On the Stability and Accuracy of the Harmonic and Biharmonic Isoneutral Mixing Operators in Ocean Models

Ocean models usually rely on a tracer mixing operator which diffuses along isoneutral directions. This requirement is imposed by the highly adiabatic nature of the oceanic interior, and a numerical simulation needs to respect these small levels of dianeutral mixing to maintain physically realistic results. For non-isopycnic models this is however non-trivial due to the non-alignment of the vertical coordinate isosurfaces with local isoneutral directions, rotated mixing operators must therefore be used. This paper considers the numerical solution of initial boundary value problems for the harmonic (Laplacian) and biharmonic rotated diffusion operators. We provide stability criteria associated with the conventional space-time discretizations of the isoneutral Laplacian operator currently in use in general circulation models. Furthermore, we propose and study possible alternatives to those schemes. A new way to handle the temporal discretization of the rotated biharmonic operator is also introduced. This scheme requires only the resolution of a simple one-dimensional tridiagonal system in the vertical direction to provide the same stability limit of the non-rotated operator. The performance of the various schemes in terms of stability and accuracy is illustrated by idealized numerical experiments of the diffusion of a passive tracer along isoneutral directions.