- Stokes drift
For a pure

wave motion influid dynamics , the**Stokes drift velocity**is theaverage velocity when following a specificfluid parcel as it travels with thefluid flow . For instance, a particle floating at thefree surface ofwater waves , experiences a net Stokes drift velocity in the direction ofwave propagation .

More generally, the Stokes drift velocity is the difference between theaverage Lagrangianflow velocity of a fluid parcel, and the average Eulerianflow velocity of thefluid at a fixed position. Thisnonlinear phenomenon is named afterGeorge Gabriel Stokes , who derived expressions for thisdrift in his 1847 study ofwater waves .The

**Stokes drift**is the difference in end positions, after a predefined amount of time (usually onewave period ), as derived from a description in theLagrangian 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 theflow 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 areinhomogeneous in space. For instance inwater waves ,tide s andatmospheric waves .[

wave length of about twice the water depth. Click for an animation (4.15 MB).

"Description (also of the animation)":

The red circles are the present positions of massless particles, moving with theflow velocity . The light-blue line gives thepath of these particles, and the light-blue circles the particle position after eachwave period . The white dots are fluid particles, also followed in time. In the case shown here, themean Eulerian horizontal velocity below the wavetrough is zero.

Observe that thewave period , experienced by a fluid particle near thefree surface , is different from thewave period at a fixed horizontal position (as indicated by the light-blue circles). This is due to theDoppler shift .]

[water waves , with awave length much longer than the water depth.Click for an animation (1.29 MB).

"Description (also of the animation)":

The red circles are the present positions of massless particles, moving with theflow velocity . The light-blue line gives thepath of these particles, and the light-blue circles the particle position after eachwave period . The white dots are fluid particles, also followed in time. In the case shown here, themean Eulerian horizontal velocity below the wavetrough is zero.

Observe that thewave period , experienced by a fluid particle near thefree surface , is different from thewave period at a fixed horizontal position (as indicated by the light-blue circles). This is due to theDoppler shift .]In the 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 and McIntyre in 1978. [*See Craik (1985), page 105–113.*]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 ofLangmuir circulation s. [*See "e.g." Craik (1985), page 120.*] Fornonlinear and periodic water waves, accurate results on the Stokes drift have been computed and tabulated. [*Solutions of the particle trajectories in fully nonlinear periodic waves and the Lagrangian wave period they experience can for instance be found in:*]

cite journal| author=J.M. Williams| title=Limiting gravity waves in water of ﬁnite depth | journal=Philosophical Transactions of the Royal Society of London, Series A | volume=302 | issue=1466 | pages=139–188 | year=1981| doi=10.1098/rsta.1981.0159

cite book| title=Tables of progressive gravity waves | author=J.M. Williams | year=1985 | publisher=Pitman | isbn=978-0273087335**Mathematical description**The Lagrangian motion of a fluid parcel with

position vector "x =**ξ**(**α**,t)" in the Eulerian coordinates is given by:See Phillips (1977), page 43.] :$dot\{\backslash boldsymbol\{xi,\; =,\; frac\{partial\; \backslash boldsymbol\{xi\{partial\; t\},\; =,\; \backslash boldsymbol\{u\}(\backslash boldsymbol\{xi\},t),$where "∂**ξ**/ ∂t" is thepartial derivative of**"ξ**(**α**,t)" with respect to "t", and:**"ξ**(**α**,t)" is the Lagrangianposition vector of a fluid parcel, inmetre ,:**"u**(**x**,t)" is the Eulerianvelocity , inmetre persecond ,:**"x**" ix theposition vector in the Eulerian coordinate system, inmetre ,:**"α**" is theposition vector in the Lagrangian coordinate system, inmetre ,:"t" is thetime , insecond .Often, the Lagrangian coordinates**"α**" are chosen to coincide with the Eulerian coordinates**"x**" at the initial time "t = t_{0}" ::$\backslash boldsymbol\{xi\}(\backslash boldsymbol\{alpha\},t\_0),\; =,\; \backslash boldsymbol\{alpha\}.$But also other ways oflabel ing the fluid parcels are possible.If the

average value of a quantity is denoted by an overbar, then the average Eulerian velocity vector**"ū**_{E}" and average Lagrangian velocity vector**"ū**_{L}" are::$egin\{align\}\; overline\{\backslash boldsymbol\{u\_E,\; =,\; overline\{\backslash boldsymbol\{u\}(\backslash boldsymbol\{x\},t)\},\; \backslash \backslash \; overline\{\backslash boldsymbol\{u\_L,\; =,\; overline\{dot\{\backslash boldsymbol\{xi(\backslash boldsymbol\{alpha\},t)\},\; =,\; overline\{left(frac\{partial\; \backslash boldsymbol\{xi\}(\backslash boldsymbol\{alpha\},t)\}\{partial\; t\}\; ight)\},\; =,\; overline\{\backslash boldsymbol\{u\}(\backslash boldsymbol\{xi\}(\backslash boldsymbol\{alpha\},t),t)\}.\; end\{align\}$ Different definitions of theaverage may be used, depending on the subject of study, see ergodic theory:

*time average,

*space average,

*ensemble average and

*phase average.Now, the Stokes drift velocity**"ū**_{S}" equals [*See "e.g." Craik (1985), page 84.*] :$overline\{\backslash boldsymbol\{u\_S,\; =,\; overline\{\backslash boldsymbol\{u\_L,\; -,\; overline\{\backslash boldsymbol\{u\_E.$In many situations, themapping 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 apath of many different Eulerian positions**"x**", it is not possible to assign**"α**" to a unique**"x**". A mathematical 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: Deep water waves**The Stokes drift was formulated for

water waves byGeorge Gabriel Stokes in 1847. For simplicity, the case ofinfinite -deep water is considered, withlinear wave propagation of asinusoidal wave on thefree surface of a fluidlayer :See "e.g." Phillips (1977), page 37.] :$eta,\; =,\; a,\; cos,\; left(\; k\; x\; -\; omega\; t\; ight),$where:"η" is theelevation of thefree surface in the "z"-direction (metre ),:"a" is the waveamplitude (metre ),:"k" is thewave number : "k = 2π / λ" (radian s permetre ),:"ω" is theangular frequency : "ω = 2π / T" (radian s persecond ),:"x" is the horizontalcoordinate and the wave propagation direction (metre ),:"z" is the verticalcoordinate , with the positive "z" direction pointing out of the fluid layer (metre ),:"λ" is thewave length (metre ), and:"T" is thewave period (second ).It is assumed that the waves are ofinfinitesimal amplitude and thefree surface oscillates around themean level "z = 0". The waves propagate under the action of gravity, with aconstant acceleration vector bygravity (pointing downward in the negative "z"-direction). Further the fluid is assumed to beinviscid [*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.*] andincompressible , with aconstant mass density . The fluid flow isirrotational . Atinfinite depth, the fluid is taken to be at rest.Now the flow may be represented by a

velocity potential "φ", satisfying theLaplace equation and:$varphi,\; =,\; frac\{omega\}\{k\},\; a;\; ext\{e\}^\{k\; z\},\; sin,\; left(\; k\; x\; -\; omega\; t\; ight).$In order to havenon-trivial solutions for thiseigenvalue problem, thewave length andwave period may not be chosen arbitrarily, but must satisfy the deep-water dispersion relation:See "e.g." Phillips (1977), page 38.] :$omega^2,\; =,\; g,\; k.$with "g" theacceleration bygravity in ("m / s^{2}"). Within the framework oflinear theory, the horizontal and vertical components, "ξ_{x}" and "ξ_{z}" respectively, of the Lagrangian position**"ξ**" are::$egin\{align\}\; xi\_x,\; =,\; x,\; +,\; int,\; frac\{partial\; varphi\}\{partial\; x\};\; ext\{d\}t,\; =,\; x,\; -,\; a,\; ext\{e\}^\{k\; z\},\; sin,\; left(\; k\; x\; -\; omega\; t\; ight),\; \backslash \backslash \; xi\_z,\; =,\; z,\; +,\; int,\; frac\{partial\; varphi\}\{partial\; z\};\; ext\{d\}t,\; =,\; z,\; +,\; a,\; ext\{e\}^\{k\; z\},\; cos,\; left(\; k\; x\; -\; omega\; t\; ight).\; end\{align\}$ The horizontal component "ū_{S}" of the Stokes drift velocity is estimated by using aTaylor expansion around**"x**" of the Eulerian horizontal-velocity component "u_{x}= ∂ξ_{x}/ ∂t" at the position**"ξ**" ::$egin\{align\}\; overline\{u\}\_S,\; =,\; overline\{u\_x(\backslash boldsymbol\{xi\},t)\},\; -,\; overline\{u\_x(\backslash boldsymbol\{x\},t)\},\; \backslash \backslash \; =,\; overline\{left\; [\; u\_x(\backslash boldsymbol\{x\},t),\; +,\; left(\; xi\_x\; -\; x\; ight),\; frac\{partial\; u\_x(\backslash boldsymbol\{x\},t)\}\{partial\; x\},\; +,\; left(\; xi\_z\; -\; z\; ight),\; frac\{partial\; u\_x(\backslash boldsymbol\{x\},t)\}\{partial\; z\},\; +,\; cdots\; ight]\; \}\; -,\; overline\{u\_x(\backslash boldsymbol\{x\},t)\}\; \backslash \backslash \; approx,\; overline\{left(\; xi\_x\; -\; x\; ight),\; frac\{partial^2\; xi\_x\}\{partial\; x,\; partial\; t\}\; \},\; +,\; overline\{left(\; xi\_z\; -\; z\; ight),\; frac\{partial^2\; xi\_x\}\{partial\; z,\; partial\; t\}\; \}\; \backslash \backslash \; =,\; overline\{\; igg\; [\; -\; a,\; ext\{e\}^\{k\; z\},\; sin,\; left(\; k\; x\; -\; omega\; t\; ight)\; igg]\; ,\; igg\; [\; -omega,\; k,\; a,\; ext\{e\}^\{k\; z\},\; sin,\; left(\; k\; x\; -\; omega\; t\; ight)\; igg]\; \},\; \backslash \backslash \; +,\; overline\{\; igg\; [\; a,\; ext\{e\}^\{k\; z\},\; cos,\; left(\; k\; x\; -\; omega\; t\; ight)\; igg]\; ,\; igg\; [\; omega,\; k,\; a,\; ext\{e\}^\{k\; z\},\; cos,\; left(\; k\; x\; -\; omega\; t\; ight)\; igg]\; \},\; \backslash \backslash \; =,\; overline\{\; omega,\; k,\; a^2,\; ext\{e\}^\{2\; k\; z\},\; igg\; [\; sin^2,\; left(\; k\; x\; -\; omega\; t\; ight)\; +\; cos^2,\; left(\; k\; x\; -\; omega\; t\; ight)\; igg]\; \}.\; end\{align\}$Performing the averaging, the horizontal component of the Stokes drift velocity for deep-water waves is approximately:See Phillips (1977), page 44. Or Craik (1985), page 110.] :$overline\{u\}\_S,\; approx,\; omega,\; k,\; a^2,\; ext\{e\}^\{2\; k\; z\},\; =,\; frac\{4pi^2,\; a^2\}\{lambda,\; T\},\; ext\{e\}^\{4pi,\; z\; /\; lambda\}.$ As can be seen, the Stokes drift velocity "ū_{S}" is anonlinear quantity in terms of the waveamplitude "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 meanfree surface , "z = 0".**ee also***

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 | 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

*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

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**Other***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

*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 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 of London, Series A | volume= 245 | pages= 535–581 | url= http://journals.royalsociety.org/content/c544r8472188711q/?p=5745742bb01e4fada6301acfe23c1c28&pi=0 | doi= 10.1098/rsta.1953.0006

*cite book| first=O.M. | last=Phillips | title=The dynamics of the upper ocean |publisher=Cambridge University Press | year=1977 | edition=2^{nd}edition | isbn=0 521 29801 6**Notes**

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