234
5 Wave Evolution in Non-uniform Currents in Deep Water
obtained by Peregrine and Smith (1975) for short waves (i.e. kzv » 1, where
zv is the scale of the vertical change of current velocity) propagating in
currents with velocity depending on the depth.
As pointed out by Skop (1987), in the case of the value V.,ff, being a continuous function of the wave number k, (5.128) provides a satisfactory approximation for the dispersion ratio between large and small wave numbers.
The applicability of (5.129) for intermediate wave number values requires
empirical testing or verification by the existing accurate solutions for depth
dependent currents.
The effective current velocity is determined with the help of (5.128) for
the case of a current velocity profile with constant vorticity (5.104) as:
[}
V.,ff = V. - 2 k th(kH) .
(5.130)
Comparing the approximate formulas (5.127), (5.130) and the accurate
solution of the problem (5.108), it can be concluded that their agreement
is quite satisfactory. There are some differences decreasing with the relative
depth. The maximum deviation in the phase velocity value is achieved at
kH-+ 0 and is equal to h/(QH) 2 + 4gH.
A comparison of the other approximate solution, namely, for the surface
jet current and laboratory measurements, is made by Kantargi (1995). The
current velocity profile is given as a piecewise-linear approximation:
V(z) = { ~Vo(1 + z/D)
-D~z~O;
z>D.
(5.131)
Substituting the current velocity (5.131) into (5.127), (5.128) gives an
approximate expression for the dispersion relation as:
w = (gkth(kD)) 1 / 2 - kVo + ;;th(kD).
(5.132)
Comparison of the wave elements (5.132) with the obtained data for different conditions including wave blocking in a countercurrent reveals good
agreement.
Some peculiarities ofthe influence of a vertical non-uniform current
on the wind wave spectrum. The presence of a non-uniform current
with velocity depending not only on the horizontal, but also on the vertical
coordinates, influences the waves in a more complex and diverse way than
a current independent of the vertical coordinate. The vertical non-uniformity
of the current velocity can significantly influence the wave blocking, redistributing the spatial location of the wave turning points and caustics, where
the wave heights reach anomalously large values. As shown above, the use of
the spectral approach removes the singularity occurring in the classical solution. The statistical averaging used in the spectral approach helps smooth
5 Wave Evolution in Non-uniform Currents in Deep Water
obtained by Peregrine and Smith (1975) for short waves (i.e. kzv » 1, where
zv is the scale of the vertical change of current velocity) propagating in
currents with velocity depending on the depth.
As pointed out by Skop (1987), in the case of the value V.,ff, being a continuous function of the wave number k, (5.128) provides a satisfactory approximation for the dispersion ratio between large and small wave numbers.
The applicability of (5.129) for intermediate wave number values requires
empirical testing or verification by the existing accurate solutions for depth
dependent currents.
The effective current velocity is determined with the help of (5.128) for
the case of a current velocity profile with constant vorticity (5.104) as:
[}
V.,ff = V. - 2 k th(kH) .
(5.130)
Comparing the approximate formulas (5.127), (5.130) and the accurate
solution of the problem (5.108), it can be concluded that their agreement
is quite satisfactory. There are some differences decreasing with the relative
depth. The maximum deviation in the phase velocity value is achieved at
kH-+ 0 and is equal to h/(QH) 2 + 4gH.
A comparison of the other approximate solution, namely, for the surface
jet current and laboratory measurements, is made by Kantargi (1995). The
current velocity profile is given as a piecewise-linear approximation:
V(z) = { ~Vo(1 + z/D)
-D~z~O;
z>D.
(5.131)
Substituting the current velocity (5.131) into (5.127), (5.128) gives an
approximate expression for the dispersion relation as:
w = (gkth(kD)) 1 / 2 - kVo + ;;th(kD).
(5.132)
Comparison of the wave elements (5.132) with the obtained data for different conditions including wave blocking in a countercurrent reveals good
agreement.
Some peculiarities ofthe influence of a vertical non-uniform current
on the wind wave spectrum. The presence of a non-uniform current
with velocity depending not only on the horizontal, but also on the vertical
coordinates, influences the waves in a more complex and diverse way than
a current independent of the vertical coordinate. The vertical non-uniformity
of the current velocity can significantly influence the wave blocking, redistributing the spatial location of the wave turning points and caustics, where
the wave heights reach anomalously large values. As shown above, the use of
the spectral approach removes the singularity occurring in the classical solution. The statistical averaging used in the spectral approach helps smooth