Energy dissipation across the wave spectrum - Breaking and Dissipation of Ocean Surface Waves - page 318

line, and the mean ratio is H B / 41. Here, we have excluded and do not show the

single data point which produced a ratio of H B /

H A = 0 .

1 and is an obvious outlier.
This ratio, however, is not a constant value and depends clearly on

H A = 3 .

α of Ardhuin et al.

( 2009a ): H B / H A versus α is plotted in Figure 7.26 (bottom). The ratio grows as a function

of , with a very high linear correlation of 96%.

Before discussing the implications of these results, we should notice that the dissipation
rate of (7.85) and (7.94) is perhaps too strong and overestimates the observed swell-height
decay. This conclusion, however, highlights an important conjecture that the dissipation
due to wave-induced turbulence is a significant player in observed attenuation of swell. Not
only do the dissipation rates identified due to such mechanisms explain the swell decay, in
the present form they over-predict it.
The present form includes empirical coefficient b 1 whose value was adopted as b 1

α

=

0 004 at the upper margin of its possible values, by assuming that the intermittent turbu-

lence in the non-breaking experiment of Babanin & Haus ( 2009 ) appears at one out of the
ten wave-phases measured. In fact, as mentioned above, this is only true for the steepest
waves, and for the smaller waves of the same length this intermittency rate is lower for
most data points of Figure 7.24 . If dependence (7.76) is divided by 20, rather than by 10,
in order to estimate mean-over-wave-period turbulence production, then

.

b 1 = 0 . 002 , (7.96)

and the corresponding outcomes are shown in Figure 7.26 with circles.
The circles, apparently, indicate higher residual H B swell amplitudes, and they are now
closer to those predicted by (7.95) , and at the upper limit are even above the one-to-one
line. The tuning can be continued, if we had a reliable reference point, but in reality both
swell-air friction and wave-turbulence production need further investigation. The former is
still to be measured experimentally for the tangential turbulent stresses.
Indeed, as already noticed, underestimation of H B with respect to H A is not random
and depends on the decay α with a very high correlation. That is, at larger values of

α , the underestimation is smaller, and in the case of b 1 (7.96) , the circles even indicate

overestimation at 10 − 7 m − 1 .

Such an observation may imply a few things. Perhaps, estimates of

α>

have a progres-
sive bias towards lower values of this decay rate. Quite likely also is that the constant-
exponential-decay model (7.60) and (7.95) is not well suited to describe the swell attenua-
tion. Indeed, while the direct quantitative applicability of the laboratory dependence (7.76)
to the ocean swells may need further justifications and elaborations, in any scenario it
indicates that qualitatively such decay rate is not constant and not exponential.
Finally, we would mention that if the (7.96) proportionality coefficient is chosen in the
swell-decay model (7.92) , it would match the estimates of Ardhuin et al. ( 2009a ) at their
values of

α

10 − 7 m − 1 ( Figure 7.26 (bottom)). For such , the observed swell decay

can be fully explained by the waves spending their enery on generating turbulence in the
upper ocean. If

α ≈ 1 . 2 · α

10 − 7 m − 1 , H B is approximately twice as high as H A , that is the
wave-induced turbulence alone is not a sufficient dissipation source. In such a case, some

α ≈ 2 ·