The rotating Green-Naghdi shallow water model: a simple framework for the investigation of mesoscale atmospheric and oceanic flows

There is a long history in geophysical fluid dynamics

of the use of the rotating shallow water equations (rSWE)

to model and understand a wide range of observed phenomena in the atmosphere

and oceans. However, the rSWE have the well-known property that

finite-amplitude gravity waves have a strong tendency to steepen and break.

In many observed mesoscale phenomena, by contrast, wave steepening is opposed

by dispersive

effects, often leading to the formation of stable

nonlinear (e.g. solitary) waves.

The Green-Naghdi (GN) equations are an extended shallow water set with many

appealing properties, such as the parcel-wise

conservation of a potential vorticity, which include dispersive effects and

consequently support stable nonlinear gravity wave motion including solitary

waves.

Here a new pseudo-spectral algorithm is introduced to solve the GN equations

numerically on a doubly periodic f-plane. The following questions,

concerning

the interaction of nonlinear gravity waves and balanced motion,

are then investigated: to what extent do solitary waves and vortices interact?

what happens when the Froude number locally exceeds unity in an initially

`balanced' flow? what controls the generation of nonlinear waves

during transcritical flow (Froude number near unity) over an obstacle of finite

height? It is argued that the rotating GN equations are the simplest set

to allow the above class of problems to be addressed.