In fluid dynamics, the Craik–Leibovich (CL) vortex force describes a forcing of the

mean flow In fluid dynamics, the fluid flow is often decomposed into a mean flow and deviations from the mean. The averaging can be done either in space or in time, or by ensemble averaging In machine learning, particularly in the creation of artificial ...

through

wave–current interaction In fluid dynamics, wave–current interaction is the interaction between surface gravity waves and a mean flow. The interaction implies an exchange of energy, so after the start of the interaction both the waves and the mean flow are affected. For ...

, specifically between the

Stokes drift For a pure wave motion (physics), motion in fluid dynamics, the Stokes drift velocity is the average velocity when following a specific fluid parcel as it travels with the fluid flow. For instance, a particle floating at the free surface of wat ...

velocity and the mean-flow

vorticity In continuum mechanics, vorticity is a pseudovector field that describes the local spinning motion of a continuum near some point (the tendency of something to rotate), as would be seen by an observer located at that point and traveling along wi ...

. The CL vortex force is used to explain the generation of

Langmuir circulation In physical oceanography, Langmuir circulation consists of a series of shallow, slow, counter-rotating vortices at the ocean's surface aligned with the wind. These circulations are developed when wind blows steadily over the sea surface. Ir ...

s 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 :

\rho\, \boldsymbol_S \times \boldsymbol,

with

\boldsymbol_S

the (

Lagrangian Lagrangian may refer to: Mathematics * Lagrangian function, used to solve constrained minimization problems in optimization theory; see Lagrange multiplier ** Lagrangian relaxation, the method of approximating a difficult constrained problem with ...

) Stokes drift

velocity Velocity is the directional speed of an object in motion as an indication of its rate of change in position as observed from a particular frame of reference and as measured by a particular standard of time (e.g. northbound). Velocity i ...

and vorticity

\boldsymbol=\nabla\times\boldsymbol

(i.e. the curl of the Eulerian mean-flow velocity

\boldsymbol

). Further

\rho

is the fluid

density Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematical ...

and

\nabla\times

is the curl operator. The CL vortex force finds its origins in the appearance of the Stokes drift in the

convective acceleration Convection is single or multiphase fluid flow that occurs spontaneously due to the combined effects of material property heterogeneity and body forces on a fluid, most commonly density and gravity (see buoyancy). When the cause of the convecti ...

terms in the mean momentum equation of the

Euler equations 200px, Leonhard Euler (1707–1783) In mathematics and physics, many topics are named in honor of Swiss mathematician Leonhard Euler (1707–1783), who made many important discoveries and innovations. Many of these items named after Euler include ...

Navier–Stokes equations In physics, the Navier–Stokes equations ( ) are partial differential equations which describe the motion of viscous fluid substances, named after French engineer and physicist Claude-Louis Navier and Anglo-Irish physicist and mathematician Geo ...

. For constant density, the momentum equation (divided by the density

\rho

) is: :

\underbrace_\text 
  + \underbrace_\text 
  + \underbrace_\text
  + \underbrace_\text
  + \underbrace_\text
  = \underbrace_\text
  + \underbrace_\text,

with * (a): temporal acceleration * (b): convective acceleration * (c): Coriolis force due to the angular velocity

\boldsymbol

of the Earth's rotation * (d): Coriolis–Stokes force * (e):

gradient In vector calculus, the gradient of a scalar-valued differentiable function of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p is the "direction and rate of fastest increase". If the gr ...

of the augmented

pressure Pressure (symbol: ''p'' or ''P'') is the force applied perpendicular to the surface of an object per unit area over which that force is distributed. Gauge pressure (also spelled ''gage'' pressure)The preferred spelling varies by country and e ...

* (f): Craik–Leibovich vortex force * (g):

viscous The viscosity of a fluid is a measure of its resistance to deformation at a given rate. For liquids, it corresponds to the informal concept of "thickness": for example, syrup has a higher viscosity than water. Viscosity quantifies the inte ...

force due to the

kinematic viscosity The viscosity of a fluid is a measure of its resistance to deformation at a given rate. For liquids, it corresponds to the informal concept of "thickness": for example, syrup has a higher viscosity than water. Viscosity quantifies the int ...

\nu

The CL vortex force can be obtained by several means. Originally, Craik and Leibovich used

perturbation theory In mathematics and applied mathematics, perturbation theory comprises methods for finding an approximate solution to a problem, by starting from the exact solution of a related, simpler problem. A critical feature of the technique is a middl ...

. An easy way to derive it is through the

generalized Lagrangian mean In continuum mechanics, the generalized Lagrangian mean (GLM) is a formalism – developed by – to unambiguously split a motion into a mean part and an oscillatory part. The method gives a mixed Eulerian–Lagrangian description for the flo ...

theory. It can also be derived through a

Hamiltonian mechanics Hamiltonian mechanics emerged in 1833 as a reformulation of Lagrangian mechanics. Introduced by Sir William Rowan Hamilton, Hamiltonian mechanics replaces (generalized) velocities \dot q^i used in Lagrangian mechanics with (generalized) ''momenta ...

description.

Notes

References

* * * * * * {{DEFAULTSORT:Craik-Leibovich vortex force Fluid dynamics Water waves