View source for Physics:Craik–Leibovich vortex force

{{short description|Forcing of the mean flow through wave–current interaction}}
In [[Physics:Fluid dynamics|fluid dynamics]], the '''Craik–Leibovich (CL) vortex force''' describes a [[Physics:Force|forcing]] of the [[Physics:Mean flow|mean flow]] through [[Physics:Wave–current interaction|wave–current interaction]], specifically between the [[Physics:Stokes drift|Stokes drift]] velocity and the mean-flow [[Vorticity|vorticity]]. The CL vortex force is used to explain the generation of [[Earth:Langmuir circulation|Langmuir circulation]]s by an [[Physics:Hydrodynamic stability|instability mechanism]]. The CL vortex-force mechanism was derived and studied by Sidney Leibovich and Alex D. D. Craik in the 1970s and 80s, in their studies of Langmuir circulations (discovered by [[Biography:Irving Langmuir|Irving Langmuir]] in the 1930s).

==Description==
The CL vortex force is

:<math>\rho\, \boldsymbol{u}_S \times \boldsymbol{\omega},</math>

with <math>\boldsymbol{u}_S</math> the (Lagrangian) Stokes drift [[Flow velocity|velocity]] and vorticity <math>\boldsymbol{\omega}=\nabla\times\boldsymbol{u}</math> (i.e. the [[Curl (mathematics)|curl]] of the Eulerian mean-flow velocity <math>\boldsymbol{u}</math>). Further <math>\rho</math> is the fluid [[Physics:Density|density]] and <math>\nabla\times</math> is the curl operator. 

The CL vortex force finds its origins in the appearance of the Stokes drift in the convective acceleration terms in the mean momentum equation of the [[Euler equations (fluid dynamics)|Euler equations]] or [[Physics:Navier–Stokes equations|Navier–Stokes equations]]. For constant density, the momentum equation (divided by the density <math>\rho</math>) is:<ref name=Leibovich1980>{{harvtxt|Leibovich|1980}}</ref>

:<math>
  \underbrace{ \partial_t \boldsymbol{u} }_\text{(a)} 
  + \underbrace{ \boldsymbol{u}\cdot\nabla\boldsymbol{u} }_\text{(b)} 
  + \underbrace{ 2\boldsymbol{\Omega}\times\boldsymbol{u} }_\text{(c)}
  + \underbrace{ 2\boldsymbol{\Omega}\times\boldsymbol{u}_S }_\text{(d)}
  + \underbrace{ \nabla ( \pi +\boldsymbol{u}\cdot\boldsymbol{u}_S ) }_\text{(e)}
  = \underbrace{ \boldsymbol{u}_S \times ( \nabla\times\boldsymbol{u} ) }_\text{(f)}
  + \underbrace{ \nu\, \nabla\cdot\nabla \boldsymbol{u} }_\text{(g)},
</math>

with
* (a): temporal acceleration
* (b): convective acceleration
* (c): [[Physics:Coriolis force|Coriolis force]] due to the [[Physics:Angular velocity|angular velocity]] <math>\boldsymbol{\Omega}</math> of the [[Earth's rotation]]
* (d): [[Physics:Coriolis–Stokes force|Coriolis–Stokes force]]
* (e): [[Gradient|gradient]] of the augmented [[Physics:Pressure|pressure]]
* (f): Craik–Leibovich vortex force
* (g): viscous force due to the [[Physics:Kinematic viscosity|kinematic viscosity]] <math>\nu</math>

The CL vortex force can be obtained by several means. Originally, Craik and Leibovich used [[Physics:Perturbation theory|perturbation theory]]. An easy way to derive it is through the [[Physics:Generalized Lagrangian mean|generalized Lagrangian mean]] theory.<ref name=Leibovich1980/> It can also be derived through a [[Physics:Hamiltonian mechanics|Hamiltonian mechanics]] description.<ref>{{harvtxt|Holm|1996}}</ref>

==Notes==
{{reflist}}

==References==
{{ref begin}}
* {{citation
| title=Wave interactions and fluid flows
| last=Craik
| first=A.D.D.
| isbn=0-521-36829-4
| lccn=lc85007803
| year=1990
| publisher=Cambridge University Press
| pages=113–122
}}
* {{citation
| title=The ideal Craik–Leibovich equations
| journal=Physica D
| volume=98
| issue=2
| pages=415–441
| year=1996
| doi=10.1016/0167-2789(96)00105-4
| first=D.D.
| last=Holm
| bibcode=1996PhyD...98..415H
}}
* {{citation
| last=Leibovich
| first=S. 
| year=1980
| title=On wave–current interaction theories of Langmuir circulations
| journal=Journal of Fluid Mechanics
| volume=99
| issue=4
| pages=715–724
| doi=10.1017/S0022112080000857
| bibcode=1980JFM....99..715L
| s2cid=14996095 
}}
* {{citation 
| title=The form and dynamics of Langmuir circulations
| journal=[[Physics:Annual Review of Fluid Mechanics|Annual Review of Fluid Mechanics]]
| volume=15
| pages=391–427
| year=1983
| doi=10.1146/annurev.fl.15.010183.002135
| first=S. 
| last=Leibovich
| bibcode=1983AnRFM..15..391L
}}
* {{citation
| title=Dynamics of winds and currents coupled to surface waves
| journal=[[Physics:Annual Review of Fluid Mechanics|Annual Review of Fluid Mechanics]]
| volume=42
| pages=19–42
| year=2010
| doi=10.1146/annurev-fluid-121108-145541
| first1=P.P.
| last1=Sullivan
| first2=J.C. 
| last2=McWilliams
| bibcode=2010AnRFM..42...19S
}}
* {{citation
| journal=[[Physics:Annual Review of Fluid Mechanics|Annual Review of Fluid Mechanics]]
| year=2004
| volume=36
| pages=55–79
| doi=10.1146/annurev.fluid.36.052203.071431
| title=Langmuir circulation
| first=S.A. 
| last=Thorpe
| bibcode=2004AnRFM..36...55T
}}
{{ref end}}

{{Physical oceanography}}

{{DEFAULTSORT:Craik-Leibovich vortex force}}
[[Category:Fluid dynamics]]
[[Category:Water waves]]

{{Sourceattribution|Craik–Leibovich vortex force}}