Nonlinear-dispersive shallow water equations on a rotating sphere

Abstract

Nonlinear-dispersive shallow water equations on a sphere are obtained. These equations can be used in simulation of large-scale propagation of long waves in the problems of atmosphere and ocean dynamics taking into account the Earth rotation and wave dispersion.

New Algorithm for Numerical Simulation of Surface Waves Within the Framework of the Full Nonlinear Dispersive Model

“…The full nonlinear dispersion (FNLD-) model on a rotating sphere based on the depthaveraged velocity was obtained in 2010 [1]. The paper [7] shows that the FNLD-model of [1] can be derived without any assumptions about the potentiality of the original 3D-flow, and that its defining equations, unlike [2], can be written in the quasi conservative form of mass and momentum balances.…”

“…The paper [7] shows that the FNLD-model of [1] can be derived without any assumptions about the potentiality of the original 3D-flow, and that its defining equations, unlike [2], can be written in the quasi conservative form of mass and momentum balances. Additionally, this model has the balance equation of total energy agreed with a similar equation of 3D-model, that confirms not only the physical consistency of the FNLD-model, but also allows an additional control in the calculations.…”

“…Additionally, this model has the balance equation of total energy agreed with a similar equation of 3D-model, that confirms not only the physical consistency of the FNLD-model, but also allows an additional control in the calculations. The equations of our FNLD-model on a rotating attracting sphere have the following form [1]:…”

“…The interest of engineers is caused by the growing demands of practice in regard to the protection and development of sea coasts, whereas theoreticians have found a lot of gaps in the justification of both the models and numerical methods for solving the respective differential equations. Among the recently published papers devoted to the development of nonlinear dispersive (NLD) hydrodynamic models, we note studies [1,2,3] which take into account Earth's sphericity and its rotation. Such complication of the NLD-models is necessary for adequate description of the surface waves behaviour in the ocean and along the coastal zone.…”

“…The purpose of this paper is to present new numerical method for solving the NLD-equations on a rotating sphere [1]. The main idea lies in special splitting of the problem allowing the successive solution of two less difficult tasks at each time step.…”