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

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.

Keywords

Bibliographic record

BibTeX Citation

@article{Khakimzyanov2018dispersiveshallow,
  author = {Khakimzyanov, G. and Dutykh, D. and Fedotova, Z.},
  title = {Dispersive shallow water wave modelling. Part III: Model derivation on a globally spherical geometry},
  journal = {Commun. Comput. Phys.},
  year = {2018},
  volume = {23},
  number = {2},
  pages = {315--360},
  doi = {10.4208/cicp.OA-2016-0179c},
  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.},
  keywords = {shallow water wave modeling, Boussinesq regime, dispersive velocity correction, Euler equations on a sphere, nonlinear dispersive waves, motion on a sphere, spherical geometry, flow on sphere, long wave models, long wave approximation, depth-averaged models, dispersive shallow water waves, weakly nonlinear models}
}