Role of wave breaking in the air–sea interaction - Breaking and Dissipation of Ocean Surface Waves - page 386

is quite different to the turbulence near the solid boundary. There is additional turbulence
production due to wind-caused shear at the mobile interface, but the main difference is
due to the powerful surface-located additional source of turbulence caused by the wave
breaking.
It is instructive and relevant to notice here, however, that for consistency the picture
lacks the wave-induced turbulence caused by the wave orbital motion, unrelated to wave
breaking. The intensity of this turbulence production may be small near the surface, if com-
pared with direct turbulence injection by the breaking, but the source of such turbulence is
distributed through the water column, rather than being localised at the surface ( Babanin ,
2006 ).
It is interesting to note that Benilov & Ly ( 2002 ) mention this turbulence, but do not
elaborate or include it into the turbulence budget (9.27) , even if implicitly. It is even more
intriguing if we notice that Benilov et al. ( 1993 ) are authors of a theory of such turbulence
and that Benilov & Ly ( 2002 ) actually explicitly point to this turbulence source:

“Another mechanism of the upper layer turbulence generation may be associated with the vortex
instability of potential surface waves ( Benilov et al. , 1993 ).”

Indeed, Benilov et al. ( 1993 ) suggested and described physically and mathematically a
mechanism of production of turbulence by potential waves. In this paper, they asked and
answered a question of

“what follows from hydrodynamic equations about vortex disturbances in the potential flow?”

They considered a problem of stability of such disturbances in the potential velocity field
of linear waves. The problem is essentially three-dimensional (even for two-dimensional
waves), as the instability develops in the dimension perpendicular to the plane of two-
dimensional waves. A second-order solution (with respect to depth-filtered steepness

=

ak as the perturbation parameter) is always unstable.

The obvious question of the origin of the vortex disturbances in the potential waves
was not approached by Benilov et al. ( 1993 ), but the answer is actually quite apparent.
First of all, the ocean is always turbulent and in any scenario perturbations of the velocity
field, even if the background mean wave motion is potential, are inevitable. Secondly, as
was discussed in Section 7.5 (see also Kinsman , 1965 ; Babanin & Haus , 2009 ), the poten-
tial wave theory is an approximate approach to begin with, based on the Euler equation
as a boundary condition which neglects the water viscosity. Since the water is a viscous
fluid, even though the viscosity is small, the shear stresses should be possible in the orbital
motion with strong vertical velocity gradients, and may serve as the source of the vorticity.
This latter issue, however, is subject to some controversy which we cannot solve here and
we address the reader to the literature ( Benilov & Lozovatskiy , 1977 ; Kitaigorodskii &
Lumley , 1983 ; Benilov et al. , 1993 ).
Thus, the theory of Benilov et al. ( 1993 ) is a consistent physical mechanism which can
explain generation of turbulence by waves whose mean orbital motion may be well approx-
imated by the potential and even linear wave theory. Following this approach, Chalikov

· exp ( − kz )