The HPM method

The subject of our research is the development and understanding of a generalization of the symplectic method above, appropriate for full-scale geophysical fluid models. Broadly, we are interested in determining what kinds of discrete models are appropriate for long time ensemble simulations of climate and weather systems, to what extent the method used influences the results obtained, and to come up with a definition of "statistical accuracy" which takes into account these issues.

A central issue is the interplay between multiscale phenomena in geophysical flows.

The Hamiltonian Particle-Mesh (HPM) method is to the best of our knowledge the first symplectic method specifically implemented for large-scale GFD. The method is derived from a Lagrangian fluid description in which the fluid is seen as continuum of particles. In HPM, we choose a finite subset of particles, distribute the fluid mass over these discrete particles, and then approximate the density field by local averaging, in concept similar to the smoothing using in the Smoothed Particle Hydrodynamics (SPH) method. The averaging operator used is a global operator, and its realization is facilitated by the use of a uniform grid. The averager smooths fluctuations on a scale of about 2 grid cells.

The HPM method has the following properties:

• Hamiltonian structure: the method is symplectic, approximately conserving energy
• Vorticity conservation: the method satisfies a circulation theorem, implying conservation of vorticity and potential vorticity
• Balance preservation: the method conserves adiabatic invariants such as geostrophic and hydrostatic balances

Results

• Barotropic instability
• A two-layer baroclinic instability
• Shallow water equations on a sphere