On short wave stability and sufficient conditions for stability in the extended Rayleigh problem of hydrodynamic stability

Abstract

We consider the extended Rayleigh problem of hydrodynamic stability dealing with the stability of inviscid homogeneous shear flows in sea straits of arbitrary cross section. We prove a short wave stability result, namely, if k > 0 is the wave number of a normal mode then k > kc (for some critical wave number kc) implies the stability of the mode for a class of basic flows. Furthermore, if \( K(z) = \frac{{ - (U''_0 - T_0 U'_0 )}} {{U_0 - U_{0s} }} \), where U0 is the basic velocity, T0 (a constant) the topography and prime denotes differentiation with respect to vertical coordinate z then we prove that a sufficient condition for the stability of basic flow is \( 0 < K(z) \leqslant \left( {\frac{{\pi ^2 }} {{D^2 }} + \frac{{T_0^2 }} {4}} \right) \), where the flow domain is 0 ≤ z ≤ D.