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.^{
[2]}

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.^{
[3]}
For
nonlinear and
periodic water waves, accurate results on the Stokes drift have been computed and tabulated.^{
[4]}

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

where *∂ ξ / ∂t* is the
partial derivative of

is the Lagrangian position vector of a fluid parcel,**ξ**(**α**,t)is the Eulerian velocity,**u**(**x**,t)is the position vector in the Eulerian coordinate system,**x**is the position vector in the Lagrangian coordinate system,**α***t*is the time.

Often, the Lagrangian coordinates * α* are chosen to coincide with the Eulerian coordinates

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

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.

The Stokes drift velocity * ū_{S}* is defined as the difference between the average Eulerian velocity and the average Lagrangian velocity:

In many situations, the
mapping of average quantities from some Eulerian position * x* to a corresponding Lagrangian position

For the Eulerian velocity as a monochromatic wave of any nature in a continuous medium: one readily obtains by the perturbation theory – with as a small parameter – for the particle position

Here the last term describes the Stokes drift velocity ^{
[7]}

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:^{
[8]}

where

*η*is the elevation of the free surface in the*z*-direction (meters),*a*is the wave amplitude (meters),*k*is the wave number:*k = 2π / λ*( radians per meter),*ω*is the angular frequency:*ω = 2π / T*( radians per second),*x*is the horizontal coordinate and the wave propagation direction (meters),*z*is the vertical coordinate, with the positive*z*direction pointing out of the fluid layer (meters),*λ*is the wave length (meters), and*T*is the wave period ( seconds).

As derived below, the horizontal component *ū _{S}*(

As can be seen, the Stokes drift velocity *ū _{S}* is a
nonlinear quantity in terms of the wave
amplitude

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^{
[10]} 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^{
[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 / s ^{2}*). Within the framework of
linear theory, the horizontal and vertical components,

The horizontal component *ū _{S}* of the Stokes drift velocity is estimated by using a
Taylor expansion around

- 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.

**^**See Kubota (1994).**^**See Craik (1985), page 105–113.**^**See*e.g.*Craik (1985), page 120.**^**Solutions of the particle trajectories in fully nonlinear periodic waves and the Lagrangian wave period they experience can for instance be found in:

J.M. Williams (1981). "Limiting gravity waves in water of finite depth".*Philosophical Transactions of the Royal Society A*.**302**(1466): 139–188. Bibcode: 1981RSPTA.302..139W. doi: 10.1098/rsta.1981.0159. S2CID 122673867.

J.M. Williams (1985).*Tables of progressive gravity waves*. Pitman. ISBN 978-0-273-08733-5.- ^
^{a}^{b}^{c}See Phillips (1977), page 43. **^**See*e.g.*Craik (1985), page 84.**^**See Falkovich (2011), pages 71–72. There is a typo in the coefficient of the superharmonic term in Eq. (2.20) on page 71, i.e instead of- ^
^{a}^{b}See*e.g.*Phillips (1977), page 37. - ^
^{a}^{b}See Phillips (1977), page 44. Or Craik (1985), page 110. **^**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.**^**See*e.g.*Phillips (1977), page 38.