Reflection of capillary-gravity waves by surface convection domains

Abstract

The detailed investigation of capillary waves is associated with the search for physical mechanisms of the formation of signals stipulated by the topography [1], internal waves [2], and other processes occurring in the ocean interior. We confront these problems, e.g., in the analysis of radar-image patterns for free ocean surface [3]. A review of the present state of the art for the linear and nonlinear theories of gravity-capillary waves in the vicinity of the phase-velocity minimum is given in [4], and the effect of viscosity on the damping and generation of short waves was considered in [5]. Under actual conditions, the temperature and concentration of substances on the ocean surface is not constant. Near-surface convection processes produce gradients of the surface-tension coefficient, which, in turn, affects the short-wave sea-way [6]. The interaction of waves with regular structures arising as a result of near-surface convection or of rainfalls is of particular interest [3]. In the present paper, we have constructed, for the first time, a model for the propagation of capillary-gravity waves in a viscous temperature-inhomogeneous medium with allowance for the corresponding near-surface boundary layers. We consider the transformation of a surface wave of frequency ω , which propagates from the left to the domain x  [0, D ] containing N identical cells of size L , D = NL , with a quasi-stationary temperature distribution. In the absence of waves, the water surface is assumed to be planar: z = 0, z being the vertical axis aligned oppositely to the gravity-force vector g . The fluid-surface temperature that determines the kinematic viscosity ν ( x ) = ν ( T ( x )) and the surface-tension coefficient α ( x ) = α ( T ( x )) is given by the spectral expansion