Comparing $\overline{w'w'}$ and $\overline{w'w'w'}$ tendency terms between ADC and LES

[BrodiePearson]

This issue lists the ADC tendency terms for $\overline{w'w'}$ and $\overline{w'w'w'}$, and the analogous LES terms (where possible).

One avenue for diagnosing the skill of the ADC closure scheme is to compare whether the tendency terms in the scheme are comparable to LES data. The tendency terms in the ADC and LES schemes represent similar physical processes, but because the ADC scheme has closure assumptions that are distinct from Reynolds averaging (which is how we analyze LES fields), the tendency terms in the ADC and LES schemes differ in their form.

$\overline{w'w'}$ budget

MPAS-Model/src/core_ocean/shared/mpas_ocn_adcReconstruct.F

Lines 565 to 566 in 5dc8d6e

	w2tend1(k,iCell) = -wumd(k,iCell)**2.0_RKIND*( &
	Entrainment(k,iCell) + Detrainment(k,iCell))

$w2tend1=$? (Entrainment/Detrainment; maybe dissipation related or this doesn't have an LES analogy)

MPAS-Model/src/core_ocean/shared/mpas_ocn_adcReconstruct.F

Lines 567 to 569 in 5dc8d6e

	w2tend2(k,iCell) = (McMid(k-1,iCell)*(1.0_RKIND - 2.0_RKIND* &
	areaFractionMid(k-1,iCell))*wumdMid(k-1,iCell)**2.0 - McMid(k,iCell)* &
	(1.0_RKIND - 2.0_RKIND*areaFractionMid(k,iCell))*wumdMid(k,iCell)**2.0) / dzmid

$w2tend2=-\frac{\partial \overline{w'w'w'}}{\partial z}$ (Turbulent transport)

MPAS-Model/src/core_ocean/shared/mpas_ocn_adcReconstruct.F

Line 570 in 5dc8d6e

w2tend3(k,iCell) = tauvVel(k,iCell)*(u2(i1,k,iCell) + v2(i1,k,iCell))/3.0_RKIND

$w2tend3=\frac{ \overline{u'u'} + \overline{v'v'} }{3\tau_{slow}}$ (Part of slow pressure; a missing $\overline{w'w'}$ term is added in time-stepping)

MPAS-Model/src/core_ocean/shared/mpas_ocn_adcReconstruct.F

Lines 571 to 572 in 5dc8d6e

	w2tend4(k,iCell) = (2.0_RKIND - 4.0_RKIND/3.0_RKIND*C_2)*Mc(k,iCell)* &
	(grav*alphaT(k,iCell)*tumd(k,iCell) - grav*betaS(k,iCell)*sumd(k,iCell))

$w2tend4= ( 2 - \frac{4}{3}C_2 ) \overline{w'b'} = ( 2 - \frac{4}{3}C_2 ) g ( \alpha_T \overline{w'T'} - \beta_S \overline{w'S'} )$ (Buoyancy plus buoyancy pressure term)

MPAS-Model/src/core_ocean/shared/mpas_ocn_adcReconstruct.F

Lines 573 to 575 in 5dc8d6e

	w2tend5(k,iCell) = (1.0_RKIND/3.0_RKIND*alpha1 - &
	alpha2)*(uw(i1,k,iCell)*Uz + vw(i1,k,iCell)*Vz) + &
	Mc(k,iCell)*(Swumd(k-1,iCell) + Swumd(k,iCell))

$w2tend5= complicated$ (Rapid pressure & subgrid terms, dissipation too(?))

MPAS-Model/src/core_ocean/shared/mpas_ocn_adcReconstruct.F

Lines 314 to 316 in 5dc8d6e

	w2tend6(k,iCell) = min(max(-config_adc_splat_tend_max, -w2(i1,k,iCell)* &
	config_adc_splat_wp2_val*tau_sfc*d_sqrt_wp2_dz**2), &
	config_adc_splat_tend_max)

$w2tend6= Splatting$ (Effectively a pressure term that converts w2 to u2 and v2 near the surface)

We could add a combined pressure term that is only used for comparison with LES: w2tend_pressure $= w2tend3 + w2tend5 + w2tend6 - \frac{2\overline{w'w'}}{3\tau_{slow}} - \frac{\frac{4}{3}C_2}{2-\frac{4}{3}C_2} w2tend4$

Then, $w2tend$_$pressure \approx -2\overline{w'\frac{\partial p'}{\partial z}}$

Since $w2tend5$ also contains dissipation, perhaps we go one step further by combining pressure and dissipation terms

$w2tend1 + w2tend$_$pressure = -2\overline{w'\frac{\partial p'}{\partial z}} - \frac{2}{3}\epsilon $

$\overline{w'w'w'}$ budget

MPAS-Model/src/core_ocean/shared/mpas_ocn_adcReconstruct.F

Line 486 in 5dc8d6e

w3tend1(k,iCell) = -wumdav**3.0*(Eav*(3.0*sigav - 2.0) + Dav*(3.0*sigav - 1.0))

$w3tend1=?$ (Entrainment/Detrainment - perhaps analogous to LES dissipation of $\overline{w'w'w'}$)

MPAS-Model/src/core_ocean/shared/mpas_ocn_adcReconstruct.F

Lines 498 to 499 in 5dc8d6e

	w3tend2(k,iCell) = -((3.0_RKIND + Swk**2.0)*(w2(i1,k,iCell)**2.0) - &
	(3.0_RKIND + Swk**2.0)*(w2(i1,k+1,iCell)**2.0) ) / dz

$w3tend2=-\frac{\partial \overline{w'w'w'w'}}{\partial z}$ (Turbulent transport, the 4th moment may not be LES output...)

MPAS-Model/src/core_ocean/shared/mpas_ocn_adcReconstruct.F

Line 500 in 5dc8d6e

w3tend3(k,iCell) = 1.5_RKIND*(w2(i1,k,iCell)**2.0 - w2(i1,k+1,iCell)**2.0) / dz

$w3tend3=\frac{3}{2}\frac{\partial \left[ \left(\overline{w'w'}\right)^2 \right]}{\partial z}$ (Source term from w2 gradients)

MPAS-Model/src/core_ocean/shared/mpas_ocn_adcReconstruct.F

Lines 504 to 505 in 5dc8d6e

	w3tend4(k,iCell) = -3.0_RKIND*(1.0_RKIND - 2.0_RKIND*sigav)*Mcav*wumdav*Swumd(k,iCell) - &
	tauw3(k,iCell)*w3(i1,k,iCell)

$w3tend4=complicated$ (Sub-plume scale transport and slow pressure term)

MPAS-Model/src/core_ocean/shared/mpas_ocn_adcReconstruct.F

Lines 506 to 507 in 5dc8d6e

	w3tend5(k,iCell) = 3.0_RKIND*(1.0_RKIND - c11)*grav*(alphaT(k,iCell)*w2t(k,iCell) - &
	betaS(k,iCell)*w2s(k,iCell))

$w3tend5=3(1-C_{11})\overline{w'w'b'}$ (Buoyancy production plus buoyancy pressure term)

MPAS-Model/src/core_ocean/shared/mpas_ocn_adcReconstruct.F

Lines 303 to 305 in 5dc8d6e

	w3tend6(1,iCell) = min(max(-config_adc_splat_tend_max, -w3(i1,1,iCell)* &
	config_adc_splat_wp2_val*tau_sfc*d_sqrt_wp2_dz**2), &
	config_adc_splat_tend_max)

$w3tend6=Splatting$ (This is effectively a pressure-driven effect)

[BrodiePearson]

@katsmith133 Let me know if you spot anything that's incorrect or doesn't make sense!

[katsmith133]

@BrodiePearson, I believe I've got correct terms plotted (splatting terms are left out for now as I am comparing to a non-splatting ADC run, the "no-splat, w2 BC off" run at 82 hours in):

Note: for the w2tend1 + w2tend_pressure term, the -2*<w'*dp'/dz> term is not available in the saved LES profiles, so I have not plotted that term for the LES. Perhaps something to discuss...

[BrodiePearson]

@katsmith133 These are interesting comparisons. I agree this we should discuss this today

[amrapallig]

LES c2 case:

[katsmith133]

@BrodiePearson and @amrapallig, per our discussion yesterday, here is the comparisons for the various terms that go into the w2t term that is used in calculating w3tend5:

w2t panel: black line - diagnosed from LES data as <w'w't'>, blue line - from ADC run, green line - diagnosed from LES data using ADC formulation:

MPAS-Model/src/core_ocean/shared/mpas_ocn_adcReconstruct.F

Line 426 in 5dc8d6e

w2tTemp = -sigav*(1.0_RKIND - sigav)*(1.0_RKIND - 2.0_RKIND*sigav)*wumdav**2.0_RKIND*tumdav

Sw - tumd panels: blue lines - from ADC run, green lines - diagnosed from LES using ADC formulations

It looks like using the ADC formulation on the LES data produces a w2t term that is still too small, indicating that something is missing from the ADC approximation of w2t.