Stokes drift

The

∂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]

${\displaystyle {\boldsymbol {\xi }}({\boldsymbol {\alpha }},t_{0})={\boldsymbol {\alpha }}.}$

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

${\displaystyle {\begin{aligned}{\bar {\mathbf {u} }}_{\text{E}}&={\overline {\mathbf {u} (\mathbf {x} ,t)}},\\{\bar {\mathbf {u} }}_{\text{L}}&={\overline {{\dot {\boldsymbol {\xi }}}({\boldsymbol {\alpha }},t)}}={\overline {\left({\frac {\partial {\boldsymbol {\xi }}({\boldsymbol {\alpha }},t)}{\partial t}}\right)}}={\overline {{\boldsymbol {u}}{\big (}{\boldsymbol {\xi }}({\boldsymbol {\alpha }},t),t{\big )}}}.\end{aligned}}}$

Different definitions of the average may be used, depending on the subject of study (see ergodic theory):

• time average,
• space average,
• ensemble average
  ,
• phase average.

The Stokes drift velocity ū_S is defined as the difference between the average Eulerian velocity and the average Lagrangian velocity:^[6]

${\displaystyle {\bar {\mathbf {u} }}_{\text{S}}={\bar {\mathbf {u} }}_{\text{L}}-{\bar {\mathbf {u} }}_{\text{E}}.}$

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: ${\displaystyle u={\hat {u}}\sin(kx-\omega t),}$ one readily obtains by the perturbation theory – with ${\displaystyle k{\hat {u}}/\omega }$ as a small parameter – for the particle position ${\displaystyle x=\xi (\xi _{0},t)}$:

${\displaystyle {\dot {\xi }}=u(\xi ,t)={\hat {u}}\sin(k\xi -\omega t),}$

${\displaystyle \xi (\xi _{0},t)\approx \xi _{0}+{\frac {\hat {u}}{\omega }}\cos(k\xi _{0}-\omega t)-{\frac {1}{4}}{\frac {k{\hat {u}}^{2}}{\omega ^{2}}}\sin 2(k\xi _{0}-\omega t)+{\frac {1}{2}}{\frac {k{\hat {u}}^{2}}{\omega }}t.}$

Here the last term describes the Stokes drift velocity ${\displaystyle {\tfrac {1}{2}}k{\hat {u}}^{2}/\omega .}$^[7]

Example: Deep water waves

The Stokes drift was formulated for

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),

T is the

wave period (seconds

).

As derived below, the horizontal component ū_S(z) of the Stokes drift velocity for deep-water waves is approximately:[9]

${\displaystyle {\bar {u}}_{\text{S}}\approx \omega ka^{2}{\text{e}}^{2kz}={\frac {4\pi ^{2}a^{2}}{\lambda T}}{\text{e}}^{4\pi z/\lambda }.}$

As can be seen, the Stokes drift velocity ū_S is a nonlinear^{[disambiguation needed]} 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

• 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.
  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.

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.
  doi:10.1175/1520-0485(1994)024<1059:AMFTAO>2.0.CO;2
  .