Geoscience Reference
In-Depth Information
channel ends in a vertical barrier at x
¼
0, the velocity there must be zero at all times.
Setting u
¼
0atx
¼
0in Equation (3.61) requires that A b ¼
A f and the expressions
for elevation and velocity can then be written as:
¼
A f sin
f
ð
kx
ot
Þ
sin
ð
kx
þ
ot
Þ
g ¼
2A f cos kx sin ot
ð
3
:
62
Þ
A f g
c
2A f g
u
¼
f
sin
ð
kx
ot
Þþ
sin
ð
kx
þ
ot
Þ
g ¼
c sin kx cos ot
:
So the incident wave moving towards the barrier from the left in Fig. 3.11 is reflected
back, with the reflected wave then interfering with the incident wave. The two
progressive waves combine to form a standing wave with double the amplitude of
the incident wave and nodes of elevation (points on the wave with zero range) for
cos kx
¼
0(i.e.atx
¼
l/4, 3l/4, 5l/4,
...
). The corresponding nodes of velocity (which
are anti-nodes, or maxima, for
).
Waves of tidal period moving on to the shelf from the deep ocean travel at the long
wave speed of c
) occur for sin kx
¼
0(x
¼
0, l/2, l,3l/2,
...
g p which decreases from c
200 m s 1 in the deep ocean (h
¼
4km)
30 m s 1 on the shelf (h
to c
100 m). Remember that a wave's wavelength, period
and speed are related by l
cT. The corresponding wavelengths at the main tidal
period (12.42 hours) are 9000 km for the deep ocean and 1400 km for the shelf, so tidal
waves certainly satisfy our initial assumption that l
¼
h. Tsunami waves, which are
forced by movements of the seabed during earthquakes, also travel at c
g p . In the
¼
4000 metres, the tsunami speed is 200 m s 1 . These waves can,
therefore, cross the ocean basins in less than a day.
deep ocean, with h
3.6.2
Long waves with rotation (Kelvin waves)
The tide generating force acting over the large extent of the ocean basins drives the
deep ocean tides which travel around the oceans in the form of long waves. The
waves bringing tidal energy on to the shelf propagate to the coastal boundaries,
where they are reflected and form standing wave patterns of the kind we have just
discussed. There is, however, an important difference in the dynamics of tidal waves
on the shelf from our simple non-rotating model. Because of the very large scales
involved in the tidal long waves, we cannot ignore the influence of Coriolis force
which modifies the structure of long waves and produces more subtle reflection
patterns. Our non-rotating waves moved in the x direction and were uniform in the
transverse y direction, so we needed only to consider the x momentum equation.
When rotation becomes important, we need also to consider the dynamical balance
in the y direction. Particle motion in the x direction involves a Coriolis force fu acting
in the y direction. For a wave travelling in the x direction parallel to a coastal
boundary, as shown in Fig. 3.12 , this Coriolis force induces a balancing pressure
gradient in the form of a sea surface slope normal to the coast, i.e.:
0 @
1
p
g @
@
¼
y ¼
y :
ð
:
Þ
fu
3
63
@
 
Search WWH ::




Custom Search