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{{short description|Average velocity of a fluid parcel in a gravity wave}}
[[File:Driftwood Expanse, Northern Washington Coast.png|thumb|350px|right|An expanse of [[driftwood]] along the northern [[coast]] of [[Washington state]].]]▼
{{Multiple image
[[File:Deep_water_wave_after_three_periods.png|thumb|350px|right|Stokes drift in deep water waves, with a [[wave length]] of about twice the water depth. ▼
| image1 = Deep water wave.gif
| direction = vertical
| width = 350
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>▼
▲
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]].]]▼
| image2 = Shallow water wave.gif
▲| 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
▲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]] phenomenon is named after [[George Gabriel Stokes]], who derived expressions for this drift in [[#Stokes1847|his 1847 study]] of [[water waves]].
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.
▲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.
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.
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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>
The Stokes drift is important for the [[mass transfer]] of
For
==Mathematical description==
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
: <math>
\dot{\boldsymbol{\xi}}
= </math>
where
: ∂'''''ξ'''
: '''''
: '''u'''('''x''', ''t'') is the Eulerian [[velocity]],
:'''''x''''' is the [[position vector]] in the [[Lagrangian and Eulerian coordinates|Eulerian coordinate system]], in meters,▼
: '''
▲: '''''
: ''t'' is
Often, the Lagrangian coordinates '''''α''''' are chosen to coincide with the Eulerian coordinates
: <math>
\boldsymbol{\xi}(\boldsymbol{\alpha}, t_0)
</math>
If the [[average]] value of a quantity is denoted by an overbar, then the average Eulerian velocity vector
:<math>
\begin{align}
\
\\
\
=
=
\end{align}
</math>
Different definitions of the [[average]] may be used, depending on the subject of study
* [[time]] average,
* [[space]] average,
* [[ensemble average]]
* [[phase (waves)|phase]] average.
:<math>
\
</math>
In many situations, the [[map (mathematics)|mapping]] of average quantities from some Eulerian position
A mathematically sound basis for an unambiguous mapping between average Lagrangian and Eulerian quantities is provided by the theory of the
==Example: A one-dimensional compressible flow==
For the Eulerian velocity as a monochromatic wave of any nature in a continuous medium: <math>u = \hat{u} \sin
: <math>\dot
: <math>
\xi(\xi_0, t) \approx \xi_0 + \frac{\hat{u}}{\omega} \cos(k\xi_0 - \omega t)
</math>
Here the last term describes the Stokes drift velocity <math>\tfrac12 k\hat{u}^2/\omega.</math><ref>See [[#Falkovich|Falkovich (2011)]], pages 71–72. There is a typo in the coefficient of the superharmonic term in Eq. (2.20) on page 71, i.e <math>-\tfrac14</math> instead of <math>+\tfrac12.</math></ref>
==Example: Deep water waves==
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{{See also|Airy wave theory|Stokes wave}}
The Stokes drift was formulated for [[water waves]] by [[George Gabriel Stokes]] in 1847. For simplicity, the case of [[Infinity|
: <math>
\eta
</math>
where
: ''η'' is the [[elevation]] of the [[free surface]] in the ''z''
: ''a'' is the wave [[amplitude]] (meters),
: ''k'' is the [[wave number]]: ''k'' =
: ''ω'' is the [[angular frequency]]: ''ω'' =
: ''x'' is the horizontal [[coordinate]] and the wave propagation direction (meters),
: ''z'' is the vertical [[coordinate]], with the positive ''z'' direction pointing out of the fluid layer (meters),
: ''λ'' is the [[wave length]] (meters),
: ''T'' is the [[wave period]] ([[second]]s).
As derived below, the horizontal component ''ū''<sub>S</sub>
:
\
=
</math>
As can be seen, the Stokes drift velocity ''ū''<sub>S</sub>
===Derivation===
It is assumed that the waves are of [[infinitesimal]] [[amplitude]] and the [[free surface]] oscillates around the [[mean]] level ''z
Now the [[flow (mathematics)|flow]] may be represented by a [[velocity potential]] ''φ'', satisfying the [[Laplace equation]] and<ref name=Phil1977p37/>
:<math>
\varphi
</math>
In order to have [[non-trivial]] solutions for this [[eigenvalue]] problem, the [[wave length]] and [[wave period]] may not be chosen arbitrarily, but
: <math>
\omega^2
</math>
with ''g'' the [[acceleration]] by [[gravity]] in (
:<math>
\begin{align}
\xi_x
\\
\xi_z
\end{align}
</math>
The horizontal component ''ū''<sub>S</sub>
:<math>
\begin{align}
\
&=
\\
&=
u_x(\
+
+
+
\right]
-
\\
&\approx
+
\\
&=
\\
&+
\\
&=
\\
&=
\end{align}
</math>
==See also==
* [[
* [[Darwin drift]]
* [[Lagrangian and Eulerian coordinates]]
* [[Material derivative]]
==References==
===Historical===
*{{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 |
*
===Other===
*
*
*
*
*
*{{cite journal | title=A mechanism for the accumulation of floating marine debris north of Hawaii | last=Kubota |first=M. |year=1994 |doi=10.1175/1520-0485(1994)024<1059:AMFTAO>2.0.CO;2 |journal=Journal of Physical Oceanography |volume=24 |issue=5 |pages=1059–1064 |ref=Kubota1994 |bibcode = 1994JPO....24.1059K }}
==Notes==
{{
{{physical oceanography}}
|