Stokes drift: Difference between revisions

More generally, the Stokes drift velocity is the difference between the [[average]] [[Lagrangian and Eulerian coordinates|Lagrangian]] [[flow velocity]] of a fluid parcel, and the average [[Lagrangian and Eulerian coordinates|Eulerian]] [[flow velocity]] of the [[fluid]] at a fixed position. This [[nonlinear system|nonlinear]] phenomenon is named after [[George Gabriel Stokes]], who derived expressions for this drift in [[#Stokes1847|his 1847 study]] of [[water waves]].

The Stokes drift is important for the [[mass transfer]] of allvarious kindkinds of materialsmaterial and organisms by oscillatory flows. FurtherIt theplays Stokesa driftcrucial isrole important forin the generation of [[Langmuir circulation]]s.<ref>See ''e.g.'' [[#Craik1985|Craik (1985)]], page 120.</ref>

For [[nonlinear]] and [[periodic function|periodic]] water waves, accurate results on the Stokes drift have been computed and tabulated.<ref>Solutions of the particle trajectories in fully nonlinear periodic waves and the Lagrangian wave period they experience can for instance be found in:<br>{{cite journal| author=J.M. Williams| title=Limiting gravity waves in water of finite depth | journal=Philosophical Transactions of the Royal Society A | volume=302 | issue=1466 | pages=139–188 | year=1981| doi=10.1098/rsta.1981.0159 |bibcode = 1981RSPTA.302..139W | s2cid=122673867 }}<br>{{cite book| title=Tables of progressive gravity waves | author=J.M. Williams | year=1985 | publisher=Pitman | isbn=978-0-273-08733-5 }}</ref>

The [[Lagrangian and Eulerian coordinates|Lagrangian motion]] of a fluid parcel with [[position vector]] ''x = '''ξ'''('''α''',&nbsp;t)'' in the Eulerian coordinates is given by:<ref name=Phil1977p43>See [[#Phillips1977|Phillips (1977)]], page 43.</ref>

: <math>

\dot{\boldsymbol{\xi}}\, =\, \frac{\partial \boldsymbol{\xi}}{\partial t}\,
=\, \boldsymbolmathbf{u}\big(\boldsymbol{\xi}(\boldsymbol{\alpha}, t), t\big),

: ∂'''''ξ'''('''α/∂''',t)'' is the Lagrangian [[positionpartial vectorderivative]] of a'''''ξ'''''('''''α''''', fluid''t'') parcel,with inrespect metersto ''t'',

: '''''uξ'''''('''x''α''''', ''t)'') is the EulerianLagrangian [[velocity]],position vector inof metersa perfluid [[second]]parcel,

: '''''α''x''' is the [[position vector]] in the [[Lagrangian and Eulerian coordinatesspecification of the flow field|LagrangianEulerian coordinate system]], in meters,

: ''t'' is the [[time]], in [[second]]s.

Often, the Lagrangian coordinates '''''α''''' are chosen to coincide with the Eulerian coordinates '''''x''''' at the initial time ''t'' = ''t''₀'' :<ref name=Phil1977p43/>

: <math>

\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:

\overline{bar\boldsymbolmathbf{u}_\text{E}_E\, &=\, \overline{\boldsymbolmathbf{u}(\boldsymbolmathbf{x}, t)},

\overline{bar\boldsymbolmathbf{u}_\text{L}_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)}.

Different definitions of the [[average]] may be used, depending on the subject of study, (see [[Ergodic theory#Ergodic theorems|ergodic theory]]):

* [[time]] average,

* [[space]] average,

* [[ensemble average]] and,

* [[phase (waves)|phase]] average.

The Stokes drift velocity '''''ū'''_S'' is defined as the difference between the average Eulerian velocity and the average Lagrangian velocity: <ref>See ''e.g.'' [[#Craik1985|Craik (1985)]], page 84.</ref>

\overline{bar\boldsymbolmathbf{u}_\text{S}_S\, =\, \overline{bar\boldsymbolmathbf{u}_\text{L}_L\, -\, \overline{bar\boldsymbolmathbf{u}_\text{E}_E.

In many situations, the [[map (mathematics)|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 (topology)|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[[generalized Lagrangian Mean''mean]] (GLM) by [[#Andrews-McIntyre1978|Andrews and McIntyre (1978)]].

==Example: OneA one-dimensional compressible flow==

For the Eulerian velocity as a monochromatic wave of any nature in a continuous medium: <math>u = \hat{u} \sin\left( kx - \omega t \right),</math> one readily obtains by the [[perturbation theory]] –{{snd}} with <math>k\hat{u}/\omega</math> as a small parameter –{{snd}} for the particle position {{nobr|<math>x = \xi(\xi_0, t):</math>:}}

: <math>\dot{{\xi}} =\, {u}({\xi}, t) = \hat{u} \sin\, \left( k \xi - \omega t \right),</math>

: <math>

\xi(\xi_0, t) \approx \xi_0 + \frac{\hat{u}}{\omega} \cos(k\xi_0 - \omega t) - \frac14 \frac{k\hat{u}^2}{\omega^2} \sin2sin 2(k\xi_0 - \omega t) + \frac12 \frac{k\hat{u}^2}{\omega} t.

Here the last term describes the Stokes drift velocity <math>\tfrac12 k\hat{u}^2/\omega.</math><ref>See [[#Falkovich|Falkovich (2011)]], pages 71–72. There is a typo in the coefficient of the superharmonic term in Eq. &nbsp;(2.20) on page &nbsp;71, i.e <math>-\tfrac14</math> instead of <math>+\tfrac12.</math></ref>

The Stokes drift was formulated for [[water waves]] by [[George Gabriel Stokes]] in 1847. For simplicity, the case of [[Infinity|infiniteinfinitely]]- deep water is considered, with [[linear]] [[wave propagation]] of a [[sinusoidal]] wave on the [[free surface]] of a fluid layer:<ref name=Phil1977p37>See ''e.g.'' [[#Phillips1977|Phillips (1977)]], page 37.</ref>

: <math>

\eta\, =\, a\, \cos\, \left( k xkx - \omega t \right),

: ''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 2''π''/ ''λ'' ([[radian]]s per meter),

: ''ω'' is the [[angular frequency]]: ''ω'' = 2π 2''π''/ ''T'' ([[radian]]s 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]] ([[second]]s).

As derived below, the horizontal component ''ū''_S''(''z'') of the Stokes drift velocity for deep-water waves is approximately:<ref name=Phil1977p44>See [[#Phillips1977|Phillips (1977)]], page 44. Or [[#Craik1985|Craik (1985)]], page 110.</ref>

:{{Equation box 1|equation=<math>

\overlinebar{u}_S_\,text{S} \approx\, \omega\, k\, a^2\, \text{e}^{2 k z2kz}\,

=\, \frac{4\pi^2\, a^2}{\lambda\, T}\, \text{e}^{4\pi\, z / \lambda}.

</math>}}

As can be seen, the Stokes drift velocity ''ū''_S'' 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 quartquarter wavelength, ''z'' = -¼ −''λ''/4, it is about 4% of its value at the mean [[free surface]], ''z ''&nbsp;= &nbsp;0''.

It is assumed that the waves are of [[infinitesimal]] [[amplitude]] and the [[free surface]] oscillates around the [[mean]] level ''z ''&nbsp;= &nbsp;0''. The waves propagate under the action of gravity, with a [[wikt:constant|constant]] [[acceleration]] [[Vector (geometric)|vector]] by [[gravity]] (pointing downward in the negative ''z''- direction). Further the fluid is assumed to be [[inviscid]]<ref>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 [[Stokes boundary layer|boundary layers]] near bed and free surface, see for instance [[#Longuet-Higgins1953|Longuet-Higgins (1953)]]. Or [[#Phillips1977|Phillips (1977)]], pages &nbsp;53–58.</ref> and [[incompressible]], with a [[wikt:constant|constant]] [[mass density]]. The fluid [[flow (mathematics)|flow]] is [[irrotational]]. At infinite depth, the fluid is taken to be at [[rest (physics)|rest]].

\varphi\, =\, \frac{\omega}{k}\, a\; \text{e}^{k zkz}\, \sin\, \left( k xkx - \omega t \right).

: <math>

\omega^2\, =\, g\, k.gk

with ''g'' the [[acceleration]] by [[gravity]] in (''m / s²''). Within the framework of [[linear]] theory, the horizontal and vertical components, ''ξ_x'' and ''ξ_z'' respectively, of the Lagrangian position '''''ξ''''' are:<ref name=Phil1977p44/>

\xi_x\, &=\, x\, +\, \int\, \frac{\partial \varphi}{\partial x}\;, \text{d}t\,

=\, x\, -\, a\, \text{e}^{k zkz}\, \sin\, \left( k xkx - \omega t \right),

\xi_z\, &=\, z\, +\, \int\, \frac{\partial \varphi}{\partial z}\;, \text{d}t\,

=\, z\, +\, a\, \text{e}^{k zkz}\, \cos\, \left( k xkx - \omega t \right).

The horizontal component ''ū''_S'' of the Stokes drift velocity is estimated by using a [[Taylor expansion]] around '''''x''''' of the Eulerian horizontal- velocity component ''u_x'' = ∂ξ∂''ξ_x'' / ∂t∂''t'' at the position '''''ξ''''' :<ref name=Phil1977p43/>

\overlinebar{u}_S_\,text{S}

&=\, \overline{u_x(\boldsymbol{\xi}, t)}\, -\, \overline{u_x(\boldsymbolmathbf{x}, t)}\,

&=\, \overline{\left[

u_x(\boldsymbolmathbf{x}, t)\,

+\, \left( \xi_x - x \right)\, \frac{\partial u_x(\boldsymbolmathbf{x}, t)}{\partial x}\,

+\, \left( \xi_z - z \right)\, \frac{\partial u_x(\boldsymbolmathbf{x}, t)}{\partial z}\,

+\, \cdots

\right] }

-\, \overline{u_x(\boldsymbolmathbf{x} ,t)}

&\approx\, \overline{\left( \xi_x - x \right)\, \frac{\partial^2 \xi_x}{\partial x\, \partial t} }\,

+\, \overline{\left( \xi_z - z \right)\, \frac{\partial^2 \xi_x}{\partial z\, \partial t} }

&=\, \overline{ \biggleft[ - a\, \text{e}^{k zkz}\, \sin\, \left( k xkx - \omega t )\right) \bigg]\,

\biggleft[ -\omega\, k\, a\,ka \text{e}^{k zkz}\, \sin\, \left( k xkx - \omega t )\right) \bigg] }\,

&+\, \overline{ \biggleft[ a\, \text{e}^{k zkz}\, \cos\, \left( k xkx - \omega t )\right) \bigg]\,

\biggleft[ \omega\, k\, a\,ka \text{e}^{k zkz}\, \cos\, \left( k xkx - \omega t )\right) \bigg] }\,

&=\, \overline{ \omega\, k\, aka^2\, \text{e}^{2 k z2kz}\,

\biggleft[ \sin^2\, \left( k xkx - \omega t \right) + \cos^2\, \left( k xkx - \omega t )\right) \bigg] }

&=\, \omega\, k\, aka^2\, \text{e}^{2 k z2kz}.

* [[Coriolis-StokesCoriolis–Stokes force]]

* [[Darwin drift]]

* [[Lagrangian and Eulerian coordinates]]

* [[Material derivative]]

*{{cite journal | author= A.D.D. Craik | year= 2005 | title= George Gabriel Stokes on water wave theory | journal= Annual Review of Fluid Mechanics | volume= 37 | issue= 1 | pages= 23–42 | doi= 10.1146/annurev.fluid.37.061903.175836 |bibcode = 2005AnRFM..37...23C }}

*{{cite journal | author=D.G. Andrews | author2=M.E. McIntyre | lastauthor2-authorlink=Michael E. McIntyre | name-amplist-style=yesamp | year= 1978 | title= An exact theory of nonlinear waves on a Lagrangian mean flow | journal= Journal of Fluid Mechanics | volume= 89 | pages= 609–646 | doi= 10.1017/S0022112078002773 |bibcode = 1978JFM....89..609A | issue=4 | s2cid=4988274 |ref=Andrews-McIntyre1978}}

*{{cite book | author= A.D.D. Craik | title=Wave interactions and fluid flows | year=1985 | publisher=Cambridge University Press | isbn=978-0-521-36829-42|ref=Craik1985}}

*{{cite journal | author= M.S. Longuet-Higgins | authorlinkauthor-link=Michael S. Longuet-Higgins | year= 1953 | title= Mass transport in water waves | journal= Philosophical Transactions of the Royal Society A | volume= 245 | pages= 535–581 | url= http://rsta.royalsocietypublishing.org/content/245/903/535 | doi= 10.1098/rsta.1953.0006 |bibcode = 1953RSPTA.245..535L | issue=903| s2cid=120420719 |ref=Longuet-Higgins1953}}

*{{cite book| first=O.M. | last=Phillips | author-link=Owen Martin Phillips |title=The dynamics of the upper ocean |publisher=Cambridge University Press | year=1977 | edition=2nd | isbn=978-0-521-29801-68 |ref=Phillips1977}}