Effect of wind and water shear on wave instabilities

The effect of vorticity or shear on the inviscid evolution and instability of surface waves is discussed. The instability of a flat surface is considered in detail. For the case of shear in the water, the instability is examined by means of Howard's semi-circle theorem, which is shown to elucidate the qualitative nature of the instability domain. This can then be determined numerically without recourse to analysis of the critical layer. The dependence of growth rate on the air profile for the case of wind blowing over water at rest is similarly considered. For bounded laminar profiles, the results are found to be sensitive to the profile shape, and Miles's approximate formula is found to be inaccurate. On the other hand, for unbounded logarithmic profiles, the growth rates are relatively insensitive to the profile shape. However, there is still disagreement between numerical calculations and Miles's formula, which overestimates the growth rate by a factor of two. A simple model is presented for the case of shear in both water and air