A Simple Direct Solver for Long's Theory Flows

David J Muraki

In a 1953 Tellus article, R. Long discovered that a steady, nonlinear streamfunction for a special case of stratified, 2D flow over topography could be obtained by the solution of a linear Helmholtz equation. Surprisingly, there are still new lessons which can be found at the margins of this now textbook example of topographic wave generation.

For example, it is generally believed that the exact boundary conditions for Long's solution introduce a hidden (and perhaps, nonlinear) complication. This turns out to be false, as general solutions can be obtained by linear means by restating the theory as a linear integral equation. In this formulation, numerical solutions are obtained by a large NxN linear solve that can be computed (on a laptop computer) over a common range of flow parameters. It is also hoped that this completely linear formulation will address the open question of whether or not multiple steady states for a given topographic scenario might be found.

AMS Mountain Meteorology, June 2004: pdf overheads (2.3M)

Figures:

: linear theory for Lilly/Klemp 1979: critical overturning over a hydrostatic bell-shaped ridge; note the mismatch between the lowest streamline (red) & the topographic profile; nonlinear solution of Lilly/Klemp 1979 case from direct solver using 3 seconds of laptop time (plot of spectral coefficients); nonhydrostatic solution matching a case from Laprise/Peltier 1988 (slightly modified scan of original figure); flow corresponding to Lyra's analytical green's function