Stability of a velocity field tangential discontinuity in a three-layer density-stratified liquid with a moving middle layer

Abstract

A dispersion relation is analytically derived for gravitational waves in an ideal incompressible threelayer liquid with a free surface in the presence of a velocity field tangential discontinuity between the layers. The discontinuity results from the motion of the middle layer. The instability of the tangential discontinuity is shown to depend on the relative velocity of contacting layers, which, in turn, depends on the ratio of their densities. The closer the density ratio to unity, the lower the moving layer velocity causing instability. In the given case, instability involves internal waves arising at the second and third interfaces in accordance with the Kelvin–Helmholtz concept of instability development. Internal waves with wavelengths far exceeding the thickness of the middle layer are found to interact with each other. Surface waves only change their frequencies.