Geostrophic flows require that the Rossby and Ekman numbers are small, Ro, Eh, Ev ≪ 1, so that the momentum balance reduces to the geostrophic balance. Quasi-geostrophic motions (or small-scale geostrophic motions) are geostrophic motions for which additionally:

• the horizontal length scale is much smaller than the radius of the Earth, γ := L/r0 ≪ 1; and

• the vertical displacement Z of isopycnals is much smaller than the vertical scale D, σ := Z/D ≪ 1.

The second condition is equivalent to requiring the vertical strain to be smaller than one. The quasi-geostrophic theory also disregards temperature–salinity effects and assumes a one-component fluid. The smallness of γ is exploited by applying the midlatitude beta-plane approximation. We will carry out the perturbation expansion with respect to all of the small parameters explicitly since the zeroth order is degenerate and one has to go to the first order. The evolution of quasi-geostrophic flows is again governed by the potential vorticity equation. The quasi-geostrophic vorticity consists of the relative vorticity, planetary vorticity, and a vertical strain contribution. The boundary conditions also include zeroth and first order contributions. Because of the inherent approximations, the quasi-geostrophic equations cannot include all dissipation and forcing processes. Linearization of the quasigeostrophic potential vorticity equation about a motionless stratified background state yields the Rossby wave solutions discussed in Chapter 8.