Stokes drift

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.

Stokes drift may occur in all instances of oscillatory flow which are inhomogeneous in space.

However, such an unambiguous description is provided by the Generalized Lagrangian Mean (GLM) theory of Andrews and McIntyre in 1978.

[2] The Stokes drift is important for the mass transfer of various kinds of material and organisms by oscillatory flows.

[3] For nonlinear and periodic water waves, accurate results on the Stokes drift have been computed and tabulated.

It is assumed that the waves are of infinitesimal amplitude and the free surface oscillates around the mean level z = 0.

Further the fluid is assumed to be inviscid[10] and incompressible, with a constant mass density.

Now the flow may be represented by a velocity potential φ, satisfying the Laplace equation and[8] 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:[11] with g the acceleration by gravity in (m/s2).

Within the framework of linear theory, the horizontal and vertical components, ξx and ξz respectively, of the Lagrangian position ξ are[9] The horizontal component ūS of the Stokes drift velocity is estimated by using a Taylor expansion around x of the Eulerian horizontal velocity component ux = ∂ξx / ∂t at the position ξ:[5]

An expanse of driftwood along the northern coast of Washington state . Stokes drift – besides e.g. Ekman drift and geostrophic currents – is one of the relevant processes in the transport of marine debris . [ 1 ]
Stokes drift under periodic waves in deep water, for a period T = 5 s and a mean water depth of 25 m. Left : instantaneous horizontal flow velocities . Right : average flow velocities. Black solid line: average Eulerian velocity; red dashed line: average Lagrangian velocity, as derived from the Generalized Lagrangian Mean (GLM).