Energy dissipation across the wave spectrum - Breaking and Dissipation of Ocean Surface Waves - page 269

wind forcing, and also the approximations (7.31) - (7.32) were based on an extrapolation of
the dependence of Donelan et al. ( 2006 ), obtained at moderately strong winds, in extreme
conditions. As discussed at the beginning of this section, such extrapolation is not straight-
forward, since the sea drag most likely tends to saturate (and perhaps even decrease) at
extreme weather. Therefore, this topic, very important for adequate sea-state predictions in
tropical cyclones, is in serious need of experimental and field observations.

7.3.6 Directional distribution of the whitecapping dissipation

The least known feature of the spectral dissipation function is its directional behaviour,
which is the main topic of this subsection. As mentioned throughout the topic (see also
Section 5.3.3 ), the waves are directional, as are all the source functions in (2.61) . Therefore,
some directional shape, usually isotropic, must be assumed for the dissipation term. There
is, however, little, if any, experimental validation of this directional shape.
The segmenting method can be used to obtain incipient-breaking and post-breaking
directional spectra, similar to that used to obtain incipient-breaking and post-breaking
omni-directional spectra in Section 7.3.3 and Figure 7.3 above. The maximum likelihood
method (MLM) developed originally by Capon ( 1969 )(seealso Young , 1994 ; Babanin &
Soloviev , 1987 , 1998b ; Young et al. , 1996 ) was used to analyse the wave-array data and
the wave directional distributions in this respect.
It was noticed that the main wave propagation direction

θ max changes from segment
to segment (in Figure 7.7 it is shown for the spectral-peak frequency f p ). This scatter
around the mean main direction appeared random, not connected to whether the segment
consisted of breaking or of non-breaking waves. Therefore, the non-normalised directional
spectra φ( f p ,θ) were obtained for each of the segments and turned to have the same

main direction ( θ max = 0in Figure 7.8 ). The connection between the non-normalised

directional spectra φ( f ,θ) and normalised directional distributions of K ( f ,θ) in (5.34) -

(5.33) is

φ( f ,θ) = A ( f ) K ( f ,θ) (7.33)

(see Babanin & Soloviev , 1987 , 1998b , for details). Since MLM does not produce
dimensional values for the spectral densities, however, the vertical scale in Figure 7.8 is
arbitrary.
In Figure 7.8 , the solid line designates the mean incipient-breaking directional spectrum
φ i ( f p ,θ) at the spectral peak, and the dotted line - the mean post-breaking directional

spectrum . Clearly, the major energy loss occurs at angles oblique to the main

propagation direction.
Figure 7.9 shows the ratio of

φ p ( f p ,θ)

spectra. Qualitatively, this ratio
reflects the directional behaviour of the dissipation at the spectral peak. Contrary to existing
assumptions, the energy loss in the main propagation direction is a minimum, with the loss
increasing away from this main direction. Since the dissipation has to start decreasing again

φ i ( f p ,θ) and φ p ( f p ,θ)