The interaction of a wave with a vortex in 2-D rotating shallow water

We explore the interaction between a fast, inertio-gravity wave and a slow, balanced vortex in a rotating shallow water system, extending the one-dimensional study of Kuo and Polvani (1999). We are interested in describing both the time-dependent state, as the wave travels through vortex, and the long-time limit of the system, after the wave has radiated away to infinity. We focus primarily on the case where the wavelength is of the same order as the size of the vortex, as this regime cannot be treated with the classical tools of asymptotic analysis.

Our approach is based on a physical space description rather than on a modal decomposition, and therefore allows an examination of the fully nonlinear regime, when the wave amplitude is comparable to the vortex strength. One aim is to establish whether the non-interaction exhibited by the long-time limit of the 1-D system is also a feature of the 2-D system. Another is to describe in detail the nature of the vortex deformation that occurs during (and after) the transient interaction.

To address these questions we compute high resolution, numerical solutions in a rectangular domain on the f-plane, and investigate the interaction in various regimes. A body forced derived from a time-dependent potential is used to generate a coherent pulse of inertio-gravity waves, which subsequently impinges on a balanced vortex. Two cases are considered, one in which the vortex Rossby number is small and where the interaction is weak, the other in which the vortex Rossby number is of order one and the interaction can be stronger. In each case the Froude number is of order one and the inertio-gravity wave amplitude is varied as an external parameter. To eliminate shock formation at large wave amplitude, we use a modified shallow water system (Buhler, 1998) that prevents nonlinear steepening of the waves.