The stability of a shear layer in an unbounded heterogeneous inviscid fluid

Abstract

This paper considers the stabilizing effect of density stratification on the horizontal shear layer between two parallel streams of uniform velocities. A simple continuous velocity distribution, $U = V\; \rm {tanh}(y|d)$, is used to represent the laminar shear layer. The density of the fluid is assumed to vary as exp (-βy), y being the vertical coordinate, with a small total change in density across the shear layer. The fluid is unbounded, and is assumed to be inviscid, incompressible and under the action of gravity.

By the methods of hydrodynamic stability theory, it is shown that a disturbance of small amplitude and wave-number α is neutrally stable if the Richardson number, defined as $J = g\beta d^2|V^2$, has the value $\alpha ^2 d^2 (1 - \alpha^2 d^2)$, and the form of the neutral disturbance is obtained. It follows that the critical Richardson number is $\frac{1}{4}$, so that the flow is stable if $J\;\; \textgreater \;\; \frac{1}{4}$. The relation between these results and Goldstein's derivation of the same critical Richardson number for a flow with discontinuous velocity and density gradients is discussed.