Stokes drift and its discontents.

The Stokes velocity [Formula: see text], defined approximately by Stokes (1847, , , 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, [Formula: see text], and a remainder that is small for waves with slowly varying amplitudes. We further show that [Formula: see text] 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, , , 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 [Formula: see text] replacing [Formula: see text], and we discuss the form of the Stokes pumping associated with both [Formula: see text] and [Formula: see text]. This article is part of the theme issue 'Mathematical problems in physical fluid dynamics (part 1)'.