42. Tracer mixing kinematics

In the process of arriving at a new formulation for the isopycnal diffusion scheme in MOM, it was useful to understand the basic kinematical properties of tracer mixing when parameterized by a second order mixing tensor. This appendix serves to document certain of these properties which, although perhaps well known by some, were not found to be readily accessible in the literature. In this appendix, only continuum equations will be discussed, and coordinates are referenced to a frame tangent to the geopotential [i.e., (x,y,z)]. The MOM code uses spherical coordinates and the coordinate transformations are straightforward (e.g., Haltiner and Williams, Chapter 1).

The basic equation to be considered here is the tracer mixing equation

$$(\partial_{t} + \vec{u} \cdot \nabla ) \; T = R(T),$$ (42.1)

where the mixing operator R(T) is given in an orthogonal coordinate frame by

$$R(T) = \partial_{m}( J^{m n} \partial_{n}T).$$ (42.2)

In these expressions, the Einstein summation convention is assumed in which repeated indices (m,n) are summed over the three spatial directions. ${\bf J}$ is a second order tensor whose contravariant components are written as J^{m n}. The three-dimensional velocity field $\vec{u}$ is assumed to be divergence-free ( $\nabla \cdot \vec{u} = 0$). T represents any tracer such as potential temperature, salinity, or a passive tracer.

In general, this equation represents a parameterization of mixing due to many different processes occurring over a wide range of scales. Therefore, the explicit form for the tensor ${\bf J}$ depends on the particular phenomenon being parametrized. Two basic forms of mixing are considered here as distinguished by the symmetry of the tensor. Either the process is purely dissipative, as represented by a symmetric positive semi-definite diffusion tensor, or purely advective, as represented by an anti-symmetric tensor.