Tides—Forces that produce them—Origin and Course of Tidal Wave—Its
Speed—Three kinds of Oscillations in the Ocean—The Semidiurnal
Tides—Equinoctial Tides—Effects of the Declination of the Sun and
Moon—Theory insufficient without Observation—Direction of the Tidal
Wave—Height of Tides—Mass of Moon obtained from her Action on the
Tides—Interference of Undulations—Impossibility of a Universal
Inundation—Currents.
ONE of the most immediate and remarkable effects of a gravitating force
external to the earth is the alternate rise and fall of the surface of
the sea twice in the course of a lunar day, or 24^h 50^m 28^s of mean
solar time. As it depends upon the action of the sun and moon, it is
classed among astronomical problems, of which it is by far the most
difficult and its explanation the least satisfactory. The form of the
surface of the ocean in equilibrio, when revolving with the earth round
its axis, is an ellipsoid flattened at the poles; but the action of the
sun and moon, especially of the moon, disturbs the equilibrium of the
ocean. If the moon attracted the centre of gravity of the earth and all
its particles with equal and parallel forces, the whole system of the
earth and the waters that cover it would yield to these forces with a
common motion, and the equilibrium of the seas would remain undisturbed.
The difference of the forces and the inequality of their directions
alone disturb the equilibrium.
The particles of water under the moon are more attracted than the centre
of gravity of the earth, in the inverse ratio of the square of the
distance. Hence they have a tendency to leave the earth, but are
retained by their gravitation, which is diminished by this tendency. On
the contrary, the moon attracts the centre of the earth more powerfully
than she attracts the particles of water in the hemisphere opposite to
her; so that the earth has a tendency to leave the waters, but is
retained by gravitation, which is again diminished by this tendency.
Thus the waters immediately under the moon are drawn from the earth, at
the same time that the earth is drawn from those which are diametrically
opposite to her, in both instances producing an elevation of the ocean
of nearly the same height above the surface of equilibrium; for the
diminution of the gravitation of the particles in each position is
almost the same, on account of the distance of the moon being great in
comparison of the radius of the earth. Were the earth entirely covered
by the sea, the waters thus attracted by the moon would assume the form
of an oblong spheroid whose greater axis would point towards the moon;
since the columns of water under the moon, and in the direction
diametrically opposite to her, are rendered lighter in consequence of
the diminution of their gravitation; and, in order to preserve the
equilibrium, the axes 90° distant would be shortened. The elevation, on
account of the smaller space to which it is confined, is twice as great
as the depression, because the contents of the spheroid always remain
the same. If the waters were capable of assuming the form of equilibrium
instantaneously, that is, the form of the spheroid, its summit would
always point to the moon notwithstanding the earth’s rotation. But, on
account of their resistance, the rapid motion produced in them by
rotation prevents them from assuming at every instant the form which the
equilibrium of the forces acting upon them requires. Hence, on account
of the inertia of the waters, if the tides be considered relatively to
the whole earth and open seas, there is a meridian about 30° eastward of
the moon, where it is always high water both in the hemisphere where the
moon is and in that which is opposite. On the west side of this circle
the tide is flowing, on the east it is ebbing, and on every part of the
meridian at 90° distant it is low water. This great wave, which follows
all the motions of the moon as far as the rotation of the earth will
permit, is modified by the action of the sun, the effects of whose
attraction are in every respect like those produced by the moon, though
greatly less in degree. Consequently a similar wave, but much smaller,
raised by the sun, tends to follow his motions, which at times combines
with the lunar wave, and at others opposes it, according to the relative
positions of the two luminaries; but as the lunar wave is only modified
a little by the solar, the tides must necessarily happen twice in a day,
since the rotation of the earth brings the same point twice under the
meridian of the moon in that time, once under the superior and once
under the inferior meridian.
The periodic motions of the waters of the ocean, on the hypothesis of an
ellipsoid of revolution, entirely covered by the sea, are, however, very
far from according with observation. This arises from the great
irregularities in the surface of the earth, which is but partially
covered by the sea, from the variety in the depths of the ocean, the
manner in which it is spread out on the earth, the position and
inclination of the shores, the currents, and the resistance which the
waters meet with: causes impossible to estimate generally, but which
modify the oscillations of the great mass of the ocean. However, amidst
all these irregularities, the ebb and flow of the sea maintain a ratio
to the forces producing them sufficient to indicate their nature, and to
verify the law of the attraction of the sun and moon on the sea. La
Place observes, that the investigation of such relations between cause
and effect is no less useful in natural philosophy than the direct
solution of problems, either to prove the existence of the causes or to
trace the laws of their effects. Like the theory of probabilities, it is
a happy supplement to the ignorance and weakness of the human mind.
Since the disturbing action of the sun and moon can only become sensible
in a very great extent of deep water, the Antarctic Ocean is the origin
and birthplace of our tides. A succession of tidal waves from that
source follow one another in a north-westerly direction down the Pacific
and Atlantic Oceans, modified as they proceed by the depth of the water
and the form of the coasts. For when the sun and moon are in the same
meridian, and pass over the mass of waters lying east from Van Diemen’s
Land, New Zealand, and the South Pole, the resulting force of their
combined attraction, penetrating to the abyss of the deep and boundless
circuit of the Southern Ocean, raises a vast wave or ridge of water,
which tends to follow the luminaries to the north and west, and
continues to flow in that direction long after the bodies cease to act
upon it; but it is so retarded by the rotation of the earth and by the
inertia of the water, that it does not arrive at the different parts of
the coasts till after the moon’s southing (N. 156). When this tidal wave
leaves the Antarctic Ocean and enters the Pacific, it rushes along the
western coast of America to its farthest end, but it is so much
obstructed by the number of islands in the middle of that ocean that it
is hardly perceptible among them; while on the east it enters the Indian
Ocean, strikes with violence on the coasts of Hindostan and the shores
at the mouths of the Ganges, and causes the terrific bore in the Hoogly.
The part of this tidal wave that enters the Atlantic passes impetuously
along the coasts of Africa and America, arriving later and later at each
place. It is modified, however, by a tide raised in the Atlantic, which
is deep and free from islands; and this combined tidal wave, still
coming northward, pours its surge into the Gulf of Fundy to the height
of fifty feet; then being deflected by the coast of America at right
angles, it rushes eastward, bringing high water to the western coasts of
Ireland and England. It then goes round Scotland, brings high water to
Aberdeen and the opposite coasts of Norway and Denmark, and, continuing
its course to the south, arrives at the mouth of the Thames and fills
the channels of that river on the morning of the third day after leaving
the Antarctic Ocean.
Thus the tides in our ports are owing to an impulse from the waters of
the Antarctic seas raised by the action of the sun and moon. No doubt a
similar action raised that tide in the North Polar Ocean which Dr. Kane
saw rolling on the northern coast of Greenland in 82° N. latitude, but
which, in the present state of the globe, is imprisoned by bars of ice
and ice-bound lands.
The tidal wave extends to the bottom of the ocean, and moves uniformly
and with great speed in very deep water, variably and slow in shallow
water; the time of propagation depends upon the depth of the sea, as
well as on the nature and form of the coasts. It varies inversely as the
square of the depth—a law which theoretically affords the means of
ascertaining the proportionate depth of the sea in different parts. It
is one of the great constants of nature, and is to fluids what the
pendulum is to solids—a connecting link between time and force.
For example: the tidal wave moves across the Southern Ocean with the
velocity of 1000 miles an hour, and in the Atlantic it is scarcely less
on account of the deep trough which runs through the centre of that
ocean; but the sea is so shallow on the British coast that it takes more
time to come from Aberdeen to London than to travel over an arc of 120°,
between 60° S. lat. and 60° N. lat.
In deep water the tidal wave is merely a rise and fall of the surface;
the water does not advance, though the wave does. Indeed, if so heavy a
body as water were to move at the rate of 1000 miles an hour, it would
cause universal destruction, since in the most violent hurricanes the
velocity of the wind is little more than 100 miles an hour. Besides, it
is evident that no ship could either sail or steam against it. When the
water is shallow, however, there is a motion of translation in the water
along with the tide.
In the deep ocean the undulating motion consists of two distinct
things—an advancing form and a molecular movement. The motion of each
particle of water is in an ellipse lying wholly in the vertical plane;
so that, after the momentary displacement during the passage of the
wave, they return to their places again. The resistance of the sea-bed
is insensible in deep water; but when the tidal wave, which extends to
the very bottom of the ocean, comes into shallow water with diminished
velocity, the particles of water moving in vertical ellipses strike the
bottom, and by reaction the wave rises higher; and that being
continually repeated, as the form moves on the wave rises higher and
higher, bends more and more forward, till at last it loses its
equilibrium, and then both form and water roll to the shore, and the
elliptical trajectories of the particles, which in deep water were
vertical, incline more and more, till at length they become horizontal.
The distance from the shore at which the water begins to be translated
depends upon the depth, the nature of the coast, and the form of the
shore. Mr. Scott Russell has demonstrated that in shallow water the
velocity of the wave is equal to that which a heavy body falling freely
by its gravity would acquire in descending through half the depth of the
fluid.
It is proved by daily experience, as well as by strict mathematical
reasoning, that, if a number of waves or oscillations be excited in a
fluid by different forces, each pursues its course and has its effect
independently of the rest. Now, in the tides there are three kinds of
oscillations, depending on different causes, and producing their effects
independently of each other, which may therefore be estimated
separately. The oscillations of the first kind, which are very small,
are independent of the rotation of the earth, and, as they depend upon
the motion of the disturbing body in its orbit, they are of long
periods. The second kind of oscillations depend upon the rotation of the
earth, therefore their period is nearly a day. The oscillations of the
third kind vary with an angle equal to twice the angular rotation of the
earth, and consequently happen twice in twenty-four hours (N. 157). The
first afford no particular interest, and are extremely small; but the
difference of two consecutive tides depends upon the second. At the time
of the solstices this difference, which ought to be very great according
to Newton’s theory, is hardly sensible on our shores. La Place has shown
that the discrepancy arises from the depth of the sea, and that if the
depth were uniform there would be no difference in the consecutive tides
but that which is occasioned by local circumstances. It follows,
therefore, that, as this difference is extremely small, the sea,
considered in a large extent, must be nearly of uniform depth, that is
to say, there is a certain mean depth from which the deviation is not
great. The mean depth of the Pacific Ocean is supposed to be about four
or five miles, that of the Atlantic only three or four, which, however,
is mere conjecture. Possibly the great extent and uniformly small depth
of the Atlantic over the telegraphic platform may prevent the difference
of the oscillations in question from being perceptible on our shores.
From the formulæ which determine the difference of these consecutive
tides it is proved that the precession of the equinoxes and the nutation
of the earth’s axis are the same as if the sea formed one solid mass
with the earth.
The oscillations of the third kind are the semi-diurnal tides so
remarkable on our coasts. In these there are two phenomena particularly
to be distinguished, one occurring twice in a month, the other twice in
a year.
The first phenomenon is, that the tides are much increased in the
syzygies (N. 158), or at the time of new and full moon: in both cases
the sun and moon are in the same meridian; for when the moon is new they
are in conjunction, and when she is full they are in opposition. In each
of these positions their action is combined to produce the highest or
spring tides under that meridian, and the lowest in those points that
are 90° distant. It is observed that the higher the sea rises in full
tide, the lower it is in the ebb. The neap tides take place when the
moon is in quadrature. They neither rise so high nor sink so low as the
spring tides. It is evident that the spring tides must happen twice in a
month, since in that time the moon is once new and once full. Theory
proves that each partial tide increases as the cube of the parallax or
apparent diameter of the body producing it, for the greater the apparent
diameter the nearer the body and the more intense its action upon the
sea; hence the spring tides are much increased when the moon is in
perigee, for then she is nearest to the earth.
The second phenomenon in the tides is the augmentation occurring at the
time of the equinoxes, when the sun’s declination is zero (N. 159),
which happens twice in every year. The spring tides which take place at
that time are often much increased by the equinoctial gales, and, on the
hypothesis of the whole earth covered by the ocean, would be the
greatest possible if the line of the moon’s nodes coincided with that of
her perigee, for then the whole action of the luminaries would be in the
plane of the equator. But since the Antarctic Ocean is the source of the
tides, it is evident that the spring tide must be greatest when the moon
is in perigee, and when both luminaries have their highest southern
declination, for then they act most directly upon the great circuit of
the south polar seas.
The sun and moon are continually making the circuit of the heavens at
different distances from the plane of the equator, on account of the
obliquity of the ecliptic and the inclination of the lunar orbit. The
moon takes about 29-1/2 days to vary through all her declinations, which
sometimes extend 28-3/4° on each side of the equator, while the sun
requires nearly 365-1/4 days to accomplish his motions through 23-1/2°
on each side of the same plane, so that their combined action causes
great variations in the tides. Both the height and time of high water
are perpetually changing, and, although the problem does not admit of a
general solution, it is necessary to analyse the phenomena which ought
to arise from the attraction of the sun and moon, but the result must be
corrected in each particular case for local circumstances, so that the
theory of the tides in each port becomes really a matter of experiment,
and can only be determined by means of a vast number of observations,
including many revolutions of the moon’s nodes.
The mean height of the tides will be increased by a very small quantity
for ages to come, in consequence of the decrease in the mean distance of
the moon from the earth; the contrary effect will take place after that
period has elapsed, and the moon’s mean distance begins to increase
again, which it will continue to do for many ages. Thus the mean
distance of the moon and the consequent minute increase in the height of
the tides will oscillate between fixed limits for ever.
The height to which the tides rise is much greater in narrow channels
than in the open sea, on account of the obstructions they meet with. The
sea is so pent up in the British Channel that the tides sometimes rise
as much as fifty feet at St. Malo, on the coast of France; whereas on
the shores of some of the South Sea islands, near the centre of the
Pacific, they do not exceed one or two feet. The winds have great
influence on the height of the tides, according as they conspire with or
oppose them. But the actual effect of the wind in exciting the waves of
the ocean extends very little below the surface. Even in the most
violent storms the water is probably calm at the depth of ninety or a
hundred fathoms. The tidal wave of the ocean does not reach the
Mediterranean nor the Baltic, partly from their position and partly from
the narrowness of the Straits of Gibraltar and of the Categat, but it is
very perceptible in the Red Sea and in Hudson’s Bay. The ebb and flow of
the sea are perceptible in rivers to a very great distance from their
estuaries. In the Narrows of Pauxis, in the river of the Amazons, more
than five hundred miles from the sea, the tides are evident. It requires
so many days for the tide to ascend this mighty stream, that the
returning tides meet a succession of those which are coming up; so that
every possible variety occurs at some part or other of its shores, both
as to magnitude and time. It requires a very wide expanse of water to
accumulate the impulse of the sun and moon, so as to render their
influence sensible; on that account the tides in the Mediterranean and
Black Sea are scarcely perceptible.
These perpetual commotions in the waters are occasioned by forces that
bear a very small proportion to terrestrial gravitation: the sun’s
action in raising the ocean is only the 1/38448000 of gravitation at the
earth’s surface, and the action of the moon is little more than twice as
much; these forces being in the ratio of 1 to 2.35333, when the sun and
moon are at their mean distances from the earth. From this ratio the
mass of the moon is found to be only the 1/75 part of that of the earth.
Had the action of the sun on the ocean been exactly equal to that of the
moon, there would have been no neap tides, and the spring tides would
have been of twice the height which the action of either the sun or moon
would have produced separately—a phenomenon depending upon the
interference of the waves or undulations.
A stone plunged into a pool of still water occasions a series of waves
to advance along the surface, though the water itself is not carried
forward, but only rises into heights and sinks into hollows, each
portion of the surface being elevated and depressed in its turn. Another
stone of the same size, thrown into the water near the first, will
occasion a similar set of undulations. Then, if an equal and similar
wave from each stone arrive at the same spot at the same time, so that
the elevation of the one exactly coincides with the elevation of the
other, their united effect will produce a wave twice the size of either.
But, if one wave precede the other by exactly half an undulation, the
elevation of the one will coincide with the hollow of the other, and the
hollow of the one with the elevation of the other; and the waves will so
entirely obliterate one another, that the surface of the water will
remain smooth and level. Hence, if the length of each wave be
represented by 1, they will destroy one another at intervals of 1/2,
3/2, 5/2, &c., and will combine their effects at the intervals 1, 2, 3,
&c. It will be found according to this principle, when still water is
disturbed by the fall of two equal stones, that there are certain lines
on its surface of a hyperbolic form, where the water is smooth in
consequence of the waves obliterating each other, and that the elevation
of the water in the adjacent parts corresponds to both the waves united
(N. 160). Now, in the spring and neap tides arising from the combination
of the simple solilunar waves, the spring tide is the joint result of
the combination when they coincide in time and place; and the neap tide
happens when they succeed each other by half an interval, so as to leave
only the effect of their difference sensible. It is, therefore, evident
that, if the solar and lunar tides were of the same height, there would
be no difference, consequently no neap tides, and the spring tides would
be twice as high as either separately. In the port of Batsha, in
Tonquin, where the tides arrive by two channels of lengths corresponding
to half an interval, there is neither high nor low water on account of
the interference of the waves.
The initial state of the ocean has no influence on the tides; for,
whatever its primitive conditions may have been, they must soon have
vanished by the friction and mobility of the fluid. One of the most
remarkable circumstances in the theory of the tides is the assurance
that, in consequence of the density of the sea being only one-fifth of
the mean density of the earth, and the earth itself increasing in
density towards the centre, the stability of the equilibrium of the
ocean never can be subverted by any physical cause. A general inundation
arising from the mere instability of the ocean is therefore impossible.
A variety of circumstances, however, tend to produce partial variations
in the equilibrium of the seas, which is restored by means of currents.
Winds and the periodical melting of the ice at the poles occasion
temporary watercourses; but by far the most important causes are the
centrifugal force induced by the velocity of the earth’s rotation, and
variations in the density of the sea.
The centrifugal force may be resolved into two forces—one perpendicular,
and another tangent to the earth’s surface (N. 161). The tangential
force, though small, is sufficient to make the fluid particles within
the polar circles tend towards the equator, and the tendency is much
increased by the immense evaporation in the equatorial regions from the
heat of the sun, which disturbs the equilibrium of the ocean. To this
may also be added the superior density of the waters near the poles,
from their low temperature. In consequence of the combination of all
these circumstances, two great currents perpetually set from each pole
towards the equator. But, as they come from latitudes where the rotatory
motion of the surface of the earth is very much less than it is between
the tropics, on account of their inertia, they do not immediately
acquire the velocity with which the solid part of the earth’s surface is
revolving at the equatorial regions; from whence it follows that, within
twenty-five or thirty degrees on each side of the line, the ocean has a
general motion from east to west, which is much increased by the action
of the trade winds. Both in the Pacific and Atlantic currents of
enormous magnitude are deflected by the continents and islands to the
north and south from this mighty mass of rushing waters, which convey
the warmth of the equator to temper the severity of the polar regions,
while to maintain the equilibrium of the seas counter currents of cold
water are poured from the polar oceans to mingle with the warm waters at
the line, so that a perpetual circulation is maintained.
Icebergs are sometimes drifted as far as the Azores from the Polar seas,
and from the south pole they have come even to the Cape of Good Hope.
But the ice which encircles the south pole extends to lower latitudes by
10° than that which surrounds the north. In consequence of the polar
current Sir Edward Parry was obliged to give up his attempt to reach the
north pole in the year 1827, because the fields of ice were drifting to
the south faster than his party could travel over them to the north.
Kotzebue and Sir James Ross found a stratum of constant temperature in
the ocean at a depth depending upon the latitude: at the equator it is
at the depth of 7200 feet, from whence it gradually rises till it comes
to the surface in both hemispheres about the latitude of 56° 26ʹ, where
the water has the same temperature at all depths; it then descends to
4500 feet below the surface about the 70th parallel both in the Arctic
and Antarctic Seas. The temperature of that aqueous zone is about 39°·5
of Fahrenheit.[7] It divides the surface of the ocean into five great
zones of temperature, namely, a medial region, in which the highest mean
temperature is 82° Fahr., two temperate zones each of 39°·5 Fahr., and
two polar basins at the freezing point of salt water.