Abstract

We develop a fluid dynamics solver for spherical domains based on the Lagrangian form of the equations of motion and a tree-structured mesh of fluid particles. Initial discretizations are based on the cubed sphere or icosahedral triangles, but these arrangements distort as the particles move with the fluid velocity. A remeshing scheme based on interpolation of the Lagrangian parameter is applied at regular intervals to maintain computational accuracy as the flow evolves. Ongoing work with adaptive mesh refinement is introduced. We present solutions of the advection equation with prescribed velocities from recent test cases (Nair and Lauritzen, 2010), and solutions of the barotropic vorticity equation where velocity is given by an approximate Biot-Savart integral and midpoint rule quadrature for test cases including Rossby-Haurwitz waves and Gaussian vortices . The discrete equations in the barotropic vorticity equation are those of point vortices on the sphere (e.g. Bogo molov, 1977). For the shallow water equations, we must also include the effects of point sources upon the fluid particles; we discuss the challenges posed by divergent flows in this Lagrangian context and strategies for evaluating nonlinear forcing terms on irregular meshes.