Nonlinear-dispersive shallow water equations on a rotating sphere and conservation laws

Fedotova, Z. I.; Khakimzyanov, G. S.

doi:10.1134/s0021894414030043

Cited by 10 publications

(9 citation statements)

References 11 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Thus, the equations of the weakly nonlinear dispersion model obtained in [7] and the classical (nondispersive) shallow water model on a sphere have the same form as the equations of the FNLD-model on a sphere. The differences appear in the expression for the kinetic energy and in the calculations of the pressure terms (6).…”

Section: Nonlinear Dispersive Hydrodynamic Modelsmentioning

confidence: 92%

“…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.…”

Section: Nonlinear Dispersive Hydrodynamic Modelsmentioning

confidence: 99%

“…Functionsη andh are connected with η and h by relations −h = η 00 −h, η = η 00 +η and describe the bottom and the free surface not in the form of deviations (4) from the surface of a sphere of radius R, but the deviationsh andη from the undisturbed free boundary, as is usually done for solving shallow water equations in plane geometry. In this case the unperturbed free boundary at rest differs from spherical [7] and is described by the equation r = R + η 00 (θ), where…”

Section: The Numerical Algorithm For Nonlinear Dispersive Equations Omentioning

confidence: 99%

See 2 more Smart Citations

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

Fedotova¹,

Gusev²,

Khakimzyanov³

2016

Proceedings of the VII European Congress on Computational Methods in Applied Sciences and Engineering (ECCOMAS Congress 2016)

View full text Add to dashboard Cite

Section: Nonlinear Dispersive Hydrodynamic Modelsmentioning

confidence: 92%

Section: Nonlinear Dispersive Hydrodynamic Modelsmentioning

confidence: 99%

Section: The Numerical Algorithm For Nonlinear Dispersive Equations Omentioning

confidence: 99%

See 1 more Smart Citation

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

Fedotova¹,

Gusev²,

Khakimzyanov³

2016

Proceedings of the VII European Congress on Computational Methods in Applied Sciences and Engineering (ECCOMAS Congress 2016)

View full text Add to dashboard Cite

“…Here we continue the recent investigations in [3][4][5][6]. It is necessary to acknowledge [7][8][9][10] and others as the first studies on this theme.…”

mentioning

confidence: 86%

“…The dimensionless variables are defined from the relations where λ 0 = , t 0 = , and ω 0 = . Writing Euler's equations (1) in dimensionless variables and rejecting the terms of the order of O(ε), we come [5] to the thin layer approximation. Adding the dimensionless boundary conditions, we obtain a problem with small parameters α and μ, which is convenient for deriving the shallow water models.…”

Section: Euler's Equations In the Thin Layer Approximationmentioning

confidence: 99%

Hierarchy of nonlinear models of the hydrodynamics of long surface waves

2015

Self Cite

View full text Add to dashboard Cite

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 ...

show abstract