Balanced models in geophysical fluid dynamics: Hamiltonian formulation, constraints and formal stability

Most fluid systems, such as the three-dimensional compressible Euler equations, are too complicated to yield general analytical solutions, and approximation methods are needed to make progress in understanding aspects of particular flows. This chapter reviews derivations of approximate or reduced geophysical fluid equations which result from combining perturbation methods with preservation of the variational or Hamiltonian structure. Preservation of this structure ensures that analogues of conservation laws in the original ``parent'' equations of motion are preserved. Although formal accuracy in terms of a small parameter may be achieved with conservative asymptotic perturbation methods, asymptotic solutions are expected to diverge on longer time scales. Nevertheless, perturbation methods combined with preservation of the variational or Hamiltonian structure are hypothesized to be useful in a climatological sense because conservation laws associated with this structure remain to constrain the reduced fluid dynamics.