Lagrangian studies of wave-induced flows in a viscous ocean

Abstract

A review of the Lagrangian approach to wave-induced drift in a rotating fluid layer of finite depth is presented. The Lagrangian description of fluid motion is (usually) more mathematically demanding than the traditional Eulerian approach. However, it yields directly the mean particle drift velocity in periodic motion. The solution to the wave-drift problem in a Lagrangian description depends crucially on the viscosity ν being non-zero, however small. It illustrates the singular nature of this problem, in which the limit of solutions as ν→0 is different from solutions obtained with ν=0. Obviously, for most oceanographic applications, the effect of an (eddy) viscosity cannot be neglected. In this survey we consider solutions for the Lagrangian mean drift in various types of surface waves influenced by viscosity. From a purely Lagrangian starting point, we demonstrate novel ways of deriving conservation equations for the mean wave momentum and the mean wave energy in a weakly viscous fluid layer of finite depth. Among several examples of mean drift in surface waves, we consider deep water gravity waves acted upon by an oscillating wind stress, short capillary-gravity waves affected by a thin elastic surface film, and friction-induced roll motion in short-crested gravity waves. Furthermore, the drift in high-frequency shallow-water gravity waves is revisited, and a new equation for the Lagrangian mean drift in temporally/spatially modulated waves is presented. Finally, we discuss the use of the Coriolis-Stokes force to implement Lagrangian properties into Eulerian numerical ocean circulation models.