208
5 Wave Evolution in Non-uniform Currents in Deep Water
components in quadrant III (see Fig. 5.20) are absent. In the area x < Xm,
the spectral components can be found in quadrants I, II, IV. But in the case
x > Xm they are present in quadrants I and IV. The solution is described by
the formula (5.68) with the kinematic conditions (5.71)-(5.81).
The tensor component of the wave slope spectral density F ( k, {3) k 2 is
estimated by the formula (5.68). Its relative values can be compared to the
optical spectra of the sea surface pattern (see Fig. 5.23), so long as the corresponding conditions for obtaining and processing are fulfilled (Grushin et
al., 1986). It is assumed in the formula (5.68) that n = 4, which is usually
used for a developed wind sea. It should be noted that the maximum slope
spectrum F (k, {3) k 2 is shifted by 1.4 times to a larger wave number area for
6-V = 0, relative to the maximum spectrum F (k,{3), kmax ~ 0.47m- 1 .
The slope spectrum can be calculated for different points of the current
velocity profile at !38 = 160°. It should be taken into account that the theoretical calculations are displaced by the angle 7t- f3 relative to full-scale data.
The results are presented in the { k, {3} plane as isolines of the slope spectrum
F (k, {3) k 2 standardized by its maximum. A two-dimensional spectral density (see Fig. 5.24a) is presented to the right of the line subdividing the two
flows, where the current velocity is equal to 0.3 ms- 1 . The corresponding
experimental spectrum is presented in Fig. 5.23b.
There are two pronounced maximuma seen in the two-dimensional spectrum. The first one deals with initial waves propagating at the angle 160° to
the horizontal axis (see Fig. 5.21). There is an angle of 40° for the second maximum. The energy distribution of reflected waves becomes narrower. Some
spectral components pass to the area of higher current velocities without
being reflected. The isoline curves are explained with the help of the superposition of the kinematic condition for the existence ofreflected waves ( 5. 78)
with the function of angular energy distribution in the initial wave spectrum.
The calculated spectrum on the left side of the subdividing line with the
difference of current velocities 6. V = 1.4 ms- 1 is shown in Fig. 5.24b. The
corresponding experimental spectrum is in Fig. 5.23d. In this case the angular
spectrum becomes narrower. One general wave direction is clearly evidently
across the flow. The wavelength of the main system is increased two-fold.
Comparison of the experimental and theoretical spectra allows the identification of a number of common features. They indicate qualitative (in some
cases quantitative) agreement of the obtained theoretical results with fullscale observations.
Thus, the frontal zone, as the boundary of two flows, appears to be a peculiar wave filter. There is a reflection of some spectral wave components
in this zone. These components create rips superimposing with the initially
propagated wave system. An intensive wave breaking appears, resulting in the
isotropic angular spectrum distribution, seen in the sea surface spectra (see
Fig. 5.23c). Some of the spectral components, without being reflected, propagate practically across the flow and manifest themselves as a quasi-regular
wave system.
5 Wave Evolution in Non-uniform Currents in Deep Water
components in quadrant III (see Fig. 5.20) are absent. In the area x < Xm,
the spectral components can be found in quadrants I, II, IV. But in the case
x > Xm they are present in quadrants I and IV. The solution is described by
the formula (5.68) with the kinematic conditions (5.71)-(5.81).
The tensor component of the wave slope spectral density F ( k, {3) k 2 is
estimated by the formula (5.68). Its relative values can be compared to the
optical spectra of the sea surface pattern (see Fig. 5.23), so long as the corresponding conditions for obtaining and processing are fulfilled (Grushin et
al., 1986). It is assumed in the formula (5.68) that n = 4, which is usually
used for a developed wind sea. It should be noted that the maximum slope
spectrum F (k, {3) k 2 is shifted by 1.4 times to a larger wave number area for
6-V = 0, relative to the maximum spectrum F (k,{3), kmax ~ 0.47m- 1 .
The slope spectrum can be calculated for different points of the current
velocity profile at !38 = 160°. It should be taken into account that the theoretical calculations are displaced by the angle 7t- f3 relative to full-scale data.
The results are presented in the { k, {3} plane as isolines of the slope spectrum
F (k, {3) k 2 standardized by its maximum. A two-dimensional spectral density (see Fig. 5.24a) is presented to the right of the line subdividing the two
flows, where the current velocity is equal to 0.3 ms- 1 . The corresponding
experimental spectrum is presented in Fig. 5.23b.
There are two pronounced maximuma seen in the two-dimensional spectrum. The first one deals with initial waves propagating at the angle 160° to
the horizontal axis (see Fig. 5.21). There is an angle of 40° for the second maximum. The energy distribution of reflected waves becomes narrower. Some
spectral components pass to the area of higher current velocities without
being reflected. The isoline curves are explained with the help of the superposition of the kinematic condition for the existence ofreflected waves ( 5. 78)
with the function of angular energy distribution in the initial wave spectrum.
The calculated spectrum on the left side of the subdividing line with the
difference of current velocities 6. V = 1.4 ms- 1 is shown in Fig. 5.24b. The
corresponding experimental spectrum is in Fig. 5.23d. In this case the angular
spectrum becomes narrower. One general wave direction is clearly evidently
across the flow. The wavelength of the main system is increased two-fold.
Comparison of the experimental and theoretical spectra allows the identification of a number of common features. They indicate qualitative (in some
cases quantitative) agreement of the obtained theoretical results with fullscale observations.
Thus, the frontal zone, as the boundary of two flows, appears to be a peculiar wave filter. There is a reflection of some spectral wave components
in this zone. These components create rips superimposing with the initially
propagated wave system. An intensive wave breaking appears, resulting in the
isotropic angular spectrum distribution, seen in the sea surface spectra (see
Fig. 5.23c). Some of the spectral components, without being reflected, propagate practically across the flow and manifest themselves as a quasi-regular
wave system.