Energy dissipation across the wave spectrum - Breaking and Dissipation of Ocean Surface Waves - page 290

is intended to describe how the directional spreading of the wave spectrum can poten-
tially be controlled through directionality of the dissipation term. These moderate alter-
ations of directional-spreading dissipation features do not impact the frequency-distributed
behaviour of the source terms in any way.
Following Tsagareli ( 2009 ), the directional-spreading function was developed as a super-
position of two Gaussian functions. According to the experimental results of Young &
Babanin ( 2006a )(see Figures 7.9 , 7.11 ), the angle of separation between the maxima
and the main wave-propagation direction can vary. Therefore, the new directional func-
tion for the dissipation term was provided with sufficient flexibility to modify the shape in
both directional and frequency spaces. The new directional-spreading function includes the
ability to symmetrically shift the locations of the directional peaks, to alter the magnitude
of the trough between the maxima, to change the cross-sectional shapes as a function of
frequency, and to vary with different wind-forcing conditions.
The spreading function V takes the form
V

A

2

( f ) exp ( − p (θ + θ p ) )θ< 0,

U 10
c p

θ, f , = (7.56)

2

A ( f ) exp ( − p (θ − θ p ) )θ ≥ 0

where p = p ( f , U 10 / c p ) is the parameter that determines the depth of the middle trough,

θ is the angle (in radians) relative to the main propagation direction of waves, θ p =

θ p ( f , U 10 / c p ) is the angle (in radians) of the maximum dissipation rates relative to this

main direction.
The term A

is the inverse integral of the directional spread defined in (5.33) above. In
order to avoid confusion, however, it has to be stressed that the directional wave-spectrum
spreading function and the directional dissipation-spreading function are different proper-
ties. For clarity, here the normalised dissipation directional spread will be designated D
rather than K which symbol is reserved for the function of normalised wave directional
spectrum in (5.34) . That is, we used

( f )

2

D = exp ( − p (θ + θ p ) ) (7.57)

in (7.56) . The normalisation condition π
− π

V ( f ,θ) d θ = 1 is then satisfied; and if the angle

θ p = 0, the directional spreading has a unimodal shape.

Now, as the angle θ p increases, the lobes of the directional spreading function move

further apart, enhancing the depth of the trough between them. Increasing the parameter
p reduces the width of the lobes, increasing the depth of the trough at the same time. As
a result, variations of parameters p and

θ p lead to different directional spreading of the
dissipation which allows the model to control the directional wave spectrum, a property
which has proved difficult in previous modelling tests (see e.g. Banner & Young , 1994 ).
Figure 7.17 demonstrates the resulting directional spreading function V (θ, f , U 10 / c p )

(7.56) with different values for the parameters p and θ p . For convenience of compari-

son, all the directional spreading functions were normalised by the maximum value at the
angle

θ p , i.e. D = V (θ p , f , U 10 / c p ) = 1.