35.1.8.1 The RM98 operator

The additional ``eddy-induced'' velocity proposed by RM98 has the components

$${\bf u}_{h}^{*} \partial_{z} \left( B \, \nabla_{h}^{2} {\bf S} \right)$$ = (35.21)

$$- \nabla_{h} \cdot \left( B \, \nabla_{h}^{2} \, {\bf S} \right),$$ w^* = (35.22)

where

$${\bf S} = - \left( \frac{\nabla_{h} \rho}{\rho_{z}} \right)$$ (35.23)

is the isoneutral slope vector, $\rho$ is the locally referenced potential density, and $B \ge 0$ is the biharmonic dissipation coefficient with units of length⁴/time. The realization of this dissipation can readily be made through the skew-flux approach of Griff ies (1998), as discussed in Appendix B and Section 34.1.6.1. The anti-symmetric tensor corresponding to the RM98 advection velocity takes the form

$${\bf A} [A^{mn}] = \left( \begin{array}{ccc} 0 & 0 & B \, \nabla_{h}^{2} ... ...\nabla_{h}^{2} \, S_{x} & -B \, \nabla_{h}^{2} \, S_{y} & 0 \end{array}\right),$$ = (35.24)

and the components to the corresponding skew-flux components for an arbitrary tracer are

$${\bf F}_{h} - \left( B \, \nabla_{h}^{2} \, {\bf S} \right) \, T_{z}$$ = (35.25)

$$\left( B \, \nabla_{h}^{2} \, {\bf S} \right) \cdot \nabla_{h} T.$$ F^z = (35.26)

In general, the skew-flux for a particular tracer is directed normal to the gradient of that tracer

$$\nabla T \cdot {\bf F}(T) = 0.$$ (35.27)

Notably, this result holds when T is the locally referenced potential density $\rho$, which reflects the adiabatic nature of the scheme. The manner in which this continuum result is implemented numerically is discussed in Griff ies (1998). That approach ensures that the numerical scheme does not alter the tracer mean and variance. It is therefore conservative in this sense.