3.3 Standard Numerical Propagation Schemes
53
where o: = 2(R/L) and L = 150km, being the constants determining the
extent of the decrease of the initial perturbation with distance.
The angular energy distribution function is assumed to be equal to:
Qo(fJ) = { ~~ cos
4
((J- fJo) with lfJ- fJol :::; ~ ,
with lfJ- fJol > ~ .
The frequency spectrum is described by the dependence:
() ( ) w;:.ax [n+1(Wmax)n]
Sow = n+ 1 mown+l exp --n- ~
'
(3.7)
(3.8)
where m 0 is the zero spectrum moment, Wmax is the frequency of the spectral
maximum and n is a parameter characterizing the frequency bandwidth. For
swell, it can be assumed that n = 5 (Davidan et al., 1985).
The propagation of the initial perturbation over the water area during
48 hours is considered. The problem is solved analytically and then numerically using two different methods. As the first method, a numerical scheme
implemented in the WAM model is used. The interpolation-ray method is
used as the second one. The numerical calculation errors are estimated by
comparison with the analytical solution.
For obtaining an analytical solution of the problem the ratios (3.2)-(3.4)
are substituted in (3.1), written in the advective form (2.1), i.e. in the form
of a complete time derivative. As noted above, the spectral energy density is
preserved along the characteristics in the case of the source function equal
to zero, G = 0. This analytical solution (2.14) of (2.1) was obtained in the
previous section.
3.3 Standard Numerical Propagation Schemes
Two numerical propagation schemes should be mentioned among the usually
used ones. They are implemented into the WAM model (1988). The first-order
upwind scheme, applied for (3.1), can be written as:
The second-order leapfrog scheme is written in the following form:
+ diffusion ,
(3.10)
where the index n is a time step number, and indices k_ and k+ are referred to
the neighbouring grid points in the upstream and downstream propagation
53
where o: = 2(R/L) and L = 150km, being the constants determining the
extent of the decrease of the initial perturbation with distance.
The angular energy distribution function is assumed to be equal to:
Qo(fJ) = { ~~ cos
4
((J- fJo) with lfJ- fJol :::; ~ ,
with lfJ- fJol > ~ .
The frequency spectrum is described by the dependence:
() ( ) w;:.ax [n+1(Wmax)n]
Sow = n+ 1 mown+l exp --n- ~
'
(3.7)
(3.8)
where m 0 is the zero spectrum moment, Wmax is the frequency of the spectral
maximum and n is a parameter characterizing the frequency bandwidth. For
swell, it can be assumed that n = 5 (Davidan et al., 1985).
The propagation of the initial perturbation over the water area during
48 hours is considered. The problem is solved analytically and then numerically using two different methods. As the first method, a numerical scheme
implemented in the WAM model is used. The interpolation-ray method is
used as the second one. The numerical calculation errors are estimated by
comparison with the analytical solution.
For obtaining an analytical solution of the problem the ratios (3.2)-(3.4)
are substituted in (3.1), written in the advective form (2.1), i.e. in the form
of a complete time derivative. As noted above, the spectral energy density is
preserved along the characteristics in the case of the source function equal
to zero, G = 0. This analytical solution (2.14) of (2.1) was obtained in the
previous section.
3.3 Standard Numerical Propagation Schemes
Two numerical propagation schemes should be mentioned among the usually
used ones. They are implemented into the WAM model (1988). The first-order
upwind scheme, applied for (3.1), can be written as:
The second-order leapfrog scheme is written in the following form:
+ diffusion ,
(3.10)
where the index n is a time step number, and indices k_ and k+ are referred to
the neighbouring grid points in the upstream and downstream propagation