J. Phys. Earth, 38, 475-491, 1990

Tidal Rhythmites: Key to the History of the Earth's

Rotation and the Lunar Orbit

George E. Williams

Department of Geology and Geophysics, University of Adelaide,

GPO Box 498, Adelaide, South Australia 5001, Australia

The recent recognition of cyclically laminated tidal rhythmites provides a new

approach to tracing the dynamic history of the Earth-Moon system. Late Proterozoic

(•`650 Ma) elastic rhythmites in South Australia represent an unsurpassed palaeotidal

record of •`60 years' duration that provides numerous palaeorotational parameters. At

•`650 Ma there were 13.1•}0.1 lunar months/year, 400•}7 solar days/year, and 30.5•}

0.5 solar days/lunar month. The lunar apsides and lunar nodal cycles were then 9.7•}

0.1 years and 19.5•}0.5 years, respectively. The indicated mean Earth-Moon distance

of 58.28•}0.30 Earth radii at •`650 Ma gives a mean rate of lunar retreat of

1.95•}0.29 cm/year since that time, about half the present rate of lunar retreat of

3.7•}0.2 cm/year obtained by lunar laser ranging. The rhythmite data imply a substantial

obliquity of the ecliptic at •`650 Ma, and indicate virtually no overall change in the

Earth's moment of inertia, which militates against significant Earth expansion since

•`650 Ma. Early Proterozoic (•`2,500 Ma) cyclic banded iron-formation in Western

Australia, that may record submarine fumarolic activity triggered by earth tides, suggests

•`14.5•}0.5 lunar months/year and a mean Earth-Moon distance of •`54.6 Earth radii

at •`2,500 Ma. The combined rhythmite data suggest a mean rate of lunar retreat of

•`1.27 cm/year during the Proterozoic (•`2,500-650 Ma); the indicated increasing mean

rate of lunar retreat since •`2,500 Ma is consistent with increasing oceanic tidal

dissipation as the Earth's rotation slows. A close approach of the Moon during earlier

time is uncertain. Continued study of tidal rhythmites promises to further illuminate

the evolving dynamics of the Earth-Moon system.

1. Introduction

Investigation into the Earth's palaeorotation and the past lunar orbit through the

analysis of growth increments in marine invertebrate fossils has made little headway in

the past decade because of uncertainty as to the tidal affinities and reliability of the

various fossil structures. Cyclically laminated and thin-bedded rhythmites of tidal origin,

recently recognized in Proterozoic (Williams, 1988, 1989 a, b, c, 1991), Phanerozoic (for

example, Broadhurst, 1988; Tessier and Gigot, 1989) and modern (Dalrymple and

Makino, 1989; Tessier et al., 1989) deposits, provide the timely prospect that sedimentary

rocks independently may record a wide range of palaeotidal and palaeorotational data.

Received November 1, 1990; Accepted February 6, 1991

475

476 G. E. Williams

Because the periodicities displayed by such rhythmites usually can be ascribed to tidal

pattern and type and because the rhythmite sequences may span many years, the study

of tidal rhythmites has rejuvenated and extended geochronometric analysis of the ancient

Earth-Moon system.

This paper reviews clastic tidal rhythmites of Late Proterozoic age from the Adelaide

Geosyncline in South Australia, and cyclic banded iron-formation of possible earth-tidal

origin from the Early Proterozoic in Western Australia. Emphasis is placed on their

encoded palaeotidal periods and the important implications they may carry for the

Earth's palaeorotation and the dynamic evolution of the Earth-Moon system.

2. Late Proterozoic Tidal Rhythmites

Late Proterozoic cyclic rhythmites of proposed tidal origin in South Australia

(Williams, 1987, 1988, 1989 a, b, c, 1991) are best seen in the Elatina Formation at Pichi

Richi Pass in the Flinders Ranges (latitude 32•‹25•ŒS, longitude 137•‹59•ŒE) and in the

correlative Reynella Siltstone along coastal sections at Hallett Cove near Adelaide some

300 km to the south (latitude 35•‹05•ŒS, longitude 138•‹29•ŒE). Both these formations were

deposited in the Adelaide Geosyncline during the Marinoan Glaciation •`650 Ma ago

(Preiss, 1987).

The rhythmites comprise graded (upward-fining) laminae •ƒ 0.2 mm to 2 cm thick

of very fine-grained sandstone, siltstone and mudstone. The laminae typically are

grouped in •glamina-cycles•h (Fig. 1) that range from •`1 mm to more than 6 cm in

thickness and contain from 8 to 26 or more laminae. Lamina thickness is maximal near

the stratigraphic centre of lamina-cycles, and minimal near cycle boundaries where

thinner, more clayey laminae may crowd together to form conspicuous darker, clayey

bands.

The laminae are interpreted as diurnal increments, some of which contain

semidiurnal sublaminae, and the lamina-cycles as groupings of such increments that

represent the lunar fortnightly cycle. Thicker laminae near the stratigraphic centre of

lamina-cycles are ascribed to the spring phase of the tidal cycle, and the clayey bands

bounding lamina-cycles are interpreted as mud drapes deposited in the quieter waters

at neap tides. Lamina-cycles commonly are abbreviated because of non-deposition of

clastic laminae near neaps. Longer tidal cycles are recorded by systematic vertical change

in lamina-cycle thickness.

The rhythmites of the Reynella Siltstone and Elatina Formation evidently were

deposited in littoral settings ranging from estuarine to distal ebb-tidal delta. Periodic

changes in the thickness of semidiurnal and diurnal laminae and of lamina-cycles reflect

variations in the amount of fine elastic material entrained and deposited by tidal currents

in response to periodic changes in the height, velocity and range of palaeo-tides. Such

tidal influence on sedimentation is clearest for thythmites of distal ebb-tidal origin, as

exemplified by the •`10-m-thick rhythmite member of the Elatina Formation at Pichi

Richi Pass. The depositional environments of the rhythmites are discussed in detail

elsewhere (Williams, 1989 a, c, 1991).

J. Phys. Earth

Tidal Rhythmites 477

Fig. 1. Late Proterozoic (•`650 Ma) tidal rhythmites, South Australia. Clayey

material appears darker than sandy to silty layers. Scale bars are

1 cm. (a) Elatina Formation from Pichi Richi Pass. Four fortnightly

lamina-cycles each comprising about 10 to 14 graded (upward-fining), diurnal

laminae are bounded by thin clayey bands. (b) Reynella Siltstone from

Hallett Cove, showing one thick, fortnightly lamina-cycle that contains 14

diurnal laminae of fine-grained sandstone each with a clayey top. Most diurnal

laminae in (b) show sublaminae of semidiurnal origin.

3. Late Proterozoic Orbital Parameters

3.1 Palaeotidal and palaeorotational values Stratigraphic series of lamina and lamina-cycle thickness measurements obtained

from drill core of the Elatina rhythmite member contain strong periodicities that are evident visually (Fig. 2) or revealed by Fourier spectral analysis (Fig. 3) (Williams, 1988, 1989 a, b, c, 1991). These data represent an unsurpassed palaeotidal record of about 60 years' duration. Supplemented by observations from the Reynella Siltstone, they give palaeotidal and palaeorotational values (Table 1) that are more accurate and numerous than values previously obtained for any geological interval.

The estimated number of lunar days per lunar month was based largely on observations of the Reynella siltstone. Diurnal laminae and semidiurnal sublaminae are readily distinguishable in the thick lamina-cycles from this formation (Fig. 1(b)). Accurate counts of identifiable diurnal laminae in apparently unabbreviated lamina-cycles were employed, and the values of 14-15 diurnal laminae per lunar

Vol. 38, No. 6, 1990

478 G. E. Williams

Fig. 2. (a-c) Thickness of fortnightly lamina-cycles from the Elatina series

(lamina-cycle number increases up-sequence). (a) Unsmoothed curve;

first-order peaks define yearly maxima in thickness. (b) Smoothed curve

(5-point filter, weighted 1, 4, 6, 4, 1), showing the solar year. (c) Residual

curve (a minus b) showing the •gtidal year•h; the vertical lines mark 180•‹

phase-reversals in the sawtooth pattern. (d-f) Tidal patterns for Townsville,

Queensland. (d) Maximum height of the fortnightly tidal cycle from October

19, 1968, to June 3, 1970. (e) Smoothed curve (5-point filter weighted 1, 4,

6, 4, 1), showing the solar year. (f) Residual curve (d minus e) showing the

•gtidal year•h; the vertical lines mark 1800 phase-reversals in the sawtooth

pattern. (Tidal data for Townsville supplied by the Tidal Laboratory of the

Flinders Institute for Atmospheric and Marine Sciences, Flinders University

of South Australia, copyright reserved.)

fortnightly cycle agreed well with the maximum value of 29 identifiable diurnal laminae

in two successive lamina-cycles, representing the lunar monthly cycle, in the Elatina

series. These figures suggest 29-30 lunar days per lunar month and in turn imply around

30.5 •} 0.5 solar days per lunar month (employing the present ratio of lunar-day to

solar-day durations of 24.8/24 h = 1.03).

A yearly, non-tidal signal, probably reflecting annual change in sea level, is evident

in the Elatina rhythmites through periodic variation in the thickness of lamina-cycles

(Fig. 2(a) and (b)). The repetition of this annual cycle indicates that the measured

rhythmite sequence spans •`60 years of continuous deposition, and the full rhythmite

member between 60-70 years. The markedly periodic, virtually noise-free nature of

these data is demonstrated by the strong, narrow spectral peaks obtained for the annual,

semi-annual and monthly signals (Fig. 3(a)). The spectrum for the maximum heights

of spring tides at Townsville, Queensland (Fig. 3(b)), likewise shows clear annual,

semi-annual and monthly peaks. The Late Proterozoic year contained 26.1 or 26.2

(depending on the method of spectral analysis; Williams, 1989 a, b) •} 0.2 lunar

J. Phys. Earth

Tidal Rhythmites 479

Fig. 3. Fast Fourier transform smoothed spectra, with power spectral densities normalized to unity for the strongest peak in each spectrum and with linear

frequency scales. (a) Spectrum for the Elatina sequence of 1,580 fortnightly lamina-cycle thickness measurements. The strong period of 26.1 lamina-cycles

represents an annual signal, with additional harmonics at 13.1 (semi-annual), 8.7, 6.6, and 5.3 lamina-cycles. The peak near 2 lamina-cycles (the Nyquist

frequency) reflects the monthly inequality of alternate thick and thin fortnightly lamina-cycles. (b) Spectrum for the maximum heights of 495 spring tides

between January 1, 1966, and December 31, 1985, for Townsville, Queensland. The periods of 24.4 and 12.5 fortnightly cycles represent annual and semi-annual signals. The peak near 2 fortnightly cycles (the Nyquist frequency) reflects the

monthly inequality of alternate high and low spring tides.

fortnightly cycles and 13.1•}0.1 lunar (synodic) monthly cycles. Hence, at •`650 Ma

there were 400 •} 7 solar days per year and 21.9 •} 0.4 h per day.

The second harmonic of the annual signal (Fig. 3(a)) has a period of 13.1 fort-

nightly cycles, and represents the semi-annual tidal cycle. This oscillation, through its

Vol. 38, No. 6, 1990

480 G. E. Williams

Table I Late Proterozoic (•`650 Ma) and modern tidal and rotational parameters.

* Values indicated by tidal rhythmites of the Late Proterozoic Elatina Formation and Reynella Siltstone , South Australia (Williams, 1987, 1988, 1989 a, b, c, 1991). The rhythmite data also display strong semi-annual and annual periods (see Figs. 2 and 3). ** Units in invariable time required for dynamical calculations.

modulation of tidal range, influenced neap-tidal pauses in the deposition of diurnal

elastic laminae in the Elatina Formation, thereby modulating the number of laminae

deposited per lamina-cycle (see Williams, 1989 a, figure 4, 1991, figure 15).

The period of the lunar nodal cycle also is recorded by the Elatina series.

Amplitude-modulation of the semi-annual palaeotidal cycle, as recorded by the

second-order peal of the annual cycle (Fig. 2(b)), indicates a long-term period of

19.5 •} 0.5 years which is interpreted as that of the palaeo-lunar nodal cycle (Williams,

1989 a, b, c).

The synodic character of the Elatina fortnightly pattern and of the Townsville tidal

pattern with which it is compared is demonstrated in Fig. 4. Modern synodic tides

display a characteristic variation in the amplitude of the lunar fortnightly cycle (Figs.

2(f) and 4(b)): (a) Alternation of high- and low-amplitude fortnightly cycles to give a

•gsawtooth•h pattern, or monthly inequality, resulting from the eccentricity of the lunar

orbit. (b) Systematic modulation of the amplitude of the sawtooth pattern, with 180•‹

changes of phase (that is, a reversal in the sequence of high- and low-amplitude fortnightly

cycles) occurring where the amplitude of the pattern is minimal. As shown in Fig. 4(a)

and (b), maximum amplitude of the sawtooth pattern (that is, maximum monthly

inequality) occurs when the Earth, Moon, and Sun all lie along the major axis of the

elliptical lunar orbit; 180•‹ changes of phase of the sawtooth pattern occur at minimal

monthly inequality when all bodies are aligned perpendicular to the major axis. The

mean period for a 360•‹ change of phase of the sawtooth pattern, which may be termed

the •gtidal year,•h is longer than the solar year (13.95 and 12.37 synodic months,

respectively) because of the prograde rotation of the lunar perigee. The same synodic

pattern (but with periods of 14.6 and 13.1 synodic months for the tidal year and solar

year, respectively; Williams, 1988, 1989 a, b, c) is displayed by the Elatina series, as

expressed by variation in the thickness of fortnightly lamina-cycles (Figs. 2(c) and 4(c));

the pattern also is shown by variation in the amplitude of such cycles.

J. Phys. Earth

Tidal Rhythmites 481

Fig. 4. Synodic fortnightly tidal patterns for one •gtidal year.•h (a) Schematic

luni-solar conjunctions: 1, 3, and 5 = minimum monthly inequality of spring

tidal amplitudes, and phase reversal in the sawtooth pattern of such amplitudes;

2 and 4= maximum monthly inequality. The •gtidal year•h (1-5) is longer than

the solar year because of prograde rotation of the lunar perigee. (b) Relative

amplitude (maximum height) of spring tides at Townsville over one •gtidal

year•h (see Fig. 2(f)); Nos. 1-5 refer to luni-solar conjunctions shown in

(a). (c) Relative thickness of fortnightly lamina-cycles in the Elatina series

over one •gtidal year•h (see Fig. 2(c); Nos. 1-5 refer to luni-solar conjunctions

shown in (a). The vertical lines in (b) and (c) mark positions of 1800

phase-reversals in the sawtooth patterns.

The duration of the lunar apsides cycle, or period of rotation of the lunar perigee

(Pp), is given by

Pp= Yt/( Yt-Ys)(1)

where Yt and Ys are the durations of the tidal year and solar year, respectively. The

mean periods for the solar and tidal years at ti •`650 Ma indicate a lunar apsides cycle

of 9.7•}0.1 years (Williams, 1989 a, b, c).

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482 G. E. Williams

Table 2. Mean Earth-Moon distance at •`650 Ma and mean rate of lunar retreat for the past

•`650 Ma indicated by tidal rhythmites of the Elatina Formation and Reynella Siltstone

(from Williams, 1989 a, c; Deubner, 1990).

3.2 Mean Earth-Moon distance

Three determinations of the mean Earth-Moon distance (semi-major axis of the

lunar orbit) at •`650 Ma employing different palaeotidal parameters are given in Table 2.

An estimate of the mean Earth-Moon distance at that time can be obtained from

the expression

P=P0(cos io/cos i)(ao/a)1.5 (2)

where P is the past lunar nodal period, Po the present lunar nodal period of 18.61

years, a the past Earth-Moon distance, ao the present Earth-Moon distance

of 60.27 Earth radii, io the inclination of the lunar orbit to the ecliptic of 5.15•‹, and i

the past lunar inclination (derived by Walker and Zahnle (1986) from the equations of

lunar motion in Kaula (1968)). This expression predicts an increased lunar nodal period

in the geological past, a prediction confirmed by the Elatina rhythmite data. Assuming

negligible evolutionary change in lunar inclination i, a past lunar nodal period of

19.5 •} 0.5 years (Table 1) gives a mean. Earth-Moon distance of 58.40 •} 1.02 Earth radii

(RE), hence a/ao=0.969 •} 0.017 (Williams, 1989 a).

Additional estimates of the mean Earth-Moon distance at •`650 Ma have been

made by Deubner (1990) by applying Kepler's third law and the principle of conservation

of angular momentum of the Earth-Moon system to the values for lunar months per

year and solar days per year given in Table 1. From Kepler's third law

(t/to)2= (a/ao)3, (3)

where to is the present length of the sidereal month, t is the length of the sidereal

month in the geological past, and ao and a are the present and past Earth-Moon

distances, respectively. As Deubner (1990) points out, his respective figures for alao of

0.967 •} 0.005 and 0.968 •} 0.007 agree closely with the above value of 0.969 •} 0.017

determined using the observed palaeo-lunar nodal period. Although these calculations

do not consider angular momentum that may be lost to the Sun, the excellent agreement

among figures for mean Earth-Moon distance derived from such widely separated and

independent rhythmite periods demonstrates the internal consistency of the palaeotidal

data and the validity of the Late Proterozoic tidal and rotational values listed in Table 1.

J. Phys. Earth

Tidal Rhythmites 483

3.3 Obliquity of the ecliptic

The rhythmites of the Elatina Formation and Reynella Siltstone record independent

features that together indicate a substantial obliquity of the ecliptic in Late Proterozoic

time. They are a clear diurnal inequality in the tides (Fig. 1(b)) and strong semi-annual

and annual signals. These features are not of local origin, but arise ultimately from the

Earth's obliquity.

The diurnal inequality is best seen in the Reynella rhythmites (Williams, 1988,

1989 a, b, c, 1991); its presence indicates that in Late Proterozoic time the Earth's poles

were inclined to the lunar. orbital plane, and that the rhythmites were not deposited

right at the palaeoequator (the diurnal inequality is minimal or absent at the equator).

The semi-annual tidal period of solar declination is well shown in the fast Fourier

transform (FFT) spectrum of the Elatina data (Fig. 3(a)). This oscillation modulated

neap-tidal ranges and thence the number of laminae deposited per fortnightly lamina-

cycle (Williams, 1989 a, 1991). The very strong annual signal in the Elatina data (Fig.

3(a); Williams, 1989 a, b; 1991) is attributable to the annual, largely non-tidal oscilla-

tion of sea level. The annual oscillation of sea level tends to be most conspicuous and

phase-coherent in low latitudes (unpublished data from the National Tidal Facility,

Flinders Institute for Atmospheric and Marine Sciences, Flinders University of South

Australia); hence its very strong, clear signature in the Elatina data accords with the

indicated low palaeolatitude of deposition of the Elatina rhythmites (palaeomagnetic

data (Embleton and Williams, 1986; Schmidt et al., 1991) indicate that the rhythmite

member of the Elatina Formation at Pichi Richi Pass was deposited between 20•‹N and

12•‹S of the palaeoequator, assuming that the axial geocentric dipole model for the

Earth's magnetic field is valid for Late Proterozoic time).

An idea of the relative power of the semi-annual and annual signals in the Elatina

palaeotidal data may be gained by comparing the FFT spectra for the Elatina data and

for modern tidal data from Townsville (Fig. 3). Comparing these spectra appears justified

because each tidal record is of mixed/synodic type and is from low latitudes (Townsville

is at 19•‹16•ŒS). The spectra in Fig. 3 are for similar, long sequences (•`20-60 years)

of fortnightly data, and show annual, semi-annual and monthly periods. Normalizing

the power spectral densities for the monthly peaks in each spectrum shows that the

annual and semi-annual signals in the Elatina spectrum (Fig. 3(a)) have •`15 times and

•`4 times more power, relative to the monthly peak, than do respective signals in the

Townsville spectrum (Fig. 3(b)). A further difference between the two spectra is the

presence, only in the Elatina spectrum, of a sequence of higher harmonics of the annual

and semi-annual periods (peaks at 8.7, 6.6, and 5.3 fortnightly cycles in Fig. 3(a)). These

higher harmonics are attributable to beating among the annual and semi-annual signals

and their beat frequencies. It would seem that the Late Proterozoic annual and

semi-annual oscillations of sea level in the Adelaide Geosyncline had sufficient power

to generate a sequence of higher harmonics in sea-level height and/or tidal range,

which was recorded by the Elatina rhythmites. The marked differences in relative power

between respective annual and semi-annual periods in the Elatina and Townsville data

cannot be explained by the regular abbreviation of fortnightly lamina-cycles in the

Elatina rhythmites (that is, the common absence of laminae near the neap part of

fortnightly cycles); such abbreviation has tended to reduce the amplitudes of all long-term

Vol. 38, No. 6, 1990

484 G. E. Williams

periods in the Elatina data. Although differences in geographic settings may well account

for some of the distinctions between these ancient and modern tidal records, the

important point is that annual and semi-annual signals are very strongly developed in

the Late Proterozoic data.

Addressing quite different evidence, Williams (1975) postulated an increased

obliquity of the ecliptic (ƒÃ•„54•‹) during Late Proterozoic time in explanation of the

indicated strongly seasonal, in situ frigid glacial and periglacial climate near sea level

in low palaeolatitudes. Subsequent geophysical and geological research has provided

strong support for such an enigmatic Late Proterozoic glacial climate in preferred low

(•ƒ30•‹) palaeolatitudes (McWilliams and McElhinny, 1980; Williams, 1986; Embleton

and Williams, 1986; Schmidt et al., 1991). Periglacial structures in Late Proterozoic

permafrost horizons within marine sedimentary basins in low palaeolatitudes indicate

that mean annual air temperatures near sea level were then as low as -12 to -20•Ž

or lower, and that the mean monthly temperature range was as great as •ƒ -35•Ž in

midwinter to +4•Ž in summer (Williams, 1986). Indeed, the nature and distribution

of the Late Proterozoic glacial climate is one of the biggest puzzles in contemporary

Earth science. It is thus of particular interest that the independent palaeogeophysical

data provided by the tidal rhythmites of the Elatina Formation and Reynella Siltstone,

which are part of the Late Proterozoic Marinoan glacial succession in South Australia,

imply a substantial obliquity of the ecliptic at that time.

3.4 Earth's moment of inertia

The values of 30.5 •} 0.5 solar days/lunar month and 400 •} 7 solar days/year for

•`650 Ma indicated by the tidal rhythmites (Table 1) can be used to test whether the

Earth's moment of inertia has changed significantly since Late Proterozoic time. From

Runcorn (1964, 1966)

1-L/Lo = [-1+(I/Io)(ƒÖ/365 .25)(Yo/Y)]/4.83(1+ƒÀ), (4)

where L/L0 is the ratio of the past to the present lunar orbital angular momentum (see

Runcorn, 1979), I and Io are the Earth's past and present moments of inertia respectively,

co is the number of sidereal days in the year, Y and Yo are the absolute lengths of the

past and present sidereal years respectively, and ƒÀ is the ratio for solar/lunar retarding

couples acting on the Earth (see Jeffreys, 1952). Assuming that Y= Yo, the palaeotida

data for 650 Ma suggest virtually no overall change in the Earth's moment of inertia

I since that time: I/Io = 0.998 •} 0.018 for ƒÀ=1/5.5, and I/Io =1.005 •} 0.018 for ƒÀ=1/3.7.

These results are of importance as they represent the only direct estimate of I/Io for

Precambrian time.

The results rule out significant Earth expansion since •`650 Ma, such as the

accelerating expansion hypothesis of Carey (1976). Nor do the rhythmite data support

slow expansion hypotheses: for example, Egyed's (1956, 1969) postulate that the Earth's

radius has increased by 0.65 •} 0.15 mm/year for at least the past 600 Ma gives

I/Io =0.91-0.94 (see Runcorn, 1964), and Creer's (1965) proposed increase in radius of

0.5-0.95 mm/year gives I/Io= 0.89-0.94. Narrowing the error limits for days/month at

•`650 Ma by further observations of the rhythmites would reduce the uncertainty in

I/Io, determined from the rhythmite data and provide tighter constraints on hypotheses

J. Phys. Earth

Tidal Rhythmites 485

of very slow Earth expansion of •ƒ0.05 mm/year (for example, Dicke, 1962). To this

end, additional coring of the Late Proterozoic rhythmites in South Australia is planned.

Overall, the Late Proterozoic palaeotidal data strongly suggest that the Earth's

moment of inertia and radius at •`650 Ma were similar to those of today. The rich

palaeotidal record of the Elatina rhythmites certainly indicates the presence of open

oceans in Late Proterozoic time.

4. Early Proterozoic Rhythmites

Cyclic rhythmites in banded iron-formation (BIF) of the •`2,500 Ma Weeli Wolli

Formation, Hamersley Basin, Western Australia, may provide palaeotidal periods for

Early Proterozoic time. These chemical deposits may be of submarine fumarolic origin

and thus may record earth-tidal rather than ocean-tidal rhythms (Williams, 1989 c).

The cyclicity in the Weeli Wolli Formation (Fig. 5) is caused by regular variations

in thickness of both the chert and haematite parts of microband couplets, giving the

rocks a characteristic striped appearance. The microbands usually are very thin (0.05 mm

thick, or less) and only the cyclic stripes are readily discernible (Fig. 5(b)). Local nodules

Fig. 5. (a) Chert nodule from Early Proterozoic (•`2,500 Ma) cyclic banded

iron-formation of the Weeli Wolli Formation, Western Australia. The nodule

contains discernible microband couplets of chert (white) and haematite (black),

and up to 28 to 30 microband couplets occur between the centres of the cyclic

stripes. (b) Strongly compacted iron-formation from the Weeli Wolli

Formation in which only the cyclic stripes are readily discernible. Scale bars

are 5 mm.

Vol. 38, No. 6, 1990

486 G. E. Williams

of early diagenetic chert have not undergone full compaction, however, thus permitting

the microbands to be more easily seen and counted (Fig. 5(a)). Counts carried out on

thin sections of chert nodules containing up to 6-8 cycles indicate as many as 28 to 30

microband couplets per cycle. Cycles containing fewer microband couplets may show

evidence that some adjacent microbands have amalgamated, and hence counts for such

cycles may underestimate the true cycle period. These observations might suggest a

cycle period near 28 to 30 microband couplets, casting doubt on previous interpretations

that the cycle period is •`23 microband couplets, or years, and records the double

sunspot cycle (Trendall, 1973) or the lunar nodal cycle (Walker and Zahnle, 1986).

Some geyser activity today is modulated by earth tides, including the lunar

fortnightly tide (Rinehart, 1972 a, b, 1974), raising the question as to whether the Weeli

Wolli cyclicity records earth-tidal rhythms that modulated the submarine discharge of

silica- and/or iron-bearing fumarolic waters. As the principal components of the solid

earth tide are semidiurnal and fortnightly, the microband couplets may be lunar

fortnightly increments that are arranged in annual cycles through seasonal influences

on sedimentation. This interpretation would imply about 28 to 30 lunar fortnights, or

about 14 to 15 lunar months, per year at •`2,500 Ma.

The earth-tidal origin for the cyclicity postulated here, as well as being consistent

with modern geyser activity, gives sedimentation rates for the compacted facies (Fig.

5(b)) that are comparable to presumed rates for other BIFs in the Hamersley Basin

whose microbanding (sometimes termed •gaftbanding•h) is regarded as annual (see

Trendall and Blockley, 1970; Trendall, 1983). Such an origin for the cyclicity also is

supported by the presence of between 15 and 27 laminae (depending on the observer)

in a thick microband, presumed to one year's accretion, from the Brockman

Iron-formation in the Hamersley Basin (Ewers and Morris, 1981). As the lunar nodal

cycle regulates modern geyser activity (Rinehart, 1972 b), the compacted Weeli Wolli

facies (Fig. 5(b)) eventually may provide information on the lunar nodal period at

•`2,500 Ma. Because of the effects of diagenesis, and the shortness of detailed cyclic

sequences in chert nodules (no more than about 8 consecutive cycles), any suggested

origin for the cyclicity in the Weeli Wolli Formation must, however, be viewed as

speculative.

5. Early Proterozoic Orbital Parameters

Additional orbital parameters that are implied by the suggested figure of 14.5•}0.5

lunar months/year at •`2,500 Ma, assuming that angular momentum of the Earth-Moon

system is conserved, are given in Table 3. Applying Kepler's third law (Eq. (3)), the

above figure indicates a mean Earth-Moon distance of 54.6•}1.2 Earth radii

(a/a0 = 0,906•}0.020) for •`2,500 Ma. The values for solar days/lunar month and solar

days/year given in Table 3 were obtained from figure 11.5 of Lambeck (1980) (which

assumes that tidal friction is the only phenomenon responsible for change in the rate

of the Earth's rotation).

J. Phys. Earth

Tidal Rhythmites 487

Table 3. Tidal and rotational parameters for •`2,500 Ma suggested by cyclic banded

iron-formation of the Early Proterozoic Weeli Wolli Formation, Western Australia.

Fig. 6. Mean rates of lunar retreat for •`2,500-650 Ma and •`650-0 Ma indicated

by the rhythmites of the Elatina Formation and Weeli Wolli Formation,

plotted with the present rate of lunar retreat obtained by lunar laser ranging.

Values and error estimates (shown here as vertical bars) are given in Tables 1

and 3.

6. History of the Moon's Orbit

6.1 Mean rates of lunar retreat

Mean rates of lunar retreat since Early Proterozoic time indicated by the palaeotidal

data of the Elatina/Reynella and Weeli Wolli rhythmites are given in Tables 1, 2, and

3 and plotted in Fig. 6. The mean Earth-Moon distance of 58.28 •} 0.30 Earth radii at

•`650 Ma, indicated by the value of 13.1 •} 0.1 lunar months/year for the Elatina series

(the best constrained parameter employed in Table 2), gives a mean rate of lunar retreat

of 1.95 •} 0.29 cm/year over the past •`650 Ma. This mean value is only •`53% of the

present rate of lunar retreat of 3.7 •} 0.2 cm/year obtained by lunar laser ranging (Dickey

et al., 1990).

The Earth-Moon distances provided by the Weeli Wolli and Elatina rhythmite

data suggest a mean rate of lunar retreat of 1.27 •} 0.52 cm/year for most of Proterozoic

time (•`2,500-650 Ma). This figure is •` 65% of the mean rate of lunar retreat for the

Vol. 38, No. 6, 1990

488 G. E. Williams

past •`650 Ma and only •`34% of the present rate (Table 1). The indicated increasing

rate of lunar retreat since •`2,500 Ma is consistent with increasing oceanic tidal

dissipation as the Earth's rotation slows (see Hansen, 1982; Webb, 1982). The present

relatively high rate of lunar retreat may reflect a near-resonance of the oceans (Brosche,

1984).

6.2 Evolution of the lunar distance

Palaeotidal data encoded in ancient tidal rhythmites eventually may allow the

Earth-Moon distance at •`4,500 or the time of a possible close approach of the Moon

to be calculated. At present, however, only one datum exists for Precambrian palaeotidal

values to which a high level of confidence can be attached-the Elatina rhythmite record

at •`650 Ma. Using the appropriate orbital parameters for •`650 Ma (Table 2), an

extrapolation of the Earth-Moon distance to earlier time may be carried out using the

ea uation

(5)

where a is the Earth-Moon distance at time T, ao the present Earth-Moon distance,

and <a0> the average rate of lunar retreat since •`650 Ma (adapted by Walker and

Zahnle (1986) from Lambeck (1980)). For a mean rate of lunar retreat of

1.95 •} 0.29 cm/year for the past 650 Ma (Table 2), an Earth-Moon collision is indicated

at •`3,030 Ma (dotted curve in Fig. 7). As the Elatina palaeotidal data indicate an

average equivalent phase lag near 3•‹ since •`650 Ma (Williams, 1988, 1989 a, b, c), this

finding accords with Lambeck's (1980, p. 352) conclusion that for an average lag of 3•‹

Fig. 7. Change in the mean Earth-Moon distance with time, as suggested by the rhythmites of the Elatina Formation and Weeli Wolli Formation. Dotted curve,

an extrapolation (Eq. (5)) based solely on the Elatina datum. Dashed curve

(Eq. (6)), shown with error limits, is based on an average Proterozoic rate of lunar retreat derived from the Elatina datum and the Weeli Wolli datum.J. Phys. Earth

Tidal Rhythmites 489

the Moon would have been within 10 Earth radii some 3,000 Ma ago. By comparison,

the extrapolation of rates of lunar retreat obtained from modern observation and

Phanerozoic palaeontological data imply a close approach of the Moon at •`1,500 Ma

(Lambeck, 1980). Although the Elatina data push back the time of a possible close

approach of the Moon, neither the geological records nor the surface featurers of the

Earth or the Moon provide evidence of such an event at •`3,000 Ma. Additional

palaeotidal data prior to •`650 Ma are required if the history of the lunar orbit is to

be traced with reasonable accuracy.

The palaeotidal data for •`2,500 Ma suggested by the Weeli Wolli Formation

(Table 3) may be used together with the Elatina data to tentatively trace the history of

the lunar orbit beyond •`3,000 Ma. The time of a possible close approach of the Moon,

using presumed Earth-Moon distances at 2,500 and 650 Ma indicated by the Weeli

Wolli and Elatina rhythmites, may be obtained from

(6)

where aT is the mean Earth-Moon distance at time T, a1 the mean Earth-Moon distance

at time T1 (650 Ma), and <a1> the average rate of lunar retreat for 2,500-650 Ma

(adapted from Walker and Zahnle, 1986). The overall low rate of lunar retreat indicated

by the Proterozoic palaeotidal data (Tables 1, 2, and 3) possibly suggests that a close

approach of the Moon did not occur during Earth history, although the wide error

limits cannot preclude a close approach near 4,500 Ma (dashed curve in Fig. 7).

The uncertainties in the data are too great to permit an estimate of the Earth-Moon

distance at 4,500 Ma. Palaeotidal data to which a high level of confidence can be attached

are required for the Archaean and Early Proterozoic to allow accurate determination

of the early history of the lunar orbit.

7. Conclusions

Geochronometric analysis of the ancient Earth-Moon system has been rejuvenated

by the study of cyclically laminated rhythmites of tidal origin. Late Proterozoic ebb-tidal

rhythmites in South Australia provide the first benchmark for Precambrian palaeotidal

and palaeorotational values: at •`650 Ma, the day was •`21.9 h long and the year

contained •`400 solar days and 13.1 lunar months. The data imply a substantial obliquity

of the ecliptic at •`650 Ma, and indicate virtually no overall change in the Earth's

moment of inertia since that time. Cyclic banded iron-formation of possible fumarolic

origin in Western Australia may record palaeotidal information for Early Proterozoic

time. The Proterozoic rhythmite data suggest an overall low rate of lunar recession that

argues against a close approach of the Moon except perhaps in early Earth history.

As palaeotidal deposits are common in the sedimentary record, many additional

examples of tidal rhythmites likely will be discovered. Geologists are encouraged to

seek such rhythmites to allow determination of palaeotidal and palaeorotational values

for numerous stratigraphic intervals. Special importance should be placed on the search

for tidal rhythmites in Precambrian deposits that might record information for very

Vol. 38, No. 6, 1990

49 0 G. E. Williams

early Earth history. Such studies promise to greatly illuminate the dynamic history of the Earth-Moon system and may throw light on the origin of the Moon itself.

I thank W. Mitchell (Flinders Institute for Atmospheric and Marine Sciences, Flinders University of South Australia, Adelaide) for helpful discussion, and F.-L. Deubner (Institute for Astronomy and Astrophysics, University of Wiirzburg, Germany) for providing me with an advance copy of his discussion in the Journal of the Geological Society of London.

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