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small-steepness waves and relatively low WRN, close to the critical number
(7.67)
.Asa
result, if averaged over the wave period, the estimates of
dis
in
(7.76)
have to be divided
at least by a factor of 10 and perhaps more.
It is interesting to notice that such intermittent turbulence, which apparently belongs to
the transitional fluid state between fully laminar and fully turbulent conditions, does not
reveal a threshold value of wave amplitude, below which it does not occur, i.e. the critical
WRN
(7.67)
, and the data fit shows a correlation of 89% for the dependence which goes
through the origin (see also
Section 9.2.2
and
Figure 9.10
). The value of
a
0
=
0
.
0182m
which corresponds to Re
wave
critical
=
3000 for the 1
.
5 Hz waves is marked in
Figure 7.24
with the
-mark on the bottom axis. The lowest wave amplitude, at which the turbulence
was still observed, i.e.
a
0
×
=
0
.
012m, corresponds to Re
wave
=
1300. If approximately
scaled to the free surface
z
attenuation of the wave amplitude,
the lowest observed turbulence-producing Reynolds number would thus be Re
wave
=
=
0byassumingexp
(
−
kz
)
2300.
This is close to the lower-margin estimate of the critical Reynolds number of
Babanin
(
2006
), and this margin for the transitional wave turbulence is even lower in experiments
and simulations of
Dai
et al.
(
2010
), down to Re
wave
1000. Such scaling of the lowest
transitional WRN, however, is quite speculative, if talking about the measurements just
below troughs of steep waves. Therefore, for the reference estimates, we chose to show
the fit going through the origin rather than trying to produce a dependence based on an
imposed threshold value for the amplitude.
Levels of the turbulent rates of
∼
10
−
3
are quite high and comparable with dissi-
pation rates measured in the presence of wave breaking (
Agrawal
et al.
,
1992
;
Young &
Babanin
,
2006a
). Thus, locally, the wave-induced turbulence can be quite intensive, but
its average rates have to be scaled down over the wave period due to the intermittency of
the observed turbulence as described above. It is also worth mentioning that in a recent
study of turbulence and energy dissipation associated with wave breaking (
Gemmrich
,
2010
) it was found that in the absence of wave breaking the turbulent dissipation per-
sists with energy dissipation levels of
dis
∼
10
−
3
. Such levels are in close agreement
with the wave-breaking onset levels measured in this study (i.e.
Figure 7.24
). Experimen-
tal approximation
(7.76)
is also consistent with
dis
∼
5
·
a
0
dependence implied by
Qiao
et al.
(
2004
,
2008
,
2010
) who introduced additional wave-induced turbulent mixing in
ocean circulation models. The very significant advances in performance of such models
can be interpreted as an indirect support of the experimental results and dependence
(7.76)
presented here.
Although treating the results obtained in a particular case of short and steep laboratory
waves has to be done with caution, this may still indicate a significance of the non-breaking
wave-induced turbulence even in a more general case. When comparing breaking and
non-breaking rates directly, our measurements revealed quite close values in the case of
steep waves. The instantaneous dissipation rates of the
a
0
dis
∼
5Hz waves
at the rear face, where the non-breaking turbulence was found in this study, and at the
front face, where the breaking-in-progress turbulence develops, show magnitudes of
=
32
.
5mm
,
1
.
dis
≈
10
−
3
and
10
−
3
, respectively.
22
.
5
·
dis
≈
14
.
6
·
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