Symmetry group analysis of the shallow water and semi-geostrophic equations

Abstract

The two-dimensional shallow water equations and their semi-geostrophic approximation that arise in meteorology and oceanography are analysed from the point of view of symmetry groups theory. A complete classification of their associated classical symmetries, potential symmetries, variational symmetries and conservation laws is found. The semi-geostrophic equations are found to lack conservation of angular momentum. We also show how the particle relabelling symmetry can be used to rewrite the semi-geostrophic equations in such a way that a well-defined formal series solution, smooth only in time, may be carried out. We show that such solutions are in the form of an 'infinite linear cascade'.

A method of determining the maximum height of localized circular waves in the proximity of shallow water

“…При цьому виникає низка фізичних та географічних параметрів, зокрема: форма початкового зміщення вільної поверхні, форми гра-ниць берегів та споруд, батиметрія дна, коефіцієнт шероховатості донної поверхні. Для розрахунку хвиль цунамі, як правило, використовують різні модифікації скінченно-різницевої схеми Мак-Кормака [6,7]. Ви-користання чисельних підходів призводить до низки проблем, пов'язаних з виникненням обчислювальних нестійкостей.…”

Multi-symplectic formulation of near-local Hamiltonian balanced models

“…is conserved following particles (see Bîlǎ et al (2006) for a thorough analysis). The total energy of a blob of fluid that occupies D at t = 0 is…”

“…(This generalizes the conservation of a single 2-form in canonical Hamiltonian dynamics.) The MS structural conservation law can be exploited to analyse stability (see Bridges (1997) for details), but in the context of balanced models, we focus on the scalar conservation laws that arise from it, namely conservation of Lagrangian momentum, energy and potential vorticity. Another advantage of the MS structure is that when it is preserved by a discretization, the resulting finite-difference methods normally have extremely good stability properties (see Bridges & Reich (2006) and Ascher & McLachlan (2005) for details).…”

Shallow water equations on a rotating attracting sphere 2. Simple stationary waves and sound characteristics

“…The system of equations written in a noninertial system of coordinates rotating together with the sphere at constant angular velocity Ω 0 has the form [3] Dv = w 2 cot θ + r 0 w cos θ + (1/4)r 2 0 sin θ cos θ − f 0 h θ , where D = ∂ t + v ∂ θ +(sin θ) −1 w ∂ ϕ is the total derivative over the surface of the sphere. Equations (1.1) are written in spherical coordinates: 0 < θ < π is the latitude, 0 ϕ < 2π is the longitude; v and w are the meridional and longitudinal velocity components, and h > 0 is the depth of the layer.…”