Shallow water equations on a movable bottom

Nonlinear-dispersive shallow water equations on a rotating sphere

“…In the derivation we suppose that the longitudinal and latitudinal components of the velocity vector of a three-dimensional water flow do not depend on the radial coordinate r, and the radial velocity component depends on r linearly. Thus, deriving NLD-equations on a sphere, we have used the approach of Green-Naghdi [5] utilized previously for obtaining NLD-equations on a plane [6]. Note that if we apply another approach to the derivation of NLD-equations on a sphere, which is based on the expansion of the three-dimensional flow velocity potential over a small parameter β (the notation is taken from [6]), then it occurs that the model obtained above belongs to the class of so-called 'complete' NLD-models of the second hydrodynamic approximation.…”

“…In derivation of the motion equations we use the same assumptions as in the derivation of the NLD motion equations on a plane [6]: we assume that the 'horizontal' velocity vector component is constant in depth (the vector u does not depend on the coordinate s) and the 'vertical' component depends on s linearly: …”

“…In [6], a unified derivation of nonlinear-dispersive equations of Green-Naghdi, Zheleznyak-Pelinovskii, and Aleshkov was performed. These equations describe surface water waves without taking into account the Earth sphericity and rotation, but taking into account the mobility of the bottom surface.…”

“…These equations describe surface water waves without taking into account the Earth sphericity and rotation, but taking into account the mobility of the bottom surface. In the present paper we apply the approach developed in [6] for the derivation of shallow water NLDequations on a rotating sphere. …”

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

“…into which the NLD-models from the papers [12,13,14] and some other NLD-models can be transformed after linearization. Finite-difference approximations of these linear equations are used to obtain necessary stability conditions of nonlinear difference schemes, and also to research their dissipative and dispersive properties.…”