Modern Applied Mathematics for the Atmospheric and Oceanic Sciences

Week 2: Asymptotics

July 21 - July 26, 2003

The same range of scales which motivate computational approaches to problems in atmospheric and oceanic sciences have long motivated asymptotic attacks on important problems. Beginning with Prandtl's boundary layer theory, through Charney's theory of quasi-geostrophy and continuing today with extensions and expansions of these ideas asymptotics is the basis of much of our rigorous understanding of atmospheric and oceanic flows. During the recent IPAM school, asymptotics arose repeatedly in the context of problems ranging from homogenization, to treatments of balanced flows. Asymptotic ideas were also the centerpiece of a recent workshop of the Institute for Mathematics and its Applications (IMA) entitled: Reduced Descriptions of Coupled GFD Systems (Slow Manifolds in the Ocean and Atmosphere), in which two of our proposed lecturers were participants.

Asymptotic descriptions of flows are often associated with large-scales or what are called balanced (slow-manifold) flows in the atmosphere and ocean. Just as the "slow-manifold'' of planetary-scale disturbances filters out fast inertia-gravity waves, the "slow-manifold'' of smaller scale motions is the low Mach number limit in which sound waves are filtered. Recent work applying asymptotic theory to the full hierarchy of atmospheric flows also has proven useful in interpreting and unifying the description of a wide class of models for atmospheric flows ¹¹, and has encouraged the development of new numerical approaches which respect these asymptotics ⁸. Asymptotics also arise repeatedly in theories of boundary layers and turbulence, and in the analysis of the advection and diffusion of passive scalars. Furthermore, numerical subtleties arising in the Monte Carlo simulation of these (turbulent transport) problems requires a new sophisticated mathematical framework, and extensive and systematic benchmarking in order to give reliable results. During the recent IPAM workshop analysis of the latter type proved to be a fruitful framework for thinking about how to construct models of mixing by eddies in the ocean and the atmosphere. Thus in this unit we hope to build on some of the energy that arose in the context of the IPAM and IMA workshops, and have chosen lecturers who can pedagogically address the state of the art on the following issues:

• Fundamentals: Dimensional, similarity and perturbation analysis; Boundary-layer flows; mixing-transport and advection-diffusion; Monte Carlo simulation.
• Phenomenology: Observations of balanced flows; balanced disturbances in the tropopause; tropospheric disturbances as sources of cyclogenesis; symmetries in balanced flows; tracer dispersion in the ocean.
• Large-scale balanced flows: Quasi-geostrophy; non-linear balance; multi-scale asymptotics; rapidly rotating shallow water asymptotics; simplified scale interaction models; multi-scale asymptotics and homogenization.
• Numerics: Asymptotically adaptive numerical methods; singularities and numerical stiffness in fluid PDEs; Monte Carlo methods in the simulation of mixing and advection-diffusion.