Instability waves on the air–sea interface

Abstract

We used a compound matrix method to integrate the Orr–Sommerfeld equation in an investigation of short instability waves (λ < 6 cm) on the coupled shear flow at the air–sea interface under suddenly imposed wind (a gust model). The method is robust and fast, so that the effects of external variables on growth rate could easily be explored. As expected from past theoretical studies, the growth rate proved sensitive to air and water viscosity, and to the curvature of the air velocity profile very close to the interface. Surface tension had less influence, growth rate increasing somewhat with decreasing surface tension. Maximum growth rate and minimum wave speed nearly coincided for some combinations of fluid properties, but not for others.The most important new finding is that, contrary to some past order of magnitude estimates made on theoretical grounds, the eigenfunctions at these short wavelengths are confined to a distance of the order of the viscous wave boundary-layer thickness from the interface. Correspondingly, the perturbation vorticity is high, the streamwise surface velocity perturbation in typical cases being five times the orbital velocity of free waves on an undisturbed water surface. The instability waves should therefore be thought of as fundamentally different flow structures from free waves: given their high vorticity, they are akin to incipient turbulent eddies. They may also be expected to break at a much lower steepness than free waves.

On the spatial linear growth of gravity-capillary water waves sheared by a laminar air flow

“…Subsequent studies incorporated some minor but also important additional factors such as surface current 4,5 and distributions of the velocity profiles. 5,6 These studies focused mainly on the application of a turbulent shear layer over a wavy surface in order to simulate the natural state of water wave growth. Comparison with the laboratory experiments of Larson and Wright 7 and Kawai 8 indicated a high probability that the wave growth can be predicted by linear instability theory in the initial stages.…”

Two-Dimensional Spatially Developing Mixing Layers

“…The solution of the OSE is accomplished in two parts: (1) the determination of the eigenvalue a using an iterative procedure, which yields the compound vector, and (2) the determination of the eigenfunction and its first three derivatives using the computed eigenvalue and the compound vector computed in part 1. The iterative procedure outlined by Wheless and Csanady [21] is used to compute an improved eigenvalue using information from three approximate eigenvalues such that the boundary conditions (given below) are satisfied to some predetermined level of accuracy. The iterative procedure used in this study is described in detail in the appendix.…”

“…Equation (25) states that the eigenfunction and its first three derivatives must be continuous at y = O. Wheless and Csanady [21] had determined such conditions in terms of the compound vector at the interface for a general system of two dissimilar fluids. For the simplified system of two similar fluids studied here, the following quadratic relationship in terms of the compound vector must be satisfied at the origin: The values for the eigenfunction and its first three derivatives at the origin are determined from Eq.…”

Simulation of Surface Instability at the Interface of Two Fluids

“…Miles's analysis did not include the shear flow in the water; it is known that the phase speeds of wind-generated gravity-capillary waves depend on the wind-induced drift in the water. Thus, the analysis was further improved by taking into account the shear flow in the water by Valenzuela [7], Kawai [8] and Wheless and Csanady [9].…”

“…He therefore concluded that the generation of the initial wavelets at the air-sea interface is caused by selective amplification of the most unstable waves in the coupled air-water shear flow. Wheless and Csanady [9] returned to the full formulation of the instability wave problem, the same as that in the investigations of Valenzuela [7] and Kawai [8], with focuses on studying the internal structure of the instability waves and extending the range of flow parameters explored. They developed a numerical technique to integrate the coupled OS equation based on the compound matrix method, a numerical technique that combines the inviscid and viscid solutions through the use of a Ricatti transformation.…”