The fundamental process of geostrophic adjustment is treated by the method of multi-scale asymptotic expansions in Rossby number and fast-time averaging (which is explained), first in the barotropic one-layer case, and then in the baroclinic two-layer case. Together with the standard quasi-geostrophic regime of parameters, the frontal (or semi-) geostrophic regime is considered. Dynamical separation of slow and fast motions is demonstrated in both regimes. The former obey quasi-geostrophic or frontal-geostrophic equations, thus providing formal justification of the heuristic derivation of Chapter 5. Fast motions are inertia-gravity waves in quasi-geostrophic case, and inertial oscillations in the frontal-geostrophic case. Geostrophic adjustment is also considered in the presence of coastal, topographic, and equatorial wave-guides, and, again, separation of fast and slow motions is demonstrated, the latter now including long Kelvin waves in the first case, long topographic waves in the second case, and long Kelvin and Rossby waves in the third case.
Nonlinear systems with fast and slow motions. Changes in the probability distribution for fast motions under the influence of slower ones
