Steady solutions of quasi-geostrophic flows in basins, gulfs and channels on a β-plane

Nonlinear steady solutions of the barotropic quasi-geostrophic equation in basins, gulfs and channels on a beta-plane are presented. The domains are rectangular with arbitrary aspect ratios. The two-dimensional solutions assume a linear relationship between the potential vorticity q and the stream function Psi. The sign of the slope in the linear q-Psi relationship defines two broad sets of solutions. For a positive slope, the solutions in a closed basin correspond to the inertial gyres derived by Fofonoff in 1954. The negative slope solutions consist of normal modes that can be resonant. For gulfs and channels, the conditions at the open boundaries are almost arbitrary flows entering or leaving the domain. Such conditions allow a great variety of solutions in the interior, characterised mainly by arrays of vortices with alternate signs. Several examples are presented and discussed.