Geoscience Reference
In-Depth Information
Notice that, if the gulf is narrow with B G
Ro, then the exponential factors tend to
unity (e y/Ro
e y/Ro
'
'
1) and (3.71) reverts to the non-rotating case (3.62).
3.6.4
Amphidromic systems
The combination of two Kelvin waves describes the motion well except in a small
region close to the head of the gulf where the reflection of the incident wave occurs
(a complication to which we shall return in a moment). Away from the head of the
gulf, we can represent the tidal regime of a gulf simply in terms of the two Kelvin
waves. The results are conveniently presented as a pattern of lines of equal amplitude
(co-range lines) and lines of equal phase (co-tidal lines). Think of the co-tidal lines as
representing all points at which each phase of the tide, e.g. high water (HW), will
occur at the same time.
We can locate the co-tidal lines by choosing a particular condition for the phase
of the tide in Equation (3.71) . The condition for HW is that there is a maximum in
which means that
0. Applying this condition in Equation (3.71) for the case
of complete reflection when A b ¼
@
/
@
t
¼
A f leads to a relation between y and x for locations
¼
at which HW occurs at time t
t HW :
tan h
ð
y
=
Ro
Þ
¼
tan ot HW
ð
Þ:
ð
3
:
72
Þ
tan kx
You can see in Fig. 3.13a the resulting pattern of co-tidal lines in the northern
hemisphere. The lines (at intervals of 1 lunar hour) radiate outwards from central
points termed amphidromes which are located where the nodal lines would be for a
standing wave without rotation. As with the standing wave nodes, the amphidromic
points are separated by l/2. Notice that the time of high water (and any other phase of
the tide) rotates around the amphidrome in an anticlockwise (clockwise) sense for the
northern (southern) hemisphere.
The range of the tide at any position within the gulf can also be found by
manipulating the expression for
in (3.71) and taking the modulus to obtain:
1 = 2
2A f cos h 2
sin h 2
jj¼
ð
y
=
Ro
Þ
sin kx
þ
ð
y
=
Ro
Þ
cos kx
:
ð
3
:
73
Þ
Contours of the range of the tide (the co-range lines) are shown in the lower panel
of Fig. 3.13a . As there is no energy loss in this frictionless case, the amplitudes of
incident and reflected waves are equal and the pattern is symmetrical about the centre
of the channel with no tide (
0) at the amphidromic points to which the co-tidal
lines converge. We can think of the Kelvin wave moving into the gulf hugging the
coast to the right (northern hemisphere), reflecting off the head of the gulf, and then
leaving the gulf still maintaining the coast on the right.
What happens if we make the gulf a little more realistic and allow friction to
extract energy from the tidal wave? With increasing friction, energy is dissipated and
the reflected wave is weaker than the incident wave so that A b <
¼
A f . This is the case
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