35.2 Schemes with nonconstant diffusivities

The isoneutral diffusion and skew-diffusion eddy coefficients have traditionally been assumed constant, with the exception of tapering discussed in Section 34.1.9 and the zero eddy flux condition at the ocean surface (Gent et al 1995). Recent studies, however, indicate the importance of moving away from a constant diffusivity towards the use of flow dependent diffusivities (e.g., Held and Larichev 1996, Visbeck et al. 1997, Treguier et al. 1997, Killworth 1997, Stammer 1998, Bryan et al. 1999).

At present, there are two schemes implemented in MOM for prescribing the isoneutral diffusion and skew-diffusion eddy coefficients according to the large-scale geostrophic flow field: the theories of Held and Larichev (1996) and Visbeck et al. (1997). The resulting diffusivities have no more depth dependence than that already implied by the discussion in Section 34.1. As such, either the eddy-induced velocity of Gent et al. (1995), or the skew-flux formulation of Griff ies (1998), are used with a diffusivity that is a function of $(\lambda,\phi,t)$. The more general ideas from Treguier et al. (1997) and Killworth (1997), which include depth dependence in the diffusivity and extra terms in the closure associated with the $\beta$-effect, have not been implemented. Both the skew-flux (Section 34.1.6.1) and advective flux (Section 34.1.6.2) formulation of GM have been extended to the nonconstant diffusivities. Again, the recommended approach is the skew-flux approach.

Held and Larichev and Visbeck et al. develop expressions for a time scale T and length scale L according to properties of the model's resolved flow field. Thereafter, a diffusivity is given by

$$\kappa = L^2/T.$$ (35.83)

The two schemes prescribe the same time scale, but different length scales.

In general, the quasi-geostrophic approximation is fundamental to all the theories used to prescribe the mixing coefficients. Since this approximation breaks down near the equator, care should be exercised when interpreting global model results using the given parameterizations.

For MOM, there are three main cases available for setting the relative values of the isoneutral diffusivity and skew-diffusivity:

1.

For the constant diffusivity case discussed in Section 34.1, the namelist parameters ahisop and athkdf set the isoneutral and skew diffusivities. When ahisop=athkdf, the code exploits the simplifications in the tracer flux discussed in Section 34.1.6.1 so long as the default option gm_skew is employed.

2.

For the nonconstant diffusivity case, setting ahisop = athkdfwill also provide for equal nonconstant diffusivities, and the code will exploit the simplifications in the tracer flux discussed in Section 34.1.6.1 so long as the default option gm_skew is employed.

3.

If either of the nonconstant diffusivity options is enabled, and if the namelist parameters $ahisop \ne athkdf$, then the isoneutral diffusive flux will be computed with the constant diffusivity ahisop, whereas the skew-flux or eddy-induced velocity will be computed with a nonconstant diffusivity.