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waves, the only source of the turbulence was surface waves, and the intensity of turbulence
indeed correlated with the magnitude of these waves.
A need for wave-induced turbulence has also been felt by the ocean modellers in their
search for mechanisms to fill the gaps in explanations of upper-ocean mixing. Jacobs
( 1978 ), Pleskachevsky et al. ( 2001 , 2005 ), Qiao et al. ( 2004 , 2008 , 2010 ) and Gayer et al.
( 2006 ) all brought in wave-induced turbulent viscosity and applied it in their circulation
models. The ocean sites ranged from the finite-depth North Sea ( Pleskachevsky et al. ,
2001 , 2005 ; Gayer et al. , 2006 ) through the open ocean ( Jacobs , 1978 ) to global applica-
tions ( Qiao et al. , 2004 , 2008 , 2010 ). In all cases, it was either impossible to describe
the observed mixing without wave turbulence, or introducing such turbulent diffusion
essentially improved the correlation between the data and numerical simulations.
The above-mentioned Ardhuin & Jenkins ( 2006 ) mechanism does not exactly belong
to the pure class of wave-induced turbulence. The approach remained within the frame of
zero-vorticity for wave solutions. The source of their turbulence is Stokes drift, the zero-
frequency solution, which then interacts with the background turbulence. The same can be
said about the mechanism of Benilov et al. ( 1993 ), except it is the mean potential wave
motion that causes the unstable infinitesimal vortices to grow. If the viscosity is allowed
in theory, however, a viscous fluid can further promote shear instability in such a system
( Balmorth , 1999 ).
Here, we will be using the linear theory to find the wave-based Reynolds number, to
describe its distribution along the water depth and to approach a possible upper-ocean
mixing mechanism due to such waves (in Section 9.2.2 below). We would like to emphasise
that this is not a compromise with the criticism of the limitations of the linear theory above.
To estimate the Reynolds numbers we only need a scaling of mean wave orbital motion at
different depths which the linear theory approximates well enough.
The hypothesis of wave Reynolds number has three important consequences. First, the
wave motion should be able to generate turbulence even in the absence of wave breaking.
As discussed above, this is not a completely unexpected conclusion as such turbulence
has been observed for a while. What has not been appreciated, however, is the potential
significance of such a turbulence source as the waves in the ocean being ever present,
unlike currents or Langmuir cells, and wave-caused speeds of water motion are at least an
order of magnitude greater than those of shear currents and Langmuir circulations which
are usually held responsible for the turbulence supply as mentioned above.
A second consequence is decoupling of the wave-induced non-breaking turbulence from
analogies with the wall-layer law tradition which are often employed to approach wave tur-
bulence (e.g. Soloviev et al. , 1988 ; Agrawal et al. , 1992 ). According to the wave-turbulence
hypothesis, the principal difference of such a turbulence is the existence of a characteristic
length scale (radius of the wave orbit) as opposed to the wall turbulence which does not
have a characteristic length other than distance to the surface.
Third, such a wave-induced turbulence would enhance the upper-ocean mixing on behalf
of the normal component of the wind stress. The wind stress plays a dual role in the upper-
ocean dynamics. A tangential component of the stress generates the surface shear currents
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