In
fluid dynamics
In physics, physical chemistry and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids – liquids and gases. It has several subdisciplines, including (the study of air and other gases in motion ...
, the Craik–Leibovich (CL) vortex force describes a
forcing of the
mean flow through
wave–current interaction, specifically between the
Stokes drift velocity and the mean-flow
vorticity
In continuum mechanics, vorticity is a pseudovector (or axial vector) 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 an ...
. 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
Irving Langmuir (; January 31, 1881 – August 16, 1957) was an American chemist, physicist, and metallurgical engineer. He was awarded the Nobel Prize in Chemistry in 1932 for his work in surface chemistry.
Langmuir's most famous publicatio ...
in the 1930s).
Description
The CL vortex force is
:
with
the (
Lagrangian) Stokes drift
velocity
Velocity is a measurement of speed in a certain direction of motion. It is a fundamental concept in kinematics, the branch of classical mechanics that describes the motion of physical objects. Velocity is a vector (geometry), vector Physical q ...
and vorticity
(i.e. the
curl
cURL (pronounced like "curl", ) is a free and open source computer program for transferring data to and from Internet servers. It can download a URL from a web server over HTTP, and supports a variety of other network protocols, URI scheme ...
of the
Eulerian mean-flow velocity
). Further
is the fluid
density
Density (volumetric mass density or specific mass) is the ratio of a substance's mass to its volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' (or ''d'') can also be u ...
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
In mathematics and physics, many topics are eponym, 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 their own unique function, e ...
or
Navier–Stokes equations
The Navier–Stokes equations ( ) are partial differential equations which describe the motion of viscous fluid substances. They were named after French engineer and physicist Claude-Louis Navier and the Irish physicist and mathematician Georg ...
. For constant density, the momentum equation (divided by the density
) is:
:
with
* (a): temporal acceleration
* (b): convective acceleration
* (c):
Coriolis force
In physics, the Coriolis force is a pseudo force that acts on objects in motion within a frame of reference that rotates with respect to an inertial frame. In a reference frame with clockwise rotation, the force acts to the left of the motio ...
due to the
angular velocity
In physics, angular velocity (symbol or \vec, the lowercase Greek letter omega), also known as the angular frequency vector,(UP1) is a pseudovector representation of how the angular position or orientation of an object changes with time, i ...
of the
Earth's rotation
Earth's rotation or Earth's spin is the rotation of planet Earth around its own Rotation around a fixed axis, axis, as well as changes in the orientation (geometry), orientation of the rotation axis in space. Earth rotates eastward, in progra ...
* (d):
Coriolis–Stokes force
* (e):
gradient
In vector calculus, the gradient of a scalar-valued differentiable function f of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p gives the direction and the rate of fastest increase. The g ...
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 eve ...
* (f): Craik–Leibovich vortex force
* (g):
viscous
Viscosity is a measure of a fluid's rate-dependent resistance to a change in shape or to movement of its neighboring portions relative to one another. For liquids, it corresponds to the informal concept of ''thickness''; for example, syrup h ...
force due to the
kinematic viscosity
Viscosity is a measure of a fluid's rate-dependent drag (physics), resistance to a change in shape or to movement of its neighboring portions relative to one another. For liquids, it corresponds to the informal concept of ''thickness''; for e ...
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 middle ...
. An easy way to derive it is through the
generalized Lagrangian mean theory.
[ It can also be derived through a ]Hamiltonian mechanics
In physics, Hamiltonian mechanics is a reformulation of Lagrangian mechanics that emerged in 1833. Introduced by Sir William Rowan Hamilton, Hamiltonian mechanics replaces (generalized) velocities \dot q^i used in Lagrangian mechanics with (gener ...
description.
Notes
References
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{{DEFAULTSORT:Craik-Leibovich vortex force
Fluid dynamics
Water waves