Stokes drift in deep water waves, with a wave length of about twice the water depth. Click here for an animation (4.15 MB). Description (also of the animation): The red circles are the present positions of massless particles, moving with the flow velocity. The light-blue line gives the path of these particles, and the light-blue circles the particle position after each wave period. The white dots are fluid particles, also followed in time. In the case shown here, the mean Eulerian horizontal velocity below the wave trough is zero. Observe that the wave period, experienced by a fluid particle near the free surface, is different from the wave period at a fixed horizontal position (as indicated by the light-blue circles). This is due to the Doppler shift.
Stokes drift in shallow water waves, with a wave length much longer than the water depth. Click here for an animation (1.29 MB). Description (also of the animation): The red circles are the present positions of massless particles, moving with the flow velocity. The light-blue line gives the path of these particles, and the light-blue circles the particle position after each wave period. The white dots are fluid particles, also followed in time. In the case shown here, the mean Eulerian horizontal velocity below the wave trough is zero. Observe that the wave period, experienced by a fluid particle near the free surface, is different from the wave period at a fixed horizontal position (as indicated by the light-blue circles). This is due to the Doppler shift.
The Stokes drift is the difference in end positions, after a predefined amount of time (usually one wave period), as derived from a description in the Lagrangian and Eulerian coordinates. The end position in the Lagrangian description is obtained by following a specific fluid parcel during the time interval. The corresponding end position in the Eulerian description is obtained by integrating the flow velocity at a fixed position—equal to the initial position in the Lagrangian description—during the same time interval.
The Stokes drift velocity equals the Stokes drift divided by the considered time interval.
Often, the Stokes drift velocity is loosely referred to as Stokes drift.
Stokes drift may occur in all instances of oscillatory flow which are inhomogeneous in space. For instance in water waves, tides and atmospheric waves.
The Stokes drift is important for the mass transfer of all kind of materials and organisms by oscillatory flows. Further the Stokes drift is important for the generation of Langmuir circulations.
For nonlinear and periodic water waves, accurate results on the Stokes drift have been computed and tabulated.
The Stokes drift velocity ūS is defined as the difference between the average Eulerian velocity and the average Lagrangian velocity: 
In many situations, the mapping of average quantities from some Eulerian position x to a corresponding Lagrangian position α forms a problem. Since a fluid parcel with label α traverses along a path of many different Eulerian positions x, it is not possible to assign α to a unique x.
A mathematically sound basis for an unambiguous mapping between average Lagrangian and Eulerian quantities is provided by the theory of the Generalized Lagrangian Mean (GLM) by Andrews and McIntyre (1978).
Example: A one-dimensional compressible flow
For the Eulerian velocity as a monochromatic wave of any nature in a continuous medium: one readily obtains by the perturbation theory – with as a small parameter – for the particle position
Here the last term describes the Stokes drift velocity 
Example: Deep water waves
Stokes drift under periodic waves in deep water, for a periodT = 5 s and a mean water depth of 25 m. Left: instantaneous horizontal flow velocities. Right: average flow velocities. Black solid line: average Eulerian velocity; red dashed line: average Lagrangian velocity, as derived from the Generalized Lagrangian Mean (GLM).
As derived below, the horizontal component ūS(z) of the Stokes drift velocity for deep-water waves is approximately:
As can be seen, the Stokes drift velocity ūS is a nonlinear quantity in terms of the wave amplitudea. Further, the Stokes drift velocity decays exponentially with depth: at a depth of a quarter wavelength, z = -¼ λ, it is about 4% of its value at the mean free surface, z = 0.
A.D.D. Craik (2005). "George Gabriel Stokes on water wave theory". Annual Review of Fluid Mechanics. 37 (1): 23–42. Bibcode:2005AnRFM..37...23C. doi:10.1146/annurev.fluid.37.061903.175836.
G.G. Stokes (1847). "On the theory of oscillatory waves". Transactions of the Cambridge Philosophical Society. 8: 441–455. Reprinted in: G.G. Stokes (1880). Mathematical and Physical Papers, Volume I. Cambridge University Press. pp. 197–229.
D.G. Andrews & M.E. McIntyre (1978). "An exact theory of nonlinear waves on a Lagrangian mean flow". Journal of Fluid Mechanics. 89 (4): 609–646. Bibcode:1978JFM....89..609A. doi:10.1017/S0022112078002773.
A.D.D. Craik (1985). Wave interactions and fluid flows. Cambridge University Press. ISBN 978-0-521-36829-2.
M.S. Longuet-Higgins (1953). "Mass transport in water waves". Philosophical Transactions of the Royal Society A. 245 (903): 535–581. Bibcode:1953RSPTA.245..535L. doi:10.1098/rsta.1953.0006.
Phillips, O.M. (1977). The dynamics of the upper ocean (2nd ed.). Cambridge University Press. ISBN 978-0-521-29801-8.
G. Falkovich (2011). Fluid Mechanics (A short course for physicists). Cambridge University Press. ISBN 978-1-107-00575-4.
Kubota, M. (1994). "A mechanism for the accumulation of floating marine debris north of Hawaii". Journal of Physical Oceanography. 24 (5): 1059–1064. Bibcode:1994JPO....24.1059K. doi:10.1175/1520-0485(1994)024<1059:AMFTAO>2.0.CO;2.
^Solutions of the particle trajectories in fully nonlinear periodic waves and the Lagrangian wave period they experience can for instance be found in: J.M. Williams (1981). "Limiting gravity waves in water of finite depth". Philosophical Transactions of the Royal Society A. 302 (1466): 139–188. Bibcode:1981RSPTA.302..139W. doi:10.1098/rsta.1981.0159. J.M. Williams (1985). Tables of progressive gravity waves. Pitman. ISBN 978-0-273-08733-5.
^Viscosity has a pronounced effect on the mean Eulerian velocity and mean Lagrangian (or mass transport) velocity, but much less on their difference: the Stokes drift outside the boundary layers near bed and free surface, see for instance Longuet-Higgins (1953). Or Phillips (1977), pages 53–58.