35.1.8.4 Effects on potential energy of an operator suggested by Gent

An alternative operator suggested by Peter Gent (personal communication; reported in RM98) is derived from the velocities

$${\bf U}_{h}^{*} -\partial_{z} \left[ \nabla_{h} \, \left( B \frac{\nabla_{h}^{2} \rho}{\rho_{z}} \right) \right]$$ = (35.42)

$$\nabla_{h}^{2} \left( B \frac{\nabla_{h}^{2} \rho}{\rho_{z}} \right).$$ W_* = (35.43)

Since the operator does not maintain the integrity of the slope vector ${\bf S} = - \nabla_{h} \rho / \rho_{z}$, its implementation in MOM is not as simple as the RM98 operator.

The anti-symmtric tensor corresponding to the Gent biharmonic operator is given by

$${\bf A} [A^{mn}] = \left( \begin{array}{ccc} 0 & 0 & -\partial_{x} \left(... ...t( B \, \frac{\nabla_{h}^{2} \, \rho}{\rho_{z}} \right) & 0 \end{array}\right),$$ = (35.44)

and the components to the skew-flux for an arbitrary tracer takes the form

$${\bf F}_{h} \nabla_{h} \left( B \, \frac{\nabla_{h}^{2} \, \rho}{\rho_{z}} \right) \, T_{z}$$ = (35.45)

$$-\nabla_{h} \left( B \, \frac{\nabla_{h}^{2} \, \rho}{\rho_{z}} \right) \cdot \nabla_{h} T.$$ F^z = (35.46)

The effects on the time tendency of potential energy arising from this skew-flux take the form

$$\int d{\bf x} \; F^{z} -\int d{\bf x} \; \nabla_{h} \left( B \, \frac{\nabla_{h}^{2} \rho}{\rho_{z}} \right) \cdot \nabla_{h} \rho$$ =

$$\int d{\bf x} \; \nabla_{h} \cdot ( B \, {\bf S} \, \nabla_{h}^{2... ...ho ) + \int d{\bf x} \; (B/\rho_{z}) \, \left( \nabla_{h}^{2} \rho \right)^{2}.$$ = (35.47)

The first term can be dropped upon assuming the isoneutral slopes vanish at the horizontal boundaries. The second term is non-positive in stably stratified water, and so represents a sink for potential energy for any density profile.

The Gent operator cannot naively be discretized using the triad technology employed by Griff ies et al. (1998) or Griff ies (1998). The reason is that the slope vector is not a fundamental piece of the Gent operator, whereas it fundamental for isoneutral diffusion, and GM90 and RM98 skew-diffusion. Indeed, results from a straightforward discretization of the Gent operator indicate the presence of numerical instabilities (RM98 and personal communication). Further research is necessary to determine the relative merits of the RM98 and Gent operators amongst other possible adiabatic biharmonic operators.