226
5 Wave Evolution in Non-uniform Currents in Deep Water
direction. Generally there are some additional difficulties in solving of this
problem. In order to determine the dispersion ratio (1.37), the marginal problem (1.35), (1.36), including the current velocity dependence on the vertical
coordinate, must be solved.
In the case of the current velocity being changed only along one horizontal
direction, the problem is connected with solving the equation of motion of
Orr-Sommerfeld type. Its precise analytical solution can be obtained only
for the simplest cases. The dispersion ratios for surface waves in water with
a simple depth profile (linear, parabolic, logarithmic, etc.) can be obtained in
explicit form (Hidy & Plate, 1966). The analytical solution for the piecewiselinear velocity profile approximation has also been obtained (Thomas, 1981).
The solutions can be found numerically or by asymptotic methods for more
general profiles (Goncharov & Leykin, 1983). Brink-Kjer & Jonsson (1975)
derived the conservation laws for waves in a current with a linear depth
velocity shear profile.
Investigations of waves in a current with a vertically non-uniform velocity profile have been published in several papers (Taylor, 1955; Peregrine
& Smith, 1975; Goncharov & Leykin, 1983; Dreyzis et al., 1986; Kantargi et
al., 1989 etc.).
In this section an attempt is undertaken to solve the problem of wave
transformation in a current with velocity depth shear. It should be noted
that the current is considered to be non-uniform along its direction.
Wave transformation in a current with linear depth velocity shear.
A wave transition from calm water to a current with linear depth velocity
profile is considered as follows:
V(z) = Vm + fl(z- H/2) ,
(5.104)
where Vm is the mean depth current velocity; fl = (Vs- VB)/ His the current vorticity; and Vs and VB are the surface and bottom current velocities,
respectively (see Fig. 5.34).
The wave evolution will be described in a non-uniform current within
the framework of the geometrical optics approximation. The parameters
Vm, Vs, VB, [l are assumed to be changed sufficiently slowly horizontally compared with the wavelength. The choice of the simple current profile (5.104) is
defined by the possibility of obtaining the dispersion ratio and the adiabatic
invariant in analytical form (Brink-Kjer & Jonsson, 1975). Only one parameter characterizing the velocity depth shear is added to the current (5.104)
compared with the uniform case.
In order to describe the wave motion in the vertically non-uniform current
V(z), the Rayleigh equation relative to the vertical component amplitude
W(z) can be applied (Peregrine, 1976):
d
2
W
(
k
d
2
V)
dz2 - k
2
+ Vk - w dz2 W = 0 ;
-H s z s 0
(5.105)
5 Wave Evolution in Non-uniform Currents in Deep Water
direction. Generally there are some additional difficulties in solving of this
problem. In order to determine the dispersion ratio (1.37), the marginal problem (1.35), (1.36), including the current velocity dependence on the vertical
coordinate, must be solved.
In the case of the current velocity being changed only along one horizontal
direction, the problem is connected with solving the equation of motion of
Orr-Sommerfeld type. Its precise analytical solution can be obtained only
for the simplest cases. The dispersion ratios for surface waves in water with
a simple depth profile (linear, parabolic, logarithmic, etc.) can be obtained in
explicit form (Hidy & Plate, 1966). The analytical solution for the piecewiselinear velocity profile approximation has also been obtained (Thomas, 1981).
The solutions can be found numerically or by asymptotic methods for more
general profiles (Goncharov & Leykin, 1983). Brink-Kjer & Jonsson (1975)
derived the conservation laws for waves in a current with a linear depth
velocity shear profile.
Investigations of waves in a current with a vertically non-uniform velocity profile have been published in several papers (Taylor, 1955; Peregrine
& Smith, 1975; Goncharov & Leykin, 1983; Dreyzis et al., 1986; Kantargi et
al., 1989 etc.).
In this section an attempt is undertaken to solve the problem of wave
transformation in a current with velocity depth shear. It should be noted
that the current is considered to be non-uniform along its direction.
Wave transformation in a current with linear depth velocity shear.
A wave transition from calm water to a current with linear depth velocity
profile is considered as follows:
V(z) = Vm + fl(z- H/2) ,
(5.104)
where Vm is the mean depth current velocity; fl = (Vs- VB)/ His the current vorticity; and Vs and VB are the surface and bottom current velocities,
respectively (see Fig. 5.34).
The wave evolution will be described in a non-uniform current within
the framework of the geometrical optics approximation. The parameters
Vm, Vs, VB, [l are assumed to be changed sufficiently slowly horizontally compared with the wavelength. The choice of the simple current profile (5.104) is
defined by the possibility of obtaining the dispersion ratio and the adiabatic
invariant in analytical form (Brink-Kjer & Jonsson, 1975). Only one parameter characterizing the velocity depth shear is added to the current (5.104)
compared with the uniform case.
In order to describe the wave motion in the vertically non-uniform current
V(z), the Rayleigh equation relative to the vertical component amplitude
W(z) can be applied (Peregrine, 1976):
d
2
W
(
k
d
2
V)
dz2 - k
2
+ Vk - w dz2 W = 0 ;
-H s z s 0
(5.105)