Simple Models for Local Instabilities in Zonally Inhomogeneous Flows

Abstract

Two very simple model equations, suggested by the form of the standard baroclinic instability dispersion relation. are studied to investigate the nature of the instability of zonally inhomogeneous flows. In particular, the simplifications of the model problems allow the consideration of examples in which the interval of instability is of the same order or even smaller than the characteristic wavelength of the free plane wave modes of the zonally homogeneous problem, a limit inaccessible to standard WKB procedures. In both model problems it is shown that the zonally gravest mode always remain no matter how small the locally unstable interval may be. Further, this gravest mode is eddy-like rather than wave-like, i.e., it has no internal nodes. When the free wave solutions are allowed to propagate, the structures of the bound modes are altered. They are squeezed against the exit edges of the unstable domain but again one mode, the bound eddy mode, is always possible. This study is complementary to the more elaborate WKB analyses recently published and, in agreement with these studies, it is shown that only local maxima in supercriticality are required to yield trapped modes. The bound eddy mode is, however,. a feature allowable only when WKB analysis fails. It is suggested that such bound eddies, anchored to local maxima in supercriticality may be regarded as the preliminary stage for the production of isolated, coherent structures.