Limits on lateral pressure gradients in the outer core from geodetic observations

280

Physics of the Earth and Planetary Interiors, 50(1988) 280—290 Elsevier Science Publishers B.V., Amsterdam — Printed in The Netherlands

Limits on lateral pressure gradients in the outer core from geodetic observations J.B. Merriam Department of Geological Sciences, University of Saskatchewan, Saskatoon, Saskatchewan S7N 0 WO (Canada) (Received March 30, 1987; accepted June 11, 1987)

Merriam, J.B., 1988. Limits on lateral pressure gradients in the outer core from geodetic observations. Phys. Earth Planet. Inter., 50: 280—290. Lateral variations in pressure associated with flow at the core—mantle boundary will produce deformations of the mantle. It is shown here that geostrophically plausible pressure gradients are sufficient to produce measurable changes in the length-of-day, polar motion, .J~,sea-level, tilt, gravity and strain. Of these, polar motion is the most discriminating and limits intra-annual variations in the lateral pressure field, at planetary wavelength, to less than — iO~Pa. Satellite observations of changes in J 2 are potentially even more useful. However, at present the time span of observations is short and the separation of a residual, which might be due to a change in the distribution of pressure at the core—mantle boundary, from other causes, such as post-glacial rebound, is ambiguous. It is suggested that not more than 10% of the decade variations in the length-of-day may be caused by pressure fluctuations at the core—mantle boundary.

I. Introduction Motions of the Earth’s fluid core are thought to produce the geomagnetic field by dynamo action, but the solution to the dynamo problem is hampered by our lack of knowledge of the motions in the fluid core. What information we do have is the result of observations of the secular variation of the magnetic field at the surface and the assumption that some of these changes monitor fluid motions at the core—mantle boundary (CMB) by virtue of the magnetic field being ‘frozen’ into the fluid flow by the high conductivity of the core. The westward drift of the non-dipole field was interpreted in this way by Bullard et al. (1950). A westward drift of 0.2° a~ implies a fluid velocity of about 4 x 10—a m s~. This figure should be used with caution since we have poor knowledge of the conductivity of the core and the ‘frozen flux’ hypothesis may be invalid on long time scales. The diffusion time for the dipole 0031-9201/88/$03.50

© 1988 Elsevier Science Publishers B.V.

component of the field in the core is about 10000 years and the longest wavelength non-dipole components would diffuse out of the core with a characteristic time of 2500 years (Merrill and McElhinny, 1985). Therefore the frozen flux hypothesis might be valid on decade time scales but on the time scales of the secular variation it begins to break down. Bloxham and Gubbins (1985) have used historical measurements of the secular variation worldwide to present evidence that there is indeed some diffusion of field lines in the core. Fluid motions in the Earth’s core must necessarily be accompanied by pressure gradients that express a departure from hydrostatic equilibrium. Because the energy supplied to the core is small, the flow is not vigorous and large pressure anomalies are not anticipated. However, surprisingly small pressure anomalies can produce observable deformations of the mantle. In subsequent sections relations between lateral variations in pressure at the CMB and deformations of the

281

mantle that result in changes in its inertia tensor, producing polar motion and changes in J2 as well as changes in the length-of-day (LOD), tilts and trends in sea level will be derived. The possibility that these geodetic data sets may contain some information on temporal changes in long wavelength pressure anomalies at the CMB will be examined. Of course, it is impossible to associate an observed variation in any of these quantities with a variation in the pressure field at the CMB, unless perhaps a correlation is found with the secular variation of the magnetic field. The purpose here is not to search for any possible correlalions, but to argue that if geodetic variations are to be attributed to fluctuations in the pressure at the CMB, then these observations imply an upper bound on the possible changes in the pressure. It will be shown that the data on changes in the LOD, J2 and polar motion, are all consistent with pressure anomalies that do not exceed io~Pa. It is probable that newer data, particularly on J2 and polar motion, will reduce this upper bound. Before examining the geodetic data there are several lines of evidence that can be used to reveal the range of pressures that may be anticipated. Whar (1987) has used satellite data on the second-degree tesseral harmonics of the Earth’s gravity field to set an upper bound on the tilt of the core’s rotation axis with respect to the mantle’s symmetry axis. This in turn sets an upper limit on the equatorial torques the core exerts on the mantle and hence an upper bound on the long wavelength pressure anomalies of 1.5 X iO~Pa. The magnitude of the pressure gradients that can exist along the core—mantle boundary can also be estimated by assuming that the fluid flow in the core is in geostrophic balance, with the Coriolis force balancing the pressure gradient. 2~Up = vP —

—

Here ~ is the rotation rate, U the fluid velocity, p the density and P the flow pressure. Using the westward drift (0.2° a Merrill and McElhinny, 1985) as an estimate 1), of and the assuming magnitudetheofpresthe velocity X i0~ ms sure field(4has a wavelength of the circumference of the core, then long wavelength anomalies in pressure of iO~ Pa might be expected. Voorhies -

—

~,

(1984, 1986) has inverted secular variation data to map fluid velocities at harmonic wavelength up to degree 10. His results indicate velocities that are similar to the velocity used above, but the shorter wavelength may imply smaller pressures. Magnetostrophic balance, in which the Lorentz force is included in the balance, could lower this estimate, but the Lorentz force depends on the current density and magnetic field in the core, both of which are uncertain. Decade-scale anomalies cannot be estimated in this way because the conductivity of the mantle masks magnetic fluctuations on time scales shorter than a couple of decades. The decade variations in the length-of-day, however, have been attributed usually to coupling with motions in the fluid core so that decade-scale motions, of some scale, must exist. Lacking evidence to the contrary, the figure of iO~Pa, which is derived from longer time-scale flows, may be viewed as at least a plausible range for decade-scale variations in pressure along the CMB. It will be shown in subsequent sections that decade-scale variations in geodetic observables also admit pressure anomalies of this magnitude.

2. Deformation of the mantle by pressure at the core—mantle boundary

Deformations of the Earth by body force potentials or gravitating surface loads, are conveniently handled by Love numbers. In this section equivalent parameters, which describe the response of the mantle to a harmonic distribution of pressure at the CMB will be produced. Approximate solutions to the problem can be obtained by examining the general problem of the thick spherical shell. It is sufficient for present purposes to consider a homogeneous, non-gravitating shell because the interest is only in establishing an upper bound on the magnitude of the pressure field and not in an accurate inversion of the deformation the pressure field at the CMB. at In the any surface case, thefor solutions to the simple elastic problem will be shown to be very good approximations because that model gives solutions to well-known problems, such as the

282

radial load on the surface, that agree with solutions from much more sophisticated models. The equations of motion in the purely elastic problem are a fourth-order set of differential equations, which are in fact a subset of the equations describing the problem of the gravitating elastic sphere (Alterman et a!., 1959). The solutions consist of two linearly independent solutions to the internal problem (where the shell is internal to the bounding surface on which a normal stress is applied) and another two to the external problem (in which the shell is external to a surface on which a normal stress is applied). The internal problem thus corresponds to a radial pressure field on the CMB. It can be verified easily by substitution that the solutions for the radial displacement, u, the tangential0rr’ displacement, v, the and the transverse radial component of stress, component of stress, Or4,~for a harmonic displacement of degree 2 are 3 + 2Br (internal) uv (7 l2Avr4v)Ar3 + Br (1) a~r/2p~= —6Avr2+2B (solution) = =

nal problem, at r c are —P~/2~t~ C (lOv 14) C D —(1 + v) ~ c5/2~ (—28 + 20v) =

=

—

=

where P~is the pressure field at r c. Calling the radial displacements in the internal and external problems u 1 and Ue, respectively, the displacements at the surface r a are =

Ueia

~i

a

[ [

=

(5 4~ / c 2 (5v 7) 3(1 + u) (~)~lCP~/2~ + (20v 28) a 6v (7 + 2v) ~aP 0/2ji (7 + 5v) (7 + Si’)

=

=

—

—

(3)

]

—

—

Using 0.3 dimensions and c/a 0.5472, so that the shell has thes’ same and average structure as the mantle, we have =

=

—

Ue~a= Uiia=

+0.191cP~,/2~i u~= +0.541cP~/2~t —0.682aPa/2~L

—0.455aP~/2j~(4)

u~=

0r~/2(7+2V)A12~

u

=

2C(5

v

=

—2C(1

~r/2P

—

4v)r2 —

—

3Dr4

2v)Cr2

+

Dr

(external) (2)

4

—2C(10 2v)r3 + 12Dr5 (solution) C(2 + 2v)r3 4Dr5 —

=

—

The accuracy of these expressions, and the justification for the simple Earth model, is easily checked by comparing the expression for Uiia with the Love number expression for the surface displacement on a more realistic Earth model, i.e. a gravitating radially inhomogeneous model. In that case we have 3h’

where v is Poisson’s ratio, p~is the rigidity and A,B,C,D are constants to be determined by the boundary conditions. I1~~a is the harmonic coefficient of the anomalous pressure field at the surface r a, then the constants A, B in the internal problem are =

B A

=

2f5 —(7+2v) (by + 14) P0/

=

Pa/2~.t (by + 14) a2

The solutions for the constants C, D in the exter-

2 Pa Ujia

=

5(m Pa~)P

=

—1.1

x 10

0

With ~u 2 X 1011 Pa as an average rigidity for the mantle we In obtain samea value in eq. 4,shell to within 3%. otherthe words, homogeneous 1011 Pa responds to av second-degree harmonic with Poisson’s ratio 0.3 and rigidity ~u 2 of X =

=

=

surface pressure in the same way (at least insofar as the surface displacement is concerned) as the Earth is observed too. It is probable that the simple model describes the response to pressures at the CMB equally as well.

283

The ratio of the external to internal solutions for u at the surface is Uei a 01 p Uiia 52-i .

So the displacement at the surface of the shell, from a pressure at the inner surface is approximately 1/7 the displacement at the surface from an equivalent pressure at the surface, and of course in the opposite direction. The solutions in eq. 4 can now be used to deduce similar relations between pressure at the core—mantle boundary and change in potential at the surface, or change in moment of inertia. For a homogeneous shell the change in potential is mainly from the radial displacement at each boundary (neglecting a small contribution from compressibility). For example, with a realistically stratified Earth, about 70% of the change in the second-degree potential is produced by the deformation of the surface and the CMB. We therefore have for the change in a second-degree potential, W ~[~sutta

~=

+

/ C PCUIC~J

]

3 is the surface density where p~4.4 2.8X XiO~kg iO~kgm3 m is the density contrast and p~ at the CMB. The first term in brackets is the contribution of the surface to the change in potential and the second term is the contribution of the CMB. Using eq. 4 we have 0.797cPj2~t 6(m2 ~—2 Pa’)P~ (5a) = 6.95 X 10 for an estimate of the change in potential at the surface from a degree 2 harmonic pressure, ~ at the CMB and

Equations 4 and 5 form the basis for deriving a range of functions relating changes in geodetic observables to changes in the distribution of pressure on the base of the mantle. By MacCullagh’s formula, the change in axial moment of inertia tiC, and the change in potential associated with it are related by

tic -a--=

1/2 (.~-)

tiw

3G \ ir Then using eq. 5 p = —0.88 x 1025(Pa kg’ m2) tiC

(6)

where tiC is in kg m2 and P~is in Pascals. With tiJ 2, where M is the mass 2 =we3 have tiC/2Ma of the Earth, p = —1.4 x 1013(Pa) tiJ 2 (7) where again P~is in Pascals. Using eq. 6 the fractional change in rotation rate tim3, is p = 7.1 x 1012 (Pa) tim3 (8) The components m,, m2 of polar motion, at penods much longer than the Chandler period, are related to the products of inertia C13, C23 and the principal components of the inertia tensor C, A by

c

13

m1, m2 =

—

A’ ~

c23 —

A

so that m1, m2 are related to changes in CMB pressure by

=

P~=—3.8X10’°(Pa)m,, m2 mass transfer in the This is the polar motion from mantle only. If it is assumed that the core also participates in the long-period polar motion then we would need to account for the change in the

W 1= =

2.23aPa/2~L 5(m~ ~—2 Pa~)P —3.55 X 10 0 —

(Sb)

for an estimate of the change in potential at the surface from a degree 2 harmonic of pressure at the surface. As a check on accuracy, the gravitating stratified model would have W1 = 3.44 X 10 ~ or about 3% smaller than the above figure. Again we may assume that eq. 5a is at least as good an approximation.

inertia tensor of the core. With the change in the inertia tensor of the core included the above would be =

1.6 X bO’°(Pa)m,, m2

Changes in the moment of inertia of the mantle are not the only way that pressures at the CMB can excite polar motion. With an elliptical CMB, fluid pressure can exert equatorial torques on the mantle. The relation between the pressure and

284

polar motion, at periods much longer than the Chandler, is in this case 3

m 1’ m 2

difficult to see how the oceans and atmosphere LOD. Forresponsible example, iffor thethe moment inertia of the could be decadeoffluctuations in mantle is constant then decade variations of 2 ms in the LOD imply changes in the angular momen-

1/2

/

(C c A) kI 15 1.0 X b0’°(Pa1)P 2

—

—

£. ~

C

Wahr (1987), where e is the ellipticity of the CMB and .P~is a fully normalized surface harmonic at pressure. Torques produced by fluid pressure on the elliptical CMB are therefore as efficient at exciting polar motion as is the deformation of the mantle produced by those same pressures. To explain a given magnitude of polar motion, we need fluid pressure of about P = 2 6 x 1010(Pa’lm m (9\ ‘

/

1’

2

‘

/

Similar arguments result in estimates of the tilt tilt = 1.6 X 10~3(Pa’)P

C

(10)

where tilt is measured in radians, the ocean tide =

9.5

x 10~(mPa~)P

where ~ is in metres, =

5.8

X

x 104PC

rent, which contains to most angular momentum, be of108them3oceans s’, axial with fluctuations of comparable magnitude. The nelative angular momentum of the current is then about 1025 kg m2 s~, or about 10% of the fluctuations in angular momentum represented by

(12)

the decade changes in LOD. Because good data on the Antarctic circumpolar current covers only about two years, one should be cautious in con-

(13)

cluding from the above that variations in the Antarctic circumpolar current can supply 10% of the observed decade vanations in the LOD. However, the implication is that fluctuations in oceamc angular momentum are not a negligible source of excitation for decade-scale changes in LOD. Similarly, the atmosphere has long been discounted as a source of the decade fluctuations because its moment of inertia is so small that conservation of angular momentum would require unlikely fluctuations in wind speed. Lambeck and Cazenave (1974) argue that 10—15% of the decade fluctuations may be atmospheric in origin. Annual variations of atmospheric angular momentum of 1026 kg m2 s1, superimposed on a d.c. angular momentum of about the same magnitude, have been observed since 1976 (Rosen and Salstein, 1983), but it is premature to judge whether or not there are decade-scale variations in atmospheric angular momentum of comparable size. A fit of a linear drift to a ALOD series induced by the

and changes in surface gravity, Ag Ag = 5.1

tum of the mantle of 1026 kg m2 s1. The moment of inertia of the oceans is much smaller than the moment of inertia of the mantle so that if this amount of (axial) angular momentum were lost or gained by the oceans the change in mean speed of zonal currents would have to increase or decrease proportionately. New data, however, reveals that at least some of the decade variations could be atmospheric/oceanic in origin. Whitworth and Peterson (1985) estimate the mean transport of the Antarctic circumpolan cur-

(11)

strains

10’3(Pa’)P~

the changes in relative angular momentum that are required are so large that it has always been

where Ag is in ~tgal, and P is in Pascals C througnout

3. Changes in the length-of-day Changes in the length-of-day (LOD) of a few milliseconds, that persist from several years to several decades, are referred to as the decade variations in the LOD. These fluctuations have been attributed usually to the exchange of angular momentum with the fluid core because of the scale of the phenomenon. The presence of lateral pressure variations of harmonic degree 2 at the CMB raises the possibility of accounting for at least part of the decade variations in LOD by the action of these pressures on the inertia tensor of the mantle. Atmospheric and oceanic variability are dominantly on time scales shorter than a few years and

.

.

.

.

.

.

285

winds, between 1976 and 1985, shows the LOD increasing by 0.03 ms a1. Over the same period the linear drift in observed ALOD was 0.17 ms a Over this restricted period the atmosphere thus appears to be able to account for 20% of the decade changes in LOD. Clearly, the atmosphere is capable of explaining at least some of the decade fluctuations, Decade variations in the LOD may also be caused by the redistribution of pressure on the core—mantle boundary. A change in the seconddegree zonal harmonic of pressure on the core— mantle boundary, that results in higher equatorial and lower polar pressure, will cause an incremental increase in the flattening of the mantle and an incremental decrease in the rotation rate of the Earth, following eq. 8. Observations of Universal Time (UT) supply the fundamental data for determining changes in the length-of-day. Since UT is not a uniform time scale some other, more uniform, time scale must be available to clock the changes in UT. Since 1956 this has been Atomic Time. For epochs prior to 1956 a dynamical time scale based on Newtoman orbits in the solar system must be used. Since the moon has the fastest mean motion its orbit is often chosen, but this choice suffers the drawback that the moons orbit was, until recently, one of the poorest known because its proximity to the Earth introduces many complications. Thus, errors in the lunar ephemeris could translate into errors in AUT. Morrison (1979) has recompiled observations of AUT, from 1861 to 1978, at yearly intervals, using the Improved Lunar Ephermeris (see The Explanatory Supplement to the American Ephemeris and Nautical Almanac). McCarthy and Babcock (1986) have performed the same service, for data at half-yearly intervals from 1656 to 1983. Unavoidably, the quality of the data decreases with time in the past. McCarthy and Babcock estimate the errors in AUT to decrease from between 10 and 20 s for the earliest data, to 1 s for most of the nineteenth century, to a small fraction of a second for the twentieth century data. The consequence of this is that the error in the derived ALOD is larger than the typical excess LOD for data collected before the late nineteenth century. Since it is ALOD that is possibly related to changes —

—

~.

—

in pressure at the CMB, and not AUT, only the data from this century is useful. A ALOD time sequence is obtained from Morrison’s data by differentiation. This is accompushed with a seven-point quadratic convolute, which has the effect of least-squares fitting a quadratic polynomial to three points either side of a central point and then evaluating the derivative of the quadratic at the central point. Superimposed on the decade variations of the ALOD sequence is a secular increase in the LOD from tidal friction. This has been estimated by Lambeck (1980) to be 27.2 ±5.6 X iO~ ms from satellite observations of ocean tides (a small contribution, of opposite sign, from atmospheric tides is included). There is also a contribution to the secular LOD from the viscoelastic rebound of the mantle to a more spherical shape following the last deglaciation. Yoder et al. (1983) have estimated this to be —6.1 ±0.5 X b0~ ms a~, in agreement with the non-tidal decrease in the LOD found by Lambeck (1980). The net secular increase in the LOD is thus 21.1 ±5.6 x iO~ ms a1. Removing tidal and post-glacial rebound terms results in a signal (Fig. 1) that must be explained by a combination of angular momentum exchange with the oceans, atmosphere and fluid core, or decade-scale changes in the degree-2 zonal harmonic of pressure on the core—mantle

30

ii ~ I 0

~ ~ ~

-3.0 I

1840

860

lisa

liii

I

I

1920

1940

I

19t0

1980

I

hoD

. Fig. 1. The excess length-of-day, from 1861 to 1979. A secular rate of2l.1±5.6X103 ms a1, a result partly of tidal friction and partly of post-glacial rebound, has been removed.

286

boundary. Using eq. 8 and assuming that all of the variations in LOD are due to core—mantle pressure, a time sequence for the degree-2 zonal cornponent of core—mantle pressure can be obtained, This results in an estimate of 5 X 10° Pa. This is an order of magnitude larger than the expected pressure variations from geostrophy. With the above estimates for the atmospheric/oceanic contribution to the decade variations in LOD the upper bound on the pressure fluctuations might be reduced to perhaps 10° Pa, or perhaps increased to 106 Pa. These are one and two orders of magnitude larger than the pressure anomalies inferred from geostrophy and the westward drift. It seems unlikely that one could invert rotation data for core—mantle pressures, and the implication is that if flow pressures at the CMB amount to only iO~ Pa, then not more than about 10% of the decade variations in LOD could be produced by pressure core—mantle coupling, —

4. Decade polar motion Polar motion is dominated by the 14-month Chandler wobble, a roughly circular, prograde motion of the instantaneous rotation axis through the body of the solid Earth. The diameter of the Chandler pole path is about 0.2 in. or 6 rad, with fluctuations of almost the same size. Accompanying the Chandler wobble is a forced annual wobble caused by seasonal shifts in atmospheric and hydrologic mass. Its amplitude is generally about half the size of the Chandler component. The instantaneous centre of polar motion defines the axis of principal moment of inertia of the solid Earth so that averaging the polar motion over several cycles of Chandler and annual motions should make the rotation pole and the principal axis almost coincident. It has long been recognized that the mean rotation pole, or the axis of principal moment of inertia, is migrating through the geographic coordinate system and this motion is commonly referred to as true polar wander or secular polar motion. During this century the secular pole position has migrated from the Conventional International Origin (CIO, the mean position of the pole

~o

in 1905) roughly in the direction 76° west of Greenwich (Dickman, 1981) at a rate of about 1.65 X iO~ rad a1 or a distance of somewhat more than a Chandler pole path diameter in 85 years. Accompanying the linear drift is an irregular decade-scale motion with an amplitude of about iO~rad. The linear part of the drift can be explained as the residual viscoelastic rebound of the mantle following the last deglaciation. Nakiboglu and Lambeck (1980) and Peltier (1983) find that the direction and rate of secular polar wander this century are compatible with the viscosity of the mantle inferred from glacio-isostatically raised beaches. The residual, after removal of a linear drift, appears to be a nearly white noise spectrum with a line peak at about 30 years (Wilson and Vicente, 1980). This motion is known as the Markowitz wobble (Markowitz, 1960) and has been described by Dickman (1981) as a highly elliptical (ellipticity = 0.823) retrograde pole path with amplitude about iO~ in either component. It is apparent as a motion nearly transverse to the linear drift. Its existence has been doubted because its small amplitude places it near the estimated accuracy of even the revised International Latitude Service (ILS) polar motion data of Yumi and Yokoyama (1980). Dickman (1981) argues that the Markowitz wobble is the impression of a free wobble of the oceans on the mantle. The polar motion service of the Bureau International de l’Heure (BIH) has been operating for barely sufficient duration to detect even one cycle of the Markowitz wobble, but indications are that the BIH series is considerably less noisy than the ILS series and a comparison of the two suggests that secular polar motion of the Markowitz type may be much smaller, or non-existent, in the BIH series. Figure 2a,b shows the homogeneous ILS series of Yumi and Yokoyama with the annual and Chandler motions removed by averaging over 6year intervals (very nearly five Chandler cycles). Using eq. 9 and assuming that changes in core— mantle pressure are responsible for the decade fluctuations, the variations in pressure are seen to be no larger than 3 x iO~Pa. This is smaller than the estimate based on geostrophy and the west-

287

00•

1900

1910

1920

1930

1941

1950

1960

910

~90Q

960

1910

920

YEAR

1930

1940

1950

1960

1910

980

YEAR

Fig. 2. (a) The decade polar motion in x

1 (i.e. the component along the Greenwich meridian). This is from the homogeneous ILS data with a linear drift, which is presumably due to post-glacial rebound, removed. The dashed line shows the BIH data for comparison. (b) The decade polar motion in x2 (900 East of Greenwich). This is from the homogeneous ILS data with a linear drift removed. The dashed line shows the BIH data.

ward drift. It may be argued then, that either a substantial fraction of decade polar motion is due to changes in the pressure regime at the CMB accompanying convection in the core, or pressure variations in the core at decade periods are smaller than the westward drift indicates, S Wahr (1987) uses estimates of the Earth’s C21, 21 gravity coefficients to argue that the rotation axes of core and mantle are misaligned. The upper bound on the misalignment then supplies an upper bound of 7 X iO~Pa for the pressure anomalies on the core—mantle boundary, if the misalignment is due to a torque alone and not to deformation of the mantle as well. This result is consistent with that found here. Wahr anticipates that with future improvements to the satellite data the upper bound on observable pressure anomalies might be only 4 x 102 Pa. The ILS data with the Markowitz wobble removed, or perhaps even the BIH data, show less than half the residual decade polar motion that is evident in the ILS data and so they suggest pressure variations, at the CMB, of less than iO° Pa. Decade-scale fluctuations in the inertia tensor of the atmosphere may cause 20% of the BIH decade polar motion variations, leaving an even smaller residual to explain, so that conceivably even iO~Pa may be too large an upper bound. —

5. Changes in 12 The coefficient J2 in a spherical harmonic expansion of the Earth is a measure of the difference between the polar and equatorial moments of inertia, C and A, respectively 1C—A~‘Ma2 ~ = 2

/1

where M is the mass of the Earth and a is the mean radius. Satellite orbits will precess at a rate proportional to J2. This has been used for nearly 30 years to measure J2, with the result that it is now known much more accurately than is the total mass of the Earth. A change in .J2 will produce an accelerated precession rate proportional to the rate of change of J2. Rubincam (1984) and Yoder et a!. (1983) have exploited this relationship to measure a secular decrease in J2 of about —10- 18 a_i. Yoder et al. (1983), Peltier (1983) and Rubincam (1984) have attributed this rate to the on-going viscoelastic adjustment of the mantle to the removal of the Pleistocene ice sheets. Inversions of this rate for mantle viscosity are roughly corroborative of the viscosity deduced from raised beaches and secular polar motion. Figure 3 shows Rubincam’s data for the changes in J2 after removal of mean and linear terms.

288

then the pressure at the core—mantle 1, boundary followwould be changing by at most 500 Pa a ing eq. 7. On a decade scale this rate would imply an upper bound on pressures similar to that inferred from decade polar motion, i.e. less than b0~ Pa. It seems likely that most of the 1018 s’ rate of change of J 2 observed over the lifetime of

1.5

=

I.0

+

+ -~

i.5

+

+

i.0

+ +

z

+

,~

05

- IS

-1.5

+ + +

+

~

42400

42800

+ ~

+

+

+

+++

++

I

43tii

+

I

44000

Lageos is in fact post-glacial simply because that from rate is consistentrebound, with secular

~+ + *

++

I + 43210

+

44400

I

I

448i0

45200

45600

MODIFIED JULIAN DAY

Fig. 3. The change in J2 from Lageos data. A linear rate has been removed,

These data have also been corrected for an 18.6year ocean tide, which Rubincam underestimates by 40%, and for the changes in J2 produced by the changes in the LOD. Both of these corrections are minor and partially cancel each other at long periods. Yoder of change of J et a!. (1983) report a secular 1 rate while 2 of —11.0 ±1.0 >< 10-19 s 9 s~ Rubincam reports —8.2the ±1.8 x i0’ just (—8.0 ±1.8(1984) x 10— 19 s’ after corrections described). The discrepancy is mostly attributable to the differences in the modelling routines that are used to reduce the satellite observations (D.P. Rubincam, personal communication, 1983). The residuals in Fig. 3 have power mostly at periods less than 2 years. The absence of longer period variations may be real, but it could also be due to the modelling process, which may absorb longer period variations into some of the parameters modelled in the orbit analysis. The A J 2 residuals, taken at face value, suggest pressure fluctuations of less than 100 Pa. However, as explained above, there may in fact be decade variations in J2, with amplitudes that would suggest much larger pressures, which have been removed in the modelling process. The difficulty with this data set is that it is so short that it is impossible to say if the ~1o 18 ~1 decrease in J2 is really secular, and therefore more likely a measure of post-glacial rebound, or part of a decade style variation, which might be attributed to fluctuations in CMB pressure. If it is the latter

polar motion, as observed and inferred from post-glacial rebound and the secular trends in sea-level. Therefore, the residual decade-scale vanations are liable to require a variation in CMB pressure of much less than b0~Pa. Many more years of Lageos data will be required to settle the issue but the promise of the data is clear. A pressure variation that will cause a change in J2 will also change the LOD. From eq. 8 a secular change in pressure of 500 Pa a will change the LOD by —6 p.s a’. During the span of the Lageos data the observed secular rate in LOD, after removal tidal and larger, wind effects, about 1,ofi.e. much and ofwas opposite + 100to p.s sign thea change in LOD implied by AJ 2. The tentative conclusion is that only a small fraction, 10% of the secular changes in LOD can be attributed to fluctuations in pressure at the CMB. This may change as more Lageos data become available. -

6. Decade trends in sea-level, tilt and gravity From eq. 11 and the preceding sections one might expect changes in sea-level of 1 cm from the admissible changes in pressure at the core— —

mantle boundary. Changes in sea-level of this magnitude are in principle detectable, but it is doubtful whether the cause could be separated from other influences, such as steric changes. Barnett (1983) reviews the difficulty in establishing the reality of a purported secular increase in global mean sea-level this century, of about 10 cm 100 a The sea-level data do not exclude the possibility of decade-scale variations of 1 cm. Tilt signals of 10 ~ rad (0.002 in.) are also expected from pressure fluctuations of io~ Pa (eq. 10) but there are no good tilt measurements (other ~,

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289

than sea-level) of long enough duration and stability to be useful. The z term is a residual of the process which determines polar motion from latitude observations. Polar motion is distinguished from azimuthal tilts at observing stations because polar motion is

The decade variations in the LOD set an upper limit of 5 x 10°Pa, but it is likely that most of the decade variations in LOD are from electromagnetic core—mantle coupling, so that these data do not contradict the upper bounds implied by the decade polar motion and J2. Rather, they imply

coherent between stations: the amplitude varies as a second-degree tesseral harmonic. Fitting a polar motion eigenfunction to the azimuthal tilt data from stations distributed over the Earth, yields an estimate the amplitude of the polar motion, a residualofthat also contains a coherent tilt ofand all

that not more than 10% of the decade variation in LOD are due to pressure core—mantle coupling. As well, the decade changes in J2, though poorly defined at present, imply1 decade variations in associated with the LOD of at most 6 p.s a changes in J 2. This is less than 10% of the decade changes in LOD during the Lageos span, further suggesting that no more than about 10% of the decade variations in LOD might be due to pressure variations at the CMB. Lageos data on A J2 will eventually prove to be even more useful than the decade polar motion data, but for the present it is difficult to draw any firm conclusion because the data span is so short. Recently, Bloxham and Gubbins (1985), Le Mouel et a!. (1985), Hide (1985), Voorhies (1984, 1986) and Whaler (1986) have made promising efforts towards using the secular variation in the geomagnetic field to map motions near the CMB. The validity of such inversions may be tested eventually by deducing the associated pressure field and examining its consequences for polar motion and J2.

the stations, but without the particular signature of polar motion. Any anomalous tilts of the observing stations will register in the z term, along with errors in nutation as well as refraction errors, The z term in the longest available polar motion data (the ILS data) has decade style variations of 10—6 rad or iO~ times larger than the tilts that would be expected to be associated with the range of pressures that the other geodetic data suggest. It must therefore be dominated by nutation and refraction errors and the possibility of extracting a tilt signal due to CMB pressure seems remote. Anticipated decade trends in gravity are 0.1 p.gal a This is much too small for absolute instruments to detect over a decade and, with the possible exception of the superconducting gravimeter, it would be impossible to separate from the drift of a relative instrument. —

~,

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Acknowledgements 7. Discussion It has been shown in the preceding sections that the decade variations in LOD, polar motion and perhaps J2, may all be used to infer an upper bound on the decade scale variations in the pressure field at the CMB. Of these, polar motion sets the most severe upper bound at 3 x iO~ Pa. By accounting for other sources of decade polar motion, such as the atmosphere and groundwater, it may be possible to reduce this bound even further. Ultimately, the possibility exists of using polar motion data to monitor the second-degree tesseral harmomc of pressure and hence acquire some control on the large-scale circulation near the CMB.

This work was supported by the Natural Science and Engineering Research Council of Canada through operating grant No. A0814.

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