Hierarchy of nonlinear models of the hydrodynamics of long surface waves

Abstract

224For numerical solution of problems of hydrody namics of long surface waves propagating in real reser voirs, it is advantageous to use models in which the region of applicability is associated with the character istic scales of the wave process [1,2]. An expansion of the circle of actual problems related to the develop ment of coastal territories stimulates the creation of new models, and this fact results in variety of the sys tems of differential equations corresponding to these models due to a variety of methods and simplifying assumptions. Among these differential equations, there are also some that formally approximate the ini tial problem without providing an adequate descrip tion of the physical process or are inconvenient for numerical implementation.The purpose of this study is formation of a uniform approach for constructing long wave approximations in order to provide a hierarchical chain of shallow water equations of the first and second approximations having a succession of physically substantial proper ties. Here we continue the recent investigations in [3][4][5][6]. It is necessary to acknowledge [7-10] and others as the first studies on this theme.The derivation of shallow water models taking into account the dispersion is based on the Euler equations for an ideal incompressible fluid on a rotating sphere, the mobility of the bottom surface being taken into account, while further passage along the hierarchy from completely nonlinear equations with dispersion towards simplifications proceeds with inheritance of the most important properties, in particular, the laws of conservation. We succeeded in writing the obtained nonlinear dispersive (NLD) equations both on the plane and on the sphere in a universal compact form, which structurally coincides with the set of gas dynamic equations. EULER'S EQUATIONS IN THE THIN LAYER APPROXIMATIONThe spherical system of coordinates Oλθr is used with the origin at the center of a sphere of radius R rotating with a constant velocity Ω. We designated the longitude by λ, and the addition to the latitude ϕ < ϕ < we denoted by θ = -ϕ, while r is the radial coordinate. It is assumed that the water layer is limited from below by an impenetrable mobile bottom r = Rh(λ, θ, t) and from above by a free surface r = R + η(λ, θ, t). As external forces, we considered only the force of Newtonian attraction g directed to the center of a rotating sphere. Considering the thickness H = η + h of the water layer as small in comparison with R, we assume that the value of g = |g| and the water density ρ are constant in the entire layer, ρ ≡ 1. The mathe matical models of the long wave hydrodynamics are derived here from Euler's equations written with sin gling out the radial direction:(1)where the vectors U = (U 1 , U 2 ) = ( , ) and V = (V 1 , V 2 ) are composed correspondingly from the contravariant and covariant components of the "horizontal" com ponent of the velocity vector; in this case, V 1 = (Ω + U 1 )r 2 sin 2 θ, V 2 = r 2 U 2 , the radial velocity component is designated by ...

Long wave interaction with a partially immersed body. Part I: Mathematical models

“…It is believed that Bouss models, such as well-known classical Peregrine [77] or Nwogu [75] systems, might be used for tsunami propagation far from the coasts, where the flow cannot be considered to be dominated by nonlinear effects solely as in the coastal areas [21]. However, some simplifications is achieved with respect to SGN model since the nonlinear effects in the dispersive terms are neglected [90]. Currently Boussinesq-type models are also used to simulate the wave run-up on the coasts [26,31,84].…”

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

“…The differences appear in the expression for the kinetic energy and in the calculations of the pressure terms (6). We provide the single form of the equations of the entire hierarchical chain of long-wave models in spherical and planar geometries [3], that made it possible to construct the uniform numerical algorithm for them.…”

“…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.…”

“…In [3,4] with using the unified approach to a construction of long-wave approximations, the hierarchy of hydrodynamic models, which possess the physically meaningful properties inherited, has been built. The hierarchical chains of shallow water models of first-and secondorder long-wave approximation enclosed in each other are constructed on a rotating sphere and on a plane.…”

Numerical Modelling of Surface Waves in the Framework of the Shallow Water Model

“…Nowadays, the methods based on the hierarchy of embedded mathematical models [1] and computational algorithms are widely used for the numerical solution of the problems on propagation of long surface waves in large water areas and their interaction with coasts. The nonlinear shallow water models hold a central position in such a hierarchy.…”