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.
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.^{[1]}
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.^{[2]} For nonlinear and periodic water waves, accurate results on the Stokes drift have been computed and tabulated.^{[3]}
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The Lagrangian motion of a fluid parcel with position vector x = ξ(α,t) in the Eulerian coordinates is given by:^{[4]}
where ∂ξ / ∂t is the partial derivative of ξ(α,t) with respect to t, and
Often, the Lagrangian coordinates α are chosen to coincide with the Eulerian coordinates x at the initial time t = t_{0} :^{[4]}
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:
Now, the Stokes drift velocity ū_{S} equals^{[5]}
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).
The Stokes drift was formulated for water waves by George Gabriel Stokes in 1847. For simplicity, the case of infinitedeep water is considered, with linear wave propagation of a sinusoidal wave on the free surface of a fluid layer:^{[6]}
where
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 zdirection). Further the fluid is assumed to be inviscid^{[7]} 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^{[6]}
In order to have nontrivial solutions for this eigenvalue problem, the wave length and wave period may not be chosen arbitrarily, but must satisfy the deepwater dispersion relation:^{[8]}
with g the acceleration by gravity in (m / s^{2}). Within the framework of linear theory, the horizontal and vertical components, ξ_{x} and ξ_{z} respectively, of the Lagrangian position ξ are:^{[9]}
The horizontal component ū_{S} of the Stokes drift velocity is estimated by using a Taylor expansion around x of the Eulerian horizontalvelocity component u_{x} = ∂ξ_{x} / ∂t at the position ξ :^{[4]}
Performing the averaging, the horizontal component of the Stokes drift velocity for deepwater waves is approximately:^{[9]}
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.
