Energy dissipation across the wave spectrum - Breaking and Dissipation of Ocean Surface Waves - page 315

x 10 −7

4

3

2

1

0

0 0.002 0.004 0.006 0.008 0.01 0.012 0.014

s

x 10 −7

4

3

2

1

0

0 2 4 6 8 10 12 14 16 18

Re wave

x 10 5

Figure 7.25 Swell decay rate α (7.60) versus (top) steepness s = H /λ . Solid line is expression
(7.80) , dashed line is expression (7.81) ; (bottom) Wave Reynolds number Re wave (7.70) . Solid line
is expression (7.82)

as shown with the dashed line in Figure 7.25 (top). In fact, the dashed line is visually
preferable.
Absence of a clear threshold in terms of steepness is not that unexpected, and the depen-
dence of dis on steepness a 0 (7.76) also goes through the origin. If the transition is due

to the wave-orbital motion becoming turbulent, then it should have a threshold in terms of
WRN, which does not convert into a steepness unambiguously. In this regard, steepness
(7.78) is not in contradiction with the observational data of Figure 7.25 (top).
Figure 7.25 (bottom) demonstrates the dependence of versus Re wave . The best corre-

lation corresponds to the linear fit. The threshold Re wave critical

α

3000 is not apparent in
the plot, due to overall values of WRN being high, but it is there and it is imposed in the
dependence

=

10 − 13

α = 1 . 77 · ( Re wave − 3000 ) (7.82)

shown with the solid line. All the data points correspond to Re wave Re wave critical , and

therefore the mechanism of turbulence production by the waves should be active.
In order to estimate the rate of turbulence production by long ocean swells of Ardhuin
et al. ( 2009a ), the laboratory short-wave dependence (7.76) , i.e.

>

a 0 should be
converted into dimensionless form first. Here, b has dimensions of m − 1 s − 3 , and therefore,

dis = b ·