zotero/storage/NW7GMBGP/.zotero-ft-cache

The Earth's Rotation
M. G. Rochester

The most ancient and funda­ mental concern of a s t r o n o m y is the orientation and motion of a ter­ restrial observer relative t o t h e stars. Its geophysical aspects date from t h e time of N e w t o n and Halley, and its mathematical foundations were laid by Euler 2 0 0 years ago. Despite t h i s honorable antiquity, the subject is far from m o r i b u n d a n d t o d a y p r e ­ sents a rich and fascinating array of challenges to observation, experi­ ment, data analysis, and theory. The many-faceted problems of the threedimensional rotation of the earth about its center of mass n o w attract astronomers and paleontologists, solid earth geophysicists a n d electri­ cal engineers, general relativists and oceanographers, and applied mathe­ maticians and scholars of classical texts.
In this review I attempt to sum­ marize, as briefly as possible, the cur­ rent state of k n o w l e d g e in a field that is complex, extensive, and resur­ gent under t h e i m p a c t of late 2 0 t h century technology. I shall begin with a survey of t h e appropriate reference frames and problems in-
This article was taken from the keynote address presented at the second GEOP Re­ search Conference on The Rotation of the Earth and Polar Motion, which was held at The Ohio State University, Columbus, February 8 - 9 , 1 9 7 3 .

volved in defining them and then outline the accuracy with which the earth's rotation can be measured rel­ ative to these frames by techniques already in use or on the threshold of realization. Following that, I shall discuss in turn the various spectral features of changes in the axis orien­ tation and spin rate of t h e so-called ' s o l i d ' e a r t h (Table 1) and t h e physical mechanisms known or likely to effect and affect them (Table 2). Copious references are given for deeper study. I shall concentrate almost exclusively on developments in the past decade or so since the appearance of the now-classic mono­ g r a p h by Munk and MacDonald [1960], the standard reference for most aspects of the subject.
Reference Frames The earth is not a rigid body, and
so the selection of a reference frame suitable for describing its rotation is n o t a completely straightforward matter. Strictly speaking, the phrase 'rotation of the earth' is shorthand for the rotation in space of a certain reference frame fixed in some pre­ scribed way t o a set of astronomical observatories distributed over the earth's crust. The difficulty of de­ fining a reference frame is increased by the fact that the observatories are located on different crustal 'plates'

t h a t are in relative m o t i o n [Vicente, 1 9 6 8 ; Markowitz, 1 9 7 0 ; Arur and Mueller, 1 9 7 1 ; Tanner, 1 9 7 2 ] . T h e current choice of astronomers and geodesists is a 'geographic' frame whose origin lies at the earth's center of mass, whose z axis points to the Conventional International Origin (CIO)—corresponding very nearly to the mean position of the rotation pole from 1900 to 1905 determined by the International Latitude Service (ILS)—and whose x axis points at right angles to this in the plane of the Greenwich meridian as determined by the Bureau Internationale de l'Heure (BIH) in Paris. At the y axis the astronomers and geophysicists part company, the former choosing a left-handed set and heading 90°W of Greenwich and the latter going 90°E (Figure 1).
Polar motion, which may be either irregular or periodic (in which case it is called 'wobble'), is the dis­ placement of the instantaneous rota­ tion axis relative to this frame, on a small scale (a < 0 " . 3 , \x\, \y\ < 10 meters). The term 'polar wander' is reserved for the large departure of the rotation pole from its mean posi­ tion at any one epoch, achieved only on the geologic time scale. Polar mo­ tion has been measured for over 70 years by the ILS and its successor, the International Polar Motion Ser-

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TABLE 1. Spectrum of Changes in Earth's Rotation

A. Inertial Orientation of Spin Axis

B. Terrestrial Orientation of Spin Axis (Polar Motion)

C. Instantaneous Spin Rate OJ about Axis

1. Steady precession: amplitude 23°.5; 1. Secular motion of pole: irregular, 1. Secular acceleration: CJ/W ~ c v

period ~ 25,7 00 years.

~ 0 " . 2 in 70 years.

10"10/yr.

~

x

2. Principal nutation: amplitude 9".20 (obliquity); period 18.6 years.
3. Other periodic contributions to nuta­ tion in obliquity and longitude: ampli­ tudes < l " ; periods 9.3 years, annual, semiannual, and fortnightly.
4. Discrepancy in secular decrease in obliquity: 0".l/century(?).

2. ' M a r k o w i t z ' w o b b l e : amplitude ~0".02(?); period 2 4 - 4 0 years(?).
3. Chandler wobble: amplitude (variable) ~ 0 " . 1 5 ; period 4 2 5 - 4 4 0 days; damping time 10-70 years(?).
4. Seasonal wobbles: annual, amplitude ~ 0 " . 0 9 ; semiannual, amplitude ~0".01.
5. Monthly and fortnightly wobbles: (theoretical) amplitudes ~0".001.

2. Irregular changes: (a) over centuries CJ/W < ±5 X 10A °/yr; (b) over l - i d years, CJ/W < ±80 X 10"1 °/yr; (c) over a few weeks or months ('abrupt') d>/co< ±500 X 10-*°/yr.
3. Short-period variations: (a) biennial amplitude ~ 9 msec; (b) annual,ampli­ tude - 2 0 - 2 5 msec; (c) semiannual, amplitude ~ 9 msec; (d) monthly and fortnightly, amplitudes ~1 msec

6. Nearly diurnal free wobble: amplitude <0".02(?); period(s) within a few minutes of a sidereal day.

7. Oppolzer terms: amplitudes ~0".02; periods as for nutations.

vice (IPMS). The five ILS observa­ tories are spaced out along the same parallel of latitude (39°8#N), and thus the effects of any systematic errors in star catalogs are eliminated. Both the IPMS and the BIH publish determinations of the pole path about the CIO (Figure 2). The IPMS pole path is based on observations at the five ILS stations, whereas the BIH pole path currently incorporates data on latitude variation from some 50 stations, and thus the effects of star catalog errors are statistically reduced. The difference between the two pole paths, which can amount to — 0 " . l , is a measure of t h e effects of errors due to local motions of the vertical, plate motions, refraction, instrumental peculiarities, and sys­ tematic differences in data reduction techniques. It is worth noting that the individual observations scatter much more widely than the pub­ lished curves suggest. The IPMS and BIH both claim errors of only ±0".01 in their published 0.5-year and 5-day means. Certainly, this sets a limit to the precision of which the basic in­ struments of optical astronomy, the photographic zenith tube (PZT) and the astrolabe, are presently capable.
New techniques, such as Doppler tracking of artificial earth satellites [ A n derle, 1 9 7 2 ] , w h i c h provides

TABLE 2. Mechanisms with Effects Now Distinguishable on the Earth's Rotation

Mechanism

Effect*

Sun Gravitational torque Solar wind torque
Moon Gravitational torque
Mantle Elasticity Earthquakes Solid friction Viscosity
Liquid core Inertial coupling Topographic coupling Electromagnetic coupling
Solid inner core Inertial coupling
Oceans Loading and inertia Friction
Groundwater Loading and inertia
Atmosphere Loading and inertia Wind stress Atmospheric tide
*Numbers refer to Table 1.

A, B7,Cl,C3c C2c(?)
A, B7,Cl,C3df
Bl,B3-4,Cl-2a,C3c-d Bl, B3 B3(?),C1 C2a
A2-3,B2,B6 C2b-cO) A4(?),B3,C2
B2(?)
B1,B3,B5,C2* B3(?),C1
B4
B4 C2C,C3J-C
CI

pole positions at 2-day intervals, have already achieved an accuracy com­ parable t o t h a t of t h e BIH [Feissel et al, 1 9 7 2 ] . In fact, t h e BIH recently began to include in its data set the observations made by the Dahlgren

Polar Monitoring Service (DFMSJ. Laser ranging to artificial satellites offers even greater accuracy together with pole positions determined at m o r e f r e q u e n t time intervals. Smith et al [ 1 9 7 2 ] report a sequence of

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CORRECTION T O SIDEREAL TIME » T C M - T'C'M' P O L A R M O T I O N = X, y
Fig. 1. Time and polar motion.

—•—

BIH IPMS

6-hour mean pole positions within ±0".03 of the BIH path from a single laser tracking system operated over 5 months. Accurate tracking of artifi­ cial satellites should also permit checking that the geographic axes of reference do pass through the earth's c e n t e r of mass [Lambeck, 1 9 7 1 ; Melchior, 1 9 7 2 ] . Gold [ 1 9 6 7 ] a n d MacDonald [ 1 9 6 7 ] first p o i n t e d o u t that very long base line interferometry (VLBI) would make possible very precise measurement of changes in t h e e a r t h ' s r o t a t i o n (see also Burke [1969]). Laser ranging to corner cube reflectors on the moon should enable polar motion and spin rate to be measured with an accuracy nearly comparable to that expected from V L B I [Alley and Bender, 1 9 6 8 ; Faller and Sampler, 1 9 7 0 ; Chollet, 1 9 7 0 ; Rosen, 1 9 7 2 ] . A l t h o u g h t h e n e w e r techniques will eventually largely replace optical astronomy, it will be important to continue the PZT observations for years to come to provide consistency and a stan­ dard of comparison during the transi­ tion period.
Orientation of the instantaneous rotation axis in space requires the selection of an inertial frame based on a catalog of fundamental stars, with due allowance for such things as proper motions and galactic rotation [Woolard and Clemence, 1 9 6 6 ] . T h e best catalog in current use is F K 4 , based on over 1500 well-studied stars, but it contains systematic er­ rors of the order of 0".l. Although these could be detected and cor­ rected by a sustained program of laser tracking and photographic ob­ servation of artificial satellites [Wil­ liams, 1 9 7 2 ] , t h e d e v e l o p m e n t of VLBI seems to promise a better solu­ tion in the long run. The ultimate resolution of the order of 0".001 ex­ pected to be obtained with VLBI raises the possibility of using radio sources at enormous distances (with negligible proper motions) as a fun­ d a m e n t a l catalog, if a sufficiently dense distribution of quasars can be found over the celestial sphere.
The orbital motion of the m o o n and changes in the obliquity of the earth's equator to the ecliptic are connected to problems of the earth's

Fig, 2. Pole paths

1969.0-1971.95. 771

rotation, and so it is useful to define

T h e third kind of time with which [ 1 9 6 9 ] . T h e m e t h o d s used by the

an intermediate reference system tied we have t o deal is ephemeris time IPMS and BIH in processing their

to the ecliptic and vernal equinox at some epoch. Lunar laser ranging (LLR) will in the course of time greatly improve the determination of the moon's orbit, the lunar ephemeris [Faller and Wampler, 1 9 7 0 ; R o s c h ,

(ET), the basis of dynamical astron­ omy. Ephemeris time is the (pre­ sumed uniform) measure of time that appears as the independent variable in Newton's laws of motion and is effectively defined by the motions of

data are described b y Yumi [1972] a n d Guinot et al [ 1 9 7 2 ] . The pos­ sibilities of the newer observational t e c h n i q u e s n o w c o m i n g into use have b e e n e x p l o r e d in t h e s t u d y edited by Kaula [ 1 9 7 0 ] .

1972]. The VLBI tracking of space­ the sun, moon, and planets over the

craft or artificial satellites placed in past few centuries. Ephemeris time is

orbit around other planets could at independent of the earth's rotation Dynamics of Changes

some future date tie the ecliptic but rests implicitly on Newtonian in Earth's Rotation

frame to an inertial frame con­ theory. Until the advent of AT, the

A change in the earth's rotation

structed from t h e quasar sources irregularities in t h e e a r t h ' s spin r a t e can be b r o u g h t a b o u t (1) by a change

[Preston et al, 1 9 7 2 ] .

were measured by ET-UT (Figure 3). in its angular m o m e n t u m due to the

Any divergences b e t w e e n E T and A T application of external torques (lunar

Time Essentially three different kinds
of time are required to discuss the earth's rotation. Sidereal and universal time (UT) are both based on the earth's diurnal rotation. Sidereal time is defined by the angle through which the Greenwich meridian has turned past the vernal equinox (Fig­

will indicate hitherto unsuspected shortcomings in the theory of the moon's motion and/or possible nonN e w t o n i a n effects [Sadler, 1 9 6 8 ; Shapiro et al, 1 9 7 1 ; Oesterwinter and Cohen, 1 9 7 2 ] .
The most recent comprehensive treatment of the measurement of t i m e and polar m o t i o n is b y Mueller

and solar gravitational torques on the e q u a t o r i a l bulge, t h e bodily and ocean t i d e s , t h e solar wind) or (2) while its angular m o m e n t u m remains constant, b y a change in its inertia tensor ( e a r t h q u a k e s , sea level fluctua­ tions, rearrangement of the geograph­ ic d i s t r i b u t i o n of air mass) or by in­ ternal r e d i s t r i b u t i o n of its angular

ure 1). Universal time is related to sidereal t i m e b y an a d o p t e d f o r m u l a Fig. 3. Changes in length of day. (a) After G u i n o t [19 70]. (bj After Stoyko

(whose origins are now of purely his­

[1970].

torical interest) and is therefore an

equivalent direct measure of the

earth's axial spin. In practice, U T is

determined by meridian transits of

stars, and so the PZT provides simul­

taneous determinations of time and

latitude. Universal time corrected for

polar motion (Figure 1) is called

UT1. Since the earth's axial spin

shows periodic, irregular, and secular

changes, UT1 does not provide a uni­

form measure of time. Such a uni­

form reproducible time scale, called

atomic time (AT), has been made

available since 1955 by atomic fre­

quency standards accurate to 1 part

in 1 0 1 3 . These atomic 'clocks,' now

widely distributed at astronomical

observatories, are presently syn­ chronized by transport and side-by-

+ 500

side comparison, or less accurately

by radio signals from quartz crystal

oscillators carried in artificial satel­ o

lites, but remote synchronization

using VLBI techniques should soon

be feasible. The PZT observations,

coupled with atomic timekeeping,

from the participating stations are

processed and disseminated by the

BIH as U T 1 - A T . Changes in the earth's axial spin rate are therefore

-500-

-+500

obtained by differentiating UT1-AT with respect to (atomic) time.

(bi

1700

1800

1900

772

momentum (winds, core-mantle
coupling). The role of the core, almost com­
pletely enigmatic at t h e t i m e when Munk and MacDonald w r o t e their book, has since become m u c h more prominent. During the past decade a number of features of t h e earth's rotation spectrum, directly affected by the existence of t h e liquid core, have been identified w i t h varying d e g r e e s of c e r t a i n t y [Rochester, 1970]. The principal effective mech­ anisms by which angular m o m e n t u m can be transferred b e t w e e n t h e c o r e and mantle are:
(1) inertial coupling due to the hydrodynamic pressure forces that act over t h e ellipsoidal c o r e - m a n t l e boundary when internal flow is in­ duced in the liquid core b y any shift in the earth's r o t a t i o n axis [Toomre, 1966];
(2) electromagnetic coupling due to the operation of Lenz's law conse­ quent upon leakage of geomagnetic secular variation (GSV) i n t o t h e elec­ trically conducting lower mantle [Rochester, 1 9 6 0 , 1 9 6 8 ] . T h e GSV in turn is associated w i t h internal motions of the highly conducting core, driven by mechanisms appropri­ ately examined in t h e c o n t e x t of geo­ magnetic dynamo theory but closely connected with the earth's rotation.
A l t h o u g h there is as yet n o seismic evidence for (or against) small-scale (not m o r e t h a n a few kilometers high) topography on the core-mantle interface, Hide [ 1 9 6 9 ] has for other reasons p r o p o s e d a coupling mechanism t h a t may be comparable to or even stronger than the electromagnetic:
(3) t o p o g r a p h i c coupling, in which a stress on t h e m a n t l e is pro­ duced by the flow of t h e rotating core over any ' b u m p s ' o r depressions on the core-mantle b o u n d a r y .
Significant viscous friction at the core-mantle interface now seems most unlikely because of the very low molecular viscosity of t h e core estimated by Gans [ 1 9 7 2 ] and t h e apparent ineffectiveness of turbulent ( e d d y ) f r i c t i o n [Toomre, 1 9 6 6 ; Rochester, 1 9 7 0 ] .

uity, and spin causes the earth to respond to the gravitational attrac­ tion of the moon and sun by a steady precession of its rotation axis in space at a rate of 5 0 3 7 " / c e n t u r y (Figure 4). T h e earth is treated as a rigid body to deduce from this rate the value of one of the fundamental geophysical constants, its dynamical ellipticity H = (C - A)/C, where C and A are its axial and equatorial moments of inertia, respectively. The effect of a liquid core, first treated imperfectly by Hopkins in 1839 and then elegantly by Poincare in 1910, has been discussed most recently by Stewartson and Roberts [ 1 9 6 3 ] , Busse [ 1 9 6 8 ] , Gans [ 1 9 6 9 ] , and
POLARIS

Suess [ 1 9 7 0 ] . Even if V L B I can r e ­ duce the error in the measured lunisolar precession rate to ±0".l/ c e n t u r y in t h e n e x t d e c a d e [Shapiro and Knight, 1 9 7 0 ] , t h e effect of t h e liquid core (4 parts in 1 0 6 ) would remain u n d e t e c t a b l e . However, Shapiro and Knight point out that a more immediate result of this refine­ ment will be the observational isola­ tion of the general relativistic contri­ bution to precession ( ^ 1 .9/cen­ tury).
Nutation The orbital motions of the earth
and m o o n give rise to ripples on t h e precessional cone (Figure 4), the

STEADY

x

PRECESSION \

Precession The combination of dynamical el-
lipticity (the equatorial bulge), obliq­

Fig. 4. Precession and nutation.

773

largest of which, the principal nuta­ tion, is associated with the regression of the lunar orbit's line of nodes with a period of 18.6 years. It is an ellipti­ cal motion of the rotation axis, the semimajor axis (nutation in obliq­ uity) having an amplitude of 9".20. I n d e p e n d e n t l y , Jeffreys and Vicente [ 1 9 5 7 ] a n d Molodenskii [ 1 9 6 1 ] showed that allowing for mantle elas­ ticity and the liquid core (inertially coupled to the mantle) removed the discrepancy of 0".02 between the observed amplitude and that derived from theory assuming the earth to be rigid.
Melchior [ 1 9 7 1 ] h a s recently reviewed the dynamical effects of the liquid core on the long-period (18.6 and 9.3 years) and short-period (an­ nual, semiannual, and fortnightly) nutations. Annual nutation in obliq­ uity is due entirely to the presence of the core but is of small amplitude ( ~ . 0 " . 0 0 6 ) . Satisfactory observa­ tional confirmation of the correc­ tions required by a deformable earth model has already been obtained by conventional astronomical tech­ niques, according to Melchior (see also Wako [ 1 9 7 0 ] ) . Clearly, t h e n e w methods of VLBI and laser ranging to the moon promise more discrimi­ nating tests of the earth models adopted by theoreticians.
'Secular' Decrease in Obliquity The observed 'secular'
(^40,000-year period) decrease in the obliquity (Figure 4), at a rate ^ 4 7 " / c e n t u r y , can be almost, if not entirely, accounted for by the gravi­ tational perturbations of the ecliptic by the other planets. Earlier analyses indicating a difference of ^0".3/century between calculated and ob­ served rates have been questioned [Lieske, 1 9 7 0 ; Fricke, 1 9 7 1 ] , a n d it now appears unlikely that any real discrepancy can exceed ^ 0 ' \ l / c e n ­ t u r y [Fricke, 1 9 7 2 ] . Shapiro and Knight [ 1 9 7 0 ] suggest t h a t a d e c a d e of VLBI and timing pulsar signals might suffice to determine whether such a discrepancy is real.
Aoki [ 1 9 6 9 ] has p r o p o s e d t h a t frictional coupling of the core to the precessing mantle can cause a rota­ tion of the equator in space, in the sense of reducing the obliquity, at approximately the rate indicated by

t h e a b o v e possible discrepancy. could exist near the Chandler peak

Kakuta and Aoki [ 1 9 7 2 ] claim t o for a n y r e a s o n a b l e m o d e l of the

have r e m o v e d certain o b j e c t i o n a b l e e a r t h ' s i n t e r i o r . T h e ILS data are suf­

features of Aoki's earlier model b y ficiently i n h o m o g e n e o u s and the rec­

taking into account electromagnetic ord length short enough that ordi­

coupling of the mantle t o a liquid nary spectral analysis cannot with

core, b u t t h e p r o b l e m is c o m p l e x a n d c o n f i d e n c e resolve t h e question of

far from being satisfactorily solved. w h e t h e r t h e y exist [Pedersen and

It is, in fact, p a r t of a m u c h larger Rochester, 1 9 7 2 ] . When they are

problem of absorbing interest in con­ analyzed b y Burg's maximum en­

nection with the possible operation tropy m e t h o d , neither the ILS data

of t h e geomagnetic d y n a m o b y stir­ [Claerbout, 1 9 6 9 ] n o r t h e BIH data

ring u p t h e core b y t h e differential [Smylie et al, 1 9 7 3 ] yield any evi­

precessional torque arising from the dence for splitting of the Chandler

25% difference between t h e elliptici- peak. T h e trouble with Q may not be

ties of t h e earth's o u t e r surface a n d real, a n y w a y , since t h e oceans have

t h e core-mantle b o u n d a r y [Malkus, n o t b e e n e l i m i n a t e d as a possible sink

1 9 6 3 ; Gans, 1 9 6 9 ; Busse, 1 9 7 1 ; for w o b b l e energy [Munk and Mac-

Stacey, 1 9 7 3 ] .

Donald, 1 9 6 0 ; Lagus and Anderson,

1 9 6 8 ; Miller, 1 9 7 3 ] . It m a y be worth

Chandler Wobble

n o t i n g t h a t Hendershott [1972] gets

T h e 7 0 years of systematic lati­ g ^ 3 5 for t h e oceans at the semi­ t u d e observations using optical diurnal period.

astronomy have not yet proved ade­

quate to resolve unambiguously the

spectrum of polar motion. The prin-

cipal features are the 14-month

(Chandler) and annual wobbles. The

Chandler wobble is the torque-free

Eulerian wobble for a uniaxial rigid

earth with the period lengthened to

^ 4 3 5 days by allowing for the liquid

core, elastic mantle, and the mobility

and loading of the oceans. The spec­

tral peak at the Chandler frequency

is broad and conventionally inter­

preted as indicating a more or less

randomly excited oscillation damped

with a relaxation time of the order of

1 0 - 2 5 years [Jeffreys, 1 9 6 8 ; Man­

delbrot and McCamy, 1 9 7 0 ] . T h e

value of Q ( ^ 3 0 - 6 0 ) thus indicated

has seemed anomalously low if t h e

damping is to be attributed entirely

to anelasticity of the mantle a n d if

Q is rather independent of frequency

[Stacey,

1 9 6 7 ] . Rochester and

Smylie [ 1 9 6 5 ] s h o w e d t h a t e l e c t r o ­

magnetic core-mantle coupling failed

by a factor of at least 1 0 4 to provide

the necessary damping.

C o n t i n u e d observation of the Chandler wobble, even at increasing amplitude from time to time, in the presence of such strong damping, p o i n t s t o an efficient excitation m e c h a n i s m . Amplification of the C h a n d l e r r e s o n a n c e b y sidebands of the annual variation in atmospheric mass d i s t r i b u t i o n is far t o o small [Munk and Hassan, 1 9 6 1 ] . Earth­ q u a k e s , dismissed by M u n k and MacD o n a l d , h a v e b e e n revived in a series of p a p e r s beginning with the one by Mansinha and Smylie [ 1 9 6 7 ] . The far-field displacements accompanying a major e a r t h q u a k e , calculated by the elasticity theory of dislocations in a spherical e a r t h , change the offdiagonal c o m p o n e n t s of the inertia tensor a n d t h u s shift t h e earth's pole of figure ( m e a n pole of epoch). In­ d e p e n d e n t f o r m u l a t i o n s of the theory for a self-gravitating earth model w i t h liquid core and realistic distributions of density and elastic properties in t h e mantle have been g i v e n b y Smylie and Mansinha [ 1 9 7 1 ] , Dahlen [ 1 9 7 1 , 1 9 7 3 ] , and

Colombo and Shapiro [ 1 9 6 8 ] have argued that the variable ampli­ tude of the Chandler wobble is strik­ ingly suggestive of a beat between t w o resonant periods within the Chandler band separated by roughly 10 days and having much sharper peaks, so as to remove the apparent p r o b l e m with Q. I t is difficult t o see h o w t w o such close frequencies

Israel et al [ ] 9 7 3 ] . T h e i r theoretical t r e a t m e n t s differ in detail and have given rise t o a small controversy over t h e p h y s i c a l principles governing static d e f o r m a t i o n of t h e liquid core ( s e e a l s o Jeffreys and Vicente [ 1 9 6 6 ] a n d Pekeris and Accad [ 1 9 7 2 ] ) . H o w e v e r , t h e effect of dif­ fering p r e s c r i p t i o n s of boundary con­ ditions a t t h e core-mantle interface is

774

likely to be small, and t h e authors enon in the observations: a change of

generally agree in concluding that a the order of 0 " . l in arc radius taking

major e a r t h q u a k e can p r o d u c e a place in a y e a r or t w o [Guinot,

polar shift of the order of 0".l. H o w ­ 1 9 7 2 ] . The details of electromag­

ever, Mansinha and Smylie [ 1 9 7 0 ] netic coupling on such a short t i m e

and Dahlen [ 1 9 7 1 ] disagree o n scale have n o t been fully w o r k e d o u t ,

whether the cumulative effect of all partly because the high-frequency

earthquakes is enough t o sustain t h e GSV is screened from o u r observa­

Chandler wobble. The sources of dis­ tion by the electrical conductivity in

agreement have not yet been con­ the lower mantle that provides the

clusively identified b u t p r o b a b l y lie coupling. B u t Kakuta [ 1 9 6 5 ] con­

primarily in the different ways in cluded that magnetohydrodynamic

which the authors relate the excita­ oscillations in the core could n o t ex­

tion due t o a p a r t i c u l a r e a r t h q u a k e c i t e d e t e c t a b l e w o b b l e . Stacey

to its seismic character.

[1970] uses a quasi-dynamical argu­

Ideally, one would like to test the . hypothesis by matching a change in the pole path with the occurrence of a major earthquake and the shift in the pole of figure predicted by elastic dislocation theory from the earth­ quake's location and associated fault geometry [Smylie and Mansinha,

ment to estimate how much energy could be fed into the mantle wobble from t h e differential precession torque on the core through a non­ linear electromagnetic coupling m e c h a n i s m . The proposal is in­ triguing but needs to be given a more rigorous formulation.

1968]. However, the data are so

In an interesting translation of

noisy that such a t t e m p t s have so far geophysics into an astronomical con­

been inconclusive [Haubrich, 1 9 7 0 ; t e x t , starquakes have b e e n offered as

Dahlen, 1 9 7 1 ] , a n d w e m u s t await a a n e x p l a n a t i o n of pulsar w o b b l e

great i m p r o v e m e n t in t h e accuracy [Pines and Shaham, 1 9 7 3 ] , a n d its

with which wobble is m o n i t o r e d .

damping has been discussed in terms

Modeling of seismic effects on the of various mechanisms in the core

inertia tensor b y s u d d e n dislocations and m a n t l e of a n e u t r o n star [Chau

leaves so far u n e x a m i n e d t h e possible and Henriksen, 1 9 7 1 ] .

effects of creep on polar m o t i o n

[Chinnery, 1 9 7 0 ] .

Seasonal Wobbles

During the past decade, excitation

The amplitude of the annual (and

of mantle w o b b l e b y e l e c t r o m a g n e t i c m u c h smaller semiannual) w o b b l e

coupling to the core has been shown can be sufficiently well explained by

to be utterly i n a d e q u a t e b y Roches­ t h e seasonal variation in t h e geo­

ter and Smylie [ 1 9 6 5 ] , w h o t o o k graphic distribution of t h e mass of

step function t o r q u e s t o be suffi­ t h e a t m o s p h e r e [Munk and Hassan,

ciently optimistic, and has been 1961], although the observed phase

resuscitated, on quite different requires some additional excitation

grounds, by Runcorn and Stacey. (snowfall, groundwater), according

Runcorn [ 1 9 7 0 a ] c o n t e n d s t h a t to Jeffreys [ 1 9 7 2 ] .

high-frequency GSV creates the core

It has been customary to separate

equivalent of sunspots at the core- the Chandler wobble from the lati­

mantle boundary and thus supplies tude data by removing an annual

an impulsive t o r q u e t o t h e m a n t l e w o b b l e t h a t is c o n s t a n t in a m p l i t u d e

that can transfer angular m o m e n t u m and phase from year to year, deter­

rapidly enough to sustain the Chan­ mined by a least squares fit. The

dler wobble. E a r t h q u a k e s leave t h e s u s p i c i o n t h a t changing weather

i n s t a n t a n e o u s r o t a t i o n pole u n ­ patterns from year t o year would dif­

changed but shift the axis of figure, ferentially drive the annual wobble is

so that t h e pole p a t h experiences a confirmed b y t h e analysis of Chollet

discontinuous change in direction. and Debarbat [ 1 9 7 2 ] , w h o find its

Impulsive torques, on the other amplitude to vary between 0".04 and

hand, leave the axis of figure un­ 0".10 over a 14-year series of obser­

changed and shift the r o t a t i o n pole, vations at Paris. The point is rein­

so that t h e radius of t h e p o l e p a t h is f o r c e d b y Wells and Chinnery

changed discontinuously. There is [ 1 9 7 2 ] , w h o find the annual wobble

some support for t h e l a t t e r p h e n o m ­ to be b u t p o o r l y d e t e r m i n e d from

the IPMS latitude data and conclude that it cannot be well separated from the Chandler wobble by the custom­ ary m e t h o d . Guinot [ 1 9 7 2 ] , h o w ­ ever, finds 'quiet' intervals of a few years over which the annual wobble is nearly constant.
Other short-period terms in the ILS data with very small amplitudes are probably forced wobbles of m e t e o r o l o g i c a l origin [Sugawa et al, 1972].

Secular Motion of the Pole

The ILS data also reveal an irregu­

lar drift of the pole from its mean

position 70 years ago in a rather slug­

gish sort of 'Brownian' motion that

h a s c a r r i e d it altogether ^ 0 " . 2

towards Newfoundland in that time

[Yumi and Wako, 1 9 7 0 ; Mandelbrot

and McCamy,

1 9 7 0 ; Mikhailov,

1972]. The observed secular motion

of the pole may be contaminated by

as yet unresolved nonpolar latitude

variations due to continental drift

[Arur and Mueller, 1 9 7 1 ] . T h e r e

seems little reason to doubt that this

secular motion is the cumulative re­

sult of changes in the inertia tensor

d u e t o sea level f l u c t u a t i o n s [Munk

and MacDonald, 1 9 6 0 ] a n d t e c t o n i c

p r o c e s s e s [Mansinha and Smylie,

1 9 7 0 ] . Batrakov [ 1 9 7 2 ] estimates

that a gigantic engineering project

proposed to turn the flow of Siberian

rivers southward will, through a re­

distribution of groundwater, shift the

pole by no more than 0".014.

Long-Period Wobbles Markowitz [ 1 9 7 0 ] adduces empir­
ical evidence from the ILS data for a 24-year period w o b b l e , which Busse [1970] has suggested may represent the response of the mantle to wobble of the solid inner core inertially coupled to the mantle via the liquid c o r e . Rykhlova [ 1 9 6 9 ] , using a longer but less homogeneous record, finds evidence for a 40-year period instead. This may be the 'Markowitz w o b b l e ' with the period poorly determined because of contamina­ tion from the secular motion. But McCarthy [ 1 9 7 2 ] also finds from latitude observations at Washington a 'period' somewhat longer than Markowitz's. If it is real, the phenomenon may well be the only observable manifestation of the presence of the

775

solid inner core in the entire spec­ trum of changes in the earth's rota­ tion.
Nearly Diurnal Free Wobble Perhaps the most intriguing wob­
ble mode is the torque-free nearly diurnal polar motion made possible in principle by the presence of the liquid core inertially coupled to the mantle, first predicted in 1896 inde­ p e n d e n t l y by both Hough and Sludskii. T h e predicted m o t i o n is ret­ rograde about the axis of figure with a period about 3 min short of a side­ real day, according to the (slightly different) e a r t h m o d e l s of Jeffreys and Vicente [ 1 9 5 7 ] and Molodenskii [ 1 9 6 1 ] . Besides giving a resonance amplification to the nearly diurnal tides, this wobble mode will appear in observations of latitude and time (UT) as a period (relative to the stars) of 4 6 4 sidereal days or 204 mean solar days, according to Molodenskii's models. The most recent discussions of the observational evi­ d e n c e are t h o s e b y Sugawa and Ooe [ 1 9 7 0 ] , Popov and Yatskiv [ 1 9 7 1 ] , and Debarbat [ 1971 ] . If this w o b b l e could be unambiguously identified, its period would serve as a fairly stringent filter for earth models. However, its amplitude ( ^ 0 " . 0 2 , ac­ cording t o Popov) is at noise level, and there is reason to be suspicious of this value, since it must be accom­ panied by a nutation hundreds of times larger (A. Toomre, private communication, 1973).
Other small (^0".02) wobbles are forced by the sun and moon. These are the Oppolzer terms due to de­ parture of the axis of rotation from the figure axis during nutation (dis­ c u s s e d b y Takagi and Murakami [1968]).
Short-Period Changes in Length of Day
Although the seasonal variations in the length of day were detected by Stoyko during the 1930's by pendu­ lum clocks, the short-period changes naturally show up much better in UT1-AT, available since 1955. The annual variation (amplitude ^20—25 m s e c ) is primarily explained by winds and the semiannual variation (amplitude ^ 9 msec) by the solar bodily tide, small additions to

changes in the axial moment of in­ ertia being contributed by the sea­ s o n a l redistribution of air mass, ocean load, groundwater, snow, and vegetation [Munk and MacDonald, I960]. The discrepancies in ampli­ tude and phase between the seasonal fluctuations deduced by different workers [Fliegel and Hawkins, 1961 \ Challinor, 1971] reflect t h e y e a r - t o year variability in the excitation m e c h a n i s m s . Frostman et al [ 1 9 6 7 ] concluded that there are still large unexplained differences in phase be­ tween theoretical and observed sea­ sonal variations. This objection appears to have been entirely re­ m o v e d by Lambeck and Cazenave [1973], who use much better mete­ orological data to calculate the sea­ sonal fluctuations in atmospheric angular momentum.
Nordtvedt and Will [ 1 9 7 2 ] p o i n t out that theories of gravitation in­ volving a preferred reference frame predict anisotropics in the gravita­ t i o n a l constant that change the earth's moment of inertia during its orbital motion and give rise to small annual and semiannual changes in the length of day. These effects are likely to be indistinguishable from the meteorological effects at the level of accuracy with which the latter are known. At present, all one can d o is use the uncertainty in the extent to which the known meteorological and hydrological excitations can account for the observed seasonal variations in the length of day to set upper limits on the relevant parameters in nongeneral relativistic theories.
A 9-msec amplitude biennial term in the length of day was first re­ ported by Iijima and Okazaki in 1966 and is presumably related to the 26-month atmospheric oscilla­ tion. For reports of other shortperiod variations (of possibly mete­ orological origin) see the papers by Korsun' and Sidorenkov [ 1971 ] a n d Iijima and Okazaki [ 1 9 7 2 ] .
Atomic timekeeping now permits unequivocal observation of the small (<l-msec amplitude) fortnightly and monthly lunar tidal variations in the length of day [Guinot, 1 9 7 0 ] .
Irregular Fluctuations in Axial Spin Rate
Subtracting from UT 1 an adopted value for the seasonal variation in the

length of day gives U T 2 . It is impor­ tant t o n o t e that UT2 will still con­ tain s o m e small meteorological ef­ fects, since the seasonal fluctuations change in amplitude and phase from year to year. Irregular changes in the length of day show up in UT2-AT since 1955 and in ET-UT over the last three centuries (Figure 3). The c o r r e s p o n d i n g rotational accelera­ t i o n s have b e e n reviewed by Marko­ witz [ 1 9 7 0 , 1 9 7 2 ] . A change of 1 msec in t h e length of day is about 1 part in 1 0 8 , so that t h e data from Figure 3 divide roughly into three categories: (1) changes of a few milli­ s e c o n d s over several decades or longer (accelerations in the spin rate of < 5 X 1 0 _ 1 0 / y r ) , ( 2 ) changes of a few milliseconds over a few years to a decade (accelerations <80 X 1 0 " 1 0 / y r ) , t h e so-called 'decade fluc­ t u a t i o n s , ' and (3) changes of a sub­ stantial fraction of a millisecond over a few weeks or months (accelerations < 5 0 0 X 1CT1 ° / y r ) , t h e most rapid of t h e s e being the 'abrupt' changes [Guinot, 1 9 7 0 ] .
Changes in this last category were not registered in UT until the atomic clocks came into use. Presumably, VLBI and laser ranging to the moon will e n a b l e t h e t i m e intervals over which such changes can be detected t o be w h i t t l e d d o w n to a fraction of a day, and improved global meteoro­ logical d a t a will be used to test w h e t h e r any appeal t o the core must be made to explain such short-term irregularities.
E l e c t r o m a g n e t i c torques have been s h o w n t o be just barely ade­ quate to transfer angular momentum b e t w e e n t h e core and the mantle at the rate necessary to account for the accelerations characteristic of the d e c a d e f l u c t u a t i o n s [Rochester, 1 9 6 0 ; Roden, 1 9 6 3 ; Kakuta, 1965; Roberts, 1 9 7 2 ] . Vestine and Kahle [1968] cite evidence for this mecha­ nism in t h e correlation of changes in the length of day with changes in the w e s t w a r d drift of a prominent geo­ magnetic field constituent during the last 80 years.
T h e m o r e r a p i d and abrupt changes in t h e length of day are m u c h m o r e likely to be explained by winds. T h e y c a n n o t be explained by electromagnetic coupling unless the e 1 e c trical conductivity approaches 1 0 3 m h o s / m at t h e very bottom of

776

the mantle and there is sufficient power in the GSV at high frequen­ cies, say, > 1 cycle/yr. T o p o g r a p h i c coupling may play some role. After a checkered history of claims for de­ tectable effects from the solar wind torque, it appears that this source can be neglected [Coleman, 1 9 7 1 ; Hirshberg, 1 9 7 2 ] .
The slower changes, of type 1, are readily accounted for b y electromag­ netic coupling to the long-period GSV in t h e core [Rochester, 1 9 7 0 ] . Braginskii [ 1 9 7 0 ] a n d Wilhelm [1970] have calculated the long-term changes in the length of day associ­ ated with particular features of t h e GSV. Munk and MacDonald [ 1 9 6 0 ] argue that changes in sea level do n o t contribute significantly to fluctua­ tions in the spin rate on this time scale because t h e r e is n o i n d i c a t i o n of the c o n c o m i t a n t p o l a r m o t i o n over the last 7 0 years. T h e same will not be true of sea level changes on a time scale of many centuries or millenia.
Secular Acceleration of Earth's R o t a t i o n
The major contributor to secular change in the length of day is tidal friction, which transfers earth's spin angular m o m e n t u m t o t h e lunar orbit and thus gives t h e m o o n an angular acceleration h in space. Until re­ cently, the accepted value for the present era was h ^ - 2 2 " / c e n t u r y 2 > determined over 30 years ago by Spencer Jones from telescopic obser­ vations of the sun, m o o n , and planets over t h e p r e v i o u s 7lA centuries (ET-UT). It now appears that errors in the poorly d e t e r m i n e d early obser­ vations could change his value by ±100%. More recent values were de­ termined by (1) Newton [ 1 9 6 8 ] , who found h ^ - 2 0 ± 3 " / c e n t u r y 2 from a few years' tidal p e r t u r b a t i o n s of artificial earth satellite o r b i t s (ac­ cording to Newton [ 1 9 7 2 A ] , t h e systematic errors here may be much larger t h a n i n d i c a t e d ) , ( 2 ) Van Flandern [ 1 9 7 0 ] , w h o o b t a i n e d n ^ - 5 2
± 16'7century2 f r 0 m 15 years of
lunar occultations t i m e d against A T and made u n u s u a l l y g e n e r o u s allow­ ances for systematic error, and (3) Oesterwinter and Cohen [ 1 9 7 2 ] , who arrived at h O=L - 3 8 ± 8"/cen~ tury2 by fitting the last 60 years of

l u n a r and planetary observations between recorded locations of solar

against UT and AT.

eclipses and those predicted by as­

suming zero accelerations between

If the sun's tide on t h e earth is t h e n and n o w . Other kinds of data,

taken into account, the total gravita­ even less trustworthy, are used to

tional tidal acceleration of the earth's separate h and co. The necessary in­

spin is given by co/co ^ 1.16/2 X t e r p r e t a t i o n of sources, almost al­

10"9/century, where the coefficient ways at second- or third-hand, has

is uncertain to within a few percent given rise to energetic controversy.

owing to uncertainty in the knowl­ U n t i l recently, a rather limited

edge of the ratio of the lunar to the b o d y of data, m u c h of it of du­

solar tidal torque (various models bious reliability, was worked over

range from 3.5 to 4.7). Thus the by different investigators in as m a n y

'modern' rate of secular deceleration different ways with varying weight­

due to tidal friction is probably close ings according to their individual

t o twice t h e value used b y Munk and assessments, so t h a t even essentially

MacDonald [ I 9 6 0 ] . T h i s in t u r n t h e s a m e data could be used t o give

nearly d o u b l e s t h e p r o b l e m of ac­ w i d e l y d i v e r g e n t results. Curott

counting for t h e a c c o m p a n y i n g ener­ [ 1 9 6 6 ] and Dicke [ 1 9 6 6 ] b o t h

gy dissipation (^3.5 X 1 0 1 2 watts, assumed Spencer Jones's value for h

according to Munk and MacDonald). to be valid over the last 3000 years in

The shallow seas have long been regarded as the chief sink for tidal energy. Miller [ 1 9 6 6 ] f o u n d t h a t t h e y could dissipate 1.7 X 1 0 1 2 watts (±50%). Degradation by scat­ tering into internal modes in the ocean is probably < 0 . 5 X 1 0 1 2 watts [Cox and Sandstrom, 1 9 6 2 ; Munk, 1966]. The contribution by bodily tides in the solid earth is probably

o r d e r t o deduce values for co f r o m eclipse data. A major contribution to t h e subject has b e e n m a d e b y New­ ton [ 1 9 7 0 , 1912b], w h o amassed and extensively discussed the relia­ bility of a much larger body of data and found n ^ - 4 2 ± 6,,/century2 over the last 2000 years, in agree­ ment with the more recent 'modern' observations cited above.

not more than a few percent of the whole [Munk, 1 9 6 8 ] . M o r e r e c e n t l y , Hendershott [ 1 9 7 2 ] and Pariiskii et al [ 1 9 7 2 ] have used cotidal charts and, taking ocean loading into ac­ count, estimated dissipation in the world ocean and thus obtained values roughly double Miller's (see also Brosche and Silndermann [ 1 9 7 2 ] ) . " An overall dissipation rate of ^3—5 X 1 0 1 2 watts can be inferred from t h e r o u g h e s t i m a t e of average (oceanic and bodily) tidal phase lag from gravimetry [Smith and Jungels, 1970]. The position appears to be

Newton's analysis gives a nontidal acceleration of the earth's spin co/co ^ 20 X 10^/century over the last 2 0 0 0 years or so. Dicke [ 1 9 6 6 ] in­ vestigated most of the presently con­ ceivable mechanisms for explaining the acceleration of about half this amount given by his analysis and found that the largest contribution was from the postglacial rise in sea level and the accompanying isostatic adjustment. After other geophysical mechanisms were examined and dis­ missed as much smaller, the bulk of the remaining acceleration was at­

that there are large uncertainties, but tributed to the effect (on ET and on

ocean tidal friction appears likely to the earth's axial m o m e n t of inertia)

meet the bulk of the dissipation re­ of a decrease in the gravitational con­

quirements posed by the lunar accel­ stant G with time predicted by cer­

eration.

t a i n theories of gravitation. The

u p p e r limit to \&\ set b y this reason­

Records of positions and times of ancient eclipses and of other events involving celestial bodies provide in­ f o r m a t i o n o n t h e average values of co and h over large segments of histori­ cal time. The best determined quan­ t i t y is an e p o c h a l average of co -0.622rc, obtained from discrepancies

ing is an order of magnitude smaller t h a n t h a t o b t a i n e d by Shapiro et al [ 1971 ] from radar and optical obser­ vations of planets since the advent of A T . Later, Dicke [ 1 9 6 9 ] reversed his argument, assumed a contribution due to G and inferred the average viscosity in the deep mantle by at-

777

tributing t h e rest of t h e n o n t i d a l co to the rise in sea level and isostatic recovery following deglaciation. O'Connell [ 1 9 7 1 ] has e s t i m a t e d t h e viscosity profile in the mantle by re­ garding the entire nontidal accelera­ tion as due to the latter processes.
The rise in sea level assumed by these authors is within the limits set by Walcott's [ 1 9 7 2 ] recent s t u d y of postglacial eustatic changes. But their interpretations neglect the possibility of substantial contributions from other geophysical effects. The longperiod GSV indicates that the core should be able to store or provide angular momentum on a millenial t i m e scale [Sekiguchi, 1 9 5 6 ; R o c h e s ­ ter, 1 9 7 0 ] . E l e c t r o m a g n e t i c corem a n t l e coupling, limited by Dicke [1966] to much smaller effects by inadequate arguments and rather c a v a l i e r l y dismissed by Newton [1970, 1972#], was studied in detail by Yukutake [ 1 9 7 2 ] . He f o u n d t h a t the 8000-year period change in the geomagnetic dipole moment (re­ vealed by archeomagnetism) would give rise to an average co/co ^ 5 X
10"9/century over the past 2000 y e a r s . A l s o , Gjevik [ 1 9 7 2 ] has argued that surface readjustments due to subcrustal phase transitions may mimic postglacial rebound in amplitude and relaxation time.
During the past decade, beginning w i t h t h e w o r k of Wells [ 1 9 6 3 ] , ef­ forts have been made to extend esti­ m a t e s of co and h back over geologic time by using the fossil clocks pro­ vided by marine organisms whose shell structures show daily, monthly, and annual ridge patterns. The data tend to support an increase in the length of day since the Precambrian at an average rate more compatible with Spencer Jones's value for h than with more recent astronomical deter­ minations of the 'modern' value [Runcorn, 1 9 7 0 6 ] . T h e r e is s o m e evidence for changes in t h e tidal de­ celeration r a t e [Pannella et al, 1 9 6 8 ; Pannella, 1 9 7 2 ] . In view of t h e pos­ sibility that the distribution of shal­ low seas was very different in the past, such changes would hardly be surprising. However, the uncertain­ ties in the determinants of ridge growth are so great that it seems pre­ mature to draw any detailed conclu­

sions from paleontological data re­ garding the history of the length of day. The most recent review is by Scrutton and Hip kin [ 1 9 7 3 ] .
Conclusion
This survey of current knowledge and problems of the earth's rotation is necessarily rough and superficial, and I can only hope that it conveys something of the compelling attrac­ tion that this global subject exerts on its devotees. There seems little doubt that the coming decade will eclipse even the enormous strides that were taken during the 1960's in the ac­ quisition, accuracy, and analysis of data and bring much closer to resolu­ tion several of the tantalizing ques­ tions still presented by the rotation of t h e earth.
Acknowledgment
Financial support of this work from a National Research Council of Canada op­ erating grant is gratefully acknowledged.
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780

Michael G. Rochester is Professor of Phys­ ics at the Memorial University of Newfound­ land, St. John's, Newfoundland, Canada. He received his B.A. in applied mathematics from the University of Toronto in 1954, his M.A. in physics from Toronto in 1956, and his Ph.D. in physics from the University of Utah in 1959. He has held faculty appoint­ ments at the University of Toronto and the University of Waterloo, and has been at Memorial since 1967. He is a member of Canada's National Research Council Associ­ ate Committee on Geodesy and Geophysics, and of the International Geodynamics Pro­ ject's Working Group on Physical Processes in the Earth's Interior.

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