Modern Applied Mathematics for the Atmospheric and Oceanic Sciences

Week 1: Numerics

July 14 - July 19, 2003

The geosciences are distinct from many other fields of basic inquiry in that the equations describing phenomena of interest are for the most part known. What is not known is the character of their solutions. Because of non-linearity and a wide range of temporal and spatial scales these equations tend to be tractable only in approximate senses, and then only numerically. Thus numerical methods for solving PDEs quite literally underlie many of our attempts to understand the behavior of the oceans and atmosphere. Historically there has been a fruitful connection between attempts to solve PDEs arising in geophysics and the development of advanced numerical methods ¹ however as the communities have matured this connection has waned.

In this unit we plan to address a variety of problems which are pressing in the geosciences, and which seem ripe for an application of techniques which have been developed by the applied mathematics community over the past decades, often with problems from other disciplines in mind. For example, sophisticated interface tracking techniques have been developed in conjunction with problems of interest to the military but have almost never been applied to the vexing problem of accurately representing cloud boundaries ^{e.g., 5, 18}. Increasingly complicated representations of chemical processes in regional and global models, or biogeochemical processes in oceanic flows, often lead to plumes with sharp boundaries whose transport by the flow is often poorly represented by numerical methods which have historically been developed for the representation of smooth fields. Sharp features are also introduced by deformation in the flow fields (fronts), or boundaries. The temporal evolution of boundary layers which develop in geophysical fluids, such as the stratocumulus topped boundary layer illustrated in Fig. 1 are very poorly represented in large-scale models whose vertical discretization is typically thousands of times coarser than the thickness of the interface. While these interface problems are not really fluid shocks, methods which have been developed for treating shocks are increasingly being applied to such problems.

Recent work on methods for representing interfaces is also widely applicable to these problems, but less widely used. For instance the level set method for computing dynamic implicit surfaces ^{14, 15} has proven to be a useful and versatile numerical device for computing moving fronts or free boundaries which may develop singularities and/or topological changes such as breaking or merging, but has yet to be extensively used in atmospheric or oceanic flow solvers.

In light of this situation the bulk of the numerics week will focus on methods for interface tracking and the representation of advection/diffusion processes where the diffusion is often associated with small scale turbulent (rather than molecular) processes and thus tends to be flow dependent, rather than fluid dependent. Applied mathematics topics to be covered will include:

• Interfaces in geophysics: fronts, boundary layers, cloud boundaries, gravity currents, entrainment, scalar transport, and the existing representation of geophysical interfaces by numerical models will be among the topics introduced to help motivate the discussion of numerical methods.
• Numerical Methods for Conservation Laws - I These lectures will cover: conservation laws: definition, difficulties, weak solutions, entropy condition, linear advection, domain of dependence, characteristics, shocks, rarefactions, contacts. Then we will cover numerical methods for linear equations, stability, convergence, CFL condition, upwind schemes.
• Numerical Methods for Conservation Laws - II These lectures will discuss high order methods: suppressing spurious oscillations, TVD, ENO, WENO, Central and/or field-by-field decomposition for systems. Additional topics involving multi-scale methods for the treatment of sound (or in some cases) gravity waves, numerical homogenization, multigrid, and far field boundary conditions will be addressed as time and interest allows.
• Level Sets: These lectures will begin by devising the relevant level set calculus and geometry toolbox. Then we shall consider multiphase compressible flow, multiphase incompressible flow, compressible/incompressible interfaces and Stefan problems. New and useful numerical techniques, such as the ghost fluid method and the particle level set method, as well as the relevant essentially nonoscillatory and weighted essentially nonoscillatory techniques will be discussed.

Students will be expected to have a background which includes basic numerical methods for solving PDE's.