Wave Radiation and Shear Instability in Rotating Stratified Flow

Abstract

The paper shows how the shear instability is suppressed by the wave radiation, and how a window of instability exists in a rotating stratified flow. Direct numerical simulations (DNS) are conducted of the nonlinear transition from the linear instability in a model of ocean current in the rotating Earth. The base flow is a jet of the hyperbolic-secant velocity profile. The depth varies across the jet initially following a hyperbolic-tangent profile for geostrophic balance between Coriolis force and the lateral pressure gradient. The geostrophic balance is maintained throughout the simulation as the shear flow is modified by the nonlinear development. An accurate, and robust, computational scheme is used to solve the governing equations for the DNS. Time integration of the equations has been carried out using the fourth-order Runge-Kutta scheme. A third-order upwind bias finite difference approximation known as QUICK (Quadratic Upstream Interpolation of Convective Kinetics) is employed for the spatial discretization. The numerical oscillations are controlled using a flux limiter for Total Variation Diminishing (TVD). The concept of radiation damping is introduced for interpretation of the simulation results. In the DNS of the shear instability in the rotating shear flow, the radiation damping is observed to approach a minimum at a critical Rossby number Ro = 4. A corresponding maximum instability occurs at the critical Rossby number. A significant fraction of available energy to the shear instability is lost due to wave radiation from the shear flow. The maximum instability, the wave radiation and the nonlinear transition from the instability are the dynamic events that leads to heat and mass exchanges and the formation of rings and eddies across the ocean currents.