Interaction between counterpropagating Rossby waves and capillary waves in planar shear flows

Abstract

A counterintuitive destabilizing effect of the surface tension in planar wakes has been observed by Tammisola et al. [“Effect of surface tension on global modes of confined wake flows,” Phys. Fluids 23, 014108 (2011)] and Biancofiore et al. [“Direct numerical simulations of two-phase immiscible wakes,” Fluid Dyn. Res. 46, 041409 (2014)] by means of linear global analyses and direct numerical simulations, respectively. In the present study, we approximate the velocity profile of a wake flow through a piecewise broken-line profile and explain the presence of temporal unstable modes using an interfacial wave interaction perspective. With this perspective, we associate to each vorticity discontinuity an individual counterpropagating Rossby wave (RW), while the introduction of a finite amount of surface tension at the interface creates two capillary waves (CWs) which propagate with respect to the interface velocity with the same relative velocity but in opposite directions. The addition of the surface tension generates a new unstable mode, which is a Rossby-capillary mode, since it is due to the interaction between one RW and one CW. Furthermore, we capture the spatio-temporal evolution of the interacting four-waves system by means of an impulse response analysis. The spreading of the wavepacket, and consequently the absolute nature of the instability, is enhanced by a moderate surface tension, especially if the interface is located close to the faster edge of the broken-line wake profile. This can be explained by the influence of the surface tension on the group velocities of the waves, taken in isolation.