Chapter 5. SPATIALLY-COHERENT STRUCTURES
responsible for the Stokes drift to take place (many times a typical wave
period), then the Stokes drift would be spatially periodic with a cross wind
wave number of 2k sin T, where k is the characteristic wave number of the
surface waves. This horizontally periodic wave drift produces a torque due
to horizontal variations of vortex force that drives the roll motions (Craik
and Leibovich, 1976). This mechanism is known in literature as the CL1
mechanism.
Leibovich and Ulrich (1972) proposed an alternate, kinematic
interpretation of the CL1 mechanism. In this interpretation, the Stokes drift
deforms the vortex lines associated with the current and produces the
streamwise vorticity periodically (in y) altering in sign.
Figure 5-54 summarizes both the dynamic and kinematic interpretations
of the CL1 mechanism. This figure also includes a schematic diagram of
idealized “cross-wave trains” assumed in the CL1 mechanism.
Leibovich (1977a) extended the CL1 theory to include time evolution of
the coupled (wind-directed) currents and circulations. He formulated the
problem as an initial-value problem, with currents and circulations initially
zero and initiated by a step function in surface stress, which resulted in a
well-posed mathematical problem. Since by assumption the wave field is
steady and invariant in the x-(wind) direction and symmetric with respect to
the x-axis, the problem is independent of x and any emerging circulations
appear in the form of rolls. This initial-value problem depends upon an angle
representing the directional properties of the waves and a single
dimensionless parameter called the Langmuir number,
1/ 2
3 2
2 2
/
T
La
k
a u
Q
V
(5.63)
where u * is the friction velocity (corresponding to a constant wind stress for
0
t t ), and Q T is the eddy viscosity; V, k, and a are the characteristic
surface waves frequency, wave number, and amplitude respectively. The
Langmuir number describes the balance between the rate of diffusion of
streamwise vorticity and the rate of production of streamwise vorticity by
the vortex stretching accomplished by the Stokes drift. This number can also
be interpreted as an inverse Reynolds number. The appropriate scalings of
the Langmuir circulations characteristics are given in Table 5-3. Since these
scalings emphasize a balance between vortex force and the applied shear
stress when x-variations are negligible, they are also appropriate to problems
involving the CL2 instability mechanism under the same set of assumptions.
In that case, however, the angle representing the directional properties of the
waves is not invoked.
377
responsible for the Stokes drift to take place (many times a typical wave
period), then the Stokes drift would be spatially periodic with a cross wind
wave number of 2k sin T, where k is the characteristic wave number of the
surface waves. This horizontally periodic wave drift produces a torque due
to horizontal variations of vortex force that drives the roll motions (Craik
and Leibovich, 1976). This mechanism is known in literature as the CL1
mechanism.
Leibovich and Ulrich (1972) proposed an alternate, kinematic
interpretation of the CL1 mechanism. In this interpretation, the Stokes drift
deforms the vortex lines associated with the current and produces the
streamwise vorticity periodically (in y) altering in sign.
Figure 5-54 summarizes both the dynamic and kinematic interpretations
of the CL1 mechanism. This figure also includes a schematic diagram of
idealized “cross-wave trains” assumed in the CL1 mechanism.
Leibovich (1977a) extended the CL1 theory to include time evolution of
the coupled (wind-directed) currents and circulations. He formulated the
problem as an initial-value problem, with currents and circulations initially
zero and initiated by a step function in surface stress, which resulted in a
well-posed mathematical problem. Since by assumption the wave field is
steady and invariant in the x-(wind) direction and symmetric with respect to
the x-axis, the problem is independent of x and any emerging circulations
appear in the form of rolls. This initial-value problem depends upon an angle
representing the directional properties of the waves and a single
dimensionless parameter called the Langmuir number,
1/ 2
3 2
2 2
/
T
La
k
a u
Q
V
(5.63)
where u * is the friction velocity (corresponding to a constant wind stress for
0
t t ), and Q T is the eddy viscosity; V, k, and a are the characteristic
surface waves frequency, wave number, and amplitude respectively. The
Langmuir number describes the balance between the rate of diffusion of
streamwise vorticity and the rate of production of streamwise vorticity by
the vortex stretching accomplished by the Stokes drift. This number can also
be interpreted as an inverse Reynolds number. The appropriate scalings of
the Langmuir circulations characteristics are given in Table 5-3. Since these
scalings emphasize a balance between vortex force and the applied shear
stress when x-variations are negligible, they are also appropriate to problems
involving the CL2 instability mechanism under the same set of assumptions.
In that case, however, the angle representing the directional properties of the
waves is not invoked.
377