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.