  The Stokes velocity $\mathbf{u}^\mathrm{S}$, 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,
$\mathbf{u}^\mathrm{S}_\mathrm{sol}$, and a remainder that is small for waves
with slowly varying amplitudes. We further show that
$\mathbf{u}^\mathrm{S}_\mathrm{sol}$ 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) which we specialise 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 equation with
$\mathbf{u}^\mathrm{S}_\mathrm{sol}$ replacing $\mathbf{u}^\mathrm{S}$, and we
discuss the form of the Stokes pumping associated with both
$\mathbf{u}^\mathrm{S}$ and $\mathbf{u}^\mathrm{S}_\mathrm{sol}$.
