Stokes drift in deepwater waves, with awave length of about twice the water depth.

Stokes drift in shallow water waves, with a wave length much longer than the water depth.

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 cases 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.

For a purewavemotion influid dynamics, theStokes drift velocity is theaveragevelocity 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 theaverageLagrangianflow velocity of a fluid parcel, and the averageEulerianflow velocity of thefluid at a fixed position. Thisnonlinear phenomenon is named afterGeorge Gabriel Stokes, who derived expressions for this drift inhis 1847 study ofwater waves.

TheStokes 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 theLagrangian description is obtained by following a specific fluid parcel during the time interval. The corresponding end position in theEulerian 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,tides andatmospheric waves.

In theLagrangian description, fluid parcels may drift far from their initial positions. As a result, the unambiguous definition of anaverage 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 theGeneralized Lagrangian Mean (GLM) theory ofAndrews and McIntyre in 1978.^[2]

The Stokes drift is important for themass transfer of various kinds of material and organisms by oscillatory flows. It plays a crucial role in the generation ofLangmuir circulations.^[3]For nonlinear andperiodic water waves, accurate results on the Stokes drift have been computed and tabulated.^[4]

TheLagrangian motion of a fluid parcel withposition vectorx =ξ(α, t) in the Eulerian coordinates is given by^[5]

${\displaystyle \dot{\mathit{\xi }}=\frac{\mathrm{\partial }\mathit{\xi }}{\mathrm{\partial }t}=\mathbf{u}(\mathit{\xi }(\mathit{\alpha },t),t),}$

where

∂ξ/∂t is thepartial derivative ofξ(α,t) with respect tot,

ξ(α,t) is the Lagrangian position vector of a fluid parcel,

u(x,t) is the Eulerianvelocity,

x is the position vector in theEulerian coordinate system,

α is the position vector in theLagrangian coordinate system,

t istime.

Often, the Lagrangian coordinatesα are chosen to coincide with the Eulerian coordinatesx at the initial timet =t₀:^[5]

${\displaystyle \mathit{\xi }(\mathit{\alpha },t_0)=\mathit{\alpha }.}$

If theaverage value of a quantity is denoted by an overbar, then the average Eulerian velocity vectorū_E and average Lagrangian velocity vectorū_L are

Different definitions of theaverage may be used, depending on the subject of study (seeergodic theory):

• time average,
• space average,
• ensemble average,
• phase average.

The Stokes drift velocityū_S is defined as the difference between the average Eulerian velocity and the average Lagrangian velocity:^[6]

${\displaystyle {\overline{\mathbf{u}}}_{\text{S}}={\overline{\mathbf{u}}}_{\text{L}}-{\overline{\mathbf{u}}}_{\text{E}}.}$

In many situations, themapping of average quantities from some Eulerian positionx to a corresponding Lagrangian positionα forms a problem. Since a fluid parcel with labelα traverses along apath of many different Eulerian positionsx, it is not possible to assignα to a uniquex.A mathematically sound basis for an unambiguous mapping between average Lagrangian and Eulerian quantities is provided by the theory of thegeneralized Lagrangian mean (GLM) byAndrews and McIntyre (1978).

For the Eulerian velocity as a monochromatic wave of any nature in a continuous medium:${\displaystyle u=\hat{u}\sin (kx-\omega t),}$ one readily obtains by theperturbation theory – with${\displaystyle k\hat{u}/\omega }$ as a small parameter – for the particle position${\displaystyle x=\xi ({\xi }_0,t)}$:

${\displaystyle \dot{\xi }=u(\xi ,t)=\hat{u}\sin (k\xi -\omega t),}$

${\displaystyle \xi ({\xi }_0,t)\approx {\xi }_0+\frac{\hat{u}}{\omega }\cos (k{\xi }_0-\omega t)-\frac{1}{4}\frac{k{\hat{u}}^2}{{\omega }^2}\sin 2(k{\xi }_0-\omega t)+\frac{1}{2}\frac{k{\hat{u}}^2}{\omega }t.}$

Here the last term describes the Stokes drift velocity${\displaystyle \frac{1}{2}k{\hat{u}}^2/\omega .}$^[7]

The Stokes drift was formulated forwater waves byGeorge Gabriel Stokes in 1847. For simplicity, the case ofinfinitely deep water is considered, withlinearwave propagation of asinusoidal wave on thefree surface of a fluid layer:^[8]

${\displaystyle \eta =a\cos (kx-\omega t),}$

where

η is theelevation of thefree surface in thez direction (meters),

a is the waveamplitude (meters),

k is thewave number:k = 2π/λ (radians per meter),

ω is theangular frequency:ω = 2π/T (radians persecond),

x is the horizontalcoordinate and the wave propagation direction (meters),

z is the verticalcoordinate, with the positivez direction pointing out of the fluid layer (meters),

λ is thewave length (meters),

T is thewave period (seconds).

As derived below, the horizontal componentū_S(z) of the Stokes drift velocity for deep-water waves is approximately:^[9]

${\displaystyle {\overline{u}}_{\text{S}}\approx \omega k{a}^2{\text{e}}^{2kz}=\frac{4{\pi }^2{a}^2}{\lambda T}{\text{e}}^{4\pi z/\lambda }.}$

As can be seen, the Stokes drift velocityū_S is a nonlinear quantity in terms of the waveamplitudea. Further, the Stokes drift velocity decays exponentially with depth: at a depth of a quarter wavelength,z = −λ/4, it is about 4% of its value at the meanfree surface,z = 0.

It is assumed that the waves are ofinfinitesimalamplitude and thefree surface oscillates around themean levelz = 0. The waves propagate under the action of gravity, with aconstantaccelerationvector bygravity (pointing downward in the negativez direction). Further the fluid is assumed to beinviscid^[10] andincompressible, with a constantmass density. The fluidflow isirrotational. At infinite depth, the fluid is taken to be atrest.

Now theflow may be represented by avelocity potentialφ, satisfying theLaplace equation and^[8]

${\displaystyle \phi =\frac{\omega }{k}a{\text{e}}^{kz}\sin (kx-\omega t).}$

In order to havenon-trivial solutions for thiseigenvalue problem, thewave length andwave period may not be chosen arbitrarily, but must satisfy the deep-waterdispersion relation:^[11]

${\displaystyle {\omega }^2=gk}$

withg theacceleration bygravity in (m/s²). Within the framework oflinear theory, the horizontal and vertical components,ξ_x andξ_z respectively, of the Lagrangian positionξ are^[9]

The horizontal componentū_S of the Stokes drift velocity is estimated by using aTaylor expansion aroundx of the Eulerian horizontal velocity componentu_x = ∂ξ_x / ∂t at the positionξ:^[5]

• Coriolis–Stokes force
• Darwin drift
• Lagrangian and Eulerian coordinates
• Material derivative

• 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.S2CID 4988274.
• 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.S2CID 120420719.
• 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.

• Fluid dynamics
• Water waves

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