Horizontal diffusion calculation

[fernandosobral]

Hi everyone,

I’m working on a cross-shelf heat transport analysis and using Huon_temp/Hvom_temp for the heat fluxes on the grid cell faces. Now, I need to add the horizontal diffusion to get the full heat transport picture.

I know I can calculate it myself, and for my case, it’s the Laplacian Horizontal Diffusion: https://www.myroms.org/wiki/Horizontal_ ... _Diffusion

Is there a way to double-check if my calculation is correct? Can it be compared somehow with the diagnostic diffusion term temp_hdiff? And last one, is it reasonable to interpolate it to u/v grid to have them matching with my flux variables Huon_temp/Hvom_temp?

Any help would be helpful.

Thanks.
Fernando

[wilkin]

Is there a way to double-check if my calculation is correct? Can it be compared somehow with the diagnostic diffusion term temp_hdiff?

Equation (10) at the link you gave to the Wiki https://www.myroms.org/wiki/Horizontal_ ... _Diffusion shows that the net diffusion is the divergence of diffusive fluxes in the two horizontal grid coordinate directions. These fluxes are naturally defined on the rho cell faces (u-, v- points) because that's where the dC/dxi and dC/deta first differences fall. The respective u-,v- faces are also where your advective terms are defined (on velocity points). Be sure to average m/n and n/m to the velocity points too.

Your sanity check would be to compute your fluxes, then take their divergence (as in eqn (10)) and compare to temp_hdiff. This calculation is linear in tracer, and m,n are constants, so you should be able to get a reasonable equivalent balance with tracer concentrations averaged over the same time as your temp_hdiff diagnostics.

[fernandosobral]

Hi John,

Many thanks for your reply, it was very helpful!

When I calculate the divergence of the temperature gradient, my units are °C·m·s⁻¹, while the temp_hdiff from diagnostics is in °C·s⁻¹. I thought the most correct way to get it right was to divide my calculated horizontal diffusion by Hz and with that I got very similar results when comparing with temp_hdiff. However, I'm not sure if this is the correct approach. Could you please clarify this issue?

Thanks

[wilkin]

I thought the most correct way to get it right was to divide my calculated horizontal diffusion by Hz

Yes, that is correct. 1/m,1/n and Hz are the dx, dy and dz of the cell volume. Where you might lose some precision in your calculation is in the fact that Hz is time varying because it depends on sea level zeta. You'd do best to use an Hz based on the average zeta during the averaging interval of your diagnostics. The divergence term that ROMS computes, temp_hdiff, is the time average of the tracer and Hz so includes any nonlinear correlation between those values <temp' Hz'>, whereas your diagnostic will miss that. But it should be close.

In shallow water tidal applications one can get a significant correlation between tracers, sea level and velocity that can make posterior calculation of advective terms tricky because of <temp' u' Hz'> correlations, but the diffusion term doesn't have the temp*u nonlinearity so you should be able to get a reasonable closure.

[fernandosobral]

Thanks for the clarification John.

One more question about the units involved.

The diffusion fluxes have units of m².s⁻¹.°C⁻¹, but to properly add them with the Huon/Hvom_temp variables, which have units of m³.s⁻¹.°C⁻¹, I am missing a meter unit. In this case, what should I do? Should I take into account the grid metrics, specifically dy for the xi-direction and dx for the eta-direction, to ensure the correct area scaling?

For example, when calculating the diffusion flux in the xi-direction, would the appropriate expression be:

(ν2​⋅Hz⋅dx​⋅1/dy⋅​∂T/∂xi​)⋅dy ?

And similarly for the eta-direction:

(ν2​⋅Hz⋅dy⋅1/dx⋅​∂T/∂eta​)⋅dx ?

Thanks

[wilkin]

The advective term in the xi-direction Huon_temp is Hz times u over n times temp
where Hz is the layer thickness (so, dz) and over n is 1/n so dy.
Hence, the elemental cell face area dz*dy is already in the flux term. To vertically integrate over a region you just add up the terms over k.

In your calculation of the diffusive flux you need to include the layer thickness, dz (or Hz), to get the meter dimension you are missing.

The subtlety here is that Hz is time varying because of the free surface. The Huon_temp etc. terms include this variability on every time step so that the nonlinearity of correlations in layer thickness and tracer transport are accumulated exactly in the average.

We don't do this for diffusion terms, but as I've said before I don't expect the nonlinearity to be as significant as in advection.