On the stability of heterogeneous shear flows

Small perturbations of a parallel shear flow U(y) in an inviscid, incompressible fluid of variable density ρ 0 (y) are considered. It is deduced that dynamic instability of statically stable flows ( $\rho ^{\prime}_0 (y)\; \textless \; 0$ ) cannot be other than exponential, in consequence of which it suffices to consider spatially periodic, travelling waves. The general solution of the resulting differential equation is considered in some detail, with special emphasis on the Reynolds stress that transfers energy from the mean flow to the travelling wave. It is proved (as originally conjectured by G. I. Taylor) that sufficient conditions for stability are $U^{\prime}(y) \not= 0$ and $J(y)\; \textgreater \frac {1} {4}$ throughout the flow, where $J(y) = -g \rho^{\prime}_0(y)|\rho (y)U^{\prime 2}(y)$ is the local Richardson number. It also is pointed out that the kinetic energy of a normal mode in an ideal fluid may be infinite if $0 \; \textless \; J(y_c) \; \textless \; \frac {1}{4}$ , where $U(y_c)$ is the wave speed.