On the Representation of Rossby Wave Critical Layers and Wave Breaking in Zonally Truncated Models

Abstract

A detailed comparison is made between the Stewartson-Warn-Warn analytical solution for a fully nonlinear, nondiffusive Rossby-wave critical layer and a new analytical solution for the corresponding zonally truncated, ?wave-mean? or ?quasi-linear? problem in which the motion is represented by the zonal mean and a single zonal harmonic only. The effect of adding harmonics one by one is also considered. The results illustrate the extent to which zonally truncated models, which inevitably miss certain aspects of the fluid behavior, nevertheless contrive to mimic some important dynamical features of the evolution of the nonlinear critical layer, particularly the absorption-reflection behavior. The zonally truncated and fully nonlinear models predict almost the same reflection coefficient up to the time Tr when a state of perfect reflection is first reached, and the predicted values of Tr itself differ by only 5%. The agreement deteriorates only subsequently, during the first overreflecting stage. The reasons for this behavior are clarified by visualizing the way in which the truncated model misrepresents the (potential) vorticity field. Solutions, analytical and numerical, are also presented for zonally truncated models of nonlinear critical layers in which Rayleigh friction, or viscous diffusion, play an important role in the vorticity balance, as they may do in some numerical models. These solutions are compared with the corresponding fully nonlinear solutions. Implications for the numerical modeling of dynamical and tracer-transport processes involving planetary or Rossby wave ?breaking? are discussed.