Response to forcing: the governing equations and some basic solutions - Introduction to the Physical and Biological Oceanography of Shelf Seas - page 85

Figure 3.16 Tidal response of a
narrow gulf showing the ocean tide
being matched by a standing wave
oscillation. A single node is located
at l/4 from the head of the gulf
and initially the amplification of the
tide is limited (grey curves). If the
nodal point was located closer to
the mouth, matching to the ocean
tide would require a large increase
in the standing wave response
(black curves).

gulf ocean

A 0

l / 4

x =0

x = L

systems are spaced at intervals of l/2 along the axis of the gulf, and the number of
amphidromes within the gulf is dictated by its length and its depth (remember - depth
controls the speed of the tidal wave, and so the tidal wavelength). At the open
boundary of the gulf, the standing wave must match the ocean tide. We can demon-
strate how this matching works for the case of a standing wave without rotation
(Equation 3.62) which would apply to a narrow gulf. Figure 3.16 shows a section along
the axis of such a narrow gulf which is long enough to contain one node of a standing
wave. If the ocean tide is a sine wave of amplitude A 0 , matching to a standing wave
at the mouth of the gulf (x

¼ L) requires that:

A 0 sin ot ¼ A sw cos kL sin ot ð 3 : 74 Þ

so that the amplitude of the standing wave within the gulf will be:

A 0
cos kL ¼

A 0
cos 2

A sw ¼

l :

ð 3 : 75 Þ

L =

Equation (3.75) implies an interesting possibility. What if we change the length of the
gulf so that it just happens that L l/4? The denominator in Equation (3.75) goes to

zero, suggesting an infinite response of the wave amplitude inside the gulf. This is
called resonance, and it suggests we might expect to see incredibly large tidal range
and currents within the gulf. In practice the resonant motions would become limited
by strong frictional effects and the ocean tide would be modified by the vigorous
motion in the gulf (Arbic and Garrett, 2010). It also seems likely that any basin
approaching resonance would experience very large sediment transport which would
modify the bathymetry and move the system away from resonance. However, there
are a number of tidal systems around the world that are close to resonance. A clear
example is that of the Bay of Fundy, where the length of the gulf is

¼

l/4 for the main
lunar tide so that there is a single node close to the ocean boundary. The result is a
near-resonant response, which is responsible for the largest tidal range (

∼

16 metres at
spring tides) in the global ocean. Note also from Equation (3.75) that similar
resonant responses will occur in gulfs whose length is close to 3l/4, 5l/4, 7l/4 etc.
At this point you can see why tides in shelf seas are often so much larger than the
open ocean tide. The tidal amplification is the result of two processes: first, the very

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