Synopses & Reviews
The complex flows in the atmosphere and oceans are believed to be accurately modeled by the Navier-Stokes equations of fluid mechanics together with classical thermodynamics. However, due to the enormous complexity of these equations, meteorologists and oceanographers have constructed approximate models of the dominant, large-scale flows that control the evolution of weather systems and that describe, for example, the dynamics of cyclones and ocean eddies. The simplifications often result in models that are amenable to solution both analytically and numerically. The volume examines and explains why such simplifications to Newton's second law produce accurate, useful models and, just as the meteorologist seeks patterns in the weather, mathematicians seek structure in the governing equations, such as groups of transformations, Hamiltonian structure and stability.
Synopsis
Numerical weather prediction, chaotic atmospheric dynamics, atmospheric modelling.
Synopsis
This book and its companion describe, in a language accessible to both mathematicians and meteorologists, the mathematics underpinning our understanding of large-scale atmosphere and ocean dynamics. Meteorologists understand 'weather' by identifying the dominant controlling mechanisms, and so mathematicians are deducing how such features can be described mathematically. They are discovering that geometry plays a key role in this process. These developments promise an important spin-off - improving numerical models by incorporating, using a geometric language, constraints that govern the optimal use of observational data and the development of typical weather systems.
Table of Contents
Introduction J. C. R. Hunt, J. Norbury and I. Roulstone; 1. Balanced models in geophysical fluid dynamics: Hamiltonian formulation, constraints and formal stability O. Bokhove; 2. The swinging spring: a simple model of atmospheric balance P. Lynch; 3. On the stationary spectra for an ensemble of plane weakly nonlinear internal gravity waves P. Caillol and V. Zeitlin; 4. Hamiltonian description of shear flow N. J. Balmforth and P. J. Morrison; 5. Some applications of transformation theory in mechanics M. J. Sewell; 6. Legendre-transformable semi-geostrophic theories R. J. Purser; 7. The Euler-Poincaréequations in geophysical fluid dynamics D. D. Holm, J. E. Marsden and T. Ratiu; 8. Are there higher-accuracy analogues of semi-geostrophic theory? M. E. McIntyre and I. Roulstone.