5.10 Wind Wave Generation in a Current
243
tangent. The character of this transfer is determined by the parameter 0.
The value o is estimated for developed wave stages as 2.0 according to the
JONSWAP experiment.
Estimation of wind wave elements in a current.
The obtained solution is used for estimating the wave element evolution along the fetch in
a current. In order to emphasize the current effect on waves the variables are
changed in the spectral expression S ( k, f3, X, i) as follows:
1 4
4
2
- 1
(
n+1 )
S = 2 u (1- i) g- Q (f3)(n + 1) moohf2k- exp --n-12 , (5.149)
where:
h ('k,/3,X,i)
{ th~ ( 2 · 72 xw-sx )
8
when 1- 2i'Vk/ cos (/3) ~ 0;
=
(1-i) 2 (1-2icos- 1 (,B)Vk)
1.0
when 1- 2)'Vk/ cos (/3) < 0;
h (i., /3, x, i) = J- ifoo 11o.34.
k (1- i)
Now the spectrum S is a function of the non-dimensional wave number
k = kJVTiJ, the angle /3, the non-dimensional fetch X and the relative
current velocity parameter i = VjU. At the value i = 0, the spectrum
(5.149) is transformed to the wind wave spectrum in the absence of current.
The ratio of the mean wave height li in a current to the height h0 in the
absence of a current can be expressed by the ratio of corresponding statistical
moments:
1
[ ~: l1 s (k,{3,X,'i) d/ 3 dkr ~ F, ( X,'i) (5.150)
where F 1 (X, i) is a function of two non-dimensional parameters: the fetch X
and the relative current velocity)'.
The ratio of the mean wavelengths is written similarly:
X y1u;.(x)(~)fiJr(1-4/n)Jmo(x) __
X =
1
F1 (X, 1)
0
[llk2 cos 2 (/3) s (i., /3, x, i) dk d/3 r
=F2(x,.:y).
(5.151)