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The
max
in
Figure 9.10
indicates the maximum dissipation in the wave-induced-
turbulence dissipation profile. In practice, this is an estimate of the volumetric dissipation
rate near the surface and above the mean water level. Since most of the wave-induced tur-
bulent energy is known to concentrate within wave crests (
Gemmrich
,
2010
), it is expected
that such a
max
(
Diss
)
dis
of
Babanin & Haus
(
2009
). In the
model, this happens because the generation of turbulence depends on the gradient of orbital
velocity
(
Diss
)
should be greater than
2
which has a maximum at the surface.
In the figure, the shaded area corresponds to the range of measurements and scatter of
the observational data of
Babanin & Haus
(
2009
). There is quantitative agreement for wave
amplitudes of
(
du
orb
/
dz
)
∼
1
.
5 cm (wavelength in the model, as in the experiment, corresponds to the
frequency 1
within the crests is greater than that
measured below the troughs. The theoretical growth of the maximal dissipation rate, as a
function of steepness, is faster than that of the background dissipation
.
5 Hz), and for the higher waves
max
(
Diss
)
dis
in
(7.76)
and
Figure 7.24
.
In
Figure 9.11
, vertical profiles of the the volumetric dissipation
dis
(denoted as
Diss
,
units of m
2
s
3
) are plotted as a function of depth
Z
, averaged over horizontal planes in
Cartesian coordinates. The thin horizontal line corresponds to the depth below troughs. The
model allows us to obtain these values also above this line within segments occupied by
the water, so there are positive depths present too. The solid line is the average dissipation,
and dashed lines indicate the standard deviation.
Different subplots show profiles for different wave amplitudes
a
and corresponding
wave Reynolds numbers
(7.70)
denoted as
RE
, as indicated within these different panels.
It is quite obvious that production of turbulence does not actually stop at low ampli-
tudes/WRNs, but the magnitude of the dissipation rates becomes so marginal (
/
(
m
)
dis
∼
10
−
8
m
2
s
3
84) that it is hardly possible to measure. The lowest dissipa-
tion which
Babanin & Haus
(
2009
) were still able to detect above the noise level with
the PIV method was of the order
/
at Re
wave
=
10
−
4
m
2
s
3
. If this is chosen as a reference, then
dis
∼
/
Re
wave
≈
2000 can be regarded as the critical WRN, close to the estimate
(7.67)
of
Babanin
(
2006
). Note that the model does not have viscosity, and therefore the physical meaning
of the critical Reynolds number, which is an external parameter linked to the wave ampli-
tude and frequency, is the capacity of viscous forces to damp the growth of these small
instabilities.
Returning back to the theory of wave-induced turbulence by
Benilov
et al.
(
1993
) and
the upper-ocean model of turbulence by
Benilov & Ly
(
2002
), the reason why
Benilov &
Ly
(
2002
) do not provide an account for such a source of turbulence in their upper-ocean
turbulence scheme rests, perhaps, with the tradition in ocean-turbulence modelling which
neglects this turbulence or is unaware of it.
Benilov & Ly
(
2002
), apparently, treat the
non-breaking wave-turbulence mechanism as hypothetical until it is explicitly confirmed
experimentally.
Ocean-circulation modellers have been facing an opposite dilemma. While having no
formal theoretical reason to introduce such turbulence, they often felt a need for an additional
source of turbulence, definitely connected with the surface waves and not necessarily with
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