THE NONLINEAR EVOLUTION OF A SLOWLY GROWING WAVE ON A LATERALLY SHEARED BAROCLINIC FLOW

Wave disturbances to baroclinic flows produce cyclones in the atmosphere and eddies in the oceans and have been extensively studied in laboratory experiments with differentially heated annuli of rotating fluid. Related analytical studies have concentrated mainly on the development of slowly growing waves on laterally uniform zonal flows. Neutral inviscid waves on such flows do not advect their own potential vorticitv field whereas neutral waves on most laterally sheared baroclinic flows do. Scaling arguments suggest that on these laterally sheared flows the harmonics generated by the neutral waves play the dominant role in arresting the initial growth of weakly unstable waves. The arrest of a wave is chiefly accomplished by fully nonlinear advection within a critical layer centred on the wave's steering level whose depth is proportional to the wave's amplitude. Explicit numerical solutions illustrating these points are presented for a case in which the critical level is non-singular and the inviscid calculations comparatively straightforward. The stability of the solutions and the effects of diffusive fluxes on them are discussed. Potential vorticity diagnostics for a numerical simulation of a wave flow in a rotating annulus near the axisymmetric transition show that distortion of the wave's potential vorticity field is mainly confined to the vicinity of the steering level. Assumptions and approximations made in the explicit calculations which are of doubtful validity for this flow are highlighted.