Particle Methods for Geophysical Flow on the Sphere.

Abstract

We present a Lagrangian Particle-Panel Method (LPPM) for geophysical fluid flow on a rotating sphere motivated by problems in atmosphere and ocean dynamics. We focus here on the barotropic vorticity equation and 2D passive scalar advection, as a step towards the development of a new dynamical core for global circulation models.

The LPPM method employs the Lagrangian form of the equations of motion. The flow map is discretized as a set of Lagrangian particles and panels. Particle velocity is computed by applying a midpoint rule/point vortex approximation to the Biot-Savart integral with quadrature weights determined by the panel areas. We consider several discretizations of the sphere including the cubed sphere mesh, icosahedral triangles, and spherical Voronoi tesselations. The ordinary differential equations for particle motion are integrated by the fourth order Runge-Kutta method. Mesh distortion is addressed using a combination of adaptive mesh refinement (AMR) and a new Lagrangian remeshing procedure. In contrast with Eulerian schemes, the LPPM method avoids explicit discretization of the advective derivative. In the case of passive scalar advection, LPPM preserves tracer ranges and both linear and nonlinear tracer correlations exactly.

We present results for the barotropic vorticity equation applied to several test cases including solid body rotation, Rossby-Haurwitz waves, Gaussian vortices, jet streams, and a model for the breakdown of the polar vortex during sudden stratospheric warming events. The combination of AMR and remeshing enables the LPPM scheme to efficiently resolve thin fronts and filaments that develop in the vorticity distribution. We validate the accuracy of LPPM by comparing with results obtained using the Eulerian based Lin-Rood advection scheme. We examine how energy and enstrophy conservation in the LPPM scheme are affected by the time step and spatial discretization. We conclude with a discussion of how the method may be extended to the shallow water equations.