5.8 Freak Wave Problem
223
Spectral solution of swell transformation in the Agulhas Current.
The swell propagation from southern latitudes, with actually no current,
towards the increasing Agulhas Current will be considered. The geometric
optics approximation can be applied in this case, because typical horizontal
scales of current velocity variations significantly exceed the horizontal wave
dimensions. Thus, the results of Sect. 5.2 can also be applied.
At a greater distance from the investigated area (see Fig. 5.28), the initial
spectrum So (w, f3o) is assumed to be prescribed at x > 1.5 x 10 6 m, where
the current speed can be neglected: Jv; + VtJ ~ 0. The spectrum can be
described with the help of the swell spectrum approximation (5.16) with
n = 5, Wmax = 0.5 rads- 1 . It should be noted that the initial value of the
angle (30 , included in So (w, (30 ), is a function of the values w, (3 and the
coordinates { x, y }.
In order to find fJo, it is necessary to solve the set of Hamiltonian equations
(5.2), which can be written using new variables:
where
dy _ Vy + ~ f sin ((3) .
dx - Vx - ~ f cos (fJ) '
d(J
sin (2(3) ~ - sin 2 ((3) ~ + cos 2 ((3) ~
dx =
v - ~ f cos ((3)
dt
1
dx
V- ~fcos((J)'
(5.103)
The Runge-Kutta method is used for a discrete set of frequencies w1 (from
w1 = 0.37 rads- 1 to w12 = 0.981 rads- 1 ) and the angles (Jj (from -75 to 75°
with a step size of 15°) to obtain the values f3oj· They correspond to the rays
arriving at the given points {x = O,y = Yn} from the region with practically
no current x > 1.5 x 10 6 m.
Typical rays y = y(x, w, (3), arriving at the point { x = 0, y = 0}, are
depicted in Fig. 5.32. They are characterized by oscillations relative to the
Ox axis. The oscillation frequencies are increased with rays approaching the
intense current area. A peculiar wave channel (i.e. a waveguide) appears in
the current with the wave rays being alternately reflected from one or other
caustics situated on different sides of the current midstream.
The velocity distribution in the Agulhas Current is not symmetric relative to the maximum velocity line. The velocity gradients at the eastern
midstream side are less than those at the western side (see Fig. 5.31). The current gradient at the western side is additionally increased due to the coastal
223
Spectral solution of swell transformation in the Agulhas Current.
The swell propagation from southern latitudes, with actually no current,
towards the increasing Agulhas Current will be considered. The geometric
optics approximation can be applied in this case, because typical horizontal
scales of current velocity variations significantly exceed the horizontal wave
dimensions. Thus, the results of Sect. 5.2 can also be applied.
At a greater distance from the investigated area (see Fig. 5.28), the initial
spectrum So (w, f3o) is assumed to be prescribed at x > 1.5 x 10 6 m, where
the current speed can be neglected: Jv; + VtJ ~ 0. The spectrum can be
described with the help of the swell spectrum approximation (5.16) with
n = 5, Wmax = 0.5 rads- 1 . It should be noted that the initial value of the
angle (30 , included in So (w, (30 ), is a function of the values w, (3 and the
coordinates { x, y }.
In order to find fJo, it is necessary to solve the set of Hamiltonian equations
(5.2), which can be written using new variables:
where
dy _ Vy + ~ f sin ((3) .
dx - Vx - ~ f cos (fJ) '
d(J
sin (2(3) ~ - sin 2 ((3) ~ + cos 2 ((3) ~
dx =
v - ~ f cos ((3)
dt
1
dx
V- ~fcos((J)'
(5.103)
The Runge-Kutta method is used for a discrete set of frequencies w1 (from
w1 = 0.37 rads- 1 to w12 = 0.981 rads- 1 ) and the angles (Jj (from -75 to 75°
with a step size of 15°) to obtain the values f3oj· They correspond to the rays
arriving at the given points {x = O,y = Yn} from the region with practically
no current x > 1.5 x 10 6 m.
Typical rays y = y(x, w, (3), arriving at the point { x = 0, y = 0}, are
depicted in Fig. 5.32. They are characterized by oscillations relative to the
Ox axis. The oscillation frequencies are increased with rays approaching the
intense current area. A peculiar wave channel (i.e. a waveguide) appears in
the current with the wave rays being alternately reflected from one or other
caustics situated on different sides of the current midstream.
The velocity distribution in the Agulhas Current is not symmetric relative to the maximum velocity line. The velocity gradients at the eastern
midstream side are less than those at the western side (see Fig. 5.31). The current gradient at the western side is additionally increased due to the coastal