Thursday, 15 January 2004: 3:30 PM

The use of a cartesian terrain-intersecting grid in a Fourier-based solution of the Helmholtz problem of an implicit nonhydrostatic forecast model

Room 607

A fully- or semi-implicit nonhydrostatic model requires that a solution to
a Helmholtz equation be obtained at every time step. In the terrain-following
coordinates employed by most models the Helmholtz operator assumes a
general non-separable form, even when the model state is arbitrarily close
to one of stably stratified rest, and the direct application of efficient
Fourier transform methods is precluded. We examine a solution strategy in
which the Helmholtz forcing is first interpolated to a Cartesian terrain-
intersecting grid, so that the interior solution operator very closely
approximates a separable, symmetric and horizontally homogeneous operator
of algebraically simple and constant coefficients, which a Fourier method
then efficiently solves. The reconciliation of the original Neumann-type
boundary conditions over the terrain surface is then accomplished, to a
very close approximation at least, by the construction, via a surface
transform of the diagnosed Neumann residual from a preliminary trial solution,
of a corrective forcing. A second iteration of the Fourier solver will now
produce a solution in the Cartesian grid essentially consistent with the
terrain boundary condition, so that a further interpolation back to the
model's native grid achieves the original objective. By permitting the
vertical portion of the grid for the solver to differ from that of the model
we are free to choose from a much more general class of model vertical
coordinates than would be the case if the choice were to be dictated by the
requirement of numerical efficiency of the Helmholtz solver constrained to use
this same grid. Implications for efficient semi-Lagrangian modeling will be
discussed.

Supplementary URL: http://sgi62.ncep.noaa.gov:8080/jpurser/NWP.25.1.purser.pdf