scholarly journals A Fast Monotone Discretization of the Rotating Shallow Water Equations

2021 ◽  
Author(s):  
Guillaume Roullet ◽  
Tugdual Gaillard
Keyword(s):  
Shallow Water ◽  
Shallow Water Equations ◽  
Rotating Shallow Water ◽  
Monotone Discretization ◽  
Rotating Shallow Water Equations

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

1991 ◽  
Vol 227 ◽  
pp. 393-406 ◽  
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.

scholarly journals Variational integrator for the rotating shallow‐water equations on the sphere

2019 ◽  
Vol 145 (720) ◽  
pp. 1070-1088 ◽  
Author(s):  
Rüdiger Brecht ◽  
Werner Bauer ◽  
Alexander Bihlo ◽  
François Gay‐Balmaz ◽  
Scott MacLachlan
Keyword(s):  
Shallow Water ◽  
Shallow Water Equations ◽  
Variational Integrator ◽  
Rotating Shallow Water ◽  
Rotating Shallow Water Equations

scholarly journals Symmetries and exact solutions of the rotating shallow-water equations

2009 ◽  
Vol 20 (5) ◽  
pp. 461-477 ◽  
Author(s):  
A. A. CHESNOKOV
Keyword(s):  
Exact Solutions ◽  
Shallow Water ◽  
Shallow Water Equations ◽  
Water Model ◽  
New Class ◽  
New Exact Solutions ◽  
Time Periodic Solutions ◽  
Rotating Shallow Water ◽  
Rotating Shallow Water Equations ◽  
Time Periodic

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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