Symmetries and exact solutions of the rotating shallow-water equations

Lie symmetry analysis is applied to study the non-linear rotating shallow-water equations. The 9-dimensional Lie algebra of point symmetries admitted by the model is found. It is shown that the rotating shallow-water equations can be transformed to the classical shallow-water model. The derived symmetries are used to generate new exact solutions of the rotating shallow-water equations. In particular, a new class of time-periodic solutions with quasi-closed particle trajectories is constructed and studied. The symmetry reduction method is also used to obtain some invariant solutions of the model. Examples of these solutions are presented with a brief physical interpretation.

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Rotating Shallow-Water Models with Full Coriolis Force

10.1093/oso/9780198804338.003.0016 ◽

2018 ◽

Author(s):

Vladimir Zeitlin

Keyword(s):

Shallow Water ◽

Coriolis Force ◽

Shallow Water Equations ◽

Vertical Component ◽

Tangent Plane ◽

Water Model ◽

Shallow Water Models ◽

Water Models ◽

Rotating Shallow Water ◽

Rotating Shallow Water Equations

The derivation of the rotating shallow-water model by vertical averaging is carried on in the tangent plane approximation without neglecting the vertical component of the Coriolis force, and contributions of the vertical component of velocity in its horizontal component (‘non-traditional’ terms), leading to one- and two-layer ‘non-traditional’ rotating shallow-water models. A similar approach on the whole sphere encounters difficulties with conservation of angular momentum. Consistent ‘non-traditional’ rotating shallow-water equations in this case are obtained from the variational principle, which is first formulated for full primitive equations. It is shown that columnar motion hypothesis should be replaced by solid-angle motion one on the sphere. Two-layer non-traditional rotating shallow-water equations are used to analyse inertial instability of jets and compare the results with Chapter 10. It is shown that non-traditional terms can increase the growth rates up to 30% in some configurations and can also change the structure of the unstable modes.

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Instabilities of coupled density fronts and their nonlinear evolution in the two-layer rotating shallow-water model: influence of the lower layer and of the topography

Journal of Fluid Mechanics ◽

10.1017/jfm.2012.556 ◽

2013 ◽

Vol 716 ◽

pp. 528-565 ◽

Cited By ~ 6

Author(s):

Bruno Ribstein ◽

Vladimir Zeitlin

Keyword(s):

Shallow Water ◽

Nonlinear Evolution ◽

Lower Layer ◽

Phase Locking ◽

Water Model ◽

Shallow Water Model ◽

Wave Instability ◽

Rotating Shallow Water ◽

New Generation ◽

Rotating Shallow Water Equations

AbstractWe undertake a detailed analysis of linear stability of geostrophically balanced double density fronts in the framework of the two-layer rotating shallow-water model on the $f$-plane with topography, the latter being represented by an escarpment beneath the fronts. We use the pseudospectral collocation method to identify and quantify different kinds of instabilities resulting from phase locking and resonances of frontal, Rossby, Poincaré and topographic waves. A swap in the leading long-wave instability from the classical barotropic form, resulting from the resonance of two frontal waves, to a baroclinic form, resulting from the resonance of Rossby and frontal waves, takes place with decreasing depth of the lower layer. Nonlinear development and saturation of these instabilities, and of an instability of topographic origin, resulting from the resonance of frontal and topographic waves, are studied and compared with the help of a new-generation well-balanced finite-volume code for multilayer rotating shallow-water equations. The results of the saturation for different instabilities are shown to produce very different secondary coherent structures. The influence of the topography on these processes is highlighted.

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Energy–enstrophy conserving compatible finite element schemes for the rotating shallow water equations with slip boundary conditions

Journal of Computational Physics ◽

10.1016/j.jcp.2018.06.071 ◽

2018 ◽

Vol 373 ◽

pp. 171-187 ◽

Cited By ~ 7

Author(s):

W. Bauer ◽

C.J. Cotter

Keyword(s):

Finite Element ◽

Boundary Conditions ◽

Shallow Water ◽

Shallow Water Equations ◽

Slip Boundary Conditions ◽

Slip Boundary ◽

Rotating Shallow Water ◽

Finite Element Schemes ◽

Rotating Shallow Water Equations

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Thermal versus isothermal rotating shallow water equations: comparison of dynamical processes by simulations with a novel well-balanced central-upwind scheme

Geophysical & Astrophysical Fluid Dynamics ◽

10.1080/03091929.2020.1774876 ◽

2020 ◽

pp. 1-30

Author(s):

Alexander Kurganov ◽

Yongle Liu ◽

Vladimir Zeitlin

Keyword(s):

Shallow Water ◽

Shallow Water Equations ◽

Upwind Scheme ◽

Dynamical Processes ◽

Rotating Shallow Water ◽

Rotating Shallow Water Equations

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Elliptical vortices and integrable Hamiltonian dynamics of the rotating shallow-water equations

Journal of Fluid Mechanics ◽

10.1017/s0022112091000162 ◽

1991 ◽

Vol 227 ◽

pp. 393-406 ◽

Cited By ~ 20

Author(s):

Darryl D. Holm

Keyword(s):

Shallow Water ◽

Shallow Water Equations ◽

Hamiltonian Dynamics ◽

Relative Equilibria ◽

Hamiltonian Mechanics ◽

Lagrangian Coordinates ◽

Vortex Solutions ◽

Rotating Shallow Water ◽

Rotating Shallow Water Equations

The problem of the dynamics of elliptical-vortex solutions of the rotating shallow-water equations is solved in Lagrangian coordinates using methods of Hamiltonian mechanics. All such solutions are shown to be quasi-periodic by reducing the problem to quadratures in terms of physically meaningful variables. All of the relative equilibria - including the well-known rodon solution - are shown to be orbitally Lyapunov stable to perturbations in the class of elliptical-vortex solutions.

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