Numerical weather prediction is a problem of mathematical physics. The complicated flows in the atmosphere and oceans are modelled by the Navier-Stokes based equations of fluid mechanics together with classical thermodynamics. However, due to the enormous complexity of these equations, meteorologists and oceanographers appeal to asymptotic methods, variational principles and conservation laws to construct models of the dominant large-scale flows that control our weather.
Simplified models are often amenable to analytical and numerical solution. The lectures in these volumes explain why such simplifications to Newton's second law produce accurate, useful models and, just as meteorologists seek patterns in the weather, mathematicians use geometrical thinking to understand the structure behind the governing equations. Here constrained Hamiltonian mechanics, transformation groups, and convex analysis are used to control the potentially chaotic dynamics in the numerical simulations, and to suggest optimal ways to exploit observational data. This book and its companion show how geometry and analysis quantify the concepts behind the fluid dynamics, and thus facilitate new solution strategies.
1. Balanced models in geophysical fluid dynamics: Hamiltonian
formulation, constraints and formal stability
2. The swinging spring" a simple model of atmospheric balancd
3. The swinging spring: a simple model of atmospheric balance
4. Hamiltonian description of shear flow
5. Some applications of transformation theory in mechanics
6. Legendre-transformable semi-geostrophic theories
7. The Euler-Poincare equations in geophysical fluid dynamics
8. Are there higher-accuracy analogues of semi-geostrophic
theory?