Gerry Browning sent in the following post:
If a time dependent equation has a solution that grows exponentially in time, then that solution is very sensitive to errors in the initial condition, i.e., any error in the initial condition will cause an exponential deviation in time from the true solution.
Using a mathematical perturbation of smooth, large-scale atmospheric flow in the presence of vertical shear (jet stream), Browning and Kreiss (1984) showed that the hydrostatic equations can lead to unbounded exponential growth (ill posedness) and the non-hydrostatic equations can lead to rapid exponential growth on the time scale of a few hours when the wave length resolved by a numerical model approaches 10 km.
I have developed inviscid approximations of both systems for a doubly periodic domain that verifies these conclusions. The growth in the hydrostatic system becomes larger and larger in the matter of a few hours as the resolution increases while the growth in the non-hydrostatic system remains bounded, but is very rapid. The bounded, fast exponential growth and rapid cascade to smaller scales has also been seen in the NCAR WRF and Clark-Hall viscous, nonhydrostatic models (Lu et al., 2006), but at a slower pace because of the dissipation.
These analytical and numerical results raise a number of troubling issues. If current global atmospheric models continue to use the hydrostatic equations and increase their resolution while reducing their dissipation accordingly, the unbounded growth will start to appear. On the other hand, if non-hydrostatic models are used at these resolutions, the growth will be bounded, but extremely fast with the solution deviating very quickly from reality due to any errors in the initial data or numerical method.
Browning, G. and H.-O. Kreiss: Numerical problems connected with weather prediction. Progress and Supercomputing in Computational Fluid Dynamics, Birkhauser.
Lu, C., W. Hall, and S. Koch: High-resolution numerical simulation of gravity wave-induced turbulence in association with an upper-level jet system. American Meteorological Society 2006 Annual Meeting, 12th Conference on Aviation Range and Aerospace Meteorology.