35.1.9.3 isotropic_mixed

In planetary boundary layers, it might be of interest to compute diagonal horizontal tracer diffusion fluxes with the same diffusivity as the vertical tracer diffusion flux. In this case, the parameterized turbulence is represented by isotropic horizontal/vertical diffusion. The physical picture is of large eddies moving parcels both horizontally and vertically in the mixed layer. If one is using some mixed layer scheme such as option kppvmix (Section 32.2.3), then the vertical diffusivities can sometimes reach up 10³ - 10⁵ cm²/sec, especially in mixed layers associated with deep convection. In these regions, the use of equal horizontal and vertical diffusivities can provide a nontrivial amount of horizontal fluxes in addition to the vertical fluxes. Option isotropic_mixed implements this idea in MOM.

For use in models with isoneutral mixing, the above ideas are implemenated inside the small angle redi diffusion scheme (the default option for redi_diffusion). In this case, the continuum horizontal diffusion fluxes take the form

$$(A_{I} + A_{D}) \, T_{x} + A_{I} \, S_{x} \, T_{z}$$ -F^x = (35.79)

$$(A_{I} + A_{D}) \, T_{y} + A_{I} \, S_{y} \, T_{z}.$$ -F^y = (35.80)

Only in regions where A_D is on the same order as A_I will its presence be significant. For regions outside the boundary layer, A_D is generally 7-8 orders of magnitude smaller than A_I, and so it is completely negligible. Inside the boundary layer in regions with steep isoneutral slopes, the effective A_I is tapered to a small value using either gkw_taper or dm_taper. The effective A_D, computed from the boundary layer scheme, is also much larger than in the interior, and so it can contribute a relatively larger amount to the horizontal fluxes.

In the code, the vertical diffusivity $diff\_cbt$ is the sum A_D + K33, as this coefficient is that which is used for the implicit solution of the vertical diffusion equation. Using a mixed continuous/numerical notation, option isotropic_mixed uses the algorithm

$$[ A_{I} + \left( \overline{diff\_cbt - K33} \right)^{z,x} ] \, T_{x} + A_{I} \, S_{x} \, T_{z}$$ -F^x = (35.81)

$$[ A_{I} + \left( \overline{diff\_cbt - K33} \right)^{z,y} ] \, T_{y} + A_{I} \, S_{y} \, T_{z},$$ -F^y = (35.82)

where the $\overline{()}^{z,x}$ and $\overline{()}^{z,y}$ represent four point averages used to bring the vertical diffusivities, defined on the bottom of a T-cell, to the relevant zonal or meridional sides.

There are no namelist parameters to set with option isotropic_mixed. Turning it on, however, requires the use of option fourth_order_memory_window. The reason is that the vertical diffusivity $A_{D} = diff\_cbt$ is computed on the bottom face of the tracer cell. To add it to the diagonal piece of the horizontal tracer fluxes, however, it must be averaged using a four point horizontal and vertical average. Such an average in the y-zplane requires option fourth_order_memory_window. Option fourth_order_memory_window is automatically enabled when option isotropic_mixed is used. Note that option isotropic_mixed can be run in parallel to setting ahsteep to some nonzero value (see discussion at beginning of Section 34.1.9).