35.1.9 Isoneutral mixing and steep sloped regions

Regions of extremely steep isoneutral slopes are typically associated with a strong amount of dianeutral mixing. These are the ocean's mixed layers. To fully resolve them requires non-hydrostatic models with nearly isotropic grid aspect ratios (e.g., Marshall et al. 1997). MOM is hydrostatic and so must parameterize the physics of these boundary layer regions through one of its mixed layer schemes. In these regions, the adiabatic stirring and isoneutral diffusion schemes can be eliminated in favor of a mixed layer scheme. This argument prompts an approach which says that when isoneutral slopes steepen, the fluxes from redi_diffusion, gent_mcwilliams, and biharmonic_rm schemes are systematically reduced in favor of the fluxes arising from the mixed layer scheme. This is the central argument motivating one to not bother using the full Redi diffusion tensor. This argument is also coincident with the numerical need to turn down the fluxes from the isoneutral schemes as the slopes steepen in order to maintain linear stability (Cox 1987, Griff ies et al. 1998). The way to turn off these schemes is simple: scale or taper the diffusivities A_I, $\kappa$, and B to zero as the magnitude of the slope increases. There are two ways in MOM to taper these coefficients to zero: in a quadratic manner as implemented in option gkw_taper (Gerdes, Köberle, and Willebrand 1991) or exponentially as with option dm_taper. The option gkw_taper is the default approach used in MOM.

When tapering the fluxes from redi_diffusion, gent_mcwilliams, and biharmonic_rm in steep sloped regions, it has traditionally been assumed that the only remaining nonzero fluxes arise from a nonzero vertical diffusivity. Namely, all horizontal fluxes are zeroed out in the steep sloped regions. However, one might suspect that horizontal mixing occurs by the unresolved eddies in the mixed layers, as argued by Treguier, Held, and Larichev (1997). Otherwise, there is no way to dissipate density structures using only vertical diffusion, since the isoneutral slopes are nearly vertical. Consequently, Treguier et al. suggest the retention of a nonzero horizontal diffusivity in the mixed layers, where the strength of this diffusivity is the same as the skew-diffusivity used in the interior. The diffusivity ahsteep serves this purpose. As the slopes steepen, the diagonal component to the two horizontal tracer fluxes employ a diffusivity which is no smaller than ahsteep. For the case of constant diffusivities, setting ahsteep to zero recovers the traditional approach with zero horizontal fluxes. When the diffusivities are nonconstant, as described in Section 34.2, the default is to have ahsteep equal the nonconstant skew-diffusivity used in the interior.

The use of a nontrivial ahsteep in steep sloped regions is restricted to models which employ the default redi_diffusion option small_tensor. For example, models without option small_tensor, but with biharmonic_rm, are meant to be eddy-permitting models, and so the eddies should explicitly provide the horizontal stirring parameterized by ahsteep. The most simple choice for coarse models is to set ahisop = athkdf = ahsteep. In this case, the algorithm becomes particularly simple in that the 13 and 23 components to the mixing tensor are identically eliminated, unless option biharmonic_rm is also enabled.

Another approach is to employ a horizontal diffusivity equal to the value of the vertical diffusivity arising from some prognostic mixed layer scheme. In this way, the steep sloped regions maintain a 3D isotropic symmetry, as appropriate for a 3D isotropic turbulent mixed later. Option isotropic_mixed implements this parameterization so long as option redi_diffusion is enabled.