52
3 Numerical Implementation of the Wave Energy Balance Equation
described by the equation used in the WAM model (The WAM model, 1988;
Komen et al., 1994) in the form:
B(S) = &S + _1_&(rpcos¢S) + &(JS) + &(~S) = G
( 3 . 1 )
&t
cos 'P
&cp
8{)
&(3
'
where B(S) is the differential operator and G is the source function. Based
on the set (1.86)-(1.90), the equations of motion for a wave packet along the
arc of a great circle can be written as follows:
. _ C sin/3 .
'P- g R '
J-c cos(3.
-
g Rcoscp'
0
cos (3
(3 = -Cg tan 'P~ ,
where Cg is the group velocity and R is the Earth's radius.
(3.2)
(3.3)
(3.4)
Further, a solution of (3.1)-(3.4) for swell wave propagation is investigated, assuming that the source function G is equal to zero.
Initial conditions. In order to formulate more or less real initial conditions, the problem of satellite data assimilation might be taken into consideration. The data are used to improve the results of calculating and predicting
wind waves (Bauer et al., 1992; Burgers et al., 1992; Lionello et al., 1992).
It i;; assumed that wind waves of large height are revealed with the help of
satellite altimeter observations in the northern area of the North Sea. Initially,
the perturbation centre is located at the point {)0 = 0°, 'Po = 72°. The initial
perturbation is distributed over space in accordance with the approximation
exp( -8r/ Lmax), where 8r is the distance between the centre of the initial
perturbation and the considered point, and Lmax is the correlation radius
of the initial perturbation. This can be taken as Lmax = 150 km for the
North Sea scale, according to Burgers et al. (1992). It is assumed that the
waves propagate southward at the initial moment of time. This means that
the general direction of wave propagation makes up the angle (30 = -90°.
A significant wave height at the centre would be 10m and a mean period 15 s.
The spectrum of the initial perturbation is approximated by the formula:
So(w, (3, cp, {), t = 0) = So(w, (3)F( ip, {)) = So(w)Qo(f3)F( cp, {)) ,
(3.5)
where F(cp, {)) is a spatial distribution function; Q0 (f3) is a function of the
angular energy distribution; and S0 (w, (3) is the initial wave frequency-angular
spectrum. The spatial distribution function is assumed to be:
F( cp, {)) = exp( -aJ[( {)- {)o)2 cos 2 'Po+ ( 'P- 'Po)2]) ,
(3.6)
3 Numerical Implementation of the Wave Energy Balance Equation
described by the equation used in the WAM model (The WAM model, 1988;
Komen et al., 1994) in the form:
B(S) = &S + _1_&(rpcos¢S) + &(JS) + &(~S) = G
( 3 . 1 )
&t
cos 'P
&cp
8{)
&(3
'
where B(S) is the differential operator and G is the source function. Based
on the set (1.86)-(1.90), the equations of motion for a wave packet along the
arc of a great circle can be written as follows:
. _ C sin/3 .
'P- g R '
J-c cos(3.
-
g Rcoscp'
0
cos (3
(3 = -Cg tan 'P~ ,
where Cg is the group velocity and R is the Earth's radius.
(3.2)
(3.3)
(3.4)
Further, a solution of (3.1)-(3.4) for swell wave propagation is investigated, assuming that the source function G is equal to zero.
Initial conditions. In order to formulate more or less real initial conditions, the problem of satellite data assimilation might be taken into consideration. The data are used to improve the results of calculating and predicting
wind waves (Bauer et al., 1992; Burgers et al., 1992; Lionello et al., 1992).
It i;; assumed that wind waves of large height are revealed with the help of
satellite altimeter observations in the northern area of the North Sea. Initially,
the perturbation centre is located at the point {)0 = 0°, 'Po = 72°. The initial
perturbation is distributed over space in accordance with the approximation
exp( -8r/ Lmax), where 8r is the distance between the centre of the initial
perturbation and the considered point, and Lmax is the correlation radius
of the initial perturbation. This can be taken as Lmax = 150 km for the
North Sea scale, according to Burgers et al. (1992). It is assumed that the
waves propagate southward at the initial moment of time. This means that
the general direction of wave propagation makes up the angle (30 = -90°.
A significant wave height at the centre would be 10m and a mean period 15 s.
The spectrum of the initial perturbation is approximated by the formula:
So(w, (3, cp, {), t = 0) = So(w, (3)F( ip, {)) = So(w)Qo(f3)F( cp, {)) ,
(3.5)
where F(cp, {)) is a spatial distribution function; Q0 (f3) is a function of the
angular energy distribution; and S0 (w, (3) is the initial wave frequency-angular
spectrum. The spatial distribution function is assumed to be:
F( cp, {)) = exp( -aJ[( {)- {)o)2 cos 2 'Po+ ( 'P- 'Po)2]) ,
(3.6)