Geophysical Fluid Dynamics

Two Weeks, Period 1
Marcel Oliver (International University Bremen, Germany)
Consultation Hours
Thursdays 11-12h Carslaw Room 635 (Weeks 1 & 2 only).
5 short assignments.
Assumed Knowledge
A first course on PDEs or equivalent exposure, a rigorous course in Analysis as well as vector calculus. The following will help, but are not essential prerequisites: functional analysis and some exposure to fluid mechanics.
Course Outline
This course is an introduction to mathematical problems in geophysical fluid dynamics. The course is structured in three parts:

Part I: The equations of geophysical fluid dynamics

  • The rotating Euler equations, the Boussinesq approximation, hydrostatic balance
  • Energy and vorticity; vorticity-streamfunction formulation in two-dimensional flows
  • Special solutions; linear and nonlinear stability
  • Shallow water and nearly-geostrophic limit equations

Part II: The quasi-geostrophic equations as a singular partial differential equations limit.

  • Review of function spaces and basic facts from Functional Analysis
  • Well-posedness of shallow water and quasi-geostrophic equations
  • Convergence theory for balanced initial data
  • Unbalanced data: Embid-Majda theory
  • Unbalanced data: Babin-Mahalov-Nikolaenko theory

Part III: Further topics (if sufficient time and interest)

  • Nonlinear dispersive waves
  • Semi-geostrophic limits
  • Variational methods
The lectures are loosely based on the book by A. Majda, Introduction to PDEs and waves for the atmosphere and ocean, American Mathematical Society, 2003, supplemented by additional reading and original research papers.