Energy dissipation across the wave spectrum - Breaking and Dissipation of Ocean Surface Waves - page 245

for coastal engineering applications which drive these efforts, but are a somewhat different,
perhaps quite different phenomenon compared to the deep-water breaking considered here.
Lubin et al. ( 2006 ) dedicated a section of their paper to investigation of the time evolu-
tion of the dissipation of a breaking wave with initial height H = 0 . 1 λ in a water depth

of d is the length of this wave and the energy is presented in (7.10) -

(7.12) terms.Thewaveperiodwas0

= 0 . 1 λ , where λ

. .

1 s preceding the breaking onset
the total energy was slightly dissipating to viscosity, while the kinetic energy was being
converted into potential energy due to steepening and growing of the wave crest. The latter
pre-breaking-onset energy behaviour is familiar from simulations by means of potential
models ( Chalikov , 2009 ).
Once the jet starts to plunge and accelerate, potential energy transforms back into kinetic
energy until it hits the surface in front of the wave. This happens within 0

339 s, and for the 0

07 s, i.e. approxi-
mately 1/5th of the wave period, which result is in excellent agreement with the conclusion
made in Section 2.4 that the developing phase of the breaking constitutes 20% of the
active-breaking duration (provided this duration is of the order of a wave period). The jet
rebounds, and the kinetic energy keeps increasing until it reaches a maximum in another
0

.

03 s which corresponds to generation of the first splash-up. By this time, more than 60%
of the original potential energy is either turned into kinetic energy or dissipated.
At t

.

2 s after the breaking onset, the splash-ups exhaust themselves, and at this stage
about 40% of the total pre-breaking wave energy is dissipated. Then, the energy continues
to dissipate at a slower rate, and in five wave periods about 35% of the total energy, 40%
of the kinetic energy and 30% of the potential energy remain in the wave.
The difference between the dissipation in two-dimensional and three-dimensional plun-
ging breakers, as concluded both by Lubin et al. ( 2006 ) and Iafrati ( 2009 ), becomes essen-
tial at these later stages of the breaking. Iafrati ( 2009 ) argues that overturning of the jet and
the first jet impact are basically two-dimensional processes. With subsequent splash-ups,
vortex generation and production of turbulence, three-dimensional effects become essen-
tial. Among them, Iafrati ( 2009 ) points out that lateral instabilities affect both the air-pocket
fragmentation and dynamics of the large vorticity structures.
These effects, not accounted for in two-dimensional models, should lead to larger lev-
els of energy dissipation in a three-dimensional case. Analogies between the deep-water
breaking such as in Iafrati ( 2009 ) and the shallow-water breaking of Lubin et al. ( 2006 )
should be drawn with caution, as the dynamics can change significantly between the for-
mer case and the latter, but still the examination of the differences between 2D and 3D
cases done by Lubin et al. ( 2006 ) is quite instructive.
Up to some half of the wave period after breaking onset, the two-dimensional and three-
dimensional total-energy curves are hardly distinguishable. The difference then
appears and grows up to 10% over some two wave periods. After that, the role of the three-
dimensional turbulence and other 3D effects apparently subsides, and while a different
amount of energy is now left in a wave, the curves remain parallel to each other.
Thus, the phase-resolvent models represent an excellent opportunity, grossly under-used
and under-developed at the present stage, for investigating dissipation behaviour in the

≈ 0 .