6.5 Numerical Model of Wind Wave Transformation in a Coastal Area
297
As shown by the calculation results, the mean wave heights and periods for the parallel and straight isobaths are not changed, at least, up to
depths where the non-linear wave transformation takes place. A comparison
of the calculated values and field measurements fulfilled in a water basin with
constant slope (Krylov et al., 1976) reveals their good conformity.
Estimation of wave transformation in a coastal area. Now the wave
transformation in a gulf with isobaths being different from straight lines (see
Fig. 6.16), will be considered. A two-dimensional grid {100 x 38 points) is
chosen for covering a natural water basin. The distance between the grid
points along the Ox axis is equal to 372 m, and along the Oy axis it is 7 43 m.
At the grid points, the water depth Hij is accepted according to the regional
bathimetric map.
As in the previous case, the spectrum is given in the form (5.16) at the
initial boundary (for x = 0). The calculation is made for the point with
coordinates: x = 20.09 x 10 3 ; y = 6.32 x 10 3 . It should be noted that this
point is situated in a hollow created by the bottom relief (see Fig. 6.16).
The characteristic equations {1.86)-(1.89) are integrated numerically. At
the same time the depth H and its gradients along the rays are determined
using interpolation.
As shown by the numerical results the spatial spectrum S(k, (3) is considerably different from the spectrum calculated in the case of a bottom of
constant slope. The spectrum becomes narrow-directed for most wave numbers, excluding the direction coinciding with the isobath directions at the
calculated point. The spectral density is sharply decreased, especially for
small wave numbers. As a result the spectral density S(k) is essentially decreased and the wave number of the maximum is significantly increased. The
relative mean values of the wave elements are as follows: h = 0.31; f = 0.85;
5. = 0.66. As can be seen, the spectrum and mean wave elements calculated
for the chosen bottom relief are principally different from the corresponding
values of the test case with the constant slope basin (6.60).
The reason of this difference can be explained as follows. Firstly, it is the
influence of the geometrical divergence of the wave energy connected with
side boundaries of the water area and limited by the area dimensions, where
the initial spectrum is given. Secondly, this difference can be caused by the
depth variations along the energy-supply ray tube of wave propagation.
Although these two factors are interconnected, an attempt is undertaken
to single out their relative effect. In order to do so some additional calculations
are made, in which the former side boundaries of the water area and the depth
are taken in accordance with the formula for depth variations with a constant
slope (6.60). The depth His accepted to be equal to the mean depth of the
natural basin at the initial point (x = 0) according to (6.60). It also coincides
with the depth at the point under consideration with x = 20.8 x 10 3 m. In
other words, in this calculation the real depth is exchanged for the depth
297
As shown by the calculation results, the mean wave heights and periods for the parallel and straight isobaths are not changed, at least, up to
depths where the non-linear wave transformation takes place. A comparison
of the calculated values and field measurements fulfilled in a water basin with
constant slope (Krylov et al., 1976) reveals their good conformity.
Estimation of wave transformation in a coastal area. Now the wave
transformation in a gulf with isobaths being different from straight lines (see
Fig. 6.16), will be considered. A two-dimensional grid {100 x 38 points) is
chosen for covering a natural water basin. The distance between the grid
points along the Ox axis is equal to 372 m, and along the Oy axis it is 7 43 m.
At the grid points, the water depth Hij is accepted according to the regional
bathimetric map.
As in the previous case, the spectrum is given in the form (5.16) at the
initial boundary (for x = 0). The calculation is made for the point with
coordinates: x = 20.09 x 10 3 ; y = 6.32 x 10 3 . It should be noted that this
point is situated in a hollow created by the bottom relief (see Fig. 6.16).
The characteristic equations {1.86)-(1.89) are integrated numerically. At
the same time the depth H and its gradients along the rays are determined
using interpolation.
As shown by the numerical results the spatial spectrum S(k, (3) is considerably different from the spectrum calculated in the case of a bottom of
constant slope. The spectrum becomes narrow-directed for most wave numbers, excluding the direction coinciding with the isobath directions at the
calculated point. The spectral density is sharply decreased, especially for
small wave numbers. As a result the spectral density S(k) is essentially decreased and the wave number of the maximum is significantly increased. The
relative mean values of the wave elements are as follows: h = 0.31; f = 0.85;
5. = 0.66. As can be seen, the spectrum and mean wave elements calculated
for the chosen bottom relief are principally different from the corresponding
values of the test case with the constant slope basin (6.60).
The reason of this difference can be explained as follows. Firstly, it is the
influence of the geometrical divergence of the wave energy connected with
side boundaries of the water area and limited by the area dimensions, where
the initial spectrum is given. Secondly, this difference can be caused by the
depth variations along the energy-supply ray tube of wave propagation.
Although these two factors are interconnected, an attempt is undertaken
to single out their relative effect. In order to do so some additional calculations
are made, in which the former side boundaries of the water area and the depth
are taken in accordance with the formula for depth variations with a constant
slope (6.60). The depth His accepted to be equal to the mean depth of the
natural basin at the initial point (x = 0) according to (6.60). It also coincides
with the depth at the point under consideration with x = 20.8 x 10 3 m. In
other words, in this calculation the real depth is exchanged for the depth