Stokes drift in deep water waves, with a
wave length of about twice the water depth. Click
here 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.
Stokes drift in shallow
water waves, with a
wave length much longer than the water depth. Click
here 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.
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]
Often, the Lagrangian coordinates α are chosen to coincide with the Eulerian coordinates x at the initial time t = t0 :[5]
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 ūE and average Lagrangian velocity vector ūL are:
Different definitions of the
average may be used, depending on the subject of study, see
ergodic theory:
The Stokes drift velocity ūS is defined as the difference between the average Eulerian velocity and the average Lagrangian velocity: [6]
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 mathematically 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: A one-dimensional compressible flow
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]
Example: Deep water waves
Stokes drift under periodic waves in deep water, for a
periodT = 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).
As derived below, the horizontal component ūS(z) of the Stokes drift velocity for deep-water waves is approximately:[9]
As can be seen, the Stokes drift velocity ūS is a
nonlinear quantity in terms of the wave
amplitudea. Further, the Stokes drift velocity decays exponentially with depth: at a depth of a quarter wavelength, z = -¼ λ, it is about 4% of its value at the mean
free surface, z = 0.
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]
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.
^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.
S2CID122673867. J.M. Williams (1985). Tables of progressive gravity waves. Pitman.
ISBN978-0-273-08733-5.
^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.