The Hamiltonian particle‐mesh method for the spherical shallow water equations

Abstract

The Hamiltonian particle-mesh (HPM) method is generalized to the spherical shallow-water equations, utilizing constrained particle dynamics on the sphere and Merilees pseudospectral method (complexity O(J 2 log J ) in the latitudinal gridsize) to approximate the inverse modified Helmholtz regularization operator. The time step for the explicit, symplectic integrator depends only on a uniform physical smoothing length.

Statistical relevance of vorticity conservation in the Hamiltonian particle-mesh method

“…The approach of Section 4 may also be useful to develop a statistical theory for the HPM method for the shallow water equations [10,12]. In this case each particle has a fixed mass M k instead of PV, and both position and momentum variables define the phase space: (X k , P k ).…”

“…The Hamiltonian particle-mesh (HPM) method was originally proposed in the context of rotating shallow water flow in periodic geometry in [10] and extended to other physical settings in [6,12,7,27]. The fluid is discretized on a finite set of Lagrangian particles that transport the mass of the fluid and persist during the flow evolution.…”

Statistical mechanics of Arakawa’s discretizations

“…are conserved, of which the most important-the second moment of vorticity C 2 -will be denoted by Z 5) and will henceforth be referred to as the enstrophy.…”

On the Rate of Convergence of the Hamiltonian Particle-Mesh Method

“…It was originally proposed in the context of the shallow water equations by Frank, Gottwald and Reich [6], and tested on a variety of two-dimensional geophysical flow problems [3,4,5]. Moreover, the HPM method was shown to be convergent [10,11] as the number of particles N tends to infinity.…”