Stokes drift and its discontents

Abstract

The Stokes velocity uS, defined approximately by Stokes (1847, Trans. Camb. Philos. Soc., 8, 441–455.), and exactly via the Generalized Lagrangian Mean, is divergent even in an incompressible fluid. We show that the Stokes velocity can be naturally decomposed into a solenoidal component, usolS, and a remainder that is small for waves with slowly varying amplitudes. We further show that usolS arises as the sole Stokes velocity when the Lagrangian mean flow is suitably redefined to ensure its exact incompressibility. The construction is an application of Soward & Roberts’s glm theory (2010, J. Fluid Mech., 661, 45–72. (doi:10.1017/S0022112010002867)) which we specialize to surface gravity waves and implement effectively using a Lie series expansion. We further show that the corresponding Lagrangian-mean momentum equation is formally identical to the Craik–Leibovich (CL) equation with usolS replacing uS, and we discuss the form of the Stokes pumping associated with both uS and usolS. This article is part of the theme issue ‘Mathematical problems in physical fluid dynamics (part 1)’.