35.1.8.2 RM98 for a special vertical profile

Consider the GM90 skew-flux discussed in Griff ies (1998)

$${\bf F}_{h} \kappa \, {\bf S} \, T_{z}$$ = (35.28)

$$-\kappa \, {\bf S} \cdot \nabla_{h} T.$$ F^z = (35.29)

Recall that in the special case of a linear equation of state, the GM90 density skew-flux takes the especially simple form

$${\bf F}_{h}(\rho) -\kappa \, \nabla_{h} \rho$$ = (35.30)

$$F^{z}(\rho) \kappa \, S^{2} \rho_{z}.$$ = (35.31)

With a stable density profile, $\rho_{z} < 0$, which means that the vertical skew-flux component is always negative. In general, the horizontal GM90 skew-flux components are directed down the density gradient, and the vertical component is upgradient.

With an always upgradient flux of density, the GM90 scheme always decreases the potential energy in the stably stratified fluid. This property is not generally respected by RM98, as discussed in Section 34.1.8.3. However, it is useful to consider a case in which these properties are shared for the purpose of illustrating the biharmonic nature of the RM98 scheme. One such profile is given by

$$\rho = \rho_{o}(z) + \rho_{1} \, \cos({\bf p} \cdot {\bf x}),$$ (35.32)

where $\rho_{o}(z)$ is some stable mean vertical profile, $\rho_{1}$is a (possibly time dependent) amplitude function, and ${\bf p} = (p_{x},p_{y},0)$ is a horizontal wave-vector. The slope vector for this profile is given by

$${\bf S} = \frac{ \rho_{1} \, {\bf p} \, \sin({\bf p} \cdot {\bf x})} { \rho_{o}'(z)},$$ (35.33)

and the horizontal Laplacian is

$$\nabla^{2}_{h} {\bf S} = - p^{2} \, {\bf S},$$ (35.34)

where $p^{2} = {\bf p} \cdot {\bf p}$. The RM98 skew-flux therefore takes the form

  

$${\bf F}_{h} (B \, p^{2}) \, {\bf S} \, T_{z}$$ = (35.35)

$$- (B \, p^{2}) \, {\bf S} \cdot \nabla_{h} T.$$ F^z = (35.36)

The RM98 skew-flux of density, linearly dependent on temperature, is given by

$${\bf F}_{h}(\rho) -(B \, p^{2}) \, \nabla_{h} \rho$$ = (35.37)

$$F^{z}(\rho) (B \, p^{2}) \, S^{2} \, \rho_{z}.$$ = (35.38)

As such, just as for the GM90 case which holds in general, the horizontal RM98 skew-flux components for density are down the horizontal density gradient, and the vertical skew-flux component is up the vertical density gradient. The effective diffusivity, however, is scale-dependent in the RM98 case, with small scales, or large p², acted on with the largest effective diffusivity. It is this sort of behaviour which is characteristic of a biharmonic mixing scheme.