            "14\n",
            "\n",
            "(B5)\n",
            "\n",
            "(B6)\n",
            "\n",
            "\fBecause |vix| is smaller than vsh, the range of |hvixi| is expected to be 1/2 <∼ |hvixi|/vsh <\n",
            "1, except maybe for extremely strong shock waves.\n",
            "In fact, if we assume a simple time\n",
            "dependence of vix\n",
            "\n",
            "vix = −vsh[1 − a1(t − t1)/(t2 − t1)]\n",
            "\n",
            "for\n",
            "\n",
            "t1 ≤ t ≤ t2,\n",
            "\n",
            "(B7)\n",
            "\n",
            "with a1 a constant (0 ≤ a1 ≤ 1), then, after integrating over time, we have hvixi = −vsh(1 −\n",
            "a1/2).\n",
            "\n",
            "The Lorentz factor for the ﬂuid ion will be close to unity. Accordingly, for instance, if\n",
            "the ion ﬂuid velocity vix is decreased to a value ∼ −vsh/2, then the value of the content in\n",
            "the parentheses would be about 1/4.\n",
            "\n",
            "The electron speed will be ∼ vsh when the electrons encounter the shock and will be close\n",
            "to the speed of light when they are at the point xm. If we assume that the electron velocity\n",
            "is nearly parallel to the ﬁeld line in the shock region, then (v · db/dt) can be neglected.\n",
            "Hence we have\n",
            "\n",
            "me(−γe2c + γe1vshBx0/B0) ≃ −eEk(t2 − t1).\n",
            "\n",
            "(B8)\n",
            "\n",
            "The second term on the left-hand side, γe1vshBx0/B0, can be neglected compared with the\n",
            "ﬁrst term γe2c. The ions and electrons will have nearly the same time period (t2 − t1) to\n",
            "keep charge neutrality. Substituting Eq. (B6) in (B8) yields\n",
            "\n",
            "γe2 ≃\n",
            "\n",
            "mivsh\n",
            "mec  \n",
            "\n",
            "γi1 +\n",
            "\n",
            "hγivixi\n",
            "\n",
            "vsh !\n",
            "\n",
            "Bx0\n",
            "B0\n",
            "\n",
            ".\n",
            "\n",
            "(B9)\n",
            "\n",
            "At x = xm, vz can be estimated as vz ≃ −cBz/B (for reﬂected electrons the sign is reversed).\n",
            "For a large-amplitude shock, the ratio Bz/B is an order-unity quantity there. The quantity\n",
            "h deﬁned by Eq. (42) is obtained as\n",
            "\n",
            "h2 ≃ mec2[1 + vshBz0/(cBx0)].\n",
            "\n",
            "If we average Eq. (43) over electrons in a small volume element, we have\n",
            "\n",
            "hhγi = e(F − F0) + mec2hγ0i.\n",
            "\n",
            "(B10)\n",
            "\n",
            "(B11)\n",
            "\n",
            "Here we used the relation hvz0i = 0. If electrons have the velocity vz ≃ −cBz/B at x = xm,\n",
            "then we ﬁnd the maximum value of F from Eq. (43) as\n",
            "\n",
            "e(Fm − F0) ≃ h2γe2 − mec2hγ0i.\n",
            "\n",
            "(B12)\n",
            "\n",
            "In many practical cases, we can take hγ0i to be order unity.\n",
            "\n",
            "Roughly speaking, the quantity e(Fm − F0) is of the order of micvsh. It is, therefore,\n",
            "T e (∼ µB) and drift kinetic energy mev2\n",
            "\n",
            "much greater than electron thermal energy mev2\n",
            "(∼ mev2\n",
            "\n",
            "d\n",
            "\n",
            "sh).\n",
            "\n",
            "15\n",
            "\n",
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            "[23] N. Bessho, K. Maruyama, and Y. Ohsawa, J. Phys. Soc. Jpn. 68, 1 (1999).\n",
            "[24] J. H. Adlam and J. E. Allen, Philos. Mag. 3, 448 (1958).\n",
            "[25] L. Davis, R. L¨ust, and A. Schl¨uter, Z. Naturforsch. Teil A13, 916 (1958).\n",
            "[26] C. S. Gardner and G. K. Morikawa, Commun. Pure Appl. Math. 18, 35 (1965).\n",
            "[27] Y. Ohsawa, Phys. Fluids 29, 2474 (1986).\n",
            "[28] T. Kakutani, H. Ono, T. Taniuti, and C. C. Wei, J. Phys. Soc. Jpn. 24, 1159 (1968).\n",
            "[29] Y. Ohsawa, Phys. Fluids 29, 1844 (1986).\n",
            "[30] Y. Ohsawa, J. Phys. Soc. Jpn. 58, 4445 (1989).\n",
            "[31] P. C. Liewer, A. T. Lin, J. M. Dawson, and M. Z. Caponi, Phys. Fluids 24, 1364 (1981).\n",
            "[32] Y. Ohsawa, J. Phys. Soc. Jpn. 57, 929 (1988).\n",
            "\n",
            "16\n",
            "\n",
            "\fFIGURES\n",
            "\n",
            "FIG. 1. Velocities and ﬁelds in the far upstream region in the wave frame. The velocity vd is\n",
            "\n",
            "Ey0 × B0 drift, and −vsh is equal to average guiding-center velocity in the x direction hvgx0i.\n",
            "\n",
            "FIG. 2. Schematic diagram of magnetic ﬁeld, electron velocity, dx, and length ds.\n",
            "FIG. 3. Plot of function Ee(vk) = (me/2)(vk − vrv)2 + K at a ﬁxed x position.\n",
            "FIG. 4. Schematic diagram of electron orbit in (x, y) plane.\n",
            "\n",
            "FIG. 5. Schematic diagram of quantity [e(F − F0) + h0γ0].\n",
            "\n",
            "In the top panel, this quantity\n",
            "is always positive. In the second one, it is negative in the dip. In the bottom one, it is always\n",
            "negative.\n",
            "\n",
            "FIG. 6. Magnetic ﬁeld proﬁles of an oblique shock at various times.\n",
            "\n",
            "FIG. 7. Snapshots of ﬁeld proﬁles. Electric and magnetic ﬁeld proﬁles at ωpet = 680 are plotted.\n",
            "\n",
            "They are normalized to B0.\n",
            "\n",
            "FIG. 9. Proﬁles of Ek, F , and ϕ at two diﬀerent times. Here, ˜F and ˜ϕ denote eF/(mec2) and\n",
            "\n",
            "FIG. 8. Phase space plots of electrons.\n",
            "\n",
            "eϕ/(mec2), respectively.\n",
            "\n",
            "FIG. 10. Time variations of (x − vsht ), y, z, and γ of electrons. The lengths are normalized to\n",
            "\n",
            "the electron skin depth c/ωpe.\n",
            "\n",
            "FIG. 11. Maximum γ versus propagation angle θ. The propagation speeds for these shocks are\n",
            "\n",
            "vsh ≃ 2.1vA.\n",
            "\n",
            "FIG. 12. Maximum γ versus shock speed vsh. The propagation angle θ is ﬁxed to be θ = 45◦\n",
            "\n",
            "17\n",
            "\n",
            "\fz\n",
            "\n",
            "E y0\n",
            "\n",
            "vd\n",
            "\n",
            "-vsh\n",
            "\n",
            "B\n",
            "\n",
            "0\n",
            "\n",
            "<v  >//0\n",
            "x\n",
            "\n",
            "y\n",
            "\n",
            "\fdx\n",
            "\n",
            "v dt\n",
            "\n",
            "g\n",
            "\n",
            "ds\n",
            "\n",
            "B\n",
            "\n",
            "\fE  (   )v//\n",
            "e\n",
            "\n",
            "vrv\n",
            "\n",
            "0\n",
            "\n",
            "K\n",
            "\n",
            "v//\n",
            "\n",
            "\fE\n",
            "\n",
            "D\n",
            "\n",
            "C\n",
            "\n",
            "AB\n",
            "y\n",
            "\n",
            "x\n",
            "\n",
            "\fe(F-F )+h0\n",
            "\n",
            "00\n",
            "\n",
            "x r\n",
            "\n",
            "x m\n",
            "\n",
            "x m\n",
            "\n",
            "x r\n",
            "x r x m\n",
            "\n",
            "x\n",
            "\n",
            "x\n",
            "\n",
            "x\n",
            "\n",
            "g\n",
            "\fBz\n",
            "Bz0\n",
            "\n",
            "4\n",
            "2\n",
            "0\n",
            "250\n",
            "\n",
            "800\n",
            "\n",
            "700\n",
            "\n",
            "600\n",
            "\n",
            "500\n",
            "\n",
            "w pet=400\n",
            "\n",
            "375\n",
            "500\n",
            "x/(c/w pe)\n",
            "\n",
            "625\n",
            "\n",
            "\fEx\n",
            "\n",
            "Ey\n",
            "\n",
            "1\n",
            "0\n",
            "-1\n",
            "2\n",
            "1\n",
            "0\n",
            "0.5\n",
            "0\n",
            "-0.5\n",
            "0.5\n",
            "0\n",
            "-0.5\n",
            "4\n",
            "\n",
            "Ez\n",
            "\n",
            "By\n",
            "\n",
            "Bz\n",
            "\n",
            "2\n",
            "\n",
            "0\n",
            "\n",
            "525\n",
            "\n",
            "550\n",
            "575\n",
            "x/(c/w pe)\n",
            "\n",
            "600\n",
            "\n",
            "\fpy\n",
            "\n",
            "px\n",
            "\n",
            "100\n",
            "50\n",
            "0\n",
            "100\n",
            "50\n",
            "0\n",
            "100\n",
            "pz\n",
            "50\n",
            "0\n",
            "\n",
            "z\n",
            "\n",
            "120\n",
            "80\n",
            "40\n",
            "0\n",
            "\n",
            "525\n",
            "\n",
            "550\n",
            "575\n",
            "x/(c/w pe)\n",
            "\n",
            "600\n",
            "\n",
            "g\n",
            "\fw pet=650\n",
            "\n",
            "w pet=680\n",
            "\n",
            "0.3\n",
            "\n",
            "0\n",
            "\n",
            "-0.3\n",
            "\n",
            "10\n",
            "\n",
            "0.3\n",
            "\n",
            "0\n",
            "\n",
            "E//\n",
            "B0\n",
            "-0.3\n",
            "\n",
            "10\n",
            "\n",
            "F\n",
            "\n",
            "0\n",
            "\n",
            "525 540 555\n",
            "x/(c/w pe)\n",
            "\n",
            "F\n",
            "\n",
            "0\n",
            "540 555 570\n",
            "x/(c/w pe)\n",
            "\n",
            "f\n",
            "f\n",
            "\ft\n",
            "h\n",
            "s\n",
            "v\n",
            "-\n",
            "x\n",
            "\n",
            "50\n",
            "\n",
            "0\n",
            "\n",
            "y\n",
            "\n",
            "10\n",
            "0\n",
            "-10\n",
            "\n",
            "z\n",
            "\n",
            "200\n",
            "100\n",
            "0\n",
            "\n",
            "40\n",
            "\n",
            "20\n",
            "\n",
            "0\n",
            "\n",
            "BCD E\n",
            "B C D E\n",
            "\n",
            "200\n",
            "\n",
            "400\n",
            "\n",
            "w pet\n",
            "\n",
            "600\n",
            "\n",
            "Fig.3\n",
            "\n",
            " \n",
            " \n",
            "g\n",
            "\f100\n",
            "\n",
            "80\n",
            "\n",
            "60\n",
            "\n",
            "40\n",
            "\n",
            "20\n",
            "40 45 50 55 60\n",
            "Angle\n",
            "\n",
            "g\n",
            "\f150\n",
            "\n",
            "100\n",
            "\n",
            "50\n",
            "0.5\n",
            "\n",
            "0.6\n",
            "\n",
            "0.7\n",
            "\n",
            "0.8\n",
            "\n",
            "vsh/c\n",
            "\n",
            "g\n",
            "\f\n"
          ]
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "## Creating text pipeline's Function"
      ],
      "metadata": {
        "id": "em4f7FtPNwnO"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "def make_text(pdf_path,save_path) :\n",
        "  \"\"\"\n",
        "  Converts a PDF file to a text file that is saved on a selected path.\n",
        "\n",
        "  Args:\n",
        "      pdf_path (str): The path to the PDF file to be converted.\n",
        "      save_path (str): The directory path where the extracted text file will be saved.\n",
        "\n",
        "  Raises:\n",
        "      OSError: If an error occurs while accessing or creating the files.\n",
        "      ValueError: If either `pdf_path` or `save_path` is an empty string.\n",
        "\n",
        "  Returns:\n",
        "      None: This function does not explicitly return a value, but it creates a text file\n",
        "             containing the extracted text from the PDF.\n",
        "\n",
        "  Prints:\n",
        "      A message indicating whether the text file was created successfully.\n",
        "\n",
        "  This function uses the `pdfminer.six` library (not included by default) to extract text\n",
        "  from the PDF and save it to a new text file\n",
        "  \"\"\"\n",
        "\n",
        "  # getting filename\n",
        "  file_name = pdf_path.split(\"/\")[-1].split(\".pdf\")[0]+\".txt\"\n",
        "  # getting text from PDF file (pdfminer.six)\n",
        "  output_string = StringIO()\n",
        "  with open(path, 'rb') as in_file:\n",
        "    parser = PDFParser(in_file)\n",
        "    doc = PDFDocument(parser)\n",
        "    rsrcmgr = PDFResourceManager()\n",
        "    device = TextConverter(rsrcmgr, output_string, laparams=LAParams())\n",
        "    interpreter = PDFPageInterpreter(rsrcmgr, device)\n",
        "    for page in PDFPage.create_pages(doc):\n",
        "      interpreter.process_page(page)\n",
        "  # getting text value\n",
        "  text = output_string.getvalue()\n",
        "  # writing the text file\n",
        "  f = open(save_path+file_name, \"w\")\n",
        "  f.write(text)\n",
        "  f.close()\n",
        "  # checking if file was created\n",
        "  if os.path.exists(save_path+file_name):\n",
        "    print(f\" '{file_name}' was created successfully!\")"
      ],
      "metadata": {
        "id": "Eqi2YsIzN8lC"
      },
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "code",
      "source": [
        "help(make_text)"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "EKAgUISAROoI",
        "outputId": "395b6a75-ed1e-499d-912f-7aa4b12e0232"
      },
      "execution_count": null,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "Help on function make_text in module __main__:\n",
            "\n",
            "make_text(pdf_path, save_path)\n",
            "    Converts a PDF file to a text file that is saved on a selected path.\n",
            "    \n",
            "    Args:\n",
            "        pdf_path (str): The path to the PDF file to be converted.\n",
            "        save_path (str): The directory path where the extracted text file will be saved.\n",
            "    \n",
            "    Raises:\n",
            "        OSError: If an error occurs while accessing or creating the files.\n",
            "        ValueError: If either `pdf_path` or `save_path` is an empty string.\n",
            "    \n",
            "    Returns:\n",
            "        None: This function does not explicitly return a value, but it creates a text file\n",
            "               containing the extracted text from the PDF.\n",
            "    \n",
            "    Prints:\n",
            "        A message indicating whether the text file was created successfully.\n",
            "    \n",
            "    This function uses the `pdfminer.six` library (not included by default) to extract text\n",
            "    from the PDF and save it to a new text file\n",
            "\n"
          ]
        }
      ]
    },
    {
      "cell_type": "code",
      "source": [
        "# testing on one file\n",
        "path =  \"/content/drive/MyDrive/UH - Final Year Project/Dataset/pdf/9905/9905061v3.pdf\"\n",
        "save_path = \"/content/drive/MyDrive/UH - Final Year Project/Dataset/text/\"\n",
        "\n",
        "make_text(path,save_path)"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "zwW7rfGHO5GN",
        "outputId": "c34ee1e0-a0c2-438b-b299-f22f41ea8cb4"
      },
      "execution_count": null,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            " '9905061v3.txt' was created successfully!\n"
          ]
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "## Applying the `make_text()` function"
      ],
      "metadata": {
        "id": "5CAqFfzUR_bn"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "save_path = \"/content/drive/MyDrive/UH - Final Year Project/Dataset/text/\"\n",
        "pdf_path = \"/content/drive/MyDrive/UH - Final Year Project/Dataset/pdf/\"\n",
        "\n",
        "for root, directories, files in os.walk(pdf_path):\n",
        "  # Access files within the current directory (root)\n",
        "  for filename in files:\n",
        "    filepath = os.path.join(root, filename)\n",
        "    make_text(filepath,save_path)"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "6aSTam2FSPXe",
        "outputId": "d40e4962-798f-413c-fe9b-8edae3011bdd"
      },
      "execution_count": null,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
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}