THE NEAR-SURFACE LAYER OF THE OCEAN
Leibovich (Section 5.7.1) and thereby explains many observable features of
the Langmuir circulations.
Leibovich (1977a, b) further extended the CL model to allow time
development of the currents and density stratification under the Boussinesq
approximation. These works, however, greatly simplified the problem by
parameterizing the effects of turbulent fluctuations by constant eddy
diffusivities. More sophisticated turbulence models like those described in
Chapter 3 were not yet available in the 1970s.
A remarkable feature of (5.58) is the presence of a vortex force:
v
s
f U Z
u
G
G G
(5.61)
where Z
G is the mean vorticity. Note that as
0
s
U o
G
(which is achieved, for
instance, when z o f ), the vortex force term (5.61) vanishes and equations
(5.58) reduce to the classical Navier-Stokes equations of a Boussinesq fluid.
In order to explore the role of the vortex force term (5.61), let us follow
Leibovich (1983) and assume that the wind blows in a fixed direction over
initially undisturbed water of unlimited horizontal extent and depth. Only
one fixed direction is involved in the problem formulation, thereby
symmetry dictates the development of a surface wave field with
unidirectional Stokes drift aligned with the wind or s
s
U U i
G
G
. Since this is a
two-dimensional hydrodynamic system, it has tendency to self-organization.
For simplicity, Coriolis acceleration is neglected; the total momentum
transferred to the surface layer of the ocean is in the same direction as the
Stokes drift velocity vector. The horizontally averaged velocity is
( , )
u u z t i
G
G
, while the positive y-vorticity is
/
u z j
Z w w
G
G
. It is remarkable
that the vortex force
/
v
s
f kU u z
w w
G G
,
(5.62)
is oriented vertically upward and is formally analogous to the buoyancy
force in (5.58).
If the Stokes drift u s would depend only on depth z, then the vortex force
could be balanced by the analog of a hydrostatic pressure gradient
(developing unidirectional current). If for any reason the Stokes drift varies
across the wind (in the y-, or transverse direction), the vortex force cannot be
balanced just by pressure gradients, and an overturning is possible.
Longuet-Higgins (1962) exploited the fact that the directional spectrum of
wind-generated surface waves in a “short-crested” sea is bimodal and
symmetric with respect to the wind. If the spectrum is bimodal (with peaks
of wave energy at angles +/-T with respect to the wind), and if the wave
spatial structure remains coherent for long enough for the rectification
376
Leibovich (Section 5.7.1) and thereby explains many observable features of
the Langmuir circulations.
Leibovich (1977a, b) further extended the CL model to allow time
development of the currents and density stratification under the Boussinesq
approximation. These works, however, greatly simplified the problem by
parameterizing the effects of turbulent fluctuations by constant eddy
diffusivities. More sophisticated turbulence models like those described in
Chapter 3 were not yet available in the 1970s.
A remarkable feature of (5.58) is the presence of a vortex force:
v
s
f U Z
u
G
G G
(5.61)
where Z
G is the mean vorticity. Note that as
0
s
U o
G
(which is achieved, for
instance, when z o f ), the vortex force term (5.61) vanishes and equations
(5.58) reduce to the classical Navier-Stokes equations of a Boussinesq fluid.
In order to explore the role of the vortex force term (5.61), let us follow
Leibovich (1983) and assume that the wind blows in a fixed direction over
initially undisturbed water of unlimited horizontal extent and depth. Only
one fixed direction is involved in the problem formulation, thereby
symmetry dictates the development of a surface wave field with
unidirectional Stokes drift aligned with the wind or s
s
U U i
G
G
. Since this is a
two-dimensional hydrodynamic system, it has tendency to self-organization.
For simplicity, Coriolis acceleration is neglected; the total momentum
transferred to the surface layer of the ocean is in the same direction as the
Stokes drift velocity vector. The horizontally averaged velocity is
( , )
u u z t i
G
G
, while the positive y-vorticity is
/
u z j
Z w w
G
G
. It is remarkable
that the vortex force
/
v
s
f kU u z
w w
G G
,
(5.62)
is oriented vertically upward and is formally analogous to the buoyancy
force in (5.58).
If the Stokes drift u s would depend only on depth z, then the vortex force
could be balanced by the analog of a hydrostatic pressure gradient
(developing unidirectional current). If for any reason the Stokes drift varies
across the wind (in the y-, or transverse direction), the vortex force cannot be
balanced just by pressure gradients, and an overturning is possible.
Longuet-Higgins (1962) exploited the fact that the directional spectrum of
wind-generated surface waves in a “short-crested” sea is bimodal and
symmetric with respect to the wind. If the spectrum is bimodal (with peaks
of wave energy at angles +/-T with respect to the wind), and if the wave
spatial structure remains coherent for long enough for the rectification
376