Energy dissipation across the wave spectrum - Breaking and Dissipation of Ocean Surface Waves - page 277

term is based on a threshold behaviour of wave breaking appeared to be mistaken. The
formulation did not consistently employ experimentally observed threshold dependences,
nor did it use experimentally obtained values for the threshold. On the contrary, the mag-
nitude of the dimensionless switch, which imitated the no-breaking threshold, appeared
to correspond to the spectra which certainly comprise breaking waves, and these waves
break at a typical for moderate conditions rate of some 5% (i.e. Babanin et al. , 2001 ). As
a result, the meaning of the switch turned to be to an extent opposite to its interpretation.
That is, the switch did not signify a breaking threshold, i.e. a lower spectral limit below
which no breaking occurs, but rather a soft upper limit for the spectrum growth, allowed by
the formulation itself. Subsequently, the dissipation function, being postulated as strongly
nonlinear, worked as quasi-linear. The role of the nonlinear option was mainly limited to
quenching the spectrum down if it happened to exceed the switch level; below this level
conventional spectral evolution, controlled by linear dissipation and input, took place and
thus regular growth rates were well reproduced.
As such, however, the Alves & Banner ( 2003 ) intention was an important step towards
adequate description of wave-field dynamics. It has been followed by a number of stud-
ies employing similar so-called 'saturation-based' dissipation terms S ds (i.e. formulations
based on a non-dimensional 'saturation' spectrum (5.31) ,see Van der Westhuysen et al.
( 2007 ) and Ardhuin et al. ( 2010 )).
Van der Westhuysen et al. ( 2007 ) initially based their simulations by means of the
SWAN model ( Booij et al. , 1999 )onthe Alves & Banner ( 2003 ) formulation, but incorpor-
ated appropriate threshold limitations and a wind-forcing dependence for the dissipation
function. As a result, in the physical sense it is essentially a new dissipation term.
First, the threshold parameter was qualitatively and quantitatively returned to its origi-
nal meaning of the bottom limit for wave breaking. Secondly, Van der Westhuysen et al.
( 2007 ) reviewed the scaling of the major source terms and made them consistent, i.e. expo-
nents of the dissipation and the wind-input functions now matched (in Alves & Banner
( 2003 ), the dissipation was declared strongly nonlinear in terms of the wave spectrum, but
then linear counterpart wind-input functions were used to reproduce regular wave-growth
dependences). Besides, gradual transition from the strongly forced sea-state cases to the
mature seas was implemented which was a novel physically sound feature. The Van der
Westhuysen et al. ( 2007 ) function is indeed nonlinear, and this nonlinearity varies in the
course of wave evolution. Also, depth dependence for the dissipation was introduced.
Ardhuin et al. ( 2010 ) further extended the saturation-based formulations. They added
the cumulative term and an S turb dissipation term (see eq. 7.26 ), responsible for wave-
turbulence interactions and based on the Ardhuin & Jenkins ( 2006 ) mechanism. Their
inherent-breaking term has an isotropic part and a direction-dependent part. The latter
allows us to use the dissipation function to control the directional spread of the wave spec-
trum (see Section 7.3.6 for a discussion on this issue). The dissipation is essentially linear
in terms of the wave spectrum F .

A separate dissipation term was reserved for swell, which does not break and exhibits
physical properties, different to wind-seas, in other regards as well. Ardhuin et al. ( 2010 )

( f ,θ)