Vortex–wave interaction in a rotating stratified fluid: WKB simulations

Abstract

In this paper we present ray-tracing results on the interaction of inertia–gravity waves with the velocity field of a vortex in a rotating stratified fluid. We consider rays that interact with a Rankine-type vortex with a Gaussian vertical distribution of vertical vorticity. The rays are traced, solving the WKB equations in cylindrical coordinates for vortices with different aspect ratios. The interactions are governed by the value of $\hbox{\it Fr} R/\lambda$ where $\hbox{\it Fr}$ is the vortex Froude number, $R$ its radius, and $\lambda$ the incident wavelength. The Froude number is defined as $ {\hbox{\it Fr}}\,{=}\,U_{max}/(NR)$ with $U_{max}$ the maximum azimuthal velocity and $N$ the buoyancy frequency. When $\hbox{\it Fr} R/\lambda\,{>}\,1$, part of the incident wave field strongly decreases in wavelength while its energy is trapped. The vortex aspect ratio, $H/R$, determines which part of this incident wave field is trapped, and where its energy accumulates in the vortex. Increasing values of $\hbox{\it Fr} R/\lambda$ are shown to be associated with a narrowing of the trapping region and an increase of the energy amplification of trapped rays. In the inviscid approximation, the infinite energy amplification predicted for unidirectional flows is retrieved in the limit $\hbox{\it Fr} R/\lambda \,{\rightarrow}\, \infty$. When viscous damping is taken into account, the maximal amplification of the wave energy becomes a function of $\hbox{\it Fr} R/\lambda$ and a Reynolds number, $Re_{wave}\,{=}\,\sqrt{U_L^2+U_H^2}/\nu k^2$, with $U_L$ and $U_H$ typical values of the shear in, respectively, the radial and vertical directions; the kinematic viscosity is $\nu$, and the wavenumber $k$, for the incident waves. In a sequel paper, we compare WKB simulations with experimental results.