A Resonant Instability of Steady Mountain Waves

An important nonlinear benchmark in our understanding of mountain waves are the exact steady solutions produced by Long's theory. These 2D flows are well-known to be unstable when the topographic heights cause the streamlines to be at, or very close to, overturning. A new mechanism for the instability of steady, hydrostatic mountain waves is identified through the analysis of the linear stability problem. When the topography has multiple peaks, it is shown that this criterion can be superceded by a resonant instability. In particular, for flow over two peaks, the threshold heights for instability are roughly half as high as required for overturning streamlines. The mechanism behind the instability is the parametric amplification of counter-propagating gravity waves. This instability mechanism identifies a new dynamical pathway to flow overturning, and the onset of turbulence.