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GE
a E
where y is the angle between EP and the line of centres and g
¼
is the gravitational
acceleration on the Earth. The magnitude of the horizontal TGF relative to g is just
j F H g ¼
d 3
a E
10 7 . The corresponding vertical component has a similar magnitude
but is negligible because it is in competition with g, whereas F H , though small
compared with g, is generally competing with the much smaller forces in the hori-
zontal. The TGF varies over the globe according to sin 2
M
E
which means that it is zero
at the sublunar point y
¼
0, on the great circle where y
¼
p/2 and at the nadir where
y
3p/4 where it is directed
towards the sublunar point and the nadir respectively, as you can see in Fig. B2.3 .
¼
p. The force has a greatest magnitude at y
¼
p/4 and at y
¼
LW
(Nadir)
HW
HW
to Moon
LW
Figure B2.3 Distribution of the Tide Generating Force and the tidal ellipse which results
from the Equilibrium model.
This pattern of forces, if acting on a uniform ocean covering the whole Earth,
would tend to move the ocean waters towards the sublunar point and the nadir and
away from the great circle where y
0. We might expect this movement to set up
slopes of the sea surface which would balance the TGF. This idea formed the basis of
the Equilibrium theory of the tides, which was originally proposed by Newton who
gave the first rational explanation of tidal forces in 1687. The water surface in the
Equilibrium theory would be an ellipsoid of revolution with its major axis directed
towards the Moon ( Fig. B2.3 ) so that high water (HW) would occur at the sublunar
point and at the nadir with low water (LW) on the great circle where y
¼
¼
p/2. In this
theory, the maximum elevation at the HW points would be
þ
35.4 cm and the
minimum
17.7 cm at LW, giving a total range of only 53.1 cm. The tide due to
the Sun would add almost half as much again. Such a range is considerably less than
is observed in most parts of the shelf seas but is not so different from what we now
know of the tides in the deep ocean, where a range of
1 metre is typical.
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