Geometry of Advection, Diffusion, and Viscosity

Efforts to parameterize mesoscale, submesoscale, and boundary layer processes are often stymied at the application phase—how to convert processes into mathematical form and code them? I will present some existing parameterizations, as written compactly, and in arbitrary coordinate systems, using tensors and differential forms. In this form, I will show how advective, diffusive, dispersive, and other transport processes are contrasted from one another, and how enforcing physical symmetries helps in first discovering parameterizations. However, the main goal is a demonstration that relaxation of symmetries is a simple way to advance parameterizations and incorporate new processes efficiently. Examples and results will be introduced from applications in coarse-resolution climate models (anisotropic mesoscale and submesoscale eddy parameterizations), mesoscale-permitting models (enstrophy and potential enstrophy cascade models), submesoscale-permitting models (mixed layer and symmetric instability parameterizations), and representations of air-sea processes suitable in all of these model classes.