Generation of ripples by shear - flow instability

We consider the linear stability of a monotonically decreasing, concave shear flow beneath a free surface with gravity and capillarity. An archetypical example, with current U = U0 exp(z/h), is solved analytically and studied in detail. Comparison with several other smooth velocity profiles indicates that the exponential profile is representative of all smooth monotonic concave velocity profiles. The modal instability is due to a resonance between gravity-capillary waves at the free surface, and a vorticial disturbance concentrated at a single sub-surface critical layer. The gravity-capillary waves participating in this resonance have negative intrinsic phase speed, but are Doppler shifted so that their actual phase speed is positive, and matches the speed of the base-state current at the critical level. The Reynolds stresses of an exponentially growing wave accelerate the below-trough Eulerian mean flow in the same direction as the base-state shear flow. The Rayleigh inflexion-point criterion does not apply in this situation because the unidirectional below-trough acceleration is balanced by recoil in the region above the troughs. Thus the total momentum of the fluid is unchanged by exponential growth of an unstable mode. Ripples are waves whose length is close to the gravity-capillary transition (wavelength 1.7cm), at which waves in still water have minimum phase speed. Ripples with negative intrinsic phase speed are most easily Doppler shifted so that their actual phase speed is positive and satisfies critical-layer resonance condition required by this instability. Growth is rapid: for typical ocean wind-drift layers the e-folding time of Doppler-shifted ripples is a fraction of a second.