Quasi-3D Nearshore Circulation Equations: a CL-Vortex Force Formulation

INTRODUCTION Recently much attention has been paid to different formulations of surface wave force in wave-driven ocean and coastal circulations (e.g., McWilliams et al., 2004, Mellor, 2003, Smith, 2006 and others). Basically, the analytical expressions for surface wave force and wave-current interaction can be classified into two types. One is the classical wave 'radiation stress' concept presented by Longuet-Higgins and Stewart (1962, 1964) and many others in depth-integrated and short wave-averaged equations. Mellor (2003) recently used the same concept to derive short wave-averaged 3-D equations with a depth-dependent wave-induced force. A direct application of this kind of depth-dependent wave-induced force was conducted by Xia et al. (2004) who related the vertical variation of current to the vertical structure of radiation stresses. Another type of wave force is the surface wave force initially derived by Garrett (1976) in the study of Langmuir circulation generation. The wave driving forces include the wave dissipation term and the wave-averaged vortex forcing term which has been identified later by Leibovich (1980) and Smith (1980) as the vertically integrated form of the 'CL vortex-force' derived by Craik and Leibovich (1976). Dingemans et al. (1987) also presented a similar formulation of this type of wave driving force, though the current refraction, that may result in the vortex-force term, was recognized to give insignificant contributions under the conditions of slowly varying wave fields. Smith (2006) extended the formulation of Garrett (1976) to include finite-depth effects and provided some insight into physical interpretation of each forcing term in depth-integrated equations. A