On the nonlinear evolution of baroclinic instability over topography

Abstract

A theory is presented for the nonlinear evolution of fluctuations in a zonal baroclinic shear flow over topography on a β-plane. It extends an earlier linearized theory on the same subject, which showed that unstable growth was possible for hybrid modes made up of a pair of primary modes with the same frequency but different wavenumbers, each stable in the absence of topography, but made unstable by topography with wavenumber that bridges the wavenumber gap between the primary modes. The slow evolution of the amplitudes of the hybrid modes can be expressed quite generally in terms of elliptic functions that fluctuate regularly between maximum and minimum values determined in a complicated way by initial amplitudes and parameters characterizing the primary modes. For small initial amplitudes, the evolution can be described in terms of even simpler function, i.e., trains of hyperbolic-secant pulses that recur with a period that depends on the logarithm of (the inverse of) the initial amplitude. Maximum fluctuation velocities comparable to, or even larger than, mean flow velocities can be achieved for disturbances of the scale of the internal deformation radius (typically 50 km in the oceans, 1000 km in the atmosphere), topography variations 10% of total fluid depth, and parameters typical of oceans and atmosphere.