Some instability issues relating to flow past complex terrain

Recent work on flow past a two-peak terrain profile shows that the wave pattern exhibits hydrodynamic instability at mountain heights far smaller than those needed to produce wave overturning at steady state. Here we explore some of the implications of this result for the problem of nonlinear flow past complex terrain.

We begin by mapping the parameter-space dependence for a test problem consisting of flow past a series of equal-height and evenly spaced Gaussian peaks. Instability is found even for a single peak, with unstable modes first appearing at height Nh/U = 0.6. However, as peaks are added to the profile the critical height for instability rapidly decreases. For two peaks the threshold height is Nh/U = 0.375, while for three peaks instability occurs at Nh/U = 0.275. In all cases the growth rate of the instability increases with increasing mountain height.

The finite-amplitude evolution is then examined by allowing the instability mode to grow in the context of nonlinear numerical model calculations. It is shown that the nonlinear evolution leads eventually to wave overturning and turbulence, with the preferred location for wave breaking being the lee slope of the last peak. However, the wave breaking is in all cases highly transient and occurs at various locations in the wave pattern at quasi-periodic intervals. The surface drag is also highly transient but is on average considerably larger than that observed for the corresponding steady-state problem.

Finally, the result that increased terrain complexity leads to enhanced wave breaking in some cases has been noted in previous studies. We end by revisiting those studies and showing that the enhancement in wave breaking results from the instability mode studied here.