Energy dissipation across the wave spectrum - Breaking and Dissipation of Ocean Surface Waves - page 249

ineffective in providing energy to long waves, whose speed is comparable with the wind
speed, the downshifting in RTE is customarily attributed to the work on nonlinear interac-
tions depicted by S nl . As discussed in Chapter 1 , this work is slow and takes place at the
scale of hundreds or thousands of wave periods. By comparison, the breaking due to mod-
ulational instability, if the background wave steepness (5.11) of these long waves can be
maintained through the energy sources, is much faster - i.e. it has a temporal scale of tens
of wave periods ( Babanin et al. , 2007a ). If so, then the dissipation term S ds may accept
responsibility for the downshifting in such wave fields.
In the case of strong modulation, Tulin & Waseda ( 1999 ) notice that the breaking is
observed not necessarily for the most unstable sideband mode, but in the broad range of
initial perturbations. This observation may be related to the dependence of the breaking
severity on the modulation depth discussed in Section 6.1 and Galchenko et al. ( 2010 ).
Such an observation also has implications for the instability breaking in wave fields with
full spectrum where a continuous set of carrier waves and their perturbations are
available.
Tulin & Waseda ( 1999 ) then followed on to suggest a theory for their observations of
downshifting and sideband behaviour, both in the breaking and non-breaking cases (see
also Section 8.1 ). In particular, in the case of breaking in a wave train with a known band-
width

δω/ω

, the energy difference between the upper and lower sidebands is a function of
energy dissipation and momentum loss. For the plunging breaking, if contributions of the
secondary sidebands are neglected (which are small), this difference can be parameterised
in terms of the dissipation D b :

∂
∂

D b
(δω/ω) t (

E − 1 − E + 1 ) = γ (7.14)

where 4 is an empirical coefficient.

The second laboratory experiment to be highlighted here is that by Meza et al. ( 2000 ).
The authors also deal with spectrum measurements before and after the breaking, in care-
fully controlled conditions, but the breaking was produced by means of frequency dis-
persion, i.e. linear superposition. Once the superposition leads to a steep wave, nonlinear
effects also start playing a role ( Brown & Jensen , 2001 ), but still the wave-breaking causes
are very different by comparison with a purely nonlinear dynamics leading to the breaking
in the case of modulational-instability development, discussed above.
Meza et al. ( 2000 ) conducted experiments in a 36m-long tank, with deep-water waves
breaking at a prescribed location either in spilling or plunging fashion depending on the
initial steepness.

γ = 0 .

“Very steep transient wave trains were formed by sequentially generating a series of waves from high
to low frequencies that superposed at a downstream location”.

Thus, the setup was essentially the same as in Rapp & Melville ( 1990 ), with some differ-
ences in the form of initial spectra produced. High-resolution resistance wire probes were