A positive-definite, down-gradient implementation of neutral diffusion

Lateral mixing of tracers, for example arising from the stirring of tracers by mesoscale eddies, is primarily directed along neutral-surfaces (locally referenced isopycnals). Large-scale numerical models typically parameterize the lateral mixing by a rotated diffusion tensor (Redi, 1982; Cox, 1987; Griffies et al., 1998). Numerical implementations of the rotated diffusion approach suffer from truncation errors that can lead to numerical instabilities and the appearance of false extrema, contrary to the down-gradient nature of diffusion. In the rotated diffusion approach, each directional component of the flux is the sum of two terms that can oppose each other, in which case the small residual can be dominated by truncation errors.

We develop a new approach to implement along-neutral-surface diffusion. We approach the problem from the perspective of a general coordinate model (MOM6) in which regridding and remapping to arbitrary vertical coordinates are implemented. By considering a model cell to be bounded by neutral surfaces, we can readily find the cells in neighboring columns that could possibly interact within that neutral density range. We construct finite volume integrals for contributions to the diffusive flux and check that the overall gradient is consistent with the gradients on the bounding neutral surfaces. Although each flux contribution is not necessarily directionally symmetric, the depth integrated fluxes seen by two adjacent columns are identical so that tracer conservation is assured. The resulting method is down-gradient by construction and thus cannot create false extrema. Moreover, in the limit that the model coordinate surfaces follow isopycnals, the conventional along-coordinate diffusion of isopycnal-coordinate models is recovered.