39.7.2 Discretization

The diagnostic option local_potential_density_terms provides a discretization of the terms on the right hand side of equations (39.79)-(39.86). Since they represent source/sink terms for the time tendency of locally referenced potential density, it is useful to diagnose them at a model tracer point. All terms, except advection, are computed inside the fortran program ptlrhoterms.F. The need to separate all of the various processes implies that none of the fluxes computed in isopyc.F can be used, since that code, for optimization reasons, combines the various processes to form a single flux vector. As such, it is very important to maintain consistency between the algorithms used in isopyc.F and ptlrhoterms.F.

Whenever discretizing continuous equations which possess certain local properties, it is not always possible to maintain those local properties on the lattice. Griffies et al. (1998) discuss how their discretization of the isoneutral diffusive flux maintains a downgradient property only over an extended ``finite volume'' rather than on each grid cell. In particular, $\vec{F}_{I}(\theta) \cdot \nabla \theta$ on the lattice is not constrained to be negative. This result means that the lattice form of cabbeling, $\nabla \theta \cdot \vec{F}_{I}(\theta) \, \rho_{a b} V^{a} V^{b}$, will not generally maintain non-negative values as it does in the continuum. The negative values for cabbeling on the lattice, however, should be minimal and localized.

Note that in the model, the computation of tracer flux components employs the opposite sign to that employed in the continuum formulation given above. Hence, the discrete equations will have a minus sign inserted to account for this convention.