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| [[File:Driftwood Expanse, Northern Washington Coast.png|thumb|350px|right|An expanse of [[driftwood]] along the northern [[coast]] of [[Washington state]].]]
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| [[File:Deep water wave after three periods.gif|thumb|350px|right|Stokes drift in deep water waves, with a [[wave length]] of about twice the water depth.
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| Click [[:Image:Deep water wave.gif|here]] for an animation (4.15 MB).<br>
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| ''Description (also of the animation)'':<br>
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| 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 case shown here, the [[mean]] Eulerian horizontal velocity below the wave [[trough (physics)|trough]] is zero.<br>
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| 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]].]]
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| [[File:Shallow water wave after three wave periods.gif|thumb|350px|right|Stokes drift in shallow [[water waves]], with a [[wave length]] much longer than the water depth.
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| Click [[:Image:Shallow water wave.gif|here]] for an animation (1.29 MB).<br>
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| ''Description (also of the animation)'':<br>
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| 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 case shown here, the [[mean]] Eulerian horizontal velocity below the wave [[trough (physics)|trough]] is zero.<br>
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| 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]].]]
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| 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]].
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| 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]] phenomenon is named after [[George Gabriel Stokes]], who derived expressions for this drift in [[#Stokes1847|his 1847 study]] of [[water waves]].
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| 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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| <br>
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| The Stokes drift velocity equals the Stokes drift divided by the considered time interval.
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| Often, the Stokes drift velocity is loosely referred to as Stokes drift.
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| Stokes drift may occur in all instances of oscillatory flow which are [[inhomogeneous]] in space. For instance in [[water waves]], [[tide]]s and [[atmospheric waves]].
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| In the [[Lagrangian and Eulerian coordinates|Lagrangian description]], fluid parcels may drift far from their initial positions. As a result, the unambiguous definition of an [[average]] Lagrangian velocity and Stokes drift velocity, which can be attributed to a certain fixed position, is by no means a trivial task. However, such an unambiguous description is provided by the ''[[Generalized Lagrangian Mean]]'' (GLM) theory of [[#Andrews-McIntyre1978|Andrews and McIntyre in 1978]].<ref>See [[#Craik1985|Craik (1985)]], page 105–113.</ref>
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| 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 circulation]]s.<ref>See ''e.g.'' [[#Craik1985|Craik (1985)]], page 120.</ref>
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| For [[nonlinear]] and [[periodic function|periodic]] water waves, accurate results on the Stokes drift have been computed and tabulated.<ref>Solutions of the particle trajectories in fully nonlinear periodic waves and the Lagrangian wave period they experience can for instance be found in:<br>{{cite journal| author=J.M. Williams| title=Limiting gravity waves in water of finite depth | journal=Philosophical Transactions of the Royal Society A | volume=302 | issue=1466 | pages=139–188 | year=1981| doi=10.1098/rsta.1981.0159 |bibcode = 1981RSPTA.302..139W }}<br>{{cite book| title=Tables of progressive gravity waves | author=J.M. Williams | year=1985 | publisher=Pitman | isbn=978-0-273-08733-5 }}</ref>
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| ==Mathematical description==
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| 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> | |
| \dot{\boldsymbol{\xi}}\, =\, \frac{\partial \boldsymbol{\xi}}{\partial t}\, =\, \boldsymbol{u}(\boldsymbol{\xi},t),
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| </math>
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| where ''∂'''ξ''' / ∂t'' is the [[partial derivative]] of '''''ξ'''('''α''',t)'' with respect to ''t'', and
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| :'''''ξ'''('''α''',t)'' is the Lagrangian [[position vector]] of a fluid parcel, in meters,
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| :'''''u'''('''x''',t)'' is the Eulerian [[velocity]], in meters per [[second]],
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| :'''''x''''' is the [[position vector]] in the [[Lagrangian and Eulerian coordinates|Eulerian coordinate system]], in meters,
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| :'''''α''''' is the [[position vector]] in the [[Lagrangian and Eulerian coordinates|Lagrangian coordinate system]], in meters,
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| :''t'' is the [[time]], in [[second]]s.
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| Often, the Lagrangian coordinates '''''α''''' are chosen to coincide with the Eulerian coordinates '''''x''''' at the initial time ''t = t<sub>0</sub>'' :<ref name=Phil1977p43/>
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| :<math>
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| \boldsymbol{\xi}(\boldsymbol{\alpha},t_0)\, =\, \boldsymbol{\alpha}.
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| </math>
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| But also other ways of [[label]]ing the fluid parcels are possible.
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| If the [[average]] value of a quantity is denoted by an overbar, then the average Eulerian velocity vector '''''ū'''<sub>E</sub>'' and average Lagrangian velocity vector '''''ū'''<sub>L</sub>'' are: | |
| :<math>
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| \begin{align}
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| \overline{\boldsymbol{u}}_E\, &=\, \overline{\boldsymbol{u}(\boldsymbol{x},t)},
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| \\
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| \overline{\boldsymbol{u}}_L\, &=\, \overline{\dot{\boldsymbol{\xi}}(\boldsymbol{\alpha},t)}\,
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| =\, \overline{\left(\frac{\partial \boldsymbol{\xi}(\boldsymbol{\alpha},t)}{\partial t}\right)}\,
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| =\, \overline{\boldsymbol{u}(\boldsymbol{\xi}(\boldsymbol{\alpha},t),t)}.
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| \end{align}
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| </math> | |
| Different definitions of the [[average]] may be used, depending on the subject of study, see [[Ergodic theory#Ergodic theorems|ergodic theory]]:
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| *[[time]] average,
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| *[[space]] average,
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| *[[ensemble average]] and
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| *[[phase (waves)|phase]] average.
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| Now, the Stokes drift velocity '''''ū'''<sub>S</sub>'' equals<ref>See ''e.g.'' [[#Craik1985|Craik (1985)]], page 84.</ref>
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| :<math>
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| \overline{\boldsymbol{u}}_S\, =\, \overline{\boldsymbol{u}}_L\, -\, \overline{\boldsymbol{u}}_E.
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| </math>
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| In many situations, the [[map (mathematics)|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 (topology)|path]] of many different Eulerian positions '''''x''''', it is not possible to assign '''''α''''' to a unique '''''x'''''.
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| 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-McIntyre1978|Andrews and McIntyre (1978)]].
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| ==Example: A one-dimensional compressible flow==
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| For the Eulerian velocity as a monochromatic wave of any nature in a continuous medium: <math>u=\hat{u}\sin\left( kx - \omega t \right),</math> one readily obtains by the perturbation theory – with <math>k\hat{u}/\omega</math> as a small parameter – for the particle position <math>x=\xi(\xi_0,t):</math>
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| :<math>\dot{{\xi}}=\, {u}({\xi},t)= \hat{u} \sin\, \left( k \xi - \omega t \right),</math>
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| :<math>
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| \xi(\xi_0,t)\approx\xi_0+(\hat{u}/\omega)\cos(k\xi_0-\omega t)+(\tfrac12 k\hat{u}^2/\omega^2)\sin2(k\xi_0-\omega t)+k\hat{u}^2t/2\omega\ .
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| </math> | |
| Here the last term describes the Stokes drift <math>\tfrac12 k\hat{u}^2/\omega.</math><ref>[[#Falkovich|Falkovich (2011)]]</ref>
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| ==Example: Deep water waves==
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| [[File:Vitesses derive.png|right|400 px|thumb|Stokes drift under periodic waves in deep water, for a [[period (physics)|period]] ''T'' = 5 s and a mean water depth of 25 m. ''Left'': instantaneous horizontal [[flow velocity|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).]]
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| {{See also|Airy wave theory|Stokes wave}}
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| The Stokes drift was formulated for [[water waves]] by [[George Gabriel Stokes]] in 1847. For simplicity, the case of [[Infinity|infinite]]-deep water is considered, with [[linear]] [[wave propagation]] of a [[sinusoidal]] wave on the [[free surface]] of a fluid layer:<ref name=Phil1977p37>See ''e.g.'' [[#Phillips1977|Phillips (1977)]], page 37.</ref>
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| :<math>
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| \eta\, =\, a\, \cos\, \left( k x - \omega t \right),
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| </math>
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| where
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| :''η'' is the [[elevation]] of the [[free surface]] in the ''z''-direction (meters),
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| :''a'' is the wave [[amplitude]] (meters),
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| :''k'' is the [[wave number]]: ''k = 2π / λ'' ([[radian]]s per meter),
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| :''ω'' is the [[angular frequency]]: ''ω = 2π / T'' ([[radian]]s per [[second]]),
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| :''x'' is the horizontal [[coordinate]] and the wave propagation direction (meters),
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| :''z'' is the vertical [[coordinate]], with the positive ''z'' direction pointing out of the fluid layer (meters),
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| :''λ'' is the [[wave length]] (meters), and
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| :''T'' is the [[wave period]] ([[second]]s).
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| As derived below, the horizontal component ''ū<sub>S</sub>''(''z'') of the Stokes drift velocity for deep-water waves is approximately:<ref name=Phil1977p44>See [[#Phillips1977|Phillips (1977)]], page 44. Or [[#Craik1985|Craik (1985)]], page 110.</ref>
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| :{{Equation box 1|equation=<math>
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| \overline{u}_S\, \approx\, \omega\, k\, a^2\, \text{e}^{2 k z}\,
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| =\, \frac{4\pi^2\, a^2}{\lambda\, T}\, \text{e}^{4\pi\, z / \lambda}.
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| </math>}}
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|
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| As can be seen, the Stokes drift velocity ''ū<sub>S</sub>'' is a [[nonlinear]] quantity in terms of the wave [[amplitude]] ''a''. Further, the Stokes drift velocity decays exponentially with depth: at a depth of a quart wavelength, ''z = -¼ λ'', it is about 4% of its value at the mean [[free surface]], ''z = 0''.
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| ===Derivation===
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| It is assumed that the waves are of [[infinitesimal]] [[amplitude]] and the [[free surface]] oscillates around the [[mean]] level ''z = 0''. The waves propagate under the action of gravity, with a [[wikt:constant|constant]] [[acceleration]] [[Vector (geometric)|vector]] by [[gravity]] (pointing downward in the negative ''z''-direction). Further the fluid is assumed to be [[inviscid]]<ref>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 [[Stokes boundary layer|boundary layers]] near bed and free surface, see for instance [[#Longuet-Higgins1953|Longuet-Higgins (1953)]]. Or [[#Phillips1977|Phillips (1977)]], pages 53–58.</ref> and [[incompressible]], with a [[wikt:constant|constant]] [[mass density]]. The fluid [[flow (mathematics)|flow]] is [[irrotational]]. At infinite depth, the fluid is taken to be at [[rest (physics)|rest]].
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| Now the [[flow (mathematics)|flow]] may be represented by a [[velocity potential]] ''φ'', satisfying the [[Laplace equation]] and<ref name=Phil1977p37/>
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| :<math>
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| \varphi\, =\, \frac{\omega}{k}\, a\; \text{e}^{k z}\, \sin\, \left( k x - \omega t \right).
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| </math>
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| In order to have [[non-trivial]] solutions for this [[eigenvalue]] problem, the [[wave length]] and [[wave period]] may not be chosen arbitrarily, but must satisfy the deep-water [[dispersion (water waves)|dispersion]] relation:<ref name=Phil1977p38>See ''e.g.'' [[#Phillips1977|Phillips (1977)]], page 38.</ref>
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| :<math>
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| \omega^2\, =\, g\, k.
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| </math>
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| with ''g'' the [[acceleration]] by [[gravity]] in (''m / s<sup>2</sup>''). Within the framework of [[linear]] theory, the horizontal and vertical components, ''ξ<sub>x</sub>'' and ''ξ<sub>z</sub>'' respectively, of the Lagrangian position '''''ξ''''' are:<ref name=Phil1977p44/>
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| :<math>
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| \begin{align}
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| \xi_x\, &=\, x\, +\, \int\, \frac{\partial \varphi}{\partial x}\; \text{d}t\,
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| =\, x\, -\, a\, \text{e}^{k z}\, \sin\, \left( k x - \omega t \right),
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| \\
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| \xi_z\, &=\, z\, +\, \int\, \frac{\partial \varphi}{\partial z}\; \text{d}t\,
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| =\, z\, +\, a\, \text{e}^{k z}\, \cos\, \left( k x - \omega t \right).
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| \end{align}
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| </math>
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| The horizontal component ''ū<sub>S</sub>'' of the Stokes drift velocity is estimated by using a [[Taylor expansion]] around '''''x''''' of the Eulerian horizontal-velocity component ''u<sub>x</sub> = ∂ξ<sub>x</sub> / ∂t'' at the position '''''ξ''''' :<ref name=Phil1977p43/>
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| :<math>
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| \begin{align}
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| \overline{u}_S\,
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| &=\, \overline{u_x(\boldsymbol{\xi},t)}\, -\, \overline{u_x(\boldsymbol{x},t)}\,
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|
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| \\
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| &=\, \overline{\left[
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| u_x(\boldsymbol{x},t)\,
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| +\, \left( \xi_x - x \right)\, \frac{\partial u_x(\boldsymbol{x},t)}{\partial x}\,
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| +\, \left( \xi_z - z \right)\, \frac{\partial u_x(\boldsymbol{x},t)}{\partial z}\,
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| +\, \cdots
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| \right] }
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| -\, \overline{u_x(\boldsymbol{x},t)}
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| \\
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| &\approx\, \overline{\left( \xi_x - x \right)\, \frac{\partial^2 \xi_x}{\partial x\, \partial t} }\,
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| +\, \overline{\left( \xi_z - z \right)\, \frac{\partial^2 \xi_x}{\partial z\, \partial t} }
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| \\
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| &=\, \overline{ \bigg[ - a\, \text{e}^{k z}\, \sin\, \left( k x - \omega t \right) \bigg]\,
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| \bigg[ -\omega\, k\, a\, \text{e}^{k z}\, \sin\, \left( k x - \omega t \right) \bigg] }\,
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| \\
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| &+\, \overline{ \bigg[ a\, \text{e}^{k z}\, \cos\, \left( k x - \omega t \right) \bigg]\,
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| \bigg[ \omega\, k\, a\, \text{e}^{k z}\, \cos\, \left( k x - \omega t \right) \bigg] }\,
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| \\
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| &=\, \overline{ \omega\, k\, a^2\, \text{e}^{2 k z}\,
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| \bigg[ \sin^2\, \left( k x - \omega t \right) + \cos^2\, \left( k x - \omega t \right) \bigg] }
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| \\
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| &=\, \omega\, k\, a^2\, \text{e}^{2 k z}.
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| \end{align}
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| </math>
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| ==See also==
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| *[[Coriolis-Stokes force]]
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| *[[Darwin drift]]
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| *[[Lagrangian and Eulerian coordinates]]
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| *[[Material derivative]]
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| ==References==
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| ===Historical===
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| *{{cite journal | author= A.D.D. Craik | year= 2005 | title= George Gabriel Stokes on water wave theory | journal= Annual Review of Fluid Mechanics | volume= 37 | pages= 23–42 | doi= 10.1146/annurev.fluid.37.061903.175836 | url= http://arjournals.annualreviews.org/doi/abs/10.1146/annurev.fluid.37.061903.175836?journalCode=fluid |bibcode = 2005AnRFM..37...23C }}
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| *<cite id=Stokes1847>{{cite journal | author= G.G. Stokes | year= 1847 | title= On the theory of oscillatory waves | journal= Transactions of the Cambridge Philosophical Society | volume= 8 | pages= 441–455 }}<br>Reprinted in: {{cite book | author= G.G. Stokes | year= 1880 | title= Mathematical and Physical Papers, Volume I | publisher= Cambridge University Press | pages= 197–229 | url= http://www.archive.org/details/mathphyspapers01stokrich }}</cite>
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| ===Other===
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| *<cite id=Andrews-McIntyre1978>{{cite journal | author=D.G. Andrews and M.E. McIntyre | year= 1978 | title= An exact theory of nonlinear waves on a Lagrangian mean flow | journal= Journal of Fluid Mechanics | volume= 89 | pages= 609–646 | url= http://journals.cambridge.org/action/displayAbstract?fromPage=online&aid=388265 | doi= 10.1017/S0022112078002773 |bibcode = 1978JFM....89..609A | issue=4 }}</cite>
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| *<cite id=Craik1985>{{cite book | author= A.D.D. Craik | title=Wave interactions and fluid flows | year=1985 | publisher=Cambridge University Press | isbn=0-521-36829-4}}</cite>
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| *<cite id=Longuet-Higgins1953>{{cite journal | author= M.S. Longuet-Higgins | authorlink=Michael S. Longuet-Higgins | year= 1953 | title= Mass transport in water waves | journal= Philosophical Transactions of the Royal Society A | volume= 245 | pages= 535–581 | url= http://rsta.royalsocietypublishing.org/content/245/903/535 | doi= 10.1098/rsta.1953.0006 |bibcode = 1953RSPTA.245..535L | issue=903}}</cite>
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| *<cite id=Phillips1977>{{cite book| first=O.M. | last=Phillips | title=The dynamics of the upper ocean |publisher=Cambridge University Press | year=1977 | edition=2nd | isbn=0-521-29801-6 }}</cite>
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| *<cite id=Falkovich>{{cite book | author=G. Falkovich| year=2011 | title=Fluid Mechanics (A short course for physicists)|url=http://www.cambridge.org/gb/knowledge/isbn/item6173728/?site_locale=en_GB | publisher=Cambridge University Press | isbn=978-1-107-00575-4 }}
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| ==Notes==
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| {{reflist|2}}
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| {{physical oceanography}}
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| [[Category:Fluid dynamics]]
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| [[Category:Water waves]]
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