56
3 Numerical Implementation of the Wave Energy Balance Equation
discretizations. It should be noted that the third and fourth terms of (3.13),
included in square brackets, are of most interest. They make up the correction
of the equation due to the finiteness of the angular and frequency spectral
resolution. These terms contain the derivatives on the spatial, angular and
frequency variables. They are proportional to the values E and 8 and the group
velocity Cg. The last terms in (3.13) are higher-order corrections O(c 2 ) and
0(8 2 ).
The correction terms in (3.13) are comparable to a similar expression derived for the case of wave propagation over a plane surface (Booij & Holtuijsen,
1987). This depends on the "wave age", which is not defined locally. In order
to determine it, an additional equation must be solved. The main advantage
of the correction term (3.13), derived in the present study, consists in determining it locally for wave propagation over a spherical surface. For a plane
surface, the kinetic equation can be written in a similar way:
fJS C fJS C fJS
C !!_ { fJ(cos(JS) _ fJ(sin(JS)}
at + gx ax + gy fJy + E g 8(3
fJy
fJx
C !!_{fJ(cos(JS) _ fJ(sin(JS)} O( 2) 0(82 )=0
+ gW fJw
fJx
fJy
+ E +
' (3.14)
where Cgx = (1 + E/2 + 8)Cgcos(3 and Cgy = (1 + E/2 + 8)Cgsin(3 are
components of the group velocity. Thus, additional terms in (3.13) and (3.14)
are dependent on the frequency-angular resolution, group velocity and spatial
and angular non-uniformity of the wave field.
Investigation of a special case.
The additional terms in the left-hand
side of the wave energy balance equation (3.13) significantly increase the
computation time in wind wave models. However, this problem can be considerably simplified in some special cases.
It should be noted that the strong influence of the "garden-sprinkler effect" is caused by a coarse angular resolution. There are 12 directions usually
used in most models (Ryvkin, 1990; Ryabinin, 1991a,b; Booij and Holtuijsen, 1987; Ocean Wave Modeling, 1985), but, actually, this number is not
sufficient. At the same time frequency spectral descritization plays a less important role. 1 Our consideration is limited only by the angular descritization
effect.
The value £ can be assumed as a typical spatial scale of wave propagation
in some basin. For the ocean, the value £ makes up the order of several
thousand kilometres while for shelf seas (such as the North Sea), it is the
1 This fact can be quantitatively confirmed by actual assessment of the parameters c: and 8. For example, for the WAM model (1988) /::;,(3 = n/6 "' 0.5 and
l::;.wjw ~ 0.1, soc:» 8.
3 Numerical Implementation of the Wave Energy Balance Equation
discretizations. It should be noted that the third and fourth terms of (3.13),
included in square brackets, are of most interest. They make up the correction
of the equation due to the finiteness of the angular and frequency spectral
resolution. These terms contain the derivatives on the spatial, angular and
frequency variables. They are proportional to the values E and 8 and the group
velocity Cg. The last terms in (3.13) are higher-order corrections O(c 2 ) and
0(8 2 ).
The correction terms in (3.13) are comparable to a similar expression derived for the case of wave propagation over a plane surface (Booij & Holtuijsen,
1987). This depends on the "wave age", which is not defined locally. In order
to determine it, an additional equation must be solved. The main advantage
of the correction term (3.13), derived in the present study, consists in determining it locally for wave propagation over a spherical surface. For a plane
surface, the kinetic equation can be written in a similar way:
fJS C fJS C fJS
C !!_ { fJ(cos(JS) _ fJ(sin(JS)}
at + gx ax + gy fJy + E g 8(3
fJy
fJx
C !!_{fJ(cos(JS) _ fJ(sin(JS)} O( 2) 0(82 )=0
+ gW fJw
fJx
fJy
+ E +
' (3.14)
where Cgx = (1 + E/2 + 8)Cgcos(3 and Cgy = (1 + E/2 + 8)Cgsin(3 are
components of the group velocity. Thus, additional terms in (3.13) and (3.14)
are dependent on the frequency-angular resolution, group velocity and spatial
and angular non-uniformity of the wave field.
Investigation of a special case.
The additional terms in the left-hand
side of the wave energy balance equation (3.13) significantly increase the
computation time in wind wave models. However, this problem can be considerably simplified in some special cases.
It should be noted that the strong influence of the "garden-sprinkler effect" is caused by a coarse angular resolution. There are 12 directions usually
used in most models (Ryvkin, 1990; Ryabinin, 1991a,b; Booij and Holtuijsen, 1987; Ocean Wave Modeling, 1985), but, actually, this number is not
sufficient. At the same time frequency spectral descritization plays a less important role. 1 Our consideration is limited only by the angular descritization
effect.
The value £ can be assumed as a typical spatial scale of wave propagation
in some basin. For the ocean, the value £ makes up the order of several
thousand kilometres while for shelf seas (such as the North Sea), it is the
1 This fact can be quantitatively confirmed by actual assessment of the parameters c: and 8. For example, for the WAM model (1988) /::;,(3 = n/6 "' 0.5 and
l::;.wjw ~ 0.1, soc:» 8.