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

Gayaz Khakimzyanov (1) , Denys Dutykh (2) , Zinaida Fedotova (1)

1 Institute of Computational Technologies
2 LAMA - Laboratoire de Mathématiques

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.

Mots clés: motion on a sphere; long wave approximation; nonlinear dispersive waves; spherical geometry; flow on sphere

Domaines: Mécanique des fluides [physics.class-ph]; Physique Atmosphérique et Océanique [physics.ao-ph]; Dynamique des Fluides [physics.flu-dyn]; Physique Classique [physics.class-ph]; Physique Numérique [physics.comp-ph]; Géophysique [physics.geo-ph]; Physique mathématique [math-ph]; Systèmes Solubles et Intégrables [nlin.SI]; Formation de Structures et Solitons [nlin.PS]; Equations aux dérivées partielles [math.AP]

Gayaz Khakimzyanov, Denys Dutykh, Zinaida Fedotova. Dispersive shallow water wave modelling. Part III: Model derivation on a globally spherical geometry. 2017. ⟨hal-01552229v1⟩