Stokes drift

For a pure wave 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 flow velocity of a fluid parcel, and the average 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 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 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 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.Stokes drift may occur in all instances of oscillatory flow which are inhomogeneous in space. For instance in water waves, tides and atmospheric 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 the flow velocity. The light-blue line gives the 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 is zero.
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.]
[
water waves, with a wave 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 the flow velocity. The light-blue line gives the 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 is zero.
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.]

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 of Langmuir circulations. [See "e.g." Craik (1985), page 120.] For nonlinear 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.] :where "∂ξ / ∂t" is the partial derivative of "ξ(α,t)" with respect to "t", and:"ξ(α,t)" is the Lagrangian position vector of a fluid parcel, in metre,:"u(x,t)" is the Eulerian velocity, in metre per second,:"x" ix the position vector in the Eulerian coordinate system, in metre,:"α" is the position vector in the Lagrangian coordinate system, in metre,:"t" is the time, in second.Often, the Lagrangian coordinates "α" are chosen to coincide with the Eulerian coordinates "x" at the initial time "t = t₀" ::But also other ways of labeling 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:: Different definitions of the average 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.] :In many situations, the 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 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 by George Gabriel Stokes in 1847. For simplicity, the case of infinite-deep water is considered, with linear wave propagation of a sinusoidal wave on the free surface of a fluid layer:See "e.g." Phillips (1977), page 37.] :$eta,\; =,\; a,\; cos,\; left(\; k\; x\; -\; omega\; t\; ight),$where:"η" is the elevation of the free surface in the "z"-direction (metre),:"a" is the wave amplitude (metre),:"k" is the wave number: "k = 2π / λ" (radians per metre),:"ω" is the angular frequency: "ω = 2π / T" (radians per second),:"x" is the horizontal coordinate and the wave propagation direction (metre),:"z" is the vertical coordinate, with the positive "z" direction pointing out of the fluid layer (metre),:"λ" is the wave length (metre), and:"T" is the wave period (second).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 constant acceleration vector by gravity (pointing downward in the negative "z"-direction). Further the fluid is assumed to be inviscid [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.] and incompressible, with a constant mass density. The fluid flow is irrotational. At infinite depth, the fluid is taken to be at rest.

Now the flow may be represented by a velocity potential "φ", satisfying the Laplace equation and:$varphi,\; =,\; frac\{omega\}\{k\},\; a;\; ext\{e\}^\{k\; z\},\; sin,\; left(\; k\; x\; -\; omega\; t\; ight).$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 relation:See "e.g." Phillips (1977), page 38.] :$omega^2,\; =,\; g,\; k.$with "g" the acceleration by gravity in ("m / s²"). Within the framework of linear theory, the horizontal and vertical components, "ξ_x" and "ξ_z" respectively, of the Lagrangian position "ξ" are:: The horizontal component "ū_S" of the Stokes drift velocity is estimated by using a Taylor expansion around "x" of the Eulerian horizontal-velocity component "u_x = ∂ξ_x / ∂t" at the position "ξ" ::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 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".

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