Energy dissipation across the wave spectrum - Breaking and Dissipation of Ocean Surface Waves - page 304

has a sharp peak, with the spectral density (wave height) dropping very rapidly away from
the peak frequency

ω p both towards smaller scales (higher frequencies) and larger scales
(lower frequencies) - see, for example, JONSWAP parameterisation (2.7) . Therefore,

ω p
and associated wave height can be chosen as characteristic wave scales which determine
MLD - z critical in the case of a wind-wave spectrum. As the representative amplitude of
spectral waves, half of the significant wave height a s =

H s /

2 (2.39) will be used.
Babanin ( 2006 ) performed qualitative and quantitative verifications of the hypothesis,
and consistency checks by testing conclusion (7.74) on the basis of known values of MLD
( z critical ) for different water bodies and different wave conditions. Additionally, a laboratory
feasibility check was conducted.
Based on the Black Sea depth of mixed layer ( z critical ≈ 25 in April ( Kara et al. , 2005 )),

we will determine the value of critical WRN Re wave critical according to (7.73) by having use
of the typical extreme values of peak frequency and amplitude of the wind-generated waves
in this sea in April. This dimensionless number should be universal for the wave motion,
according to the hypothesis (7.70) , and therefore be equally applicable to such outermost
extremes as the deep ocean and much smaller laboratory mechanically-generated waves.
Thus, once this critical Reynolds number is known, we should be able to predict the tran-
sition of non-forced wave motion from laminarity to turbulence in the laboratory and to
predict MLD in the ocean for different wave circumstances on the surface in cases when
wind stresses (waves) dominate over other mixing mechanisms.
April in the Northern hemisphere was chosen for our estimates because the combined
effect of surface cooling and heating on MLD is expected to be minimal in early spring
(e.g. Martin , 1985 ), and therefore April data will provide the cleanest material for inves-
tigation of mixing due to wind (waves). An extensive wind-wave data set collected in
a deep water region of the Black Sea throughout April by Babanin & Soloviev ( 1998a )
was used to determine the relevant wave climate. The minimal value of the peak fre-
quency f p = .

175 Hz is encountered three times in Table I of Babanin & Soloviev
( 1998a ) and was thus chosen as representative of the typical extreme wave conditions in
the Black Sea in April. Corresponding values of variance m 0 (2.10) ranged from 0

0

379m 2 .

500m 2 .
Water temperatures recorded at the measurement site in April were around 10 ◦ C, and
therefore, for the Black Sea whose salinity is half of that in the ocean, the kinematic vis-
cosity was chosen as

to 0 .

10 − 6 m 2 ν = 1 . 35 · / s. The wave amplitude used was a s = H s / 2 =

2 √ m 0 . Finally, (7.73) leads to the critical Reynolds number in the range Re wave critical =

2602-3433. Given the approximate nature of the estimates, we chose Re wave critical 3000

as was already indicated in (7.67) , close to the centre of the range. This is the critical
Reynolds number for wave orbital motion.
Such a Reynolds number is in very good accordance with the critical numbers for
other fluid flows. Typical Reynolds numbers for a great variety of engineering applications
outside the boundary layer are Re critical =

=

2000-4000 (e.g. Cengel & Cimbala , 2006 ).
A laboratory test was conducted in the wave tank of Monash University to check the
feasibility of the calculated critical number. Regardless of its relation with the upper ocean
mixing, critical WRN, Re wave critical = 3000 should be able to predict the onset of turbulence