An explicit potential-vorticity-conserving approach to modelling nonlinear internal gravity waves

Abstract

This paper discusses a potential-vorticity-conserving approach to modelling nonlinear internal gravity waves in a rotating Boussinesq fluid. The focus of the work is on the pseudo-plane motion (motion in the x, z-plane), for which we present a broad range of numerical results. In this case there are two material coordinates, the density and the y-component of the velocity in the inertial frame of reference, which are related to the x and z displacements of fluid particles relative to a reference configuration. The amount of potential vorticity within a fluid region bounded by isosurfaces of these material coordinates is proportional to the area within this region, and is therefore conserved as well. Two new potentials, defined in terms of the displacements and combining the vorticity and density fields, are introduced as new dependent variables. These potentials entirely govern the dynamics of internal gravity waves for the linearized system when the basic state has uniform potential vorticity. The final system of equations consists of three prognostic equations (for the potential vorticity and the Laplacians of the two potentials) and one diagnostic equation, of Monge–Ampère type, for a third potential. This diagnostic equation arises from the nonlinear definition of potential vorticity. The ellipticity of the Monge–Ampère equation implies both inertial and static stability. In three dimensions, the three potentials form a vector, whose (three-dimensional) Laplacian is equal to the vorticity plus the gradient of the perturbation density.

Numerical simulations are carried out using a novel algorithm which directly evolves the potential vorticity, in a Lagrangian manner (following fluid particles), without diffusion. We present results which emphasize the way in which potential vorticity anomalies modify the characteristics of internal gravity waves, e.g. the propagation of internal wave packets, including reflection, refraction, and amplification. We also show how potential vorticity anomalies may generate internal gravity waves, along with the subsequent ‘geostrophic adjustment’ of the flow to a ‘balanced’ wave-less state. These examples, and the straightforward extension of the theoretical and numerical approach to three dimensions, point to a direct and accurate means to elucidate the role of potential vorticity in internal gravity wave interactions. As such, this approach may help a better understanding of the observed characteristics of internal gravity waves in the oceans.