30
1 General Problem Formulation
dk19 = _ {taH + av"' k"' + av19 k19 }
dt
{){)
{){) R
{){} R cos( cp)
(1.74)
d1-l
dw
k"' av'P
k19
av"'
- = - = - - +
-
dt
dt
R 8t
Rcos(cp) at '
(1. 75)
where
(1. 76)
and
8k
k'P
8k
k19
8k'P
kR 2 ' 8k19
kR 2 cos 2 ( cp)
(1. 77)
8k
k'P tan( cp)
!=
gk
k
8cp
kR 2 cos 2 ( cp)
th(kH) ch 2 (kH)
(1. 78)
c = ~Jgth(kH) ( 1
2kH )
g
2
k
+ sh(2kH) '
where V19, V'P are the zonal and meridional components of the current velocity.
Equations (1. 71 )-(1. 75) describe the motion of the wave packet in a spherical
surface under the influence of the non-uniform current velocity V( cp, {), t) and
depth H( cp,{J).
The wave number k = I k I (or the frequency w) and the angle (3 between
the wave vector direction and the x axis of local rectangular coordinates are
usually used in numerical simulations of wind waves. The wave number k is
connected with the former variables k'P and k19 by the ratio (1. 76), and the
angle (3 can be determined as:
(3 _ k'P cos( cp)
tan -
k 19
.
(1. 79)
Using the ratios (1.73), (1.74), it can be shown that variations of the new
variables k and (3 in time are connected with the previous variables by:
.
1 [·
k19k19]
k = kR2 k'Pk'P + cos2 cp ;
(1.80)
(3 . = cos cp cos2 (3 [k k - k k ]
k2
19 'P
19 'P .
19
(1.81)
The relation between the value N, introduced by the aforementioned
method, and the spectral density of the wave action N(k), usually used in
local rectangular plane coordinates {x,y}, will now be determined. It should
be noted that the typically used spectral density N ( k) is determined as