Resonant wave-wave instability in rotating and nonhydrostatic mountain waves

The analysis of mountain-wave dynamics is often cast in the steady-state framework. However, recent work has shown that steady-state mountain waves are often unstable, even for mountain heights far smaller than those needed for steady-state wave overturning. The growth rates of the instability are generally small but are nonetheless relevant on synoptic timescales.

The present study revisits the classical case of flow past an isolated 2D ridge, with the goal of examining stability for the steady states. The steady solutions are obtained using either Long's theory for f = 0 or else using a new iterative method (described in a companion study) for rotating cases. A wide range of parameters is considered, including nonhydrostatic flows and flows with significant background rotation. The control parameters for the flow are then the dynamical aspect ratio (U/NL), the Rossby number (U/fL) and the nondimensional mountain height (Nh/U).

Analysis of the steady-state solutions shows that instability occurs over a wide range of U/NL and U/fL. However, compared to previous work with multiple terrain peaks, the range of unstable mountain heights is much reduced. The unstable domain is largest for nonhydrostatic flows and remains significant for hydrostatic cases with large Rossby number. For smaller U/fL the instability effectively disappears, and solutions are stable almost to the point wave overturning.

As in the previous multi-peak studies, the mechanism for the instability is shown to be resonant triad interaction.