Role of wave breaking in the air–sea interaction - Breaking and Dissipation of Ocean Surface Waves - page 378

The energy transfer is only a little bit more complicated, as it additionally includes
the viscous wave-energy dissipation D . In the lower part of Figure 9.7 , the energy E ml
transferred to the mixed layer per unit of time is less than the energy E transferred to the
waves, by this dissipation D and by the full derivative of the wave energy flux F :

dF
dx −

E ml = E

− .

D (9.18)

In the figure, long-wave modulation of the short-wave generation is also indicated.
In this scheme, the

“breaking-wave-generated turbulence may deepen the mixed layer directly by penetrating to the bot-
tom of the layer to entrain fluid or by increasing the mean flow and shear across the bottom of the
layer through Reynolds stress interactions.”

For that to happen, however, the depth of the mixed layer should be relatively shallow as
the direct penetration of wave-breaking turbulence occurs at the scale of the wave height
(see discussions in Section 9.2.2 ).
A much more complex scheme was presented by Chalikov & Belevich ( 1993 ) and is
shown in Figure 9.8 . It pictures the momentum and energy transfer all the way from the
free atmosphere down to the thermocline. With respect to the role of the waves in this
transfer, the authors also point out that part of the momentum goes back from the waves to
the atmosphere.
While the wall-turbulence analogy is often assumed for descriptions of WBL, and for a
good reason - since the logarithmic constant-flux layer (9.1) and (3.19) is indeed observed,
the difference between the WBL and solid-wall boundary layer is in fact substantial. The
near-water turbulent momentum flux is dominated by the wave-induced fluctuations of
pressure, velocity and turbulent stress (see Section 7.4 and (7.36) ).
Because of these oscillations, the total flux (7.36) is constant only on average. Dynami-
cally, the constant-flux idealised regime is not maintained where the wave-caused oscilla-
tions become significant. As a result, deviations of the wind profile from the logarithmic
occur inside WBL and cause changes to the profile outside this boundary layer. Chalikov &
Belevich ( 1993 ) suggest that it is convenient, within the quasi-logarithmic boundary layer,
to parameterise these effects by still using the logarithmic profile but adjusting the total
roughness in such a way that it takes into account contributions of both the dominant waves
and spectrum tail.
Figure 9.8 demonstrates the general scheme of dynamic interactions in the system of
wave boundary layer and wave-mixed layer. A lower boundary condition for the momen-
tum balance equation in WBL is applied at some height z

= z r (see Chalikov & Belevich ,

1993 , for details):

ν t ∂

u

z =

T r = C r | u r | u r (9.19)

∂

where ν t is the coefficient of turbulent diffusion, u is a vector of wind speed, and T r and

C r are local momentum-flux vector and drag coefficient, respectively. T r defines viscous
tangential stress and direct momentum exchange between WBL and mixed layer currents.