On a theory of amplitude vacillation in baroclinic waves

Abstract

This study contributes to the theory of amplitude vacillation for finite amplitude baroclinic waves in a two-layer, quasi-geostrophic, zonal flow as worked out by Pedlosky. In a recent paper the author has shown that Pedlosky omitted a certain side-wall boundary condition on the mean zonal flow. The neglect of this boundary condition results in an unspecified energy source at the side-wall boundaries and the physical problem is incorrectly posed.In this paper, Pedlosky's analysis is repeated but with the side-wall boundary condition included. It is shown that the side-wall energy source is negligible only when the zonal wavenumber of the disturbance is large compared with the meridional wavenumber, and not otherwise. Moreover, the energy conversions to and from mean zonal kinetic energy corresponding to Pedlosky's calculations and those given here have essential differences, although for fixed meridional wavenumber, these differences become less pronounced as the zonal wavenumber increases.It is also shown that, when the side-wall condition is included, the mean flow distortion associated with the wave is different in structure to that which occurs when the condition is omitted. However, as the total disturbance wavenumber a increases, the influence of the side wall on the mean flow structure is confined to a boundary layer of width comparable with the internal deformation radius 2½a−1.Even when the deformation radius is comparable with the channel width, the conclusions of Pedlosky (1972) concerning the existence of stable periodic solutions are correct, providing viscous effects are vanishingly small, and the criteria for stability of the steady solutions obtained herein are not significantly different from those given by Pedlosky. In this viscous regime, we have also studied the evolution to limit-cycle solutions.Evolution in the case where viscous effects are small on the time scale for the initial growth of an incipient wave, but not vanishingly small, will be discussed in a subsequent paper.