Kelvin-Helmholtz instability of a slowly varying flow

Abstract

The linear stability of a basic flow of two homogeneous inviscid incompressible fluids under the action of gravity is treated mathematically. In the basic state, one fluid is at rest below a horizontal planez= 0; and the other flows above in thexdirection, its speed varying slowly with the lateral co-ordinatey. The eigenvalue problem for normal modes is derived; its equation is a partial differential one, the co-ordinatesyandznot being separable. The problem is solved approximately by taking the modeslocallyas if the basic velocity were independent ofy, though the lateral wavenumber is allowed to vary slowly withy. This leads to an ordinary differential equation inywhich is solved by the JWKB method. Detailed calculations are made for a parabolic profile, representing the blowing of air over water in a wide channel, and for other profiles.