Stokes drift
Observe that the
For a pure
More generally, the Stokes drift velocity is the difference between the
The Stokes drift is the difference in end positions, after a predefined amount of time (usually one
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
In the
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.[3] For nonlinear and periodic water waves, accurate results on the Stokes drift have been computed and tabulated.[4]
Mathematical description
The
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:[5]
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
The Stokes drift was formulated for
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 (radiansper meter),
- ω is the angular frequency: ω = 2π/T (radians per second),
- x is the horizontal coordinateand 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),
- T is the wave period (seconds).
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 amplitude a. Further, the Stokes drift velocity decays exponentially with depth: at a depth of a quarter wavelength, z = −λ/4, it is about 4% of its value at the mean free surface, z = 0.
Derivation
It is assumed that the waves are of
Now the
In order to have
with g the
The horizontal component ūS of the Stokes drift velocity is estimated by using a
![{\displaystyle {\begin{aligned}{\bar {u}}_{\text{S}}&={\overline {u_{x}({\boldsymbol {\xi }},t)}}-{\overline {u_{x}(\mathbf {x} ,t)}}\\&={\overline {\left[u_{x}(\mathbf {x} ,t)+(\xi _{x}-x){\frac {\partial u_{x}(\mathbf {x} ,t)}{\partial x}}+(\xi _{z}-z){\frac {\partial u_{x}(\mathbf {x} ,t)}{\partial z}}+\cdots \right]}}-{\overline {u_{x}(\mathbf {x} ,t)}}\\&\approx {\overline {(\xi _{x}-x){\frac {\partial ^{2}\xi _{x}}{\partial x\,\partial t}}}}+{\overline {(\xi _{z}-z){\frac {\partial ^{2}\xi _{x}}{\partial z\,\partial t}}}}\\&={\overline {\left[-a{\text{e}}^{kz}\sin(kx-\omega t)\right]\left[-\omega ka{\text{e}}^{kz}\sin(kx-\omega t)\right]}}\\&+{\overline {\left[a{\text{e}}^{kz}\cos(kx-\omega t)\right]\left[\omega ka{\text{e}}^{kz}\cos(kx-\omega t)\right]}}\\&={\overline {\omega ka^{2}{\text{e}}^{2kz}\left[\sin ^{2}(kx-\omega t)+\cos ^{2}(kx-\omega t)\right]}}\\&=\omega ka^{2}{\text{e}}^{2kz}.\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2e4b479a44dd7c441336dd45b55fabe977097a07)
See also
- Coriolis–Stokes force
- Darwin drift
- Lagrangian and Eulerian coordinates
- Material derivative
References
Historical
- A.D.D. Craik (2005). "George Gabriel Stokes on water wave theory". Annual Review of Fluid Mechanics. 37 (1): 23–42. .
- 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.
Other
- D.G. Andrews & S2CID 4988274.
- A.D.D. Craik (1985). Wave interactions and fluid flows. Cambridge University Press. ISBN 978-0-521-36829-2.
- S2CID 120420719.
- 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. .
Notes
- ^ See Kubota (1994).
- ^ See Craik (1985), page 105–113.
- ^ See e.g. Craik (1985), page 120.
- 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.
- 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.
