A Simple Model of the Resonant Interaction between Vortex Rossby and Gravity Waves

Abstract

 A simple conceptual model of the resonant interaction in a typhoon-like vortex between vortex Rossby waves (VRWs) and gravity waves (GWs) that are caused by the VRWs is presented. It is well-known that the VRWs in the central region of the vortex can grow by the interaction with the GWs in the outer region, but a simple conceptual model for their interaction has not yet been proposed. The proposed conceptual model is based on the buoyancy-vorticity formulation (BV-thinking), and is different from that for the barotropic and baroclinic instabilities based on PV interactions (PV-thinking).

 We consider disturbances of the first baroclinic mode on a basic barotropic vortex. The disturbance vertical vorticity ζ of the VRW in the central region has a large amplitude on the upper and lower levels, whereas the disturbance buoyancy b and radial vorticity η of the GW have a large amplitude on the middle level. The central VRW propagates (relative to the fluid) anticyclonically, but moves cyclonically because of the strong cyclonic advection by the vortex. The outer cyclonically-propagating GW is also weakly advected cyclonically by the vortex. As a result, the counter-propagating VRW and GW (satisfying Rayleigh's condition) may be phase-locked with each other (satisfying Fjørtoft's condition).

 By the counter-propagation and phase-lock, the circulation around ζ of the VRW enhances b of the GW, which in turn enhances η. Simultaneously, the circulation around η of the GW enhances ζ of the VRW. As a result, the VRW and GW grow simultaneously.

 We analytically show the possibility of the resonant interaction, and numerically obtain the growing solution in the system linearized about the basic vortex.