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{{short description|Average velocity of a fluid parcel in a gravity wave}}
{{Multiple image▼
| image1 = Deep water wave.gif▼
| direction = vertical▼
| width = 350▼
| caption1 = Stokes drift in deep [[water waves]], with a [[wave length]] of about twice the water depth.▼
| image2 = Shallow water wave.gif▼
| caption2 = Stokes drift in shallow water waves, with a wave length much longer than the water depth.▼
| footer = The red circles are the present positions of massless particles, moving with the [[flow velocity]]. The light-blue line gives the [[path (topology)|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 cases shown here, the [[mean]] Eulerian horizontal velocity below the wave [[trough (physics)|trough]] is zero.<br>▼
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]].▼
}}▼
[[File:Driftwood Expanse, Northern Washington Coast.png|thumb|350px|right|An expanse of [[driftwood]] along the northern [[coast]] of [[Washington state]]. Stokes drift – besides e.g. [[Ekman drift]] and [[geostrophic current]]s – is one of the relevant processes in the transport of [[marine debris]].<ref>See [[#Kubota1994|Kubota (1994)]].</ref>]]
For a pure [[wave]] [[motion (physics)|motion]] in [[fluid dynamics]], the '''Stokes drift velocity''' is the [[average]] [[velocity]] when following a specific [[fluid]] parcel as it travels with the [[fluid flow]]. For instance, a particle floating at the [[free surface]] of [[water waves]], experiences a net Stokes drift velocity in the direction of [[wave propagation]].
More generally, the Stokes drift velocity is the difference between the [[average]] [[Lagrangian and Eulerian coordinates|Lagrangian]] [[flow velocity]] of a fluid parcel, and the average [[Lagrangian and Eulerian coordinates|Eulerian]] [[flow velocity]] of the [[fluid]] at a fixed position. This [[nonlinear system|nonlinear]]
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 and Eulerian coordinates|Lagrangian description]] is obtained by following a specific fluid parcel during the time interval. The corresponding end position in the [[Lagrangian and Eulerian coordinates|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.
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The Stokes drift is important for the [[mass transfer]] of various kinds of material and organisms by oscillatory flows. It plays a crucial role in the generation of [[Langmuir circulation]]s.<ref>See ''e.g.'' [[#Craik1985|Craik (1985)]], page 120.</ref>
For
==Mathematical description==
▲{{Multiple image
▲| image1 = Deep water wave.gif
▲| direction = vertical
▲| width = 350
▲| caption1 = Stokes drift in deep [[water waves]], with a [[wave length]] of about twice the water depth.
▲| image2 = Shallow water wave.gif
▲| caption2 = Stokes drift in shallow water waves, with a wave length much longer than the water depth.
▲| footer = The red circles are the present positions of massless particles, moving with the [[flow velocity]]. The light-blue line gives the [[path (topology)|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 cases shown here, the [[mean]] Eulerian horizontal velocity below the wave [[trough (physics)|trough]] is zero.<br>
▲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 [[Lagrangian and Eulerian coordinates|Lagrangian motion]] of a fluid parcel with [[position vector]] ''x = '''ξ'''('''α''', t)'' in the Eulerian coordinates is given by<ref name=Phil1977p43>See [[#Phillips1977|Phillips (1977)]], page 43.</ref>
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</math>
As can be seen, the Stokes drift velocity ''ū''<sub>S</sub> is a
===Derivation===
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