35.1.8.3 Effects on potential energy of the RM98 operator

Now consider how the RM98 operator affects the potential energy for the case when density is a linear function of potential temperature. A similar discussion was given in the RM98 paper, where they employ the advective flux formulation rather than the skew-flux formulation. A few speculative remarks are added here as well. Focusing just on the biharmonic operator, the time tendency for potential energy is given by

$$g \int d{\bf x} \; z \, \rho_{t}$$ P_t =

$$- g\int d{\bf x} \; z \, \nabla \cdot {\bf F}$$ =

$$g\int d{\bf x} \; F^{z},$$ = (35.39)

where the no-normal flux condition at the sides was assumed. Note that for simplicity, a rigid lid was also assumed, although this assumption has no bearing on the effects the $g\int d{\bf x} \; F^{z}$term has on the evolution of the total potential energy. Using the RM98 skew-diffusion leads to

$$\int d{\bf x} \; F^{z} - \int d{\bf x} \; B \, \rho_{z} \, {\bf S} \cdot \nabla_{h}^{2} {\bf S}$$ =

$$- \int d{\bf x} \; \nabla_{h} \cdot ( B \, \rho_{z} \, S_{i} \, \... ...+ \int d{\bf x} \; \nabla_{h} ( B \, \rho_{z} \, S_{i}) \cdot \nabla_{h} S_{i},$$ = (35.40)

where i=1,2 is summed. Assuming the isopycnal slopes vanish next to the lateral ocean boundaries allows for the total derivative term to be dropped. The result is

$$\int d{\bf x} \; F^{z} (1/2) \int d{\bf x} \; \nabla_{h} \, (B \, \rho_{z}) \cdot \nabla... ...{\bf x} \; B \, \rho_{z} \, (\partial_{j} {\bf S} \cdot \partial_{j} {\bf S} ),$$ = (35.41)

where $S^{2} = {\bf S} \cdot {\bf S}$, and j=1,2 is summed in the second term over the horizontal spatial dimensions. The second term is non-positive in stably stratified fluids, for which $\rho_{z} \le 0$. It therefore represents a potential energy sink. The first term, however, is sign indefinite even when B is a constant. For the special case of $B \, \rho_{z}$ independent of horizontal position, the first term vanishes and so potential energy is reduced. For example, the special density profile considered previously $\rho = \rho_{o}(z) + \rho_{1} \, \cos({\bf p} \cdot {\bf x})$ has $B \, \rho_{z}$ horizontally constant if B is constant, and so the potential energy is reduced. In the slightly more general case of constant B and with $\rho_{z} = \rho_{z}^{0}(z) + \rho_{z}^{1}({\bf x})$, where $\vert\rho_{z}^{0}\vert >> \vert\rho_{z}^{1}\vert$, the first term in the expression for potential energy is nonzero, but subdominant to the second term. So potential energy is again reduced in this case. It is unclear what happens in the more general case.

It might be speculated that the inability to prove that the potential energy is generally reduced may indicate that the RM98 operator is unstable. However, if numerically implemented according to the triad approach of Griff ies et al. (1998), the discretized RM98 skew-fluxes will preserve the density variance. As variance growth is typically associated with linearly unstable numerical schemes, any potential instability of the RM98 scheme will likely be nonlinear. So far, no such instabilities have been encountered. Rather, the operator appears to be quite stable in a wide suite of both coarse and fine models.