6.5 Numerical Model of Wind Wave Transformation in a Coastal Area
293
roughness. The spectrum F(k) is considered to be a monotonic decreasing
function of the wave number k. Two cases can be distinguished: the spectrum
F(k) is wide enough to correspond to a small correlation scale; and the spectrum F(k) is narrow enough (i.e., a large correlation length). The first case
is considered to be scattering in the medium with small-scale fluctuations.
The second one is considered to be scattering in the medium with large-scale
fluctuations. But in both cases the scattering is effected by the same wave
numbers. It is obvious that for the same value (c 2 ) the function F(O) is
greater for large-scale than for small-scale non-uniformity, as the integral of
the function F(k) is equal to (c 2 ). That is why the wave absorption coefficient is greater for large bottom roughness than for a small one. At the same
time the value J.L is mainly determined by forward scattering for large-scale
roughness; but as for small-scale roughness it is determined by scattering
forward and backward. The scattering is practically equal in both directions.
As a wave propagates at some angle to the bottom unevenness ({3 f. 0), the
absorption coefficient depends on the incidence angle, and at the same time
backward scattering disappears at {3 = 45°. This can be explained by noting
that for the angle of 45°, the scattered wave propagates perpendicularly to
the main one, without affecting its amplitude. It is interesting to note that
the scattered wave also propagates forward for {3 > 45°. The value J.L is unlimitedly increased as {3 -+ 90°. This result is easily explained by the resonance
character of scattering. The effective correlation scale is increased sharply
(Leff = L/ cosf3) for a wave propagating under the small angle n/2- {3, so
the unevenness becomes more large-scaled, and the scattering is increased
sharply.
Now the two-dimensional isotropic roughness of the bottom relief with
the spectrum:
3 L
2
2 • It is seen that the power dependence of the
l+l/3k2H 0
absorption coefficient is changed (k 3 instead of k 2 ) in the case of small-scale
roughness. At large scale roughness the spectrum
,f9HOk2L2
can be found that J.L rv - y'~
2 •
l+l/3k 2 H 0
Thus, with this approach, the classic problem of wave scattering by
a rough bottom can be solved more precisely taking into account wave dispersion.
6.5 Numerical Model of Wind Wave Transformation
in a Coastal Area
Formulation of the problem. Some cases, when the solution of the problem of wave refraction in shallow water and in non-uniform currents can be
obtained analytically, are given in the previous sections. It is impossible to
293
roughness. The spectrum F(k) is considered to be a monotonic decreasing
function of the wave number k. Two cases can be distinguished: the spectrum
F(k) is wide enough to correspond to a small correlation scale; and the spectrum F(k) is narrow enough (i.e., a large correlation length). The first case
is considered to be scattering in the medium with small-scale fluctuations.
The second one is considered to be scattering in the medium with large-scale
fluctuations. But in both cases the scattering is effected by the same wave
numbers. It is obvious that for the same value (c 2 ) the function F(O) is
greater for large-scale than for small-scale non-uniformity, as the integral of
the function F(k) is equal to (c 2 ). That is why the wave absorption coefficient is greater for large bottom roughness than for a small one. At the same
time the value J.L is mainly determined by forward scattering for large-scale
roughness; but as for small-scale roughness it is determined by scattering
forward and backward. The scattering is practically equal in both directions.
As a wave propagates at some angle to the bottom unevenness ({3 f. 0), the
absorption coefficient depends on the incidence angle, and at the same time
backward scattering disappears at {3 = 45°. This can be explained by noting
that for the angle of 45°, the scattered wave propagates perpendicularly to
the main one, without affecting its amplitude. It is interesting to note that
the scattered wave also propagates forward for {3 > 45°. The value J.L is unlimitedly increased as {3 -+ 90°. This result is easily explained by the resonance
character of scattering. The effective correlation scale is increased sharply
(Leff = L/ cosf3) for a wave propagating under the small angle n/2- {3, so
the unevenness becomes more large-scaled, and the scattering is increased
sharply.
Now the two-dimensional isotropic roughness of the bottom relief with
the spectrum:
3 L
2
2 • It is seen that the power dependence of the
l+l/3k2H 0
absorption coefficient is changed (k 3 instead of k 2 ) in the case of small-scale
roughness. At large scale roughness the spectrum
,f9HOk2L2
can be found that J.L rv - y'~
2 •
l+l/3k 2 H 0
Thus, with this approach, the classic problem of wave scattering by
a rough bottom can be solved more precisely taking into account wave dispersion.
6.5 Numerical Model of Wind Wave Transformation
in a Coastal Area
Formulation of the problem. Some cases, when the solution of the problem of wave refraction in shallow water and in non-uniform currents can be
obtained analytically, are given in the previous sections. It is impossible to