In this paper we discuss an adaptive mesh refinement algorithm for modeling a low Mach number approximation to atmospheric flow. The techniques described in this paper were developed to solve the incompressible Navier-Stokes equations, but have been extended to model atmospheric flow as governed by the anelastic approximation and a relatively simple set of equations for conservation of mass and momentum. In this simple case we omit temperature from the equations, and compute density from the continuity equation rather than from the equation of state. We solve these equations on a composite (i.e. multilevel) grid structure, which allows for different degrees of refinement in different regions of the flow. The method is based on a projection formulation in which we first solve advection-diffusion equations to predict intermediate velocities, and then project these velocities onto a space of vector fields satisfying the divergence constraint. The advection-diffusion step uses a specialized second-order upwind method for differencing the nonlinear advection terms that provides a robust treatment of these terms; the diffusion step uses a Crank-Nicholson discretization with standard spatial approximations.
Our approach to adaptive refinement uses a nested hierarchy of grids with simultaneous refinement of the grids in both space and time. The integration algorithm on the grid hierarchy is a recursive procedure in which coarse grids are advanced, finer grids are advanced multiple steps to reach the same time as the coarse grids and the fine and coarse grid data are then synchronized. Second-order accuracy of the method is demonstrated elsewhere for incompressible flow; here we show results using the anelastic approximation for three-dimensional calculations of a hot gas released into a wind-sheared adiabatically stratified atmosphere.