#
A High-Resolution Adaptive Projection Method for
Regional Atmospheric Modeling

### Michael L. Welcome

### Abstract

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.