Dispersive shallow water wave modelling. Part III: Model derivation on a globally spherical geometry

TL;DR

This paper derives a family of long wave shallow water models on a spherical geometry, including a new fully nonlinear weakly dispersive base model and Boussinesq regime models, using a systematic derivation from Euler equations.

Contribution

It introduces a novel formulation of the Euler equations on a sphere and derives new dispersive shallow water models, expanding the theoretical framework for wave modeling on spherical surfaces.

Findings

Derived a fully nonlinear weakly dispersive base model.

Formulated weakly nonlinear Boussinesq models on a sphere.

Proposed a dispersive correction velocity variable with closure relation.

Abstract

The present article is the third part of a series of papers devoted to the shallow water wave modelling. In this part, we investigate the derivation of some long wave models on a deformed sphere. We propose first a suitable for our purposes formulation of the full Euler equations on a sphere. Then, by applying the depth-averaging procedure we derive first a new fully nonlinear weakly dispersive base model. After this step, we show how to obtain some weakly nonlinear models on the sphere in the so-called Boussinesq regime. We have to say that the proposed base model contains an additional velocity variable which has to be specified by a closure relation. Physically, it represents a dispersive correction to the velocity vector. So, the main outcome of our article should be rather considered as a whole family of long wave models.