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. 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. The Stokes drift is important for the mass transfer of various kinds of material and organisms by oscillatory flows. It plays a crucial role in the generation of Langmuir circulations. For nonlinear and periodic water waves, accurate results on the Stokes drift have been computed and tabulated.

Mathematical description The Lagrangian motion of a fluid parcel with position vector x = ξ(α, t) in the Eulerian coordinates is given by

ξ ˙ = ∂ ξ ∂ t = u ( ξ ( α , t ) , t ) , {\displaystyle {\dot {\boldsymbol {\xi }}}={\frac {\partial {\boldsymbol {\xi }}}{\partial t}}=\mathbf {u} {\big (}{\boldsymbol {\xi }}({\boldsymbol {\alpha }},t),t{\big )},}

where

∂ξ/∂t is the partial derivative of ξ(α, t) with respect to t, ξ(α, t) is the Lagrangian position vector of a fluid parcel, u(x, t) is the Eulerian velocity, x is the position vector in the Eulerian coordinate system, α is the position vector in the Lagrangian coordinate system, t is time. Often, the Lagrangian coordinates α are chosen to coincide with the Eulerian coordinates x at the initial time t = t0:

ξ ( α , t 0 ) = α . {\displaystyle {\boldsymbol {\xi }}({\boldsymbol {\alpha }},t_{0})={\boldsymbol {\alpha }}.}

If the average value of a quantity is denoted by an overbar, then the average Eulerian velocity vector ūE and average Lagrangian velocity vector ūL are

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