Local Energy Generation in Barotropic Flows

Abstract

The local growth of disturbances to a steady, nondivergent shear flow is investigated in the context of the barotropic vorticity equation (BVE). A new expression for the instantaneous energy generation rate is derived by using a local coordinate frame defined by the directions parallel and perpendicular to the basic flow. The equilibrium information enters this expression through two scalar coefficients, which can be interpreted as local components of an ?effective shear? vector S (S coincides with the shear vector for irrotational flows). A simple picture of local instability results: the magnitude of S gives the maximum possible energy growth rate for localized unstable disturbances, while its orientation determines the direction of maximum energy growth, which is given by the perpendicular to the bisector of the angle between S and the equilibrium velocity. This allows extension of the familiar ?leaning against the shear? instability picture, developed for zonal currents, to generic, two-dimensional (2D), steady-state solutions of the BVE. Applications of the new approach to some simple basic flows of interest for atmospheric dynamics are discussed.