Energy dissipation across the wave spectrum - Breaking and Dissipation of Ocean Surface Waves - page 242

where the initial steepness ranged very broadly, from 65. A Navier-

Stokes solver was employed for two-phase flow of incompressible air and water, with a
level-set method to capture the surface.
In the paper it was implicitly assumed that the breaking intensity (severity) will depend
on the initial steepness. As discussed in Section 4.1 , based on the Chalikov & Sheinin
( 2005 ) fully nonlinear model, strictly two-dimensional waves should break within one
period if steepness

= 0 . 3 through = 0 .

≥ 0 . 3. Iafrati ( 2009 ) finds this steepness to be

≥ 0 . 33 (7.9)

which is in good accord with the above conclusion and thus can serve as an independent
corroboration of his model. The highest steepness of

65 used by Iafrati ( 2009 )is
unrealistic (see Babanin et al. , 2007a , 2010a ; Toffol i et al. , 2010a ) unless such a wave
is produced by some artificial means, which is perhaps why the outcome of numerical
simulations for waves deviate steeply from the trend established for steepnesses in the
range

= 0 .

= . . = .

0 33-0 60. In the discussion further down, we omit results for the 0 65

steepness.
In between, Iafrati ( 2009 ) found the breaking to be of a spilling type if steepness is in
the range 0 37. According to this model,

the type of breaking and the spilling/plunging behaviour difference (see Section 2.8 )is
determined by the role of surface tension:

. 33 ≤ < 0 . 37, and of a plunging type for ≥ 0 .

“For small amplitude breaking waves, the velocity of the jet tip is not strong enough and thus surface
tension forces prevent the formation of the jet which is replaced by a bulge developing about the
wave crest.”

This bulge starts to slide down the wave's front crest and so the spilling breaking develops.
For larger amplitudes, the surface tension goes round the tip of the jet, but cannot stop it,
and the plunging breaking is produced (see also Iafrati , 2011 ).
It has to be noted that the steepnesses mentioned above are background values for the
initial wave slopes. Iafrati ( 2009 ) does not mention what the value of the steepness was at
the breaking onset, but he does point out that in the cases that resulted in wave breaking,
progressive steepening of the wave profile was observed before the breaking started. As we
know (see Sections 4.1.2 , 5.1.2 ), for two-dimensional waves the breaking onset steepness
should be Hk 44.

The dissipation was investigated in terms of the behaviour of the kinetic energy E K and
potential energy E P :

/ 2 ≈ 0 .

ρ u 2

2 dxdz

1
2

E K = + w , (7.10)

1
2

E P = ρ zdxdz , (7.11)