Finite-amplitude stability of a topographically forced wave with Ekman dissipation

The linear and finite-amplitude stability characteristics of a topographically forced wave are examined in a quasigeostrophic, viscous, barotropic model on a beta-plane. The linear analysis reveals that amplification of a superimposed disturbance is favored when the topographically forced wave is approximately 90° out of phase with the topography. This phase relationship corresponds to flow near topographic resonance. The amplitude evolution of a weakly unstable disturbance is characterized by a monotonic approach to a steady value as it reaches finite amplitude. The nonlinear equilibrating mechanism is the convergence of momentum flux which results from the disturbance interacting with the correction to the topographically forced wave, its higher harmonics, and the form drag induced correction to the mean flow. In the absence of a secondary zonal wavenumber in the disturbance field, the stability threshold for a given set of parameter values is minimized, and equilibration at finite amplitude results solely from the interactions involving the higher harmonics. Finite amplitude instability is shown to exist in certain restricted regions of parameter space.