58
3 Numerical Implementation of the Wave Energy Balance Equation
The elementary f)-periodical solution of (3.17) can be written as:
(3.19)
where B(m) is a function of the parameter m determined by the initial conditions of the problem. In time duration the solution of (3.19) tends to the
isotropic spectrum:
2n
lim S(;J, T) = B(m) = __!__ J S(f), 0) d;J .
r--+oo
2n
(3.20)
0
The characteristic time scale of the process relaxation is estimated as:
where Tis the average wave period. Since in reality the initial non-isotropic
wave spectrum should not become quasi-isotropic, it can be assumed that
the validity of the approximation (3.16) is limited by the condition:
(3.22)
where tmax is the maximum duration of wave propagation. This condition
gives an upper limit to the value A:
(3.23)
Thus, for an average wave period of 10 s, an angular resolution /1;3 = n/12,
m = 5 and a maximum duration of 36 hours, it is estimated that A :::; 10 2 .
A more precise value of the parameter A can be obtained by a numerical
simulation.
Numerical algorithm.
The numerical implementation of the proposed
correction of the finite angular resolution effects can be presented by a simple
finite difference approximation:
sn+l(wk, f3z) = vSn(wk, f3z-I) + (1- 2v)Sn(wk, f3z)
+vSn(wk,f3l+l),
(3.24)
where n is an indicator of the time step number, v = ACgf1t/12 R and
A = R/ £ are dimensionless parameters depending on typical wave scales.
As shown in (3.21), the value v is not explicitly dependent on the angular
resolution /1;3. It should be noted that (3.16) describes an angular smoothing
of the wave spectrum or a weak "energy exchange" between the angular
components, taking place at every time step of wave propagation.
One of the algorithm testing methods consists in controlling whether the
total wave energy is conserved or not. In order to estimate the total energy
change for all directions, the spectrum is integrated over all angles:
3 Numerical Implementation of the Wave Energy Balance Equation
The elementary f)-periodical solution of (3.17) can be written as:
(3.19)
where B(m) is a function of the parameter m determined by the initial conditions of the problem. In time duration the solution of (3.19) tends to the
isotropic spectrum:
2n
lim S(;J, T) = B(m) = __!__ J S(f), 0) d;J .
r--+oo
2n
(3.20)
0
The characteristic time scale of the process relaxation is estimated as:
where Tis the average wave period. Since in reality the initial non-isotropic
wave spectrum should not become quasi-isotropic, it can be assumed that
the validity of the approximation (3.16) is limited by the condition:
(3.22)
where tmax is the maximum duration of wave propagation. This condition
gives an upper limit to the value A:
(3.23)
Thus, for an average wave period of 10 s, an angular resolution /1;3 = n/12,
m = 5 and a maximum duration of 36 hours, it is estimated that A :::; 10 2 .
A more precise value of the parameter A can be obtained by a numerical
simulation.
Numerical algorithm.
The numerical implementation of the proposed
correction of the finite angular resolution effects can be presented by a simple
finite difference approximation:
sn+l(wk, f3z) = vSn(wk, f3z-I) + (1- 2v)Sn(wk, f3z)
+vSn(wk,f3l+l),
(3.24)
where n is an indicator of the time step number, v = ACgf1t/12 R and
A = R/ £ are dimensionless parameters depending on typical wave scales.
As shown in (3.21), the value v is not explicitly dependent on the angular
resolution /1;3. It should be noted that (3.16) describes an angular smoothing
of the wave spectrum or a weak "energy exchange" between the angular
components, taking place at every time step of wave propagation.
One of the algorithm testing methods consists in controlling whether the
total wave energy is conserved or not. In order to estimate the total energy
change for all directions, the spectrum is integrated over all angles: