**Article**

*CraikâLeibovich vortex force*In
fluid dynamics, the **CraikâLeibovich (CL) vortex force** describes a
forcing of the
mean flow through
waveâcurrent interaction, specifically between the
Stokes drift velocity and the mean-flow
vorticity. The CL vortex force is used to explain the generation of
Langmuir circulations by an
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
Irving Langmuir in the 1930s).

## Description

The CL vortex force is

with the ( Lagrangian) Stokes drift velocity and vorticity (i.e. the curl of the Eulerian mean-flow velocity ). Further is the fluid density and 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 or
NavierâStokes equations. For constant density, the momentum equation (divided by the density ) is:^{
[1]}

with

- (a): temporal acceleration
- (b): convective acceleration
- (c): Coriolis force due to the angular velocity of the Earth's rotation
- (d): CoriolisâStokes force
- (e): gradient of the augmented pressure
- (f): CraikâLeibovich vortex force
- (g): viscous force due to the kinematic viscosity

The CL vortex force can be obtained by several means. Originally, Craik and Leibovich used
perturbation theory. An easy way to derive it is through the
generalized Lagrangian mean theory.^{
[1]} It can also be derived through a
Hamiltonian mechanics description.^{
[2]}

## Notes

## References

- Craik, A.D.D. (1990),
*Wave interactions and fluid flows*, Cambridge University Press, pp. 113â122, ISBN 0-521-36829-4, LCCN lc85007803 - Holm, D.D. (1996), "The ideal CraikâLeibovich equations",
*Physica D*,**98**(2): 415â441, Bibcode: 1996PhyD...98..415H, doi: 10.1016/0167-2789(96)00105-4 - Leibovich, S. (1980), "On waveâcurrent interaction theories of Langmuir circulations",
*Journal of Fluid Mechanics*,**99**(4): 715â724, Bibcode: 1980JFM....99..715L, doi: 10.1017/S0022112080000857 - Leibovich, S. (1983), "The form and dynamics of Langmuir circulations",
*Annual Review of Fluid Mechanics*,**15**: 391â427, Bibcode: 1983AnRFM..15..391L, doi: 10.1146/annurev.fl.15.010183.002135 - Sullivan, P.P.; McWilliams, J.C. (2010), "Dynamics of winds and currents coupled to surface waves",
*Annual Review of Fluid Mechanics*,**42**: 19â42, Bibcode: 2010AnRFM..42...19S, doi: 10.1146/annurev-fluid-121108-145541 - Thorpe, S.A. (2004), "Langmuir circulation",
*Annual Review of Fluid Mechanics*,**36**: 55â79, Bibcode: 2004AnRFM..36...55T, doi: 10.1146/annurev.fluid.36.052203.071431