Barotropic instability due to Kelvin wave-Rossby wave coupling

The stability of a vertically bounded, piecewise linear shear profile in a channel is analyzed using both a quasigeostrophic (QG) and primitive equation (PE) model. The choice of a finite depth domain thus allows us to consider more realistic flows in which the jet is vertically bounded. A potential vorticity discontinuity in the QG model can give rise to an isolated shear-generated Rossby mode that remains stable in the absence of other mean flow discontinuities. The finite depth assumption in the QG model is of little consequence as the vertical scale of the basic state enters the dynamical equations in a trivial manner. Solving this problem in the PE model, on the other hand, leads to unstable modes not present in the QG limit. Using semigeostrophic (SG) dynamics the authors are able to identify two modes of instability. One occurs as a result of a Kelvin wave-Kelvin wave coupling and the other is a product of Kelvin wave-Rossby wave coupling. It is also found that resonance between the Rossby mode and an unphysical "mirage wave" takes place in SG theory, causing a spurious instability not present in the PE case in regions of parameter space where the depth of the domain tends to infinity.