An asymptotic theory for the interaction of waves and currents in coastal waters
JAMES C. McWILLIAMS a1, JUAN M. RESTREPO a2andEMILY M. LANE a3 a1 Department of Atmospheric and Oceanic Sciences and Institute of Geophysics and Planetary Physics, University of California, Los Angeles, CA 90095-1565, USA a2 Department of Mathematics and Department of Physics, University of Arizona, Tucson, AZ 85721, USA a3 Program in Applied Mathematics, University of Arizona, Tucson, AZ 85721, USA
A multi-scale asymptotic theory is derived for the evolution and interaction of currents and surface gravity waves in water of finite depth, under conditions typical of coastal shelf waters outside the surf zone. The theory provides a practical and useful model with which wave–current coupling may be explored without the necessity of resolving features of the flow on space and time scales of the primary gravity-wave oscillations. The essential nature of the dynamical interaction is currents modulating the slowly evolving phase of the wave field and waves providing both phase-averaged forcing of long infra-gravity waves and wave-averaged vortex and Bernoulli-head forces and hydrostatic static set-up for the low-frequency current and sea-level evolution equations. Analogous relations are derived for wave-averaged material tracers and density stratification that include advection by horizontal Stokes drift and by a vertical Stokes pseudo-velocity that is the incompressible companion to the horizontal Stokes velocity. Illustrative solutions are analysed for the special case of depth-independent currents and tracers associated with an incident surface wave field and a vortex with O(1) Rossby number above continental shelf topography.
(Received March 12 2003) (Revised February 13 2004)