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OCR for page 81
C:ETAPTER VI
TIDAI) FRICTION
NV. D. LAMBERT
U. S. Coast and Geodetic Survey
Friction represents conversion of mass motion into molecular motion,
that is, into heat. Obviously, wherever there is a tidal current there is
friction between adjacent particles of water, also between the moving
water and the bottom. The earth tides involve a strain or relative move-
ment of particles in the solid earth, and in this relative movement. there
is friction also.
It is I'amiliar to students of hydraulics that the coefficient of' viscosity
of water determined in the laboratory (0.018 c.~.s. units at 0° C.) ap-
plies only to velocities so small that regular flow is maintained. When
the velocity exceeds some critical velocity, turbulence sets in and the mass
energy ultimately convertecl into heat is many times greater than would
be inferred from the laboratory coefficient of viscosity a.ncl from the
effective velocity of the current. The usual procedure for treating motion
with velocities such as are ordinarily met with in the case of tides is to
replace the laboratory coe-~i.cient of viscosity by a virtual coefficient, or
coefficient of turbulence. This its, however, only a makeshift, as the
coefficient of' turbulence depends on the velocities attained
To what extent friction a.fI:'ects the amplitudes and phases of the tide
is not easy to say. At one time it was common to attribute many familiar
tidal phenomena mainly to I'riction; for instance: the interval between
the transit of the moon and the occurrence of high water; the interval
between syzygy and the occurrence of spring tides or between quadrature
and Heap tides, and the interval between the time of the moon's maximun~
declination and the occurrence of tropic tides; the fact that in a river
the duration of fall exceeds that of rise, a difference that in general in-
creases with the distance from the mouth of the river and which when
extremely exaggerated becomes the tidal bore (eager, a.gger or eyt,re).
The present tendency is to attribute these phenomena mainly to causes
other than friction. The part played by friction is most naturally treated
in connection with the phenomena in question.
The main subject of' this chapter is a particular effect of' friction,
the interest of which is primarily astronomical and cosmogonic. The
81
\
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82
FI CURE OF THE EAR TH
energy dissipated in friction obviously comes from somewhere; it comes
in fact from the rotational energy, both axial and orbital, of the earth-
moon system.
Before considering the consequences of this loss of rotational energy,
let us consider in a general way how it comes about and how it may be
estimated. A very natural but entirely incorrect mental picture of the
phenomenon is that of an ocean enveloping the rotating earth, the ocean
tendin, to remain stationary in space and so acting directly as a brake
'1 ~1 ~· ~1 1
1 V V
upon tne earths axial rotation. This mental picture is commonly associ-
ated with the idea of tides as equilibrium phenomena and the inadequacy,
already pointed out, of this simple conception of the tides should put us
on our guard against any such over-simplified idea.
Frictional dissipation of energy must be reckoned as such regardless
of the direction of the tidal current. At some given point, say point ~L,
the current might be along a meridian and thus its friction would con-
tribute no component to a torque about the earth's axis; or the current
Night be from west to east, that is, in the direction of the earth's rotation,
so that its friction along the bottom, or between the water particles them-
selves, would seemingly contribute a torque that would accelerate the
earth's rotation rather than retard it. Nevertheless the energy dissipated
in both these cases must be counted in along with the rest. Again, a
current near the pole seems to exert but little torque, because the lever
arm, or distance from the point of application of the force to the axis of
rotation is so short; nevertheless the friction even in such a case must
be counted at its full value. The key to this seemingly paradoxical state
of affairs lies in the fact that no small region call it A, for convenience-
may be considered by itself; the whole ocean must be included and the
currents at regions like A that have no component of torque about the
earth's axis, or that have an accelerating rather than a retarding com-
ponent, are not isolated but affect the tides and currents elsewhere, at
other regions, as B. C, D, etc. If we arbitrarily separate the region A
from the whole complex phenomena of tidal movement and consider
what happens at A as to some extent the cause of what happens at
B. C, D, etc., we may say that if the phenomena at A did not exist the
phenomena at B. C, D, would be different. Suppose the friction appears
to accelerate the rotation at A. The current, though dissipatin, energy
in the process, may well cause the retarding action at B. C or D to be
reinforced. In short, the whole phenomenon is so complex that we can
not just say where and how the real braking action of a current occurs.
We merely put our faith in the principle of the conservation of energy
and in other general principles of mechanics, and try to find out, if we
can, just how much energy is dissipated in friction, anal we accept it
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TIDAL FRICTION
1 ~1 c~
83
as a fact that this energy cones out of the energy stored up in the axial
and orbital rotation of the earth-moon system.
Various attempts to estimate the amount of friction have been made
from time to time but all have involved so much assumption that little
confidence could be placed in the results obtained. Taylor,9 availing,
himself of our recently acquired knowledge of fluid friction, evaluated
the energy dissipated in the Irish Sea. Jeffreys'0 using such data as
were available, extended Taylor's method to the entire world. His result
was dissipation during spring, tides ~ at the rate of 2.2 x i0~9 ergs per
seconcl. ~
The available data are rather meager, so that there is a considerable
element of estimate in this result, as is illustrated by the fact that
Heiskanen,5 using, essentially the same data, found 3.6 xlOi9 ergs per
second (4.83 billion horsepower). Perhaps the most striking result of
these investigations is that frictional dissipation occurs almost entirely
in the shallow seas, the part contributed by the oceans, although they
vastly exceed the shallow seas in area, being almost negligible. Jeff'reys
finds for the different bodies of water the following values for the rate
of' dissipation of energy.
Irish Sea 0.6 X 10~S ergs per sec.
English Channel . . 1.1
North Sea 0.7
South China Sea Small
Yellow Sea 1.1
Sea of Okhotsk. . . 0.4
Bering Sea ....15.0~<10lS ergs per sec.
Malacca Str. . . 1.1
Hudson Strait.. 0.2
Hudson Bay . . Small
Fox Strait .... 1.4
Bay of Fundy. . 0.4
The contribution of the <,reat ocean basins is negligible when the
uncertainty of the data is considered. The figures {'or the waters driven
ad] up to 22.0 xlOiS, or 2.2 Slot eras per seconcl, as previously merl-
tioned.
Since the friction increases rapidly with the range of tide, the factor
to reduce the rate of dissipation of energy at spring tides to an average
figure is much less than unity. Jeff'reys obtains a value 0.51 so that
the average rate of dissipation of energy due to the lunar tide alone is
1.1 x 10~9 ergs per second, or 1.5 billion horsepower.
Merely from the principle of the conservation of energy and other
fundamental principles of mechanics we can calculate the rate at which
the axial rotation of the earth and the orbital revolution of the moon
are being slowed down.
' The tide tables usually give the information for spring tides rather than for
mean tides. Hence first estimates are most cons eniently made for spring tides.
~ (2.95 billion horsepower.) The Cord billion is here used in the sense of one
thousand million or 109.
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81
FI CURE OF THE EAR TH
NOTATION
E=mass of the earth.
11!=mass of the moon.
C=moment of inertia of the earth about its axis of rotation.
a=equatorial radius of the earth.
r= radius of the moon's orbit, assumed to be circular.
o.=angular velocity of the earth's axial rotation.
Q-. angular orbital velocity of tile moon.
D=length of sidereal clay, or o.= ~g .
ret = length of sidereal month, or Q =-.
r'',
t=time in general.
h=total moment of angular momentum o:t earth-moon system.
T=total energy (potential plus }kinetic) o1: earth-moon system.
Assume the moots to be the only ticle-producing body and to move in
a circular orbit in the plane of the earth's equator. Let us neglect the
energy of the moor's rotation. The following expressions may be easily
derivecl:
Kinetic energy of rotation = '°'C(, 2.
Kinetic energy of orbital motion about center of :< ~ E.ll All
,ravity of earth-moon system.
Potential Oll en lo,> = ~; _ 7cE lI
-
where K is an a,rbitrary constant,, depencling on the assumecl confi~ura
tion of %ero potential ener,~y and k is the Newtonian gravitat,ion constant.
The total energy T is then
But we have
or
T= '-C(,, +' ~E~,~ r~Q3+K_ O'ER
7~ ( E + .11! ) _ it
_ ,
ret
q~,Q,= 7~(E+lI)
r
(1)
(2)
This corresponds to Kepler's third law in the solar system and merely
states that the attraction toward the center of the circular orbit balances
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TIDAL FRICTION 85
the centri]:'u~t,al force away from the center. By using (2 ), equation
~1 ) becomes
T = -~-Co. 2 - -4; + 7t . ~ 3
The total Monet of rotation is found to be
+ L' - ~ ~ ~
The quantity 7~, is constant regardless of frictional resistance; this I'act
is due to the principle that action and reaction are equal and opposite.
Differentiate (3) and (4) with respect to the time. TVe know that
it =0 and that 2~ has a value estimated from oceanic tidal friction;
:t'or purposes of' numerical illustration we shall adopt Jeff'rey's value of'
1.l x ~ Oi9 eras per second. Alter some rather easy manipulation we find:
dT
d(,)_ dt
dt C((,)-Q)
dQ_3C do E+
d-l- dt JU~1Jr~
(is)
Instead of' thin};in~ in terms ol' it and 2~ ~ the rates of change
in the angular velocities, it seems a little more natural to evaluate the
total accumulated chant,es in the lengths of the day and month over
some ion<, period of' time, say a century. If' we treat the change as
uniform over this period and let At represent one century expressed in
seconcls, I'ro~n the first part of' equation (I) eve At i'or the change, ~D,
ire the length of the sidereal day
-D -~ t
fit
Its) ( 0)-Q )
(6)
By calculation with this formula and JeE'reys' value of IT, we find
that the sidereal day increases 0.00072 second in a century. The cor-
respondin~, calculation for the month from the second part of equation
(I) gives an increase off:' 0.012 second. An increasing period means an
increasing, distance; the corresponding increase in r, the radius of' the
lunar orbit, is 14~- meters per century.
If it were possible to construct a clocl: that would run at an invariable
rate and to regulate it exactly according to the earth's velocity: of' rotations
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86
,.
FIGURE OF THE EARTH
at the beginning of a century, then at the end of the century it would
be 13.2 seconds last according to the stars. But in practice we keep time
by the stars. If we waited for this 13.2 seconds to elapse, so that the earth
might assume what we would consider its proper position with respect
to the stellar universe, the moon would have advanced in its orbit by
rY."2, which would represent an apparent acceleration of the moon's
Otis. But tidal friction has been at work on the moon too, and during
the century has retarded its motion by 4."4. The net apparent accelera-
tion of tidal friction on the assumptions made is thus 2."8 per century.
This is a quantity that increases as the square of the time.* Some
approximations made in the calculation tend to reduce somewhat the
computed acceleration. The actual acceleration may be determined by
~ study of the records of ancient solar eclipses, which, though not very
precise, are valuable because of their relative remoteness in time, and of
the occultations of stars by the moon, of which we have accurate records
from the seventeenth century on; for more modern times we have also
good observations of the meridian transits of the moon. These records
show an acceleration of some 11" or Id" per century. Part of this is
provided for in the ordinary lunar theory, quite apart from any con-
siderations of tidal friction. The remainder is plausibly attributed to
tidal friction and mainly, as has been said, to tidal friction in the shallow
seas, for Jeffreys' estimate of a frictional dissipation of 1.! x i0~9 ergs
per second provides for about 80 per cent of the needed acceleration.
The figures here given do not correspond to the full 80 per cent.
This is mainly due to omitting the tidal friction caused by the sun.
The tidal friction caused by the sun, like that caused by the moon,
tends to lengthen the day; but unlike the friction caused by the moon
has little effect on the length of the month. In view of the uncertainty
of the whole estimate, tidal friction might well provide for the whole
of the otherwise unexplained portion of the secular acceleration. To show
the uncertainty of the process of estimating we have only to note that
Heiskanen 5 using essentially the same data as Jeffreys, found a dissi~a
tion of 1.9 x 10~9 ergs per second, or about 70 per cent more than
Jeffreys ~ did.
* This is what the astronomers call the acceleration. It is half of what a physicist
would be inclined to take as the measure of the acceleration. Thus the distance s
travelled by a falling body is -hit, where t is the time and g the acceleration of
gravity in the physicist's sense. According to analogy of the practice of astrono-
mers in connection with the lunar theory, the acceleration would be fig. In read-
ing Jeffreys' book,7 note that he uses acceleration in the physicist's sense.
~ Both Jeffreys and Heiskanen made the preliminary calculation for the diss.i-
pation at spring tide. This is because most tide tables give data for spring tide.
Jeffreys used a factor 0.51 to reduce from conditions at spring tide to a mean
figure. The value 1.9 X 1019 ergs is obtained by applying this factor to Heiska.nen's
OCR for page 87
I/~[ F~Io~
87
Ibus, as we bare seen; tidal friction in Me sb~lIQw seas is by itself
capable of accounting for abut portion of the secular ~ccele~tion Off
the moon not provided for hy We ordinary lunar theory. Ibus Iriction~1
resistance in the Bogy tides of the earth ~pp~en~ seems to be crowded
out AS OD impact element in the case; ~ at Mast there ~ no urgent
uccessit~; even on the showing gluey manes for assuming that it now
plays an important part. Obese is indeed ~ reason of hotbed sort:
presently to be giveD; for believing tat friction in the bogy tides is
resistively small It is Jesir~ble, bovever, to consider at Last the theory
of friction of this sort, for it is easier to form some Sort of ~ meIlt~1
picture of bow friction acts; wbere~s the preceding resUlts, tbougb SOUGH
Cough seem to have beck derived ty ~ kin] of m~tbem~tic~1 jUgglery
that gives US gut little UPON which our intuitions of space add of mecb~Di-
Cal action can My bold. Moreover; the theory of friction in the body tides
of Me ebb ma be used as ~ basis on which to develop ~notber method
of estimating friction in the ocean tides ~ metbo] tat bus the double
advantage of enabling ~n estimate to he made from data of ~ Ji~ereDt
nature from those Just mentioned, and of lending itself better to the
formation of ~ mental idea of the process.
Suppose we leave the ocean out of consiJerndon for the moment ~]
consider only Me body tides. If the substance of Ibid the ebb is
compose] were frictionless there would he tidal swells abut reached
their maximum at two points; Norma, the point fibers the moon I ~
in the zenith RD] the ~ntipoisl point where ~ is in the nadir. If, bow-
ever, the e~rtb-sUtst~nce is not Irictionless, friction will tend to make
it act to some extent hke ED inveigle body; 8~] the tidal hUlge tat
it bus acquired wig he Carrie] fly the rbtad~ of the ebb to ~ position
past the meridian of the moon. Ibe two maximum tiff bulges OCCUT
in time after the moon bus passed the meridian of ~ given place (eitbeT
at Upper or lower truant) RD] are four] in sp~cc along au axis in
advance of the moons position in its ortiL See Figure 1. It is easy to
see abut in this position the moon exerts ~ force on the tiff Bulges of
the ebb tedding to retard its rotations Dud conversely the tiff bulges
tend to increase the linear orbital velocity of the moon.: Ibis Act
result for spring Odes. If me use Heisk~nen's reduction factor, ~bicb is lamer
HID Jerseys) Me discrepant between the two ~im~te~ed ~ =~aud~ly
the same data becomes even greater.
* ~in) calculation dealt mosOy gab f~ct10n in We body tides of the egg
and must now be read in the [gh1 of more modern developments
t For (mplichy we ~E speak of the moon as if ~ were the only dde-producing
body.
t It may seem p~do~ic~I that an increase ig the Wider velocity of the moon
tends to lengthen the month, but such is the case. This apparent paradox ~iH be
. . . . .
exp^ulDea in due course.
OCR for page 88
88
FIGURE OF THE EARTH
appears whether we consider the total acceleration acting on the bulges
~ and B in Figure 1, or whether we consider merely the tide-producing
acceleration, which is the resultant of the total acceleration due to the
tide-producing I'orce of' the moon and the centri:f'ugal acceleration of the
center of the earth due to the orbital motion. (See Figure 2.) If we
/
consider total force only
~ I
~ I
l
l
it\
-
1
l
/ 1
1
~ M t M
1
l
l
l
l
l
:~?IG. 1.
J 1
l
FIG. 2.
/
( Figure 1 ), and l'or simplicity replace the
bulges A and B by hearty particles at these points, then it is easy to
see that the component of' the :f'orce on A tending to retard the rotations
of' the earth is greater than the correspoT~clin~ component at B. both
because A is nearer the moon than B is, so that the force along, A)/I is
greater than aloe,, BN1, and also 1~eca.use the line AM is more nearly
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Representative terms from entire chapter:
central force
TIDAL FRICTION
89
perpendicular to AB than the line BM, and hence gives a greater compo-
nent perpendicular to AB, which is the component tending to retard
rotation.
We must get the same result if we consider tidal forces only, as in
Figure 2, for the tidal acceleration is merely the total acceleration com-
bined with the centr;:tugal acceleration of the center of the earth; this
latter component is the same for A and B and since these are on opposite
sides of the center its effect on the rotation is null. For many purposes,
moreover, it is more convenient to use the tidal force. By evaluating
the expressions for the tidal force in the prime vertical we see that the
forces on A and B (Figure 2) are numerically equal and that both teal
to retard the rotation of the earth.
The elevation due to the tidal bulge (y), positive or negative, for
latitude ~ and longitude * A is given by the expression
y=H cost ~ cos 2A.
If now the angle A()M ~ between the meridian of the snoop and the
meridian of magnum bulge be denoted by c, the couple, 1V, tending to
retard the rotation is given by
N ~ 1M (tat H sin 2
Here p is the density of the matter composing the bulge; ~ the other
symbols have either just been explained or have the same meaning as
in. equations (~) to (6), pares 84 and So. To get the rate at which e~.-
ergy is being expended we must multiply N by ~-Q. the arlą,ular velocity
of the moon relative to a point on the earth. We thus get
AT = ~ ~76~f 't [Ip sin he (~-Q). (7)
Let us attempt to follow out the consequences of assuming that the
tidal retardation of the earth's rotation and the consequent apparent
acceleration of the moon are entirely due to friction in the body tides
of the earth. For GG we shall
90
FIGURE OF THE EARTH
observed acceleration of the moon. The quantities E, p, and ~ depend
on the physical properties of the earth, that is, on how the density,
elasticity and viscosity vary from center to surface. If the viscosity is
zero, ~ is zero, the tidal bulges A and B are in line with the moon, and no
energy is dissipated. For simplicity we shall assume the physical proper-
ties of the exotic constant throughout its mass.
The mathematical developments for even the simplest cases are too
lofts, and complicated to be given here; they will be -found in the world
oil Darwin. The combination of elasticity Title viscosity is assumed to be
according, to the Maxwell-Butcher law o-? elastico-viscosity. The coe~-
cient of viscosity (v) tales the forms
V IT
(8)
where ,u is the modulus of rigidity and ~ is the so-called time of relaxa-
tio~-~; this ~ is the time in which the stresses sink to i/e part of their
initial value.*
Of this law Darwin remarries: ~
" The term elastico-viscous is used to denote that the stresses requisite
to maintain the body in a given strained configuration decrease the longer
the body is thus constrained, and this is undoubtedly the case with many
solids. In the particular case which is here treated, it is assumed that the
stresses diminish in geometrical progression, as the time increases ire
arithmetical progression. If, for example, a cubical block of the substance
be strained to a given amount by a shearing stress T. and maintained in
that: nn~it:inn then after a time t. the shearin,, stress is Texp ( - - ).
The time T measures the rate at which the stress falls off, and is called
(I believe by Professor Maxwell) ' the modulus of the time o:t the
:elaxation of rigidity '; it is the time in which the stress leas been
reduced to c-i or .3679 of its initial value. ~] I do not suppose, however,
that any solid conforms exactly to this law; but I conceive that it is often
use:tu.1 in physical problems to discuss mathematically an ideal case, which
presents ~ sufficiently marked likeness to the reality, where we are unable
to determine exactly what that reality may be." Darwin also used the
term "semi-elastic" as synonymous with "elastico-viscous."
According to this law all substances to which a constant stress is applied
:Ior a tinge and then removed would receive a constant " set," the magni-
tude o:l: which would increase with the intensity of stress and lerlt,tI~
of: time for which it was applied, that is, such substances exhibit plasticity
e is the base of the natural logarithm = 2.71828 .... 1/e = 0.368.
The base of the natural logarithms is here called c.
TI~[ F~zcTIo~
91
Under the action of Any atreaa, bomber appall Ibere is also the bypotbesis
of Brmo-viacoaity; which may Be combine] him that of el~stico-viacoait>.
[or ~ fuller discussion of the aUBject, reference May he acme to the work
of Jeerers/ Trowel even frmo-Viacosi~ RD] el~stico-viscosity co~hinc]
Jo not represent the full complexity of the n~tUr~1 pbenomen~ Jeerers;
hovever, bus given reasons Ior believing Abut BTmo-viscosity is not im-
port~nt in the case of We secular acceleration of Me moon, ~] so me
mall confine the calculations to Me tr~0ition~1 case of el~stico-viscosity.
Let us assume ~ modulus of rigidity aqua to 16.0 xlOll c.g.s. units,
which is twice abut of steel and Ibid represents Ronald the mean e~ec-
tive rigidity ic~uced from observations of ebb tides; and calculate ty
Delvings formulas go ~ modulus of the time of religion of rigidity ~
(or mom trieby Me <; time of relaxation ~) corresponding to ~ dissipation
of energy at the rate of 1.1 x TOlg ergs per second. Ibe details cannot
be given here But ~ is Ionn] tat the maximum equ~101i~1 tiff protutcr-
~ncD is about 12 cm., the Anne between the axis of equ~tori~1 bulges ~]
the meridian Plane of Me moon is 0° J53 and the time ~ r~ion for
file Crib is about 1.8 x 105 seconds; or ~ little over go Alas.
As far as the earth tides alone ale collcerneJ, Mere is at present r~
reason Baby we sold not be gibing to accept this result Ibe tangle
between the axis of tiff protuter~Dces ~d the meridian plane of the
moon implies ~ lag in the obese of the observed ebb tides Rebind the
corresponding please of the tidy forces. But this lag in pose Stolid in
the present i~st~nce'5 be only 25~, ~] this is too small ~ quantity to
LO detected vitb cert~iD~. Abe secondary erects of the owed Udrs
Disturb the abase t~ amounts In larger than Aim
Ibe most conclusive argument against attributing And great portion
of the ret~rJ~tior of the Cribs rotation to tiff friction in the bogy
of the earth comes Idiom the variation oI latitude. Wb~tever may be the
C]USC of the 14-montb Cb~ndleri~n or prolonged nutrition of the Doles,
it is IoulId fly otser~tion to Persist with little ii~blution of amplitude
our periods of sever~1 dears. If; bowed the tiff dissipation of energy
hos entirely in the body of go clothe which implies, as bus been scon, ~
Hmc of rcl~tion of the c~lth-suDst~re cqn~1 to gout two dam than
~ The brad 1~ ~ not the same as twice the angle bet~ecu the axis of hdu1
~ ~ ~ ~ ~ ~ _ ~
protuberances and the moons merlulun plane. -lnls IS fine 1~ in Vellum of Lag
oc6~Z earn [de, Me favor 2 ending hiccup the tide ~ s~idiu=~, but abet
unsaved is the tidal defo~ion of the level surfaces near the ebb. On ~c-
count of the method Of observation, this deformation is referred to the arty and
~ bigoted by the tide) in the soLd earth. Far restively large values of the time
of relaxation, such as are bare considered, the observed lag in please is less than
the rectum lag in pose of the earth tides; for subduer values, such as are used by
Darwin the observed lag in phase may be greater than the rectum lag.
92
FI GURE OF THE EAR Y'H
Lo the forces concerned in the Chandlerian Mutation, which is, ire fact, ~
sort of tide of long period, the earth would behave as if it were practi-
cally fluid. The Chandleria.n Mutation could not exist; on the a.ssump-
tions previously made, calculation shows that its amplitude would be
reduced in the ratio 1: e in about eleven days.
Thus the arguments in favor of the thesis that tidal dissipation of
energy takes place chiefly in the hydrosphere and above all in shallow
seas are: 1 ) that the best available estimates of the dissipation of energy
ire the shallower seas give just about enough dissipation of energy to account
for the apparent secular acceleration of the moon; 2) that the results
of the assumptions of any considerable dissipation in the body of the
earth would be in contradiction with the observed behavior of the Char~d-
lerian nutation in the variation of latitude.
The conception of the tidal distortion of the earth, elongation, the
earth with the elongation in the general direction of the moon, but some-
what in advance of it in space (reckoned in the direction of orbital
revolutions ), is too useful in forming a. mental picture of how ticla1
friction acts to be summarily put asicle, in spite of the fact that friction
ill the body tides of the earth appears to be relatively unimportant. The
conception may be adapted to the case of the ocean tides and enables us
to base an estimate of tidal friction on observational data different from
those used in deducing the estimate so far used.
The semidiurnal tides in the solid earth conform approximately to
the equilibrium theory, that is, the two high tides lie in the line of the
meridian plane of the moon. Except for friction the high waters would be
exactly in this plane. But in the case of the ocean tides neither the equilib-
~iu~ theory nor any letdown modification of it is even a first ouch
a~proxi~natioT~ to the truth. If we consider the tidally perturbed surface
,. . .
v
of the ocean at any instant, instead of its being, an ellipsoid it is an
exceedingly irregular surface, A, B. C, D (Figure 3). But within this
irregular surface, and in the mathematical sense a part of it, lies hidden,
so to spear<, a re~,ula.r ellipsoid, E, F. G. H. This ellipsoid may be sepa-
rated from the irregular tidal surface of the sea by the process of spherical
harmonic analysis; furthermore, it appears that only this ellipsoid is
needed for the calculation of the couple tending to retard the rotation
of the earth and to increase the linear velocity of the moon; the remaining
components which the spherical harmonic analysis requires to represent
mat.hematicallyr the irregular surface of the sea are without effect on this
couple.*
* This is because the ellipsoid corresponds to a s,pherica,1 harmonic of the second
degree. In the calculation of the moment the tidal height of the sea is multiplied
by another second-degree harmonic and integrated over the surface of the globe.
TIDAL FRI C TI O1V
93
If we lo the form of' the tidal surface of the sea at any instant
eve can deduce the ellipsoid that gives rise to this couple alla thus calcu-
late'the rate at which work is beings, done. A picture of the tidal surface
-H
\W
An\
-
-
-
By
FIC. 3.
F
-
-
TIC
ma:: be deduced frolic a map of' the cotiLlal lines of the globe. Noxious
sets of cotidal lines have been drawn which differ considerably in cletai.1
According to one of the fundamental properties of spherical harmonics only the
second-degree component of the tidal height that is, the ellipsoid, contributes to
the moment. All other spherical-harmonic components, when thus multiplied and
integrated, give a zero contribution.
94
FIGURE OF THE EARTH
and at various points exhibit more fundamental differences; still it
may be said that, taking the world as a whole, there is a general re-
sembla.nce among the more modern sets of cotidal lines. If we accepted
any one of these we could calculate the rate of dissipation of energy at
any instant during the lunar day by a process of mechanical quadrature.
NVe are not, however, so much interested in the rate of dissipation at
one or another instant as in. the average rate over a lunar day and this
may be calculated once for all by mechanical quadrature. What is needed
Is the amplitude of the tide and the lunitidal interval. If we denote
the amplitude by A, and the lunitidal interval expressed in lunar time
by c, we have for the average rate at which energy is dissipated
cat = 3-7cplll 3 (0-Q) iiA cost ~ sin be dS.
Here dS denotes an element of area and the two integral signs mean
that the integration covers the entire globe. On land, A is zero.*
The necessary quadrates have been made by Heiskanen 5 using Ster-
~eck's cotidal lines as a basis. Heiskanen found 2.0bx10~9 ergs per
second as the rate at which energy is being dissipated at sprint, tides.:
Obviously this :~<,ure is somewhat less for the mean tide than for the
spring tide; the necessary reduction factor may be estimated at 3-. Still
another reduction must be applied to allow for the yielding of the earth.
Formula (8) is based on a calculation of the gravitational potential due
to a load of tidal water on an unyielding earth beneath. But the actual
earth yields and a given load produces less gravitational effect than if
it stood upon an unyielding earth. The reduction factor depends on
the constitution of the earth, in red ard both to density and elastic
properties and on the extent ot the ioacl. nor a load represented by
second-de~,ree spherical harmonic, and for an earth of uniform density
and rigidity, the reduction factor is 0.8. For the actual earth it is even
mnil~r r)~.rh~.r)s as small as 0.6.~ Takirlt, both reduction factors together
~ ~ 1~ ·- ''
* The formula may be derived from quite a different method, that of consider-
ing the work done against the tidal forces by the vertical rise and fall of the water.
This is the method followed by Taylor and Heiskanen.5
~ Heiskanen made the quadratures, as stated, but treated the result not as ~
different method of estimating the same quantity as had been derived from a
study of the currents; but as a correction to the latter quantity. The subject is
further discussed by Lambert.8
~ See in this connection paper by L. Rosenhea.d on the Annual Variation of
Latitude, in the Geophysical Supplement to' the Monthly Notice of the Royal
Astronomical Society, Sol. 11, no. 2, May 1929. This same question of a. reductions
factor for the yielding of the earth comes into the problem of the variation of
latitude as in the problem of the tides; moreover, the spherical harmonics are of
the second degree in both problems.
95
~e sce tb~t roUghly ~e ~e~n r~te of dissIp~tio~ of energy ~s csti~tc]
from tbe cotid~1 m~p of tbe glote is ~bo~t 1.0 x 101# in good ~greement
~ith the Ogure ott~ine] 1rom tid~1 cUlrents in sh~llov se~s. In viev of
tbe uncert~int~ unic~l~ing thc d~t~ of toth mctbo~s thc closcncss of
tbe ~g~ccmcnt mUst tc ~ttribUtc] to cb~nce. Hovever; tbe ~greement
m~kcs US ~ [ttlc more con0dent of tbe ~>p~D~i~te gener~1 correctness
of OUT cotid~1 m~ps. Ibe dissip~tion c~lcul~te] Irom ore~n curreuts c~n
be only of one ~l~ebr~ic sT~n thc co~rcct one. bo~c~er erronro~ thr
~ ~ ~ ~ ~ ~ ~ O _, _ ~ ~ ~ v v ~ v ~ v ~ ~ v . . v . ~ ^ ~ ~ ~ ~ v ~ ~ ~ ~
~~gnitude ~ight be; ~bere~s; ~dtb ~n erroneous ~oti]~1 m~p it voul] he
entirely possitle to act ~ 1csult vitb ~ i~possiblc ~1gctf~ic sign.
Tn thc mccb~Dic~1 qu~dr~ture c~lle] Ior ty fDrmul~ (8) so~e oI 81C
elcmc~ts ~re posi6>e HD] s~me neg~tive. If tbe tides ever>~bere con-
forme] to tbe cquilibriu~ tbeory; sll ele~ents voul] be zero Ber~use
voul] tc zcro ~] tb~ zero result ~oul] inJic~tc thc ~tsencc of friction.
zero result coul] ~lso tc rc~cb~] By ex~rt countert~l~ncing of poeitive
d Ile~tive elemcnts. ~ sct of coti]~1 lincs tb~t ~oUl] ~ccomplisb
tbis counterb~l~ncing might look just ~s complic~te], ~D] be ~s Ji~erent
fro~ ~ set conformi~ to tbe equiLtriUm tbeory as ~re Sterneck~s cotid~1
[nes or ~ny oI thc othcr ~odcrn ones. BUt tbe ex~ct cou~terb~l~ncing
of positivo ~n] neg~ti~e clcmcnts ~o~l] indic~tc ~c ~tsc~cc of friction.
Tbis vould inJic~te tb~t thc m~I~ pecuIi~rities of the tiJes (if ve t~kc
the eqniL#](u~ tbeory ~s ~ ~orm j: n~mel>: the irregUl~r v~ri~tions in
Lbe r~: the 1~Tge v~Tues oF the lul~tid~l interv~ls; ~e <: ~s ~ of ~e
v~rious tida1 ine-~l~ies ~n] tbe di~erc~ce Bet~een tbe aCtU~ r~ho
of ~c v~rious tid~1 co~poncnts ~nd tbe thcorctic~1 r~tio of tbe cor-
respondiDg Iorccs, ~11 thcsc could Be Juc; ~nd perh~ps i~ thc ~in ~re
]uc, to c~uscs other ~u tid~1 friction, n~mcl>; to thc ir~cgul~rities of
tb~ oce~n t~sins hoth i~ Iorm oI peri~eter ~n] T~ Jeptb; ~n] to the
rot~tion oI tbe e~rtb.
Tbus f~r in tbe csl~l~ion of the hd~1 dissip~ti~ of enc~y ~c h~vc
becn roDcerne] ~ith the r~tcs ~t ~bicb thc lc~gtb oI thc sidore~1 d~> ~d
siicre~1 ~ontb ~re cb~nging. B~ either oT thc two ~il~Blc mctbods
tbo r~te ~t ~hicb energy js Deing iissip~te] or vork Jone ty tbe tides
is ~tout I.0 x IO10 ergs ~cr scconi, or; s~y Bet~een o~e ~0 two hibion ~
horsepover. The r~tes ~t ~hicb tbe i~y ~nd monib ~re cb~nging in
consequence of tbTs iissip~Iion of cnc~g~ voUld b~ve ~ ncgligiLlc c~ert
witbin histolic or C~Cn ~cccnt gcologic~1 timc. But it is intC~CStiDg to
considcr ~b~t ~ight h~vc b~ppcnc] ~s ~ ~csult of [d~1 1riction iD thc
remoter geologic~1 p~st. Oere thc~c is room o~l~ for ~ vcry tri~f t~c~t
A trillion here mcuns onc thousu.nd ~~iIlion, or 10'}.
7
96
FIGURE OF THE EARTH
ment of the subject. For a fuller discussion reference must be made
to the writings of Darwin ~ ~ ~ 3' 4 and Jeff'reys.7
Friction obviously depends very largely on the outline and depth of
the ocean basins. In the past this has been quite different from what it
is now, especially as regards the shallow coastal seas, where, as we have
seen, tidal friction is comparatively large. Geologists have reconstructed
for us the map as it looked in past geological time, but our mathematical
powers are entirely inadequate to the problem of determining a prior,
from such a map what the tides and tidal friction were during some past
geological era. Geologists tell us that at one time areas of shallow sea
were much more extensive there now. This condition would make I'or
increased tidal friction. But shallow seas by themselves are not sufficient
to produce a great amount of friction. Conditions must be such that
there must be a considerable tidal current in these seas, and free com-
munication with an open ocean where large tides are found would gen-
erally be required. How the rate of frictional dissipation of tidal energy
during the era of extensive shallow seas compared with the corresponding
rate today we are, therefore, unable to say.
But some tidal friction must have been present always; accordingly
as in imagination we go further and further back into the past we find
the day and the month getting shorter and the moon getting nearer the
earth. The decreasing distance of the moon implies that tidal friction
would tend to be greater, other things being equal, in the past than it is
now., for the f'riction-producing effect varies inversely as the sixth power
of' the distance. The more rapid rotation of' the earth would diminish the
period of tidal forces, that is, the time of a tidal oscillation. If the
tidal load of water to be shifted from one place to another and back
to its original position within the tidal period remains the same, the
shortened period means a more rapid transfer and therefore greater
friction. The general tendency would thus be for the effects of tidal
friction to develop at a more rapid rate in the remote past than now.
If the moon in the past was much closer to the earth than it is now,
might it not have been at one time almost in contact with it, thus
raising enormous tides? ~ If' we look still farther bacl<, might not the
* In. reading Darwin it must be remembered that he wrote with the idea that
most of the tidal friction would be found in the body of the earth rather than in
the ocean waters; this is the opposite of the view now generally accepted. But
friction, wherever it may be found, is still friction, and affects the length of the
day and month. Darwin's calculation of the heat developed by tidal friction in the
interior of the earth must, however, be accepted with appropriate reservations.
~ When the moon is very near the earth the customary approximation by whirls
the tidal forces and the tidal protuberances (of the earth tides) on opposite sides
of the earth are equal becomes inadequate. Tidal forces depending on the fourth
TIDAL FRICTION
97
moon and the earth have formed a single body'? These ideas are the
germ of Darwin's theory of the origin of the moon. He supposes that
the rotation of the earth was originally so rapid that the solar sem;-
diurnal tides had a period approximating to the natural period of vibra-
tion of the earth for a disturbance of the tidal type, that is, a period o-t'
about two hours. The near coincidence in period of the free vibration
of the earth with the forced tidal vibration raised such enormous solar
tides that the earth was disruptecl. A fragment separated from the
parent earth and formed the moon. Since the separation there have been
lunar tides also; and tidal i'riction leas done the rest, lengthening the
day and month and malting the moon recede from the earth. This is
merely a very brie:t' and inadequate statement of:' a plausible but not
ur~iversall~r accepted speculation.
For I'urther discussion of' this theory End for the tracing out of:' the
supposed future evolution of the earth-moon system under the influence
of' tidal 'friction, reference must again be made to the authors already
mentioned.
Another interesting speculation regarding the effects of tidal friction
arises in connection with the variation of latitude. If the earth in the
past rotated more rapidly it would have been more flattened, its natural
free period for the motion of the pole, now fourteen months, would have
been shorter arid might have coir~;ded at a~ epoch. sufficiently remote
with the year, the period o:t' forced oscillation due to seasonal effects.
With the free and -i'oreed periods coinciding, resonance Plight be expected;
that is, large excursions of the pole :L'rom its mean position arid in con-
sequerlce large stresses in the earth, perhaps large enough to cause rupture,
also a large latit.ude-va.riation tide.
The change in the velocity of rotation needed to reduce the free period
to one year on the supposition that the constitution of' the earth remains
unaltered is about six per cent, corresponding to a decrease of 1.4 hours
in the length of day. If' we talkie an average decrease at the rate of 0.001
sec. per century, about half a billion years would be needed to decrease
the clay by this 1..l hours. According to currently accepted estimates
of the age of the earth an era half a billion years ago is rather ancient
geological history, but it would belong to geology rather than to
cosmogony.
In order to account for climatic and other phenomena of past geologi-
cal time, geologists have sometimes postulated large displacements of'
power of the moon's distance and involving third-degree spherical harmonics can
no longer be neglected? as in the customary approximation, and these tides made
the protuberance under the lagoon higher than that on the opposite side of the
earth.
98
FIGURE OF THE EAR TH
the pole. But such shiftings of the pole as the geologists clesire, in general
require more extensive displacements of matter on or within the earth
than neologists are willing to accept. The phenomena of tidal friction
suggest, however, that the pole might on one occasion have undergone a
fairly large displacement and that without any great displacement of
mass.
In the preceding discussion the inclination of the lunar orbit to the
equator has been neglected. The fact that the moon does not move in
the plane of the equator, but in an orbit the average inclination of which
to the equator is nearly equal to the obliquity of the ecliptic, introduces
other effects of' tidal friction. The most important of these is a tendency
for the obliquity of the ecliptic to increase. The existence of this tendency
seems to be contradicted by the fact that obliquity of the ecliptic is
now decreasing at the rate of' some 47'" per century. This clecrease, how-
ever, is due to the complicated gravitational interactions among the
members of the solar system and does not represent a permanent state
of affairs; for the ~resent. however. the tendency of the obliq u-itv to
decrease much
tidal friction.
~.,
more than counterbalances the contrary small effect of
NVe now return to the postponed question as to why the moment due
to I'riction which obviously retards the axial rotation of the earth should
also slow clown the Notion of' the moon and tend to push the moon farther
away from the earth. At -first ,la~-~ce it should appear that the moment
would make the snooty move -['aster and would have no effect on the
distance.
Suppose the orbit of' the moon to be originally circular and a tangential
impulse to act at A for an instant in the direction of motion, thus increas-
ing the velocity; what is the effect on the shape of the orbit'? Evidently,
because of the inertia of the moon, the new orbit is no longer circular
but elliptic with the apogee lying in a line parallel to the direction ol'
the impulse and in the sane direction. The major axis is thus increased
by the impulse and according to K:epler~s third law this increase is
accompanied by an increase in the time of revolution or a decrease in
the mean angular velocity. The effect of' tidal friction may be compared
to a succession of very minute impulses acting, in succession all around
the orbit. The effect of these impulses-is not to elongate the originally
circular orbit into an ellipse with its major axis in a fixed direction (as
a single impulse would have done) but on account of inertia gradually
to open out the orbit into a very slowly increasing spiral. At any given
point in the spiral there is a difference in direction between the direction
of' a tangent to the spiral at the center of' the moon and a tangent at the
sense point to a circle the center of' wl~icl~ is at tl~e center of' the earth.
TIDAL FRICTION
99
The central force due to the earth leas Rio component whatever in the
direction of the tangent to the circle and hence no effect either of accelera-
tion or retardation. But. in the direction of Else ta,n`~ent to the spiral
the central force evidently acts to retard the orbital motion. The angle
between circle and spiral is small so that this retarding effect of the
central force is very minute; but the central force is so much larger than
the pull of the tidal protuberance, which acts to accelerate the orl~ital
~notion, that the central force prevails and the net effect is ~ retardation
o-L' the orbital notion.
REFERENCES
1. Darwin, Sir G. H. On the bodily tides of viscous and semi-elastic spheriods
and on the ocean tides upon 3~ yielding nucleus. Phil. Trans. Roy. Soc.,
170: 1 (1879); Ol, Scientific Papers' Cambridge (Eng.), 2: 1 (1908).
2. . On the procession of a viscous spheroid, and on the remote history of
the earth. Phil. Trans. Roy. Soc.' sol. 170, pt. 2: 447; or, Scientific Paper,
2:36.
3. . Problems connected with the tides of a viscous spheroid. Phil. Trans.
Roy. Soc., vol. 170, pt. 2: 539; or, Scientific Papers, 2: 140.
4. . The tides and kindred phenomena of the solar system. 3rd ea., London,
1911.
5. Heiskanen, W. Uber den Einfluss der Gezeiten auf die sakulare Acceleration
des Mondes. Annales Academiae Scientiarum Fennicae, 18A: 1 (1921).
6. Jeffreys, Harold. Tidal friction in shallow seas. Phil. Trans. Roy. Soc., 221A:
239 (1920).
7. . The earth, its origin, history and physical constitution. 2nd ea., Cam-
bridge (Eng.), 19291. (Particularly Chap. 14.)
8. Lambert, W. U. The importance from a geophysical point of view of a knowI-
edge of tides in the open sea. Union Geodesique et Geophysique Inter-
nationale. Section d'Oceanographie. Bulletin No. 11: 52 (1928).
9. Taylor, G. I. Tidal friction in the Irish Sea. Phil. Trans. Rov. Soc. 220A: 1
(1919).
