43.0.1 Summary and Caveats

The new scheme does the following:

1.

It produces a zero flux of locally referenced potential density. The Cox (1987) scheme did not respect this property, and this led to the instability of that scheme.

2.

It reduces tracer variance, and produces downgradient oriented tracer fluxes when considering a particular finite volume.

3.

It computes the best approximation to the neutral directions within the limitations of the discrete lattice.

4.

It requires zero background diffusion to remain stable; the Cox (1987) scheme blew-up without this diffusion.

Some limitations of the new scheme:

1.

Because it only guarantees downgradient oriented tracer fluxes over finite volumes, rather than individual tracer cells, tracers can move outside their physically constrained range. As such, the new scheme does not guarantee positive-definiteness of tracers. This problem appears most pernicious for passive tracers in steep sloped regions. The temperature and salinity, being constrained by the need to produce a zero flux of locally referenced potential density, appear less problematic, though they too can move outside physical bounds. In order to do the pure isoneutral diffusion problem for all slopes, it appears necessary to employ a positive-definite isoneutral diffusion scheme. Speculations about the requirements of such a scheme were given by Beckers et al. (1998). Currently, no scheme exists. Note that since the problems appear mostly in steep sloped regions, it might be physically justifiable to apply horizontal fluxes in these regions (e.g., see Treguier, Held, and Larichev 1997). If this is the case, then the issue becomes much less problematic, and one perhaps is not constrained by this technical limitation of the current diffusion scheme.

2.

Upon stabilizing the diffusion scheme, it is possible to remove the horizontal background diffusion previously required with the Cox (1987) scheme. In so doing, one is now exposed to problems with dispersive advection schemes. These problems are most apparent next to topography in regions where the tracers are aligned with isoneutrals, and so diffusive fluxes are weak. This ``Peclet grid noise'' problem is fundamental to dispersive advection schemes. It is not a problem with the rotated diffusion scheme.