Resonant instability in steady mountain waves: nonconstant background states and nonhydrostatic effects

Mesoscale flow over topography is responsible for phenomena such as clear-air turbulence and downslope windstorms, which may be hazardous to both human life and property. These features are associated with the breaking of mountain waves, a consequence of a shear instability in which the Richardson number decreases below the critical value of 0.25. From a historical standpoint, wave breaking has been associated with the nondimensional mountain height (Nh/U) exceeding some critical value. However, recent work shows that breaking may also occur at smaller mountain heights, due to the onset and growth of nonlinear wave-wave instabilities. In a companion study, a Newton solver is developed for finding nonlinear mountain-wave solutions in a general 2D background flow, including both rotation and nonhydrostatic effects. Here the solver is extended to examine stability of the solutions, with a focus on wave-wave instability at subcritical mountain heights. The instability is explored over a wide range of parameter space and for various profiles of shear and static stability. For constant background N and U the dimensional growth rates of the instability are shown to increase with decreasing horizontal scale, maximizing in the most nonhydrostatic cases with an e-folding time on the order of one hour.