Short surface waves on surface shear

Abstract

The problem of short wind waves propagating on surface wind drift is considered here. Under the assumption of small monochromatic surface waves on a steady and horizontally uniform surface shear of an inviscid fluid, the governing equation becomes the well-known Rayleigh instability equation. Perturbation solutions exist for the surface-wave problem; however, the conditions for these approximations are violated in the case of short wind waves on wind drift shear. As an alternative approach, the piecewise-linear approximation (PLA) is explored. A proof is given for the rate of convergence of the piecewise-linear approximation for solving the Rayleigh equation without limitations on boundary conditions. The artificial modes of the piecewise-linear flow system are also discussed. The method is numerically efficient and highly accurate. Applying this method, the linear instability of various boundary-layer flows is examined. Short waves propagating with surface shear-flows are stable, while it is possible for waves that are travelling against a shear current to become unstable when the surface speed of the shear is greater than the wavespeed in stagnant fluid. PLA is also applied to examine the applicability of other perturbation approaches to the problem of propagation of short waves on wind drift shear. It is found that the existing approximations cannot fit the whole range of short wind waves. To bridge the gap, new approximations are derived from an implicit form of the exact dispersion relation based upon the variational principle.