Changes in the amount of reflected sunlight correlate with local roughness and wave steepness, thus revealing wave patterns on photographs. Even internal waves can be detected by photography because of their effect on surface roughness (Section 1.2.1). The photographs in Figure 1.23 were taken from a manned spacecraft with a hand-held camera, and show the surface manifestations of some internal waves in the South China Sea and in the Mediterranean.
Figure 1.23 (a) Internal waves in the South China Sea (Hainan Island visible beneath clouds on lower left). Four wave packets are visible.
(b) Tidally generated internal waves propagating into the Mediterranean from the Straits of Gibraltar. The internal waves (which have amplitudes of the order of 15 m) are visible because of variations in the rouahness of the sea-surface (see p. 20). Area = 74 km x 74 kin.
4/
1 Idealized waves of sinusoidal form have wavelength (length between successive crests), height (vertical difference between trough and crest), steepness (ratio of height to length), amplitude (half the wave height), 9eriod (length of time between successive waves passing a fixed point) and
~requency (reciprocal of period). Water waves show cyclical variations in water level (displacement), f r o m - a (amplitude) in the trough to +a at the crest. Displacement varies not only in space (one wavelength between successive crests) but also in time (one period between crests at one
ocation). Steeper waves depart from the simple sinusoidal model, and more closely resemble a trochoidal wave form.
2 Waves transfer energy across/through material without significant
9verall
motion of the material itself, but individual particles are displaced~rom, and return to, equilibrium positions as each wave passes. Surface waves occur at interfaces between fluids, either because of relative
novement between the fluids, or because the fluids are disturbed by an external force (e.g. wind). Waves occurring at interfaces between oceanic water layers are called internal waves. Wind-generated waves, once
"nitiated, are maintained by surface tension and gravity, although only the atter is significant for water waves over 1.7 cm wavelength.
3 Most sea-surface waves are wind-generated. The stronger the wind, the arger the wave, so variable winds produce a range of wave sizes. A constant wind speed produces a fully developed sea, with waves of
H1/3
~average height of highest 33% of the waves) characteristic of that wind speed. The Beaufort Scale relates sea state and
H1/3
to the causative wind speed.4 Water particles in waves in deep water follow almost circular paths, but with a small net forward drift. Path diameters at the surface correspond to wave heights, but decrease exponentially with depth. In shallow water, the
?rbits become flattened near thesea-bed. For waves in water deeper than 1/2 wavelength, wave speed equals wavelength/period (c =
L/T)
and is 9roportional to the square root of the wavelength (c =~/gL /
2to); it is maffected by depth. For waves in water shallower than 1/20 wavelength, wave speed is proportional to the square root of the depth (c =~fgd)
and _!oes not depend upon the wavelength. For idealized water waves, the three zharacteristics, c, L arid T, are related by the equation c =L/T.
In addition, each can be expressed in terms of each of the other two. For example,= 1.56T and L = 1.56T 2.
5 Waves of different wavelengths become dispersed, because those with greater wavelengths and longer periods travel faster than smaller waVes. If :wo wave trains of similar wavelength and amplitude travel over the same sea area, they interact. Where they are in phase, displacement is doubled,
~hereas where they are out of phase, displacement is zero. A single wave :rain results, travelling as a series of wave groups, each separated from adjacent groups by an almost wave,free region. Wave group speed in deep
~ater is half the wave (phase) speed. In shallowing water, wave speed approaches group speed, until the two coincide at depths less than 1/20 of -he wavelength, where c =
~/gd.
6 Wave energy is proportional to the square of the wave height, and travels at the group speed. Wave power is rate of supply of wave energy, and so it is wave energy multiplied by wave (or group) speed, i.e. it is wave energy propagated per second per unit length of wave crest (or wave speed multiplied by wave energy per unit area). Total wave power is conserved, so waves entering shallowing water and/or funnelled into a bay or estuary (see also 7 below) increase in height as their group speed falls. Wave energy has been successfully harnessed on a small scale, but large-scale utilization involves environmental and navigational problems, and huge capital outlay.
7 Dissipation of wave energy (attenuation of waves) results from white- capping, friction between water molecules, air resistance, and non-linear wave-wave interaction (exchange of energy between waves of differing frequencies). Most attenuation takes place in and near the storm area. Swell waves are storm-generated waves that have travelled far from their place of origin, and are little affected by wind or by shorter, high-frequency waves.
The wave energy associated with a given length of wave crest decreases with increasing distance from the storm, as the wave energy is spread over an ever-increasing length of wave front.
8 Waves in shallow water may be refracted. Variations in depth cause variations in speed of different parts of the wave crest; the resulting refraction causes wave crests to become increasingly parallel with bottom contours. The energy of refracted waves is conserved, so converging waves tend to increase, and diverging waves to diminish, in height. Waves in shallow water dissipate energy by frictional interaction with the sea-bed, and by breaking. In general, the steeper the wave and the shallower the beach, the further offshore dissipation begins. Breakers form a continuous series from steep spilling types to long-period surging breakers.
9 Waves propagating with a current have diminished heights, whereas a counter-current increases wave height, unless current speed exceeds half the wave group speed. If so, waves no longer propagate, but increase in height until they become unstable and break. Tsunamis are caused by earthquakes or by slumping of sediments, and their great wavelength means their speed is always governed by the ocean depth. Wave height is small in the open ocean, but can become destructively large near the shore. Seiches (standing waves) are oscillations of water bodies, such that at antinodes there are great variations of water level but little lateral water movement, whereas at nodes the converse is true. The period of oscillation is proportional to basin length and inversely proportional to the square root of the depth. A seiche is readily established when the wavelength of incoming waves is four times the length of the basin.
1 0 Waves are measured by a variety of methods, e.g. pressure gauges on the sea-floor, accelerometers in buoys on the sea-surface, and via remote- sensing from satellites.
4~
.~ow try the following questions to consolidate your understanding of this
~hapter.
50
' ... being governed by the watery Moon ... ' Richard III, Act II, Scene II.
The longest oceanic waves are those associated with the tides, and are characterized by the rhythmic rise and fall of sea-level over a period of half a day or a day (Figure 1.2). The rise and fall result from horizontal
movements of water (tidal currents) in the tidal wave. The rising tide is usually referred to as the flow (or flood), whereas the falling tide is called the ebb. The tides are commonly regarded as a coastal phenomenon, and those who see tidal fluctuations only on beaches and in estuaries tend to think (and speak) of the tide as 'coming in' and 'going out'. However, it is important to realize that the ebb and flow of the tide at the coast is a manifestation of the general rise and fall in sea-level caused by a long- wavelength wave motion that affects the oceans as well as shallow coastal waters. Nonetheless, because of their long period and wavelength (Figure
1.2), tidal waves behave as shallow-water waves. Do bear in mind also, from Section 1.6.3, that the destructive waves generated by earthquakes are not 'tidal waves' as so often reported in the press - they are tsunamis, which also behave as shallow-water waves because of their long wavelength.
From the earliest times, it has been realized that there is some connection between the tides and the Moon. High tides are highest and low tides are lowest when the Moon is full or new, and the times of high tide at any given location can be approximately (but not exactly) related to the position of the Moon in the sky; and, as we shall see, the Sun also influences the tides.
Before discussing these relationships, we shall first describe some principal features of tidal wave motions. Figure 2.1 is a tidal record, showing regular vertical movements of the water surface relative to a mean level, over a period of about a month.
Figure 2.1 A typical 30-day tidal record showing oscillations in water level with a period of about 12.5 hours, at a station in the Tay estuary, Scotland.
51 If you compare Figures 2.1 and 1.5, you will see two important differences between wave motions resulting from the tides and those associated with wind-generated waves. These are:
1 The period of the oscillations of wind-generated waves (Figure 1.5) is typically in the order of seconds to a few tens of seconds, and both period and amplitude of the oscillations can be quite irregular. In contrast, Figure 2.1 shows the period of the tides to be about 12.5 hours, i.e. high and low tides occur twice a day, and both period and amplitude vary in a systematic way. (Figure 2.1 illustrates a semi-diurnal tide; we shall consider the different types of tide later.)
2 Although the amplitude (and height) of tidal and wind-generated wave motions is of the same order in both Figures 1.5 and 2.1, we have seen that the heights of wind waves can range from virtually zero to 30 m or more (Section 1.6.2). By contrast, in most places the tidal range is typically of the order of a few metres, and tidal ranges of more than about 10 m are known only at a few locations. Tidal range nearly always varies within the same limits at any particular location (Figure 2.1), and because the cause of tidal wave motion is both continuous and regular, so that the periodicities that result are pre-determined and fixed (as you will see shortly), tidal range can be very reliably predicted. Wind-generated waves, on the other hand, are much less predictable, because of the inherent variability of the winds.
Tidal waves are what are known as 'forced waves'because they are
generated by regular (periodic) external forces, and therefore do not behave exactly like the gravity waves considered in Chapter 1. For practical purposes, however, they can be treated as gravity waves, especially in the deep oceans.
A 7-8-day periodicity can also be seen: around days 9 to 11 and 23 to 26, :he tidal range is more than twice what it is around days 0 to 2 and 16 to 18, This 7-8-day alternation of high and low tidal range (spring and neap tides, respectively) can also be predicted with great accuracy and characterizes
iides all over the world (see Section 2.2.1).
Spring tides have an amplitude of nearly 3 rn (i.e. above and below the nean water level), so the spring tidal range is close to 6 m. In contrast, the leap tides have a range of little more than 2 m.
Where there is urban or industrial development in coastal areas, it is common
"or high and low tidal levels to be quite rigorously identified, because along gently sloping shorelines a tidal range of even a couple of metres results in substantial areas of ground being alternately covered and exposed by the
tooding and ebbing tides. In coastal areas, maps and plans commonly
"ndicate Mean High Water and Mean Low Water, as well as the Mean Tide _~evel. The Mean Tide Level is often used as a datum or baseline for
;opographic survey work, i.e. it is the baseline for all measurements of elevation and depth on maps and charts. For example, in Britain, this baseline
~known as the Ordnance Datum) is the Mean Tide Level at a specific location at Newlvn in Cornwall.
52
This discussion of tidal levels raises an important general point about people's perception of the tides. As mentioned earlier, those who see tidal fluctuations only on beaches or in estuaries tend to perceive the tide as
'coming in' and 'going out'. In fact, the sea advances over and retreats from the land only because the water level is rising and falling with the passage of tidal waves like those illustrated in Figure 2.1.
So much for some basic descriptions of the tides. We must now consider the forces that cause them. The relative motions of the Earth, Sun and Moon are complicated, and so their influence on tidal events results in an equally complex pattern. Nevertheless, as we have just seen, the actual motions of the tides are quite regular, and the magnitudes of the fide-generating forces can be precisely formulated. Although the response of the oceans to these forces is modified by topography and by the transient effects of weather patterns, it is possible to make reliable predictions of the tides for centuries ahead (and indeed to relate specific historical events to tidal states many centuries in the past).
The Earth and the Moon behave as a single system, rotating about a common centre of mass, with a period of 27.3 days. The orbits are in fact elliptical, but to simplify matters we will treat them as circular for the time being. The Earth rotates eccentrically about the common centre of mass (centre of gravity), which is within the Earth and lies about 4700 km from its centre. Figure 2.2 illustrates the motions that result. The principal consequence of the eccentric motion about the Earth-Moon centre of mass is this: All points on and within the Earth must also rotate about the
common centre of mass and so they must all follow the same elliptical path.
So each point must have the same angular velocity (2rc/27.3 days), and hence will experience the same centrifugal force (which is proportional to acceleration towards the centre, i.e. to the product of the radius and the square of the angular velocity).
The eccentric motion described above has nothing whatsoever to do with the Earth's rotation (spin) upon its own axis, and should not be confused with it (we have shown the Earth's rotation axis on Figure 2.2 for the situation where the Moon is directly above the Equator, which happens only twice every 27.3 d a y s - see Section 2.1.1 and Figure 2.8). Nor should the centrifugal force resulting from the eccentric motion (which is equal at all points on Earth) be confused with the centrifugal force caused by the Earth's spin (which increases with distance from the rotation axis).
If you find these concepts difficult, the following simple analogy may help.
Imagine you are whirling a small bunch of keys on a short length (say 25 cm) of chain. The keys represent the Moon, and your hand represents the Earth. You are rotating your hand eccentrically (but unlike the Earth it is not spinning as well), and all points on and within your hand are experiencing the same angular velocity and the same centrifugal force. Provided your bunch of keys is not too large, the centre of mass of the 'hand-and-key' system lies within your hand.
b3
Figure 2.2 Rotation of the Earth-Moon system (not to scale). The Moon orbits the Earth about their common centre of mass (located within the Earth) once every 27.3 days. The centre of the Earth also rotates about this centre of mass once every 27.3 days, describing a very much smaller orbit (fine black line), as do all other points on and within the Earth. Note that the orbits are shown as circular for simplicity, whereas in fact they are elliptical (see later text); note also that the Earth's own central rotation axis is shown here as perpendicular to the plane of the Moon's orbit, which happens twice every 27.3 days - see Figure 2.8.
In the text which follows, you do not need to understand the details of the explanation related to Figures 2.3 and 2.4. However, you do need to be aware of the relationship embodied in Equation 2.2 on p. 55, i.e. that tide- producing forces are inversely proportional to the cube of the Earth-Moon distance, and that the tide-producing forces are greatest along the small circles shown in Figure 2.4(a).
The total centrifugal force acting on the Earth-Moon system exactly balances the forces of gravitational attraction between the two bodies, so the system is in equilibrium, i.e. we should neither lose the Moon, nor collide with it, in the near future. The centrifugal forces are directed parallel to a line joining the centres of the Earth and the Moon (see red arrows on Figure 2.3, overleaf). Now consider the gravitational force exerted by the Moon on the Earth. Its magnitude will not be the same at all points on the Earth's surface, because they are not at the same distance from the Moon.
Points nearest the Moon will experience a greater gravitational pull from the Moon than those on the opposite side of the Earth. Moreover, the direction of the Moon's gravitational pull at all points will be directed towards its centre (see blue arrows on Figure 2.3), so it will not be exactly parallel to the direction of the centrifugal forces, except along the line joining the centres of the Earth and Moon.
The resultant (i.e. the composite effect) of the two forces is known as the title-producing force. Depending upon its position on the Earth's surface with respect to the Moon, this force is directed into, parallel to, or away from, the Earth's surface. Its direction and relative strength (not strictly to scale) is shown by thick purple arrows on Fieure 2.3.
54
:igure 2.3 The derivation of the tide-producing :orces (not to scale), for a hypothetical water- :overed Earth. The centrifugal force has exactly :he same magnitude and direction at all points, Jvhereas the gravitational force exerted by the
Aoon on the Earth varies in both magnitude inversely with the square of the distance from :he Moon) and direction (directed towards the
Aoon's centre, but shown with the angles
~xaggerated for clarity). The tide-producing force at any point (thick purple arrows) is the resultant _~f the gravitational and centrifugal force at that )oint, and varies inversely with the cube of the :listance from the Moon (see text).
The gravitational force (Fg) between two bodies is given by:
GMIM2 Fg= R2
where M 1 and M2 are the masses of the two bodies, R is the distance between their centres, and G is the universal gravitational constant (whose value is 6.672 x 10 -11N m 2 kg-2).
However, we need to reconcile Equation 2.1 with the statement in the caption to Figure 2.3 that the magnitude of the tide-producing force exerted by the Moon on the Earth varies inversely with the cube of the distance.
Consider the point marked G on Figure 2.3. The gravitational attraction of the Moon at G (/'go) is greater there than that at the Earth's centre, because G is nearer to the Moon by the distance of the Earth's radius (a). The gravitational force exerted by the Moon at the Earth's centre is exactly equal and opposite to the centrifugal force there, so the fide-producing force at the centre of the Earth is zero. Now as the centrifugal force is equal at all points on Earth, and at the Earth's centre is equal to the gravitational force exerted there by the Moon, it follows that we can substitute the expression on the fight-hand side of Equation 2.1 (i.e.
GMaMz/R 2)
for the centrifugal force.(2.1;
bb I'he fide-producing force at point G ( T P F 6 ) is given by the force due to gravitational attraction of the Moon at G (Fgc) minus the centrifugal force at G, i.e.
GMIM2 GM1M2
TP F G = ( R - a )---7 - - - - T - - R which simplifies to:
T P F G =
GMIM2a ( 2 R - a) R 2 ( R - a) 2
Now a is very small compared to R, so ( 2 R - a) can be approximated to 2R, and ( R - a) 2 to R 2, giving the relationship:
T P F c = G M 1 M z 2 a
R3 (2.2)
In other words, the fide-producing force is proportional to I l R 3.
You may have considered point G as your answer. Certainly, G is nearest to
~:he Moon, and hence is one of the two points where the difference between :he centrifugal force and the gravitational force exerted by the Moon is greatest. However, at point G all the resultant tide-producing force is acting vertically against the pull of the Earth's own gravity, which happens to be about 9 x 10 6 greater than the fide-producing force. Hence the local effect of -he fide-producing forces at point G is negligible. Similar arguments apply at )oint A, except that the gravitational attraction of the Moon at point A (FgA) :s less than the centrifugal force, and consequently the fide-producing force at A is equal in magnitude to that at G, but directed a w a y from the Moon (Figure 2.3).
The points we need to identify are those where the horizontal component of the fide-producing force, i.e. the tractive force, is at a maximum. Such points do not lie directly on a line joining the centres of the Earth and Moon, and so Equation 2.2 becomes slightly more complex. For example, at point P on Figure 2.4(a) the gravitational attraction (Fgp) would be, to a first approximation:
GMIM2
= ( 2 . 3 )
Fgp ( R - a cos l/t) 2
The length a cos ~ is marked on Figure 2.4(a) ( ~ is the Greek letter 'psi').
Equations such as 2.3 can be used to show that the tractive force is greatest at )oints along the small circles defined in Figure 2.4(a), w h i c h have n o t h i n g to do with latitude or longitude.
It is the tractive force that causes the water to move, because this horizontal zomponent (by definition parallel to - i.e. tangential to - the Earth's surface at the location concerned) is unopposed by any other lateral force (apart from friction at the sea-bed, which is negligible in this context). The gravitational force due
~o the Earth is much greater than the tractive force but acts at fight angles to it and so has no effect. The longest arrows on Figure 2.4(b) show where on the Earth the tractive forces are at a maximum when the Moon is over the Equator.
b6
Figure 2.4 (a) The effect of the gravitational force of the Moon at three positions on the Earth. The gravitational force is greatest at G (nearest the Moon) and least at A (furthest from the Moon). At P the gravitational force is less than at G, and can be calculated from Equation 2.3. The tide-producing forces are smallest at A and G, but greatest at P, and all other points on the two small circles. The value for the angle for these circles is 54 ~ 41 ". The circles have nothing to do with latitude and longitude. For
~xplanation, see text.
~b) The relative magnitudes of the tractive orces (i.e. of the horizontal components of the :ide-producing forces, shown as purple arrows )n Figure 2.3) at various points on the Earth's
~urface. The Moon is assumed to be directly _~ver the Equator (i.e. at zero declination, see Section 2.1.1). Points A and G correspond to
:hose on (a) and in Figure 2.3.
In this simplified case, the tractive forces would result in movement of water towards points A and G on Figure 2.4(b). In other words, an equilibrium state would be reached (called the equilibrium tide), producing an ellipsoid with its two bulges directed towards and away from the Moon. So, paradoxically, although the fide-producing forces are minimal at A and G, those are the points towards which the water would tend to go. Figure 2.5 shows how such an equilibrium tidal ellipsoid would look in the simplified case we have been considering, i.e. a completely water-covered Earth with the Moon directly above the Equator and the distribution of tractive forces as in Figure 2.4(b).
If you found Figures 2.3 and 2.4 and related text and equations difficult to follow, here is a shorter explanation of why there are two equilibrium tidal bulges (Figure 2.5). The centrifugal force acts in the same direction all over the Earth, i.e. away from the Moon (Figure 2.3). Moreover, on the side of the Earth away from the Moon, the gravitational attraction due to the Moon is less than it is on the side of the Earth facingthe Moon. The resultant fide- producing force thus acts away from the Moon at points such as A on
Figure 2.3. That is why there is a tidal bulge away from the Moon as well as a bulge towards it (Figure 2.5). The mathematics of the relationship is such that theoretically the corresponding tide-producing forces on either side of the Earth are equal and opposite.
b/
Figure 2.5 The equilibrium tidal ellipsoid (not to scale) as it would appear on a water-covered Earth with the Moon directly above the Equator.
In practice, the equilibrium ellipsoid does not develop, partly because the Earth is not of course entirely water-covered, but chiefly because the Earth rotates about its own axis. If the two bulges were to maintain their positions relative to the Moon, they would have to travel around the world at the same rate (but in the opposite direction) as the Earth rotates about its axis.
Any point on the Earth's surface would thus encounter two high and two low tides during each complete rotation of the Earth (i.e. each day), as illustrated in Figure 2.6.
Figure 2.6 shows both Moon and tidal bulges remaining stationary during a complete rotation of the Earth. That cannot be the case, for the Moon continues to travel in its orbit as the Earth rotates. Because the Moon revolves about the Earth-Moon centre of mass once every 27.3 days, in the same direction as the Earth rotates upon its own axis (which is once every 24 hours), the period of the Earth's rotation with respect to the Moon is 24 hours and 50 minutes. This is the l u n a r day.
-igure 2.6 Rotation of the Earth within the ]quilibrium tidal bulge (seen from above the
~lorth Pole and not to scale), showing how a )oint on the Earth's surface would experience 9 wo high tides (1 and 3) and two low tides
"2 and 4) during each complete rotation of the -arth about its axis.
The interval between successive high (and low) tides would be about 12 hours 25 minutes - and the interval between high and low tide would be close to 6 hours 121 minutes. This is the reason why the times of high tides at many locations are almost an hour later each successive day (Figure 2.7, overleaf).
The equilibrium tidal concept also brings out another very important aspect of tidal wave motions.