OCEAN TIDES AND THE ROTATION OF EARTH - · PDF fileOCEAN TIDES AND THE ROTATION OF EARTH Dr. (Prof.) ... tides except taking that it is the work of Almighty and ... In 730 another

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OCEAN TIDES AND THE ROTATION OF EARTH

Dr. (Prof.) V.C.A. NAIR1

1Educational Physicist and Research Guide at JJT University, Rajasthan-333001, India

[email protected]

Abstract The Paper begins with a detailed historical aspect on Tides. The Physics of Tides is extensively dealt

with classification of different types of tides. Mathematical theory of tides giving a derivation for the

velocity of tidal wave.The Laplace Tidal Equation has been briefly dealt with. The effects of tides with

special attention to Rotation of Earth form a major part of the Paper. A personal observation of the

author in the drowning of swimmers during low tides is added by a free hand diagram. The author,

even though, has not referred any standard text book, the matter in the paper forms almost a chapter

or at least a major topic in Geophysics.

Keywords Centrifugal force,, Effects of Tides, Effects of atmospheric pressure on tides, Gravitational

force, Laplace Tidal Equation, Mathematical theory of tides, Neap Tides, Prediction of tides, Roche

Limit, Rotation of Earth, Spring Tides, Tidal Bore, Tidal Friction, Tidal Heating, Tractive Force.

I. HISTORICAL

The word, „Tide‟ derived from the low German „tiet‟ = time is a planetary phenomenon creating

rise and fall of sea levels which is due to the combined effects of gravitational forces of the Sun and the

Moon in conjunction with the rotation of Earth. There are many legends attached to tides. In one of the

legends, the tides are ascribed to the breathing cycle of a giant whale.

I would like to give below a brief history of Tides almost in a chronological order. The tides

were observed and studied by the ancients even before Christ. As early as 2300 BC inthe ruins of coastal

cities along the Gulf of Cambay in India. The coastal tribes in order to keep their boats floating, used to

trap the tidal water in an enclosure fitted with gates during high tides and close the gate before the tides

recede. Ancients in India during the 4th

and 3rd

century BC believed of a link between tides and phases

of the Moon. Tides play a very important role in the culture and commerce of the coastal areas of any

region. Pytheas travelled to the British Isles sometime during 325 BC and it is believed that he was the

first to have related Spring Tides to the phase of the Moon. The Babylonian astronomer, Seleucus of

Seleucia in the 2nd

century BC correctly theorized that the tides were caused by the Moon and they

varied in time and strength in different parts of the world. Strabo another thinker and philosopher

supported the ideas of Seleucus specially regarding the heights of tides depending on the position of

Moon relative to the Sun. Ancients were not interested in the „Cause and Effect‟ formalism regarding

tides except taking that it is the work of Almighty and something divine. Ignorance of tides had an

impact on history. The war galleys of Julius Caesar (100 -44BC) were devastated on the British shore

because he failed to pull them high enough out of the water to avoid the returning tide. In China, Wang

Chong (27-100) published a book entitled, “Lunheng”in which he has mentioned that the tides rise and

fall follow the Moon and vary in magnitude. In 730 another thinker from Europe explained the rise of

International Journal of Applied and Pure Science and Agriculture (IJAPSA) Volume 03, Issue 2, [February- 2017] e-ISSN: 2394-5532, p-ISSN: 2394-823X


tide water on one coast of British Isles and the fall on the other coast. In the 9th

century, Al-Kindi, an

Arabian earth scientist wrote a treatise entitled: “The Efficient Cause of the Flow and Ebb” in which he

described a laboratory experiment to prove that tides are temperature-dependent. In the 10th

century the

Arabs had already begun to relate the timings of tides to the cycle of the Moon.In 1056, China was the

first to prepare a Tide Table for visitors to see the tidal bore in the Qiantang River In the year 1216,

King John of England (1167-1216) was caught in a high tide, lost his treasure and part of his army and

was so enraged, he died a week later.

A detailed study of tides, from the point of view of Physics, started in the year 1632 with Galileo

(1564-1642). He published an article, „Dialogue on the Tides‟ in his „Dialogue concerning the two Chief

World Systems‟. In that article he attributed the cause of tides to the motion of Earth around the Sun and

the theory was disputed and finally discarded. Later, Johannes Kepler (1571-1630) a German

mathematician and founder of Celestial Mechanics, based upon ancient observations and correlations,

correctly suggested that Moon caused the tides. Sir Isaac Newton (1642-1727) in the year 1687

published his Principia in which he mentioned that the tide generating forces are due to the lunar and

solar attractions. After the scientific period of Newton, in the year 1740 the Academic Royale des

Sciences Paris declared a prize for the best theoretical essay on Tides. Eminent scientists of the time,

Daniel Bernoulli, Leonhard Euler, Colin Maclaurin and Antoine Cavalleri shared the prize. Colin

Maclaurin (1698-1746) the Scottish mathematician was the first to include the rotational effect of the

Earth in the mathematical treatment on tides. Later in the year 1776, Pierre Simon Marquis de

Laplace(1749-1827) gave the theory of tides a firm mathematical basis by his famous „Laplace Tidal

Wave Equations‟. Sir William Thomson (Later named Lord Kelvin) (1824-1907) re-wrote Laplace‟s

equations in terms of vorticity which allowed for solutions describing tidally driven coastally trapped

waves called Kelvin Waves. Later Henri Poincare (1854-1912) the French mathematician and

theoretical physicist joined Lord Kelvin in modifying Laplace‟s theory and the same was used by Arthur

Thomas in 1921 following E.W. Brown‟s Lunar theory.Kelvin‟s work was further developed and

extended by George Darwin (1845-1912). Tidal theory related to the instability of the Sun and formation

of the planets was proposed by physicist Sir James H Jeans (1877-1946) and mathematician Sir Harold

Jeffreys (1891-1989). A.T. Doodson (1890-1968) following the lunar theory has distinguished nearly

388 tidal frequencies some of which are still in use today. Thus, the history continues with the

researchers still pondering over the solutions of Laplace Tidal Equations which are trivial.

II. INTRODUCTION

This paper essentially deals with one of the very important forces of nature, that is „Gravity‟

which, in fact, rules the entire universe and appropriate to the situation, I would like to give a quotation

from “Letters to Bentley” by Sir Isaac Newton:

“You sometimes speak of Gravity as essential and inherent to matter. Pray do not ascribe that notion

to me; for the cause of gravity is what I do not pretend to know, and therefore would take more time to

consider of it…….. “

Newton further adds below thus:

“Gravity must be caused by some agent acting constantly according to certain laws: but, whether this

agent be material or immaterial – I have left to the consideration of my readers.”

Newton was very humble, generous and considerate in making the above statements giving

enoughmargin for future researchers in the field.

This research paper, following strictly its title, is divided intoseven parts. I Historical aspect of

Tides, II Introduction, III The Ocean Tides, IV The Mathematical Theory of Tides, V The Effects of



Tides, VI The Rotation of Earthand VII Conclusion. A majority of the material in this Paper is from the

Ph.D. Thesis of the author [7]. In order to maintain the originality of certain authors, some of the

material is reproduced in their name and style by giving them due credits.

REVIEW OF LITERATURE

III. THE OCEAN TIDES

3.1Tides – General: Tide is a planetary phenomenon creating rise and fall of sea levels due to the

combined effects of gravitational forces exerted by the Sun and Moonin conjunction with the rotation of

the Earth. The physical processes of formation of tides take place continuously day in and day out. But

the effect is seen in majority of places only twice a day (High Tide and Low Tide). Nobody can stop the

formation of tides which are harmless. As the saying goes, “Time and Tide wait for no man” and they

go on. Tides are very much required for life of aquatic animals and other organisms in the coastal areas

so as to get periodically exposed to the Sun and air during low tides and at times sink in water during

high tides. It is the work of almighty to synchronize their lives with the motion of tides.

As already mentioned in the history of tides, we go by Newton‟s proposition that the tide-generating

forces are due to the lunar and solar attractions exerted on the Earth. The fundamental tide generating

force on the Earth has two interactive but distinctive components. The tide generating forces are

differential forces between the gravitational attraction of the bodies (Earth-Sun and Earth-Moon) and the

centrifugal forces on the Earth produced by the Earth‟s orbit around the Sun and the Moon‟s orbit

around the Earth. Thus the Newton‟s law of gravitation and his second law of motion have to be

combined to develop formulations for the differential force at any point on the Earth, as the direction

and magnitude are dependent on the position on the Earth. Taking the Earth-Moon system, the

gravitational force, Fm between the Earth and the Moon is

Fm = G(M𝑚 ME

r2 ). . . . . . . . . . . . . . . . . . . .. . . (1)

where G is Newton‟s constant of gravitation the value of which is 6.67 x 10−11 Nm2

kg 2 , Mm , the mass of

Moon, ME the mass of Earth and r the distance between Earth and Moon. If ME/m the mass of either

Earth or Moon and v the speed of Earth or Moon, then the centrifugal forceF𝑐 can be written as

Fc= ME /m v2

r . . . .. . . . . . . . . . . . . . . . . . . (2)

As a result of these differential forces, the Tide-generating force which we may call as the Tractive force

T is inversely proportional to the cube of distance between the bodies. That is,

T ≈ G (2 Mm ME

r3 ) a . . . . . . . . . . . . . . . . . . . (3)

where a is the radius of Earth. Now, coming to the Sun which is 27 million times more massive than the

Moon, but 390 times farther away, we may write maintaining proper suffix for Sun and Moon as s and

m respectively. Tm

Ts =

Mm

Ms (

rs3

r𝑚3 ) = (

1

2.7x107) x (390)3 = 2.2 . . . . (4)

Thus Moon is more than twice as important to create tides. In other words, the tide-generating force of

the Sun is only 1

2.2 = 0.46 or 46% of the tide-generating force of the Moon even though the Sun is much

more massive, it is also much far away. Mac Donald [6] gives actually the values of the forces that exist

in the Sun-Earth-Moon system.



Moon feels from Earth . . . . . 1.82 x 1018 N

Earth feels from Moon . . . . . 6.69 x 1018 N

Earth feels from the Sun . . . . . 3.02 x 1018 N

That is, the Earth feels 3.7 times more tidal force than the Moon from their relationship because the

Earth is bigger and thus has a bigger differential. Hence, in the literaturethat follow we shall give more

importance to lunar tides rather than the solar tides.

The mechanism of formation of tides is based on the Equilibrium Theory of Tides [5]according to

which the entire surface of the spherical Earth is assumed to be covered with water of the oceans. The

rise and fall of water at any point on the Earth‟s surface is imagined to be produced by two disturbing

bodies the Moon and the other anti-Moon (as quoted by the author for brevity) revolve round the Earth‟s

axis once in the lunar 24 hours, with the line joining them always inclined to the Earth‟s equator at an

angle equal to the Moon‟s declination. If it is assumed that at each moment the condition of hydrostatic

equilibrium is fulfilled; that is the free water surface is perpendicular to the resultant force, is the

Equilibrium Theory of Tides in a nutshell.

The formation of tides isschematically shown in Fig.1.[7]The gravitational force exerted by the

Moonpulls the water of the oceans on Earth creating a bulge of water as shown in Fig.1 (a).

Fig. 1 (a) Effect of Gravitational Force Fig.1(b) Effect of Centrifugal Force

Now, the rotation ofthe Earth responsible for the centrifugal force, comes into picture and it is

considerable. The rotational velocity of Earth at the equator is 465 m

s and the centrifugal force at the

center of mass of the Earth-Moon system creating a bulge of water on the opposite side as shown in

Fig.1(b). This is but natural and shows that even Nature lovessymmetry. The bulging on one side of the

globe creates a similar bulge on the opposite side. The resultant of the gravitational and centrifugal

forces is as shown in Fig.1(c).The bulging of water is symmetrical with high tides at A and B and low

tides at C and D.

Fig. 1 (c) Resultant of Gravitational and Centrifugal Forces



The explanation given by George Gamow [3] appears to be more satisfactory and logical as he

adds more Physics to his arguments. The explanation given by him is illustrated in Fig.2.

Gamow considered solar tides and applied principles of celestial mechanics. When the Earth

revolves around the Sun, its linear velocity v1 of the side turned towards the Sun (point F in the Fig.2 is

less than the linear velocity v2 of the center C of the Earth which in turn is lower than the linear velocity

v3 of the rear side R. But, according to celestial mechanics, the linear velocities of circular orbital

motion under the action of solar gravity must decrease with the distance from the Sun. Thus the point F

has less linear velocity V3 than is needed to maintain circular motion, and so will have the tendency to

be deflected towards the Sun as indicated by a dotted line at F in the Fig. Similarly the point R has

higher linear velocity V1 than needed for its circular orbit, and will have a tendency to move farther

away from the Sun (dotted line at R). This might create an imbalance and rupture the Earth into pieces.

This does not happen as the gravitational forces G will try to hold the Earth together and as a

compromise, the globe becomes elongated in the direction of the orbital radius with two bulges on each

side.

Fig.2 Origin of the Tidal Force

The explanation for the lunar tides is exactly the same as in the case of solar tides given above.

As the Earth and Moon move around the same center of gravity and the Moon being about 80 times

lighter than the Earth, the common center of gravity lies at about 1

80

th of the distance from the center of

Earth and this distance equals 60 Earth radii, the conclusion is that the center of gravity of the Earth-

Moon system is located at 60

80 =

3

4 of the Earth‟s radius from the center. Ignoring any quantitative



difference in geometry, the physical argument remains the same. The waters of the Earth‟s oceans must

form two bulges, one directed to the common center of gravity (which is also the direction towards the

Moon) and the other in the opposite direction. The second bulge is just an inertia counter balance taking

place at the center of the Earth.

I give due credit to Gamow for the above extraordinarily great and satisfactory explanation and

hence quoted the same in his own words.

What we gather from the above is that due to tidal forces the Earth takes the shape of an ellipsoid

in its orbit.

3.2Types of Tides:The tidal forces affect the entire Earth. In the case of ocean waters, the tide is the

ocean tide where the surface moves in meters. It affects the solid Earth too, i.e. the Earth tide which is in

few centimeters. The atmosphere is much more fluid and compressible and the tide in that case is the

atmospheric tide, where the surface moves in kilometers. The timing of tidal events is related to the

Earth‟s rotation and revolution of the Moon around the Earth.

Fig. 3 High tide, Alma, New Brunswick In the

Bay of Fundy

Low tide at the same fishing port in the Bay of

Fundy

Effect of formation of high tide and low tide at Alma, New Brunswick in the Bay of Fundy is

shown in Fig.3. Assuming the Moon to be stationary, the cycle of tides will be 24 hours long i.e. for the

entire day, but that is not the case. The Moon revolves round the Earth and one revolution takes 27 days

and adds about 50 minutes to the tidal cycle and hence the tidal period is 24 hours and 50 minutes in

length.

Now, let us take into consideration the effect of solar gravity. The height of the average solar tide

is about half the average lunar tide. When the Sun and Moon are aligned with the Earth(Moon SYZYGY

– Derived from „Sysigia’ means Union)„ as shown in Fig.4. The two tide producing bodies (Sun and

Moon) act together to create the highest and lowest tides of the year. These are Spring Tides that occur

every fortnight during full Moon and New Moon. Pytheas travelled the British Isles about 325 BC and

seems to be the first to have related spring tides to the phase of the Moon. The name Spring Tide has no

connection with the Spring Season. Theword,‟Spring‟ comes from the word, Springen meaning to rise

up. It is like a mechanical spring which jumps and bumps and rises.

Now, when the gravitational pull of the Sun and Moon are at right angles to each other (Moon

QUADRATURE – Quadra means Four) as shown in Fig.5, the resultant variation on the Earth is the

least. The tides in this situation are called Neap Tides. (Word, Neap is shortened to „Nep’ meaning

scarcely or barely touching). Neap tides occur during the first and last quarter of the Moon, i.e. there is

about a seven-day interval between Spring tides and Neap tides.



Fig. 4 Formation of Spring tides Fig. 5 Formation of Neap tide

The formation of tides is highly rhythmic and strictly follows the position of the Moon in

conjunction with the Sun. In South-East Asia and parts of northern Gulf of Mexico, a high tide is

followed by a low tide every day. These are called Diurnal Tides and their tidal cycle is illustrated in

Fig.6.

Fig. 6 Tide cycles of Diurnal Tides

Fig. 7 Tide cycles of Semi Diurnal Tides

Fig. 8 Tidal Cycle for Mixed Tides



There are some Semi-diurnal Tides having two high and two low waters per tidal day. Their tidal cycles

are shown in Fig. 8. These tides are common on the Atlantic coasts of the United States and Europe.The

tides in the western coastal regions of Canada and United States are of a different type. The high and

low tides in these regions differ considerably. In these regions, there is higher high water and lower high

water as well as higher low water and lower low water. Such tides are called Mixed Tides. In Fig. 8 is

illustrated the tide cycles of the mixed type.

The lunar day is 24 hours 50 minutes. This gives 2 high (Flow) and 2 low (Ebb) tides each 12.42 hours

apart. Thus the tidal period is 12.42 hours which we call as the M2 Tide. Taking the Sun into

consideration, the period is 12 hours and the same is called S2 Tide.

3.3Other Tides:- In addition to the ocean tides, there are Lake Tides, Atmospheric Tides and Earth

Tides. Some of the lakes like Lake Superior, Lake Michigan Also experience tides, but hardly few cm.

Lake Superior experiences a tide of maximum 4 cm. Atmospheric tides are both gravitational and

thermal origin. The atmosphere being a mixture gases is pulled to a height ranging from 80 to 120 km.

Earth tides cause the crust being pulled due to gravity of the Sun and Moon to a height of 55 cm at the

equator. We standing on the Earth, cannot notice this.In addition, the bottom topography of the ocean is

uneven and the oscillating tidal currents in the stratified ocean flow over this uneven stretch thereby

generating internal waves with frequencies matching with the tides. These tides are called internal tides.

Keeping in view of the title of the Paper, we shall stick on to ocean tides in a generalized form.

A.M. Gorlov [4] has quoted some of the highest tides of the global ocean and the same is shown in the

following Table:

Table

Country Site Height of Tide (m)

Canada Bay of Fundy 16.2

France Port of Ganville 14.7

England Seven Estuary 14.5

France La Rance 13.5

Russia Penzhenskaya Guba (Sea of

Okhotsk)

13.4

Argentina Puerto Rio Gallagos 13.3

Russia Bay of Mezen (White Sea) 10



3.4 Declination:[2] The Earth‟s axis is tilted 23.50 relative to its rotation about the Sun and this is

called Declination due to which there is a diurnal inequality creating unequal tides at different latitudes

and the same is illustrated in Fig.9. Moon orbits Earth every 27.3 days and the half cycle starts every

3.6 days.

Fig.9 Illustration of Diurnal inequality(See description below) [Observer at A sees 2 high tides of equal size (Semi diurnal)

Observer at B sees 2 high tides of different size (Mixed tides)

Observer at C sees only 1 high tide (Diurnal)]

IV. MATHEMATICAL THEORY OF TIDES

4.1 Velocity of Tidal Wave: Tidal theory is one of the biggest messes in contemporary Physics. To start

with, we have to consider a progressive tidal wave Fig.10 in shallow water as the waves are formed

when the water meets with the land as shown in the figure. The characteristic of the wave is its velocity

which we shall find now. We follow Louis A Pipes [10]. Tidal waves are such that their vertical

displacement is small compared to their wavelengths. Consider the crest

Fig. 10 Progressive Tidal Waves in Shallow Water

of the tidal wave at some point P (x,y) in Fig.(11) and let the wave move in the x-direction in the sea

water of depth H. If h is the height of the wave the free surface of water in the x-y plane is given by



Fig. 11 Graph to obtain formula for velocity of a Tidal Wave

ys = H + h . . . . . . . . . . . . . . . . . . . . . . . (5) Let us assume that the water moves smoothly with velocity, v in the x-y plane of unit width and depth H.

Further, let

p0 → Pressure at the surface of water

ρ → Density of water and

g → Acceleration due to gravity

As h≪H, we may neglect the vertical acceleration of water above the surface and write the following expression for pressure, p at any point such as P (x,y) as

p = p0 + gρ (ys – y) = p0 + gρ (H+h – y) . . . . .(6)

following equation (5). Now, consider a vertical strip bounded by the planes x and x + dx just at

the start of the wave motion. The volume of the strip is H dx andy𝑠 = H + h is the elevation of the surface of the strip. When the motion has started with velocity v, the planes have moved to x + v and x

+ dx + v +(𝜕𝑣

∂x)dx. The volume of the strip becomes (H + h) [ dx + (

𝜕v

𝜕𝑥)dx]. As sea water is in-

compressible, H dx = (H + h) [dx + 𝜕𝑣

𝜕𝑥 dx]. That is,

h+ H(∂v

𝜕𝑥) = 0 . . . . . . . . . … …. . . . . .. . . . . .(7)

as h(𝜕𝑣

𝜕𝑥) being ignored being a quantity of higher order

We shall now work out the resultant force on the strip. If dy is an element of height parallel to the

bottom, then the force on this element is

p dy – [p +( ∂p

∂x) dx] dy = − (

𝜕𝑝

𝜕𝑥) dx dy . . . . . . . . . .(8)

Following equation (6), this becomes 𝜕𝑝

𝜕𝑥 = g ρ

𝜕h

∂x s . . . . . . . . . . . . . . . . . . . (9)

since h is the only function of x in equation (6)Thus the force on the element becomes

− (𝜕p

∂x) dx dy =− gρ(

𝜕h

∂x) dx dy . . . . . . . . . . . (10)



Now, to get the force F, let us integrate on all the elements such as dy

∴ F = − g ρ∂h

∂x dx dy

H

0 = − g ρ H

∂h

∂x dx . . . . . .(11)

But, this is the rate of change of increase in momentum and hence by Newton‟s second law, we can

write

F = Hρ dx ∂2v

∂t2 = −gρ H∂h

∂x dx . . . . . . . . . (12)

Differentiating (7) we have ∂h

∂x = − H

∂2v

∂x2 . . . . . . . . . . . . . . (13)

From equations (12) and (13) we get

H ρdx∂2v

∂t2 = − g ρ H (−H∂2v

∂x2) . . . . . . (14)

∴ ∂2v

∂t2 = g H ∂2v

∂x2

or ∂2v

∂x2 = 1

gH

∂2v

∂t2 . . . . . . . . . . . . . . . . .(15)

This is a one-dimensional wave equation, ∂2v

∂x2 = 1

c2

∂2v

∂t2 where c is the velocity so that

c = gH . . . . . . . . . . . . . . . . .. (16)

Thus the speed of a tidal wave is proportional to the square root of the depth of the sea. Taking the

square root of the acceleration due to gravity as 3.13, for a sea of depth 100 m, the speed of tide will be

about 31 m

𝑠.

4.2Closed and Open Harbors: The formula (16) for speed is useful to find the period T of a tidal wave

in closed and open harbors (Basins) as shown in Fig.12 [2]In a closed harbor, the antinodes are at the

walls so that the

Fig. 12 Tidal Waves in Closed and Open Harbor

distance between the walls is L = λ

2, where λ is the wavelength of tide. But, however in an open harbor

the distance L is one-quarter wavelength. i.e. 𝜆

4. We may work out the time period for an open harbor. If

T is the time period of wavelength 𝜆, the tidal speed is

√gH = 𝜆

T=

4L

T

or T = 4L

√gH . . . . . . . . . . . . . . . . . (17)

We can calculate and find the typical value for an open harbor 250 km long and 50 m deep.

T = 4L

gH =

4 x 250000

√(9.8x50) =

1000000

√490 =

1000000

22.14 = 41567 second = 12.55 hour = 12 hours 33 minutes

which is very close to the period of a standard M2 tide.

4.3 Laplace Tidal Equations:Before we deal with the Laplace Tidal Equations, let us review the Equilibrium Theory of Tides that have been dealt with at the start of the paper wherein the theory is built



on a hypothetical water-covered planet.In the year 1776, Pierre Simon Laplace conceived the idea that

tides are forced oscillations of the ocean having the same frequency as the forcing. He showed that the

tide to be mathematically separable into three different kinds of tides: long-periodical, diurnal and semi-

diurnal. This separation has since then been a corner stone in tidal theory. He was also first to treat

ocean tides as a problem of water in motion instead of water in equilibrium. His hydro dynamical

equations describing the propagation of tidal waves and related findings were first published in French

in the year 1799 and the same translated in English in the year 1832. His 200-year old paper is very

relevant and valid even today.

As he considered water in motion, it is the dynamical theory which is the most realistic model

which takes into account the effect of continents, shallow water and partially enclosed ocean basins on

tide formation. It is the most accepted theory which describes the ocean‟s real reaction to tidal forces.

The theory has taken into account friction, resonance and natural periods of ocean basins and the same

are confirmed from measurements done by satellite observations.

Laplace formulated a single set of Partial Differential Equations [17] for tidal flow described as a

barotropic∗ two-dimensional sheet flow. Both coriolis and gravitational effects are accounted for. Laplace obtained these equations by simplifying equations of fluid dynamics keeping in mind that the

horizontal forces are:

Acceleration + Coriolis Force = Pressure Gradient Force + Tractive Force.

The set of equations obtained by him is given below: ∂ζ

∂t +

1

a Cos (φ) [

𝜕

𝜕𝜆 (u D) +

𝜕

𝜕𝜑 (v D Cos (𝜑)] = 0

∂u

∂t– v [2 ΩSin (𝜑)] +

1

𝑎 Cos (φ)

𝜕

𝜕𝜆 (g𝜁 + U) = 0. . . . . . . . . . . . . . . . . . .(18)

∂v

∂t + u [2Ω Sin (𝜑)]

1

a

1

∂φ (g𝜁 + U) = 0

The symbols in the above set of equations (18) have the following significance.

D → Average thickness of the fluid sheet

𝜁 → Vertical tidal elevation

u → Horizontal velocity component in the Latitude 𝜑

v → Horizontal velocity component in the Longitude λ

Ω → Angular frequency of the planet‟s rotation

g → Acceleration due to gravity of the planet at the mean ocean surface

a → Radius of the planet and

U → External tidal forcing potential. His hydro dynamical equations describing the propagation of tidal waves through ocean, could not be

solved in practice till the invention of the Computer. Only co-tidal charts which were un-reliable were

used.

V. EFFECTS OF TIDES

5.1Tidal Bore:- [9] The „Bore‟ (Fig.13) standing for Crest or Waste is a wall of water that moves up

certainlow lying rivers due to an in-coming tide. It is a large wave caused by the constriction of the

spring tide as it enters a long narrow shallow inlet. As it is a wave created by tides, it is a true tidal wave.

When an in-coming tide rushes up a river, it develops a steep forward slope because the flow of the river

resists the advance of the tide. This creates a tidal bore which may reach heights of 5 meters (16.4 feet)

or more and move at speeds 24 km (15 miles) per hour. Among many rivers in the world, the Qiantang

river in China has the highest tidal bore often reaching a height of 8 meter (26 feet)



Fig.13 Illustration of Tidal Bore

5.2The Roche Limit:-[6] In the year 1848 the astronomer, Eduard Albert Roche (1820-1883) found

that, if a satellite was held together mainly by its own gravitational attraction, there would be a

minimum distance from the primary inside which the tidal forces of the primary would exceed the

satellite‟s binding forces and would tear it apart. The Roche limit L for two bodies is approximated by a

function of their densities and is given by

L = 2.456 R (rplanet

rsatellite)

1

3

where R is the radius of the planet and the r values are the densities. For typical satellites the Roche

limit is approximately 2.5 times the radius of the primary planet. The Roche limit for Earth is about

18,470 km. It is of interest to note that the Roche limit for planet Jupiter which is 175,000 km and the

comet Shoemaker-Levy discovered in 1993 passed within 21,000 km of Jupiter only to break up the

comet into more than 20 pieces which spiraled and fell into Jupiter.

5.3Tidal Heating:-[6]

Tidal forces can also internally heat a satellite. Satellites rotate at constant speeds, but in an

eccentric orbit according to Kepler‟s second law, the orbital speed varies with the distance from the

primary. Thus the tidal bulge of a tidally locked satellite cannot always point precisely at its primary. At

some points in the orbit the bulge will be pointed ahead or behind where it should be, and the primary

will be exerting a small pull on the bulge. The friction of the bulge being pulled back and forth through

the solid body of the satellite heats the interior of its body. The innermost satellite Io of planet Jupiter

whose orbit is almost circular frequently encounters Europa and Ganymede and experience heating.

5.4Effect of atmospheric pressure on Tides:[15]

When the atmospheric pressure is low, the increase in elevation can be found by the formula

R0 = 0.01 (1010 – P) . . . . . . . ..(19)

where R0 is the increase in elevation in meter and P the atmospheric pressure in hectopascals∗ *1 hectopascal = 100 pascal

This is equal approximately to 1 cm per hecto pascal depression or 13.6 inche per inch depression. For a

moving low, the increase in elevation is given by the formula,



R = R0

1− C 2

gH

where R is the increase in elevation in feet, R0 is the increase in meters for a stationary low. C is

the rate of motion of the low in feet per second and g the acceleration due to gravity (32.2 feet per sec2

and H the depth of water in feet.

5.5 Tides affect the Salinity and freezing of Sea Water:C.F.Postlethwaite, et al [11] in their research

paper titled, “The effect of tides on dense water formation in Arctic shelf areas”, have used a 25 km × 25

km 3-D ocean circulation model coupled to a dynamic and thermodynamic sea ice model. The tidal

amplitudes from the model are compared with the tide gauge data. The sea ice extent is compared with

the tide gauge data and the sea ice extent is compared with the satellite data. In tidally active coastal

regions, it is observed that there is a delay in the freezing. The impact of tides on the salt-content is,

however, found to vary from region to region. In the Pechora Sea and around Svalbard, where the tides

are strong, the vertically integrated salt budget is dominated by lateral advection. It is found that tides

increase the salt flux by 50% in the Pechora and White Seas. The authors have stressed in their paper

that ocean tides and sea ice should be considered while modeling the Arctic.

Tidal mixing within a water column and at the base of the sea ice cover can increase the flow of

heat from the deep interior to the surface thereby decreasing the ice formation and ice melting. The tidal

currents cause a sort of stress and strain on the ice bed. There is a loos of heat in winter and creating an

additional fall in temperature with the ice floating on top and the high density salt water (brine)

remaining at the bottom.

The interaction between the tides and the sea ice bed is schematically illustrated in Fig. (14)

Process-1 is the tides bringing more heat from the bottom to the surface creating melting of ice.

Process-2 represents the heat loss and increased brine rejection. The alternation of convergence and

divergence periods during the tidal cycle creates a mechanical redistribution of ice itself and this is

shown in Process-3.

Fig. 14 Interaction of sea ice and tides

(Credits. 8.11) The authors have dealt in details the following. But, however, to maintain brevity of this Paper,

only the following headlines shall be mentionedThe authors have dealt in details the following:

Model description

Surface and boundary forcing

Description of the modeled tides

Impact of tides on sea ice distribution

Impact of tides on salinity distribution

Salt Budget

Lateral advection of salt

Surface salt fluxes from ice free areas

Surface salt fluxes from ice covered areas.



The authors have concluded that tides can create strong vertical current shear that increases

vertical mixing thereby facilitating to homogenize the water column in order to bring warm water to the

surface. It is of great interest to study increased brine rejection due to tide ice interaction.

5.6Tides affect Navigation: . Sailors should have up to date information regarding high and low tides at various places in the sea on

the globe. Many rivers and harbors have a shallow bar at the entrance which prevents boats with

significant draft from entering at low tide. They carry charts having this information. Tidal flow timings

and velocities appear in tidal charts also called tidal atlas. The procedure adopted is as follows:

Calculate a “dead reckoning” position (DR) from travel distance and direction.

Mark the chart with a vertical cross like a plus sign and

Draw a line from the DR in the direction of the tide. The distance the tide moves the boat along this line is computed by the tidal speed and this gives

an “estimated position” (EP) traditionally marked with a dot in a triangle.

(iii) Useful for fishing. Some of the deep sea fish appear on the surface during high tides and

hence beneficial to fishermen.

(iv) Some aquatic animals and other insects are more aware regarding tides than humans. They

follow the tide cycle for gestation and egg laying and hatching. Some of them lay eggs having a

hatching time of a fortnight or so immediately after end of a high tide. The eggs get hatched before the

arrival of the next high tide in that region. Some aquatic animals and some of the vertebrates requiring

sunlight and atmospheric air automatically get them during low tides. The menstrual cycle in women

almost follow a lunar month which is an even multiple of the tidal period indicating a common descent

of man from a marine ancestor. This is known as intertidal ecology.

VI. ROTATION OF EARTH

6.1 General:Other than the above effects, an important effect from the point of view of Geology and

Astronomy is the slowing down of Earth‟s rotation by Tidal Friction and we shall now deal with it.

Slowing down of rotation of Earth is one of the many effects of Tides often go un-noticed. It is

caused by the tidal acceleration which is an effect of the tidal forces between an orbiting natural satellite

such as the Moon and the primary planet such as the Earth. The acceleration causes a gradual recession

of a satellite in a prograde orbit away from the primary, and a corresponding slowing down of the

primary‟s rotation. The lunar tidal acceleration along the Moon-Earth axis is about 1.1x10−7g whereas

the Solar tidal acceleration along the Sun-Earth axis is only 0.52x10−7 g where g is the acceleration due to gravity on the Earth.

Due to the continuous formation of tides throughout the globe at all times, the tidal oscillations

introduce dissipation of energy at the rate of 3.75 Tera Watt (1012

W). A majority of this rate of

dissipation is due to marine tidal movement, which in turn causes friction known as tidal friction on the

surface of the Earth. The word, „friction‟ is general and may refer to many physical mechanisms. It can

be bottom friction induced by tidal currents flowing across the ocean floor, various kinds of wave

breaking, and scattering of tidal waves into oceanic internal waves are all thought to play a role. This

tidal drag creates a torque on the Moon which in turn transfers angular momentum to its orbit and a

gradual increase in Earth-Moon separation. An equal and opposite torque on the Earth creates a decrease

in its rotational velocity which otherwise means that the Earth is slowing down in its rotation. On the

slowing down of the rotation of Earth, the exaggeration given by George Gamow [3] is worth noting. He

compares the slowing down of Earth‟s rotation to the slowing down of an automobile‟s wheels when

the brakes are applied. It has been worked out and found that the Moon recedes from the Earth by about

3.8 cm (1.5 inch) per year, This recession of Moon creates an increase in the terrestrial day by about 2



hours in the last 500 million years. As a point of interest, it can be said that if the recession rate to be

constant, 70 million years ago, the length of the day was shorter adding to 4 more days per year!

The gravitational torque between the Moon and the tidal bulge of Earth causes the Moon to be

constantly promoted to a slightly higher orbit and Earth to be decelerated in its rotation. As in any

physical process within an isolated system, total energy and angular momentum are conserved.

Effectively, energy and angular momentum are transferred from the rotation of Earth to the orbital

motion of the Moon (however, most of the energy lost by Earth (-3.321 TW) is converted to heat by

frictional losses in the oceans and their interaction with the solid Earth, and only about 1/30th (+0.121

TW) is transferred to the Moon). The Moon moves farther away from Earth (+38.247±0.004 mm/y), so

its potential energy (in Earth's gravity well) increases. It stays in orbit, and from Kepler's 3rd law it

follows that its angular velocity actually decreases, so the tidal action on the Moon actually causes an

angular deceleration, i.e. a negative acceleration (-25.858±0.003 "/century) of its rotation around Earth.

The actual speed of the Moon also decreases. Although its kinetic energy decreases, its potential energy

increases by a larger amount.

The rotational angular momentum of Earth decreases and consequently the length of the day

increases. During the last 620 million years the period of rotation of the Earth (length of day) has

increased from 21.9 hours to 24 hours as on today losing about 17% of its rotational energy.

George Gamow [3] has explained the conservation of angular momentum by using simple algebraic

terms of Classical Mechanics. The angular momentum I of the Moon‟s orbital motion is:

I = Mm v r . . . . . . . . . . . . . . . . . (20)

Where Mm is the mass of the Moon, v its velocity and r the radius of the orbit. On the other hand,

Newton‟s law of gravity combined with the formula for centrifugal force, gives us:

G(ME Mm

r2 ) = Mm v2

r . . . . . . . . . . . . . . . . . . (21)

where ME is the mass of the Earth. Thus from the above,

v2 = G ( ME

r) . . . . . . . . . . . . . . . . . . .(22)

From this and the expression for I given by ( ), we get

r = I2

GME M𝑚2 . . . . . . . . . . . . . . . . . . . . (23)

and v = G(ME M𝑚

I) . . . . . . . . . . . . . . . . . . (24)

From equations (23) and (24) it follows that the increase of the angular momentum of the Moon

in its motion around the Earth must result in the increase of its distance from the Earth and decrease of

its linear velocity.

P. Brosche [1] has a different view. According to him, “Modern computers enable us to obtain

realistic values for the present tidal torque between the Moon and the oceans; those values agree with

the observations. In principle, computations for distant geological epochs are possible as well and have

been performed. However, the very complex eigen period spectrum of the oceans today precludes a

continuous reconstruction of the Tidal torque for an essential part of the Earth‟s history. Hence, the

original state of the Earth-Moon system is still uncertain. We emphasize the importance of results for

intermediate time scales”.

6.2 Formation of the Moon: At this stage it is worth considering the formation of the Moon as shown

in Fig. ( a ) [8] which is an artist‟s conception of a planetary smash-up whose debris was spotted by

NASA‟S Spitzer Space Telescope in 2009 gives an impression of the carnage that would have been

caused when a similarimpact created the present Moon. Some 4 to 5 billion years ago as put by George

Darwin, the Earth and Moon must have been very close together as a single body (Earthoon or Moorth).

The break-up into two

https://en.wikipedia.org/wiki/Energy

https://en.wikipedia.org/wiki/Angular_momentum

https://en.wikipedia.org/wiki/Potential_energy

https://en.wikipedia.org/wiki/Gravity_well

https://en.wikipedia.org/wiki/Laws_of_Kepler

https://en.wikipedia.org/wiki/Angular_velocity

https://en.wikipedia.org/wiki/Kinetic_energy



Parts must have been caused by the tidal force of solar gravity leaving the Earth as Anti-Moon

Fig. (b)a word used by Lord Kelvin in his famous classical book [5]. When the Moon once formed, the

recession

(a) (b)

Fig.15 (a) The present Earth initially called Earthoon or Moorth being smashed up by

a planet-like body. (b) The Moon just formed acted upon by tides of solar

gravity leaving the present Earth as AntiMoon.

of the Moon was so fast due to the large rotation of the Earth and the Moon appears in the sky as on today Fig.(16)Richard Ray [12] has shown in Fig.17 rapid variations in the rotation rate in terms of

Universal Time (UT) as observed by hourly measurements and as predicted by a numerical model of

ocean tides. The measurements are carried out in January 1994 with Very Long Baseline Interferometry

(VLBI) and provided by IERS special Bureau for Tides.

6.3Observation and Prediction of Tides: In the year 1867, Lord Kelvin built a tide predicting machine

using a system of pulleys with which he could add together six harmonic time functions. It was the first

systematic harmonic analysis of tidal records. The machine can be programmed by resetting gears and

chains to adjust phasing and amplitudes. Such machines were in use till 1960.

EARTH

H

External Planetary Body

Earth (Anti Moon)

Moon

http://star-trek.fl-ink.com/st-sstory.htm



Fig. 16 The fully formed Moon seen in the sky as on today

Fig. 17 Rapid variations in Universal Time as measured by VLBI and predicted by Ocean Model

In Fig. [18] is shown a tidal indicator in the Delaware River, US fitted sometime in 1897. It is

marked in feet. As shown in the diagram, the height of the tide is slightly more than1-foot above mean

sea level and is still falling as shown by the arrow. The indicator is powered by a system of pulleys,

cables and float.

On the Navy Dock in the Thames on the Thames Estuary an entire Spring-Neap cycle was made

in the year 1831. By 1850 many large ports had automatic tide gauge stations. William Whewell

prepared a global chart by mapping co

Tidal lines and hypothesized the existence ofno-tide zones which were confirmed in 1840 by

CaptainHewett from the surroundings in the North Sea.

MOON

NO

MOON

E A R T H

https://www.colourbox.com/image/the-earth-the-moon-and-stars-in-a-free-space-image-2637984



An interesting and personal observation on Tides by Author of the Paper:

It is safer to swim in the sea during a high hide as the tidal waves of high amplitude and short

wavelength have a tendency of approaching the shore and rarely the swimmer is dragged into the sea as

shown Fig.19.(a). The situation, however, is different and dangerous during a low-tide when the waves

of low amplitude and large wavelength are pulled towards the sea with such gravitational pull that

sometimes the swimmer loses control and might get drowned. This situation is shown if Fig.19 (b). The

curvature of the ocean bed acts like an inclined plane with the fast moving shallow water of the low tide

virtually pulls the swimmer right inside the sea.

Fig. 18 Tidal Indicator in the Delaware River

https://en.wikipedia.org/wiki/File:Tidal_Indicator_Delaware_River_ca1897.jpg



Fig. 19 (a) Swimmer during high tide Fig. 19 (b) Swimmer during low tide (Picture illustration exclusively by author of the Paper

VII. CONCLUSION

Tides! What a rhythmic physical process formed in the Earth as per perfect time-table in a

geological scale related with the Sun and Moon and seen by the humans day in and day out. Studying

tides is a good lesson for the scientist with the Earth never getting tired of the process. But, however the

tiresome nature is revealed in the tidal friction creating a loss of power and slowing down its rotation. I

take the Earth as a running athlete slowly getting tired of its activity. Whether it will take a final rest will

depend on the existence of gravity which will continue for generations to maintain life on this planet.

BIBLIOGRAPHY

[1] Brosche. P, Tidal Friction in the Earth-Moon System – Philosophical Transactions, Royal Society, London, A.313,

p.71 (1984).

[2] Dr. Cherryl Ann Blain, Tidal Hydrodynamics and Modeling – Stennin‟s Space Center, MS, USA, Pan-American

Studies Institute, Pasi Univeridad Tecnica Federico, Santa Maria 2-13,Jan. 2013, Valparaiso, Chile.

[3] George Gamow. Gravity, Ch. 9. The Science Study Series. Pub. By Vakil Feffer and Simons Pvt. Ltd., Bombay, 3rd

India Reprint 1968, p. 79-91.

[4] Gorlov M. Tidal Energy-North-Eastern Univ. Boston, Massachusetts,US, 2001..p. 2955- 2960,

[5] Kelvin Lord – Principles of Mechanics and Dynamics by Sir William Thomson (Lord Kelvin) and Peter Guthrie

Tait, Dover Edition 1962, Sec.805, p.375

[6] Mac Donald – Tidal Forces and Their Effects in the Solar System, Sept.10, 2005

[7] Nair V,C.A. –A Perspective View of Physical Processes and their Effects taking place in the Earth Right from its

Formation-Shodhganga@INFLIBNET, Ch.8

[8] NASA/JPL - Caltech

[9] Pearson, Tides, Ch.9, p.276-299, 30 Dec.2006.

[10] Pipes Louis A.- Applied Mathematics for Engineers and Physicists, MacGraw Hill Book Co.Inc, 1958, Ch.16,

p.454-456.

[11] Postlethwaite.C.F.. M.A. Morales Maqueda, V.L.Fouest, G.R. Tattersall, .H. Holt and A.J. Willmott The Effect of

Tides on Dense Water Formation in Arctic Shelf Seas, Pu. By Ocean Science, 24 Mar. 2011, p.203-217

[12] Ray Richard, Website Author and Curator, GGFC, Ocean Tides and Earth‟s Rotation – An Introductory Discussion

– Courtesy of IERS, Special Bureau for Tides., May 15, 2001.

[13] Steacy Dopp Hicks- Understanding Tides- Physical Oceanography, US Dept. of Commerce, National Oceanic and

Atmospheric Administration, Dec.2006, p.26

[14] Thompson/Ocean 420/Winter, Tide Dynamics-Dynamic Theory of Tides.

[15] Wikipedia – Tide

[16] Wikipedia – Tidal Power

[17] Wikipedia – Tidal Theory

Author’s Profile *



*Dr.(Prof.) V.C.A. Nair (b.15th Aug. 1939) is an Educational Physicist,

Counselor, Research Guide and Consultant. He did his Masters in Physics from

Mumbai University, India and Ph.D. from JJT University, Rajasthan also in

India. He has to his credit over 4 decades of teaching Applied Physics in eminent

Polytechnics in Mumbai and having taught nearly 16,000 students since 1965.

He has published a number of research papers in Physics and Geophysics. He is a

Life Member of Indian Society for Technical Education which is an all India

body. He had been to USA a number of times and visited eminent Universities

such as Stanford, Harvard, MIT, University of California both at Berkeley and

Los Angeles and University of San Francisco. At present Dr. Nair is a Research

Guide at JJT University, Rajasthan-333001, India . His Ph.D. Thesis is in

Geophysics and he is working on topics such as Tides, Clouds, Global Warming

and Climate Change.

*[email protected]

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