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   The Stokes velocity u 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, u sol S , and a remainder that is small for waves with slowly varying amplitudes. We further show that u sol S 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 u sol S replacing u S , and we discuss the form of the Stokes pumping associated with both u S and u sol S . This article is part of the theme issue ‘Mathematical problems in physical fluid dynamics (part 1)’. 

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