Barotropic and baroclinic instability in rotating stratified fluids

Abstract

Abstract The linear normal mode instabilities of a parallel shear flow which varies both vertically ( z ) and meridionally ( y ) in a quasigeostrophic, rotating, stratified fluid are considered. The β effect (variation of Coriolis parameter with y ) is included. Both two-layer and continuous fluids are treated. Attention is concentrated on the types of instability possible for a given shear flow. It is found that the instability can be described adequately by three nondimensional parameters: Λ, the ratio of the horizontal length scale of the shear to the internal deformation radius: δ, which is either the ratio of layer depths in the two-layer fluid or the fractional depth of variation of the stratification in the continuous fluid; and β, suitably nondimensionalized. Asymptotic analyses, confirmed by direct numerical solutions, are performed for conditions in which various parameters become large or small. The β effect is essentially quantitative, whereas Λ and δ define the type of instability as barotropic (if the kinetic energy of the mean flow feeds the growing perturbations), baroclinic (if the available potential energy of the mean flow feeds the perturbations) or mixed (a combination of the two). The case of large Λ (the most relevant for oceanographic applications) is treated in detail. It is shown that y -independent problems have only limited relevance. For a fixed deformation radius and the y scale of the mean flow increasing without limit, the asymptote is not the case of no y variation in the mean flow.