284
6 Wave Transformation in Shallow Water
depth. The described tendency of wave element variations is preserved with
decreasing initial depth. A joint decrease in depth and increase in current
velocity results in increasing the maximum wave height (h ~ 2.5 at 0: ~ 0.1
and ii ~ 10- 3 , see Fig. 6.14b).
The wave element transformation will be considered in the section of
a counterflow' with the current velocity spatial variation: av I ax # 0. The
peculiarity of the wave behaviour is connected with reflecting wave packets
propagating countercurrent at this precise section. After being reflected, the
wave packets are carried away backwards by the current in the form of reverse
waves, increasing their heights and decreasing their lengths. The total wave
pattern in this case is represented by a superposition of straight and reverse
waves.
The appearance of reverse waves in a flow is determined by the blocking
condition (6.33). If it is fulfilled, the spectral components of reverse waves
make a significant contribution to the total wave spectrum. Limitations of the
spectrum value connected with possible wave breaking are introduced, using
the notion of the equilibrium range. Its presence in the solution results in the
appearance of the parameter characterizing the ratio of the initial spectrum
value relative to the equilibrium range value. In Sect. 5.5, this value is expressed by the mean steepness ho/5.0 of the initial wave spectrum. The initial
steepness value can significantly affect the wave element changes in a current.
The values of wave elements along the increasing flow ( 0: = 5; ii = 0, 1) for
waves with steepness 1/36 (corresponding to the mean wind wave steepness)
are shown in Fig. 6.15, where the straight and reverse waves as well as their
breaking are taken into account. The behaviour of wave elements is similar to
that described above (see Fig. 5.16). It is seen that the presence of the reverse
wave and its breaking do not result in a large increase of mean heights.
In conclusion it should be noted that the evolution of mean wave elements
occurs under the effect of two mechanisms. They are, firstly, a wave transformation by the action of a horizontally non-uniform current and, secondly, by
changing depth of the basin. Although these mechanisms are described using the same equations (within the framework of the initial approximation),
their effects are different. The current influences are greater on the shortwave spectrum components, while the changing depth affects the long wave
components. They penetrate into deeper water and are subjected to depth
variations.
As shown by the obtained results the effect of non-uniform depth and
current velocity lead to such changes of wave elements, which are impossible
to predict using the mechanisms of transformation separately. In some cases,
the combined effect of the changing depth and current essentially increases the
wave transformation, while decreasing it in some other cases. In shallow water
areas with spatial non-uniform currents, such situations can be observed,
when the wave heights and other wave elements are increased greatly (by
several times). But there are also opposite situations, with waves decrease up
to their almost complete disappearance.
6 Wave Transformation in Shallow Water
depth. The described tendency of wave element variations is preserved with
decreasing initial depth. A joint decrease in depth and increase in current
velocity results in increasing the maximum wave height (h ~ 2.5 at 0: ~ 0.1
and ii ~ 10- 3 , see Fig. 6.14b).
The wave element transformation will be considered in the section of
a counterflow' with the current velocity spatial variation: av I ax # 0. The
peculiarity of the wave behaviour is connected with reflecting wave packets
propagating countercurrent at this precise section. After being reflected, the
wave packets are carried away backwards by the current in the form of reverse
waves, increasing their heights and decreasing their lengths. The total wave
pattern in this case is represented by a superposition of straight and reverse
waves.
The appearance of reverse waves in a flow is determined by the blocking
condition (6.33). If it is fulfilled, the spectral components of reverse waves
make a significant contribution to the total wave spectrum. Limitations of the
spectrum value connected with possible wave breaking are introduced, using
the notion of the equilibrium range. Its presence in the solution results in the
appearance of the parameter characterizing the ratio of the initial spectrum
value relative to the equilibrium range value. In Sect. 5.5, this value is expressed by the mean steepness ho/5.0 of the initial wave spectrum. The initial
steepness value can significantly affect the wave element changes in a current.
The values of wave elements along the increasing flow ( 0: = 5; ii = 0, 1) for
waves with steepness 1/36 (corresponding to the mean wind wave steepness)
are shown in Fig. 6.15, where the straight and reverse waves as well as their
breaking are taken into account. The behaviour of wave elements is similar to
that described above (see Fig. 5.16). It is seen that the presence of the reverse
wave and its breaking do not result in a large increase of mean heights.
In conclusion it should be noted that the evolution of mean wave elements
occurs under the effect of two mechanisms. They are, firstly, a wave transformation by the action of a horizontally non-uniform current and, secondly, by
changing depth of the basin. Although these mechanisms are described using the same equations (within the framework of the initial approximation),
their effects are different. The current influences are greater on the shortwave spectrum components, while the changing depth affects the long wave
components. They penetrate into deeper water and are subjected to depth
variations.
As shown by the obtained results the effect of non-uniform depth and
current velocity lead to such changes of wave elements, which are impossible
to predict using the mechanisms of transformation separately. In some cases,
the combined effect of the changing depth and current essentially increases the
wave transformation, while decreasing it in some other cases. In shallow water
areas with spatial non-uniform currents, such situations can be observed,
when the wave heights and other wave elements are increased greatly (by
several times). But there are also opposite situations, with waves decrease up
to their almost complete disappearance.