Geoscience Reference
In-Depth Information
analogous to (5.50) to be relevant. Parameter A (5.53) can be used for this purpose as it
has the proper physical meaning of the inverse relative width of the directional spectrum
whose peak is normalised to be 1.
A t the spectral peak, a relative steepness (as the wave spectrum develops) is defined by
γ
is the peak enhancement of the JONSWAP spectrum (2.7) . That is, for the
peak, we can define a directional analogue of M I as
where
γ
A γ.
M Id =
(5.57)
Now, it is informative to look at how this index evolves over the wave development.
From (eq. 19) of Babanin & Soloviev ( 1998b ), at the spectral peak
12 U 10
c p
0 . 50
π) 1
A
=
1
.
+ (
2
,
(5.58)
and from (eq. 44) of Babanin & Soloviev ( 1998a )
7
.
6
U 10
c p ,
γ =
(5.59)
2
π
that is
10 U 10
c p
0 . 50
γ =
1
.
.
(5.60)
Therefore,
U 10
c p
0 . 50
1
2
M Id =
1
.
23
+
(5.61)
π
is a weak function of the wind forcing, and its value at the spectral peak varies from 1.40
to 1.79 for U 10
c p
in the range from 0.89 to 10 where U 10
c p
=
0
.
89 signifies the limit of full
development ( Pierson & Moskowitz , 1964 ).
Now, if the M Id assumption is valid and the critical value for this index is in the range
of M Id =
8, the 'de-focusing' effect of directionality can be overcome by a stronger
nonlinearity if waves grow steeper. It is worth noting here that the directional spectra
broaden towards frequencies above the peak (e.g. Babanin & Soloviev , 1998b ). This means
that, even if applicable at the peak, the directional modulational instability may not be
working at higher frequencies and some other causes of breaking and dissipation will have
to be found in that spectral band. In this regard, two-phase behaviour of breaking has
indeed been observed in field experiments of Babanin & Young ( 2005 ), Manasseh et al.
( 2006 ) and Babanin et al. ( 2007c )(see Sections 5.3.2 and 7.3.4 for a detailed discussion)
- i.e. the direct dependence of breaking on spectral density at the peak and an induced
breaking/dissipation at higher frequencies.
If the steepness of waves is known, then the directional modulational index (5.57) can
be formulated explicitly:
1
.
4-1
.
M Id =
A
·
ak
.
(5.62)
 
Search WWH ::




Custom Search