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TIDAL THEORY
Introduction
Most commonly the word ‘tide’ refers to the cyclic rise and fall of the Earth’s Oceans
due to tidal forces created by the Moon and Sun. However, tidal phenomena is not
limited to the Ocean; affecting any object subjected to a gravitational field that varies
in space and time. The tidal forces of the moon and sun create tides in the crust of the
Earth and also in the atmosphere.
How Tides Are Generated
The tides created by the moon and sun are known as Lunar and Solar tides
respectively. There are several theories, of varying complexity, as to how tides are
created; two of the most widely accepted theories will be discussed here. In reality it
is most likely that tides are generated as a result of a combination of the effects
outlined in both theories. Initially the lunar tide will be examined to explain how tides
form. The most important factor in both tidal theories is that the gravitational
attraction between the earth and the moon varies with distance. The relationship
between gravitational force and distance is given by Newton’s inverse square law:
F = G memm/r2
G = Universal Gravitation Constant
me= Mass of the earth
mm= Mass of the moon
r = Distance between the Earth and moon centres
It is apparent, therefore, that the side of the earth closest to the moon experiences a
stronger attractive force to the moon than the side farthest away from it. By
calculation it can be shown that the gravitational attraction is approximately seven
percent stronger on the moonward side of the earth; this difference is the basis of the
first theory to be examined. The figure below illustrates the relative displacements of
three points on the earth; the further the point is from the moon, the less it is
displaced. As a result the earth’s crust and oceans are stretched toward the moon
along an axis that runs through the earth and moon’s centres. This creates a bulge of
water on the moon side of the earth. On the opposite side of the earth a bulge is also
created as the earth has been pulled away from the oceans. The bulges in the diagram
are greatly exaggerated; the theoretical increase in depth of open sea water due to the
lunar tide is around fifty centimetres. This theory assumes a uniform, spherical earth
with no land masses and does not take the earth’s rotation or friction into account.
Figure 1 Variation of the moon’s gravitational pull with distance
The second theory is known as the equilibrium theory and is based on the rotation of
the earth-moon system. A common misconception is that the moon simply orbits
around the centre of the earth. Due to the earth’s much greater mass, when compared
to the moon, the orbital centre of mass, or barycentre, of the earth-moon system is
located at approximately three-quarters of the distance from the centre of the earth to
its surface. As a result the centrifugal force applied to the oceans, due to the earth’s
rotation about the barycentre, varies with position. The side of the earth opposite the
moon feels the strongest centrifugal force, creating a tidal bulge in the oceans on this
side. The gravitational pull of the moon on the earth causes the oceans to swell on the
opposite side of the earth, that closest to the moon. When the centrifugal forces
exerted by the earth and gravitational forces from the moon are combined it results in
two swells in the oceans, of approximately equal amplitude, directly opposite each
other.
Figure 2 Representation of the equilibrium theory
It must now be remembered that the Earth rotates about its own axis, its surface
moving beneath the oceans. As a result, coasts generally experience two high tides a
day coinciding with when they line up with each of the two bulges of water. High tide
occurs in intervals of twelve hours and twenty-five minutes, this is due to the rotation
of the moon relative to the earth. It takes the earth twelve hours to rotate 180°; in this
time the moon has advanced by 6°, which is why there is an extra, ‘unexpected’,
twenty-five minutes between high tides.
The solar tide works using the same principles, but its magnitude is less than half that
of the lunar tide. The solar and lunar tides interact to give noticeable variation in the
overall tides of the oceans. These facts are detailed the sections that follow.
Tidal Forces
Tidal forces are differential forces; they are the result of the difference in gravitational
force between two points. As a result it is useful to introduce the potential function, V,
to help analyse the tidal forces of the sun and moon:
F total  V total , where V total  V m oon  V sun
The relationship between the tidal potentials, and therefore the tidal forces produced
by the sun and moon is given by an inverse cube relation:

V sun
m r
 sun  moon 
V moon mmoon  rsun 
3
Body
Moon
Sun
m (kg)
(Mass)
7.36 x 1022
1.99 x 1030
r (m)
(Mean distance to Earth)
3.84 x 108
1.5 x 1011
Substituting the values from the table into the relation yields a ratio of 0.454; i.e. solar
tidal forces are less than half the magnitude of the lunar tidal forces. This correlates
with observations of the oceans tides; their variation very much synchronised with the
movement of the moon.
Tidal Force Components
Tidal forces have both vertical and horizontal components; it the latter that is the
major contributor to the earth’s tides, and is known as the tractive force. The tractive
force is at a maximum when the force lines are parallel to the surface of the earth;
shown by the red arrows in the following figure. The tractive force is zero at the point
directly below the moon, and directly opposite that point (points A and B
respectively). These points represent the positions where the vertical tidal force
component is a maximum. 1As a result objects directly below the moon decrease in
weight by a factor of approximately 1/107. The Queen Mary 2 ocean liner, for
example, would be 14.6kg lighter than normal when at point A.
A
B
Figure 3 Tractive forces exerted on the Earth by the Moon
Tidal Predictions on a Global Scale
The production of tides lends itself to being predicted due to its apparently periodic
nature. Tides are constantly generated by the sun and moon, but the magnitude of the
tide varies due to interactions between the cycles of the earth, moon and sun. As a
result a variety of tidal cycles, with different periods, are created. Because the
astronomical behaviour of the earth-moon-sun system is well understood, it is
possible to predict tidal behaviour, on a global scale, well into the future.
Spring / Neap Tides
Spring tides are created when the Sun and Moon are both in line with the Earth, i.e.
during full or new moon, resulting in Lunar and Solar tides collaborating. This results
in high tide being higher than normal, and low tide being lower than normal. Neap
tides occur when the Sun is at 90° to the Moon; this causes the lunar high tide to occur
at the same point as the solar low tide, and Lunar low tide to coincide with the solar
high tide. The results are lower high tides, and higher low tides. Neap tides occur at
the first and third quarters of the Moon’s cycle. The spring tidal range is
approximately twice that of the neap tide.
Figure 4 Orientation of the sun and moon during spring and neap tides
Lunar Variations
The distance between the moon and earth varies due to the moons elliptic, 27.3 day
orbit. The moon is said to be at perigee at its closest point to earth, 363,104km, and
apogee at its farthest, 405,696km. The amplitude of the tide varies as a result of the
changes in separation between the two bodies. The effect is greatest when perigee
coincides with a spring tide, giving the largest spring tide, or when apogee coincides
with a neap tide, giving the smallest neap tide.
Equinoctial Tides
Close to the Spring Equinox of March 21st, and the Vernal equinox of September 23rd
spring tides are higher than normal due to the alignment of the moon’s orbit to that of
the earth. Similarly, smaller spring tides than average are experienced close to the
summer and winter solstices.
The Age of the Tide
When making tidal predictions it is important to note that there is a delay between the
astronomical cycle and the resulting tidal cycle. This phenomenon is known as the age
of the tide, and the length of the delay varies with the positioning and characteristics
of each coast. In the North Sea, for example, the delay between a new or full moon
and the spring tide is approximately two days.
Predictions of Local Tides
Up to this stage, tides have been
considered on a global scale, looking at
the theoretical behaviour of the oceans as a
whole. For engineers, it is localised tides
that are of the greatest use in the
identification of the best areas to harness
tidal power. In order to predict tides at a
particular coastal point, the sea level must
be observed and recorded for a prolonged
period using a tide gauge. Tide gauges are
mechanical or electronic devices used to
measure sea level at designated time
intervals. The data from a tide gauge is Figure 5 Tidal staff used to measure extreme high
tides at the Severn Estuary
used to plot a continuous curve of sea
height against time, as shown in the figures below.
Categorisation of Tides
Due to the inclination of the lunar and solar orbits relative to the earth, two distinct
types of tide are experienced on coastlines depending on their location. Semi-diurnal
tides behave according to the theoretical tidal model discussed earlier, producing two
high and low tides each day. Diurnal tides produce only one high tide and low tide per
day. A variation of the semi-diurnal tide is the mixed tide; in this case there are two
high tides a day, but of different amplitudes. The map below displays which type of
tide is experienced by each coastal location.
Figure 6 The semi-diurnal, diurnal and mixed tidal cycles (from top to bottom)
Figure 7 Summary of the tides experienced by the Earth’s coasts
Tidal Range
The open sea tidal range, the difference between high and low tide sea levels,
averages approximately one metre. The tidal range experienced by certain coastal
areas, however, greatly exceeds this figure; the world’s largest tidal range occurs at
the Bay of Fundy, Canada; recorded at over sixteen metres. The second largest tidal
range is that of the Severn Estuary and Bristol Channel; reaching up to fifteen metres.
The average tidal range experienced by the earth’s coastlines is similar to the open sea
range at around one metre. The key factor that affects the tidal range of a coastal point
is its topography; the surface features, both natural and man made, of the sea bed and
coastline. With the right topography, tidal resonance can occur; the process
responsible for the huge tides of the Bay of Fundy and the Bristol Channel. Tidal
resonance occurs when the movement of the tide excites one of the resonant modes of
the ocean. It is of greatest effect when the ocean meets with a continental shelf whose
width is approximately one quarter of the oceans wavelength. This situation leads to
superposition of the incident tidal waves with those reflected between the shelf edge
and the coast; creating much larger tidal ranges as a result. Estuaries often experience
increased tidal range due to their funnel like shape, getting thinner as the sea comes
inland whilst also getting shallower. The largest tidal range is usually observed in the
vicinity of the Severn crossings.
Figure 8 Schematic of a generalised coastline
Surges and Weather Effects
The movements of ocean water due to meteorological effects, such as changes in
atmospheric pressure and winds, are known as surges. Accurate tidal prediction is
made more difficult by surges because weather forecasting can only predict
meteorological events for the very near future. Even with the most advanced computer
software, accurate predictions can only be made about the next thirty-six hours of
weather. The sea levels given in tide tables only take the astronomical cycles into
consideration and not surges. For this reason there is usually a difference between the
values stated in tide tables and those actually measured. Tidal surges act
independently to the astronomical tides. If
a surge is large enough, and coincides
with the astronomical tides it can result in
mass flooding. In November 2007 the east
coast of England and the Netherlands were
warned of a three-metre surge tide in the
North Sea. The surge, created by storm
winds, was expected to coincide with high
tide and, therefore, flood defences were in
danger of being breached. Fortunately,
when the surge reached the coast it had
reduced in size and did not cause any Figure 9 Coasts threatened by the surge
major damage.
The figure below shows how a surge tide can be created by a storm and brought
inland. The low atmospheric pressure associated with a storm creates a bulge in the
sea surface. Sea levels rise by approximately 1cm for every 1 millibar decrease in
pressure; a depression of 960 millibar, which is 50 millibars below average barometric
pressure, would raise sea levels by approximately half a metre. A tidal surge is created
when the powerful winds of a storm push the bulge towards the coast.
Figure 10 Production of a storm tidal surge
Tidal Streams
Tidal streams, or currents, are the horizontal motion of ocean waters due to the tide.
Flood flow refers to tidal streams when high tide is impending, i.e. the tide is ‘coming
in’. Ebb flow refers to tidal streams when low tide is impending, i.e. the tide is ‘going
out’. Tidal streams have both direction and magnitude, which makes them more
complex to analyse compared to tidal ranges, which are scalars. Collecting tidal
stream data is more difficult than for tidal range as the strength and direction of
currents can vary greatly over short distances, whereas tidal range is relatively
constant over a wide area. Tidal streams vary with a location’s depth, and are also
strongly influenced by the shape of the area from which they originated. Due to these
facts, even the most simple of assumptions cannot be made about real life tidal
streams. In a simplistic model, however, tidal streams will be in one direction during
flood flow and in another direction during ebb flow. The two directions are not
necessarily reciprocal. Velocities along the flood direction are taken to be positive and
those along the ebb direction, negative. In many situations tidal streams are not totally
dominated by the basic ebb and flow directions. In this case, an analysis should take
both a primary flow direction and a secondary, orthogonal direction in to account.
Figure 11 Example of a Tidal Stream Atlas
The behaviour of tidal streams at a particular coastal location can be looked up
using a tidal stream atlas, or chart, which generally consist of twelve diagrams; one
for every hour of the tidal cycle. Each of the diagrams uses arrows and numbers to
represent the direction and speed of streams at various points in the local waters. At
certain points in the tidal cycle there may be no measurable tidal streams; these areas
are known as slack water. Slack water usually occurs near the time of high or low
tide, and also at instances in between when the current changes direction.
The energy in tidal streams,
and therefore the power
available, is proportional to
the cube of flow velocity. The
cubic relationship between
velocity and power means
that a large change in the
power density can be caused
by a tiny variation in tidal
stream
velocity.
As
mentioned previously, spring
tides have roughly twice the
magnitude of neap tides. Due
to the cube law there is
approximately eight (i.e. 23)
times more tidal stream
power during spring tides
than neaps.
Figure 12 Variation of tidal power during spring/neap tides
Analysing the Tidal Resource of the UK
The UK is made up of islands and as such has a large quantity of coastline for its area.
For this reason tidal power shows great promise for the UK in the future, as the
country seeks sustainable energy. The maps of the UK below display the relative
quantities of tidal power available around the coastlines, from both tidal streams and
tidal range. The edge of the coloured border marks the UK’s continental shelf and
Channel Island territorial sea limit.
Figure 13 UK Tidal Stream Resource
Figure 14 UK Tidal Range Resource