Some catastrophe properties of two-layer shear flow

Abstract

In this paper a simple current system which consists of two stratified incompressible layers is examined. For the basic equations of the motion of fluid a lower order spectrum model is established by means of Galerkin method. Adopting the difference of wind velocity between the upper and lower layers,$\Delta \bar u = \frac{{\rho _1 \bar u_1 - \rho _2 \bar u_2 }}{{\rho _2 }}$ as a control parameter, the bifurcation and stability of the solution of the dynamical system are discussed. It is found that the flow states in the lower layer will have a catastrophe, when$\left| {\Delta \bar u} \right| > 2\frac{{\delta ^2 }}{{f^2 }}C_g^3$ where C_g is the phase velocity of the internal inertio-gravitational wave in a geostrophic current. These results may give a reasonable explanation for the mechanism of the catastrophe phenomena, including the “pressure-jump” in the atmosphere.