Energy dissipation across the wave spectrum - Breaking and Dissipation of Ocean Surface Waves - page 314

Now, let us try to apply the estimates of transition to turbulent wave motion (7.67) and
of the wave-turbulence dissipation rates (7.76) to the swell-attenuation measurements of
Ardhuin et al. ( 2010 ). As said above, these observations provide a unique data set for
testing theories and verifying mechanisms of surface-wave dissipative phenomena which
are weak and slow compared to wave breaking.
First of all, we will check whether the steepness (7.61) - (7.62) is consistent with WRN
(7.70) indicative of the turbulent wave motion. It should be noticed that Re wave cannot be
unambiguously converted into the wave steepness, as for waves of the same steepness but
different wavelengths or amplitudes WRN can be different.
The kinematic viscosity ν of the sea water, with three significant digits of precision, can

be calculated as follows:

10 − 3 10 − 6

ν = (( 0 . 659 · · ( T s − 1 ) − 0 . 05076 )( T s − 1 ) + 1 . 7688 ) · (7.77)

where T s is the surface water temperature in degrees of Celsius (this empirical formula
is taken from http://ittc.sname.org/2002_recomm_proc/7.5-02-01-03.pdf). Ardhuin et al. 's
( 2009a ) data (their online auxiliary material) cover swell periods of T

=

13-19 s. For

waves with T = 16 s, in the middle of this range, k = 0 . 0157 rad / m and, if we choose

10 ◦ over the swell path, then 10 − 6 m 2 constant T s = ν = 1 . 37 · / s and, based on the critical

WRN (7.67) , the critical amplitude is a 0 = 0 . 102m. This leads us to

a k = 0 . 0016 , (7.78)

way below the steepness (7.62) . Therefore, for such a wavelength and for the steepness
(7.62) , WRN is quite high: Re wave ≈

29000 and thus the wave motion is certainly turbulent
and should lead to an excessive swell dissipation compared to the linear case (7.60) .
Before estimating this dissipation, we should look in more detail into the issue of the crit-
ical steepness (7.61) - (7.62) .In Figure 7.25 (top), a swell-decay rate α of (7.60) is plotted

versus swell steepness s based on the online data of Ardhuin et al. ( 2009a ).

When analysing these data points, the best correlation was observed not for the linear fit
(corr

= H /λ

= 0 . 86), but for a quadratic dependence with the intercept at

s = 0 . 0028 (7.79)

rather than at s = 0 . 005 as in (7.61) :

10 − 3

2

α = 2 . 9 · ( s − 0 . 0028 ) . (7.80)

This dependence is shown with the solid line in Figure 7.25 (top). To obtain dependence
(7.80) , the obvious outlier at s

004 was removed, but in any scenario, the correlation
is higher for the quadratic expression.
This correlation, with intercept (7.79) is corr

≈ 0 .

≈ 0 . 932, but it is only reduced marginally

to corr ≈ 0 . 930 if the intercept is set to 0, that is

10 − 3 s 2

α = 1 . 6 · (7.81)