Rotational deformation of the earth

200

Physics of the Earth and Planetary Interiors, 11(1976) 200—206 © Elsevier Scientific Publishing Company, Amsterdam — Printed in The Netherlands

ROTATIONAL DEFORMATION OF THE EARTH L. MANSINHA and PO-YU SHEN Department of Geophysics, University of Western Ontario, London, Ont. (Canada)

(Received October 6, 1975; accepted for publication November 19, 1975)

Mansinha, L. and Shen, P.-Y., 1976. Rotational deformation of the earth. Phys. Eaith Planet. Inter., 11: 200—206. The effect of rotation on two elastic real earth models and the equivalent fluid earth models has been investigated. The present angular velocity has resulted in the increase of the surface radius by about 0.7 km for a real earth and 1.02 km for a liquid earth. The corresponding ellipticities are 0.1052 . 10—2 and 0.3329 . 10—2, respectively. A surprising result is that the solid inner core of a real-earth model becomes slightly prolate under uniform rotation while the rest of the earth is oblate.

1. Introduction The effect of rotation on a spherical body is twofold. There is a small overall radial expansion, and, a departure from sphericity by flattening at the poles. Similarly, for a slowly decelerating body there is radial contraction and a decrease of oblateness. The present oblateness of the earth is essentially due to rotation. Determination of the shape of a hypothetical rotating fluid earth has been a subject of interest to mathematicians and astrophysicists ever since the formulation of the equation governing the shape of a rotating fluid body by A.C. Clairaut in the 18th century. An approximate solution to Clairaut’s equation is (O’Keefe, 1966): 1Urn a = ~ + 25 1 3C/ 2Ma 2\12 (1) where: ratio of centrifugal force at the equator to the

mean surface gravitational acceleration M = mass of the earth C = polar moment of inertia a = mean radius of the earth The quantity C/Ma2 is determined from the precessional constant H = (C — A)/C and J

2 = 3(C A)! 2 has been determined

2. Since 1958 the value of J 2Ma

-

culations are not possible with Clairaut’s equation.

—

m

from satellite-orbital data. Because of this the value of hydrostatic flattening of the earth has been revised to 1/299.67 (Jeffreys, 1963) and about 1/300 (O’Keefe, 1966). Our approach to the problem is different from eq. 1. We solve the static equations of equilibrium witlj the centrifugal force as the inhomogeneous term. Physically, the problem consists of slowly accelerating a spherical earth to the present rotation rate of &~l= 7.29211 10~rad/sec. Under slow acceleratioli or deceleration the static equations are appropriate. There is no explicit direct dependence of our solution on the polar moment of inertia. There is, however, an mdirect dependence through the earth model, especially the density. Our approach enables us to calculate the radial expansion as well as the ellipticity for an elastic earth and the equivalent fluid earth. Elastic-earth cal-

—

In the eighteenth century there was a controvçrsy between proponents of a prolate earth and those be. lieving in an oblate earth (Chandrasekhar, 1963~ . . . . Fischer, 1975). J. Cassini had determined the elhptici-

ty to be —1/95 from measurements in the south of France (Kmg-Hele, 1967). Subsequent data showed .

.

conclusively wrong. results show that that Cassini the solidwas inner core Interestingly, of an elastic our earth is slightly prolate. A change of the rat~ofrotation will have the oppo-

RO~~AT1ONAL DEFORMATION OF THE EARTH

site effect on the inner core of an elastic earth as cornpared to the shell and outer core. An increase in the rate of rotation will make the inner core more prolate while the rest of the earth oblate. Since the long-term behaviour ofbecomes the earthmore is that of a fluid, with time the inner core will become oblate. The earliest numerical calculation on the shrinking of the crust due to retardation of the earth’s rotation was by Stoneley (1924). His calculations were based on relatively simple earth models. He estimated that the radius of the earth decreased by 0.5 km due to the change of the length of day from 12 to 24 hours.

201

specified by: 0 = 2/3 2 U

0f W rP 13w 2 rp 0 u2f=—2

0 = v 2f

1’3

(5)

2

~ rp The zeroth-degree term causes change in the radius of the earth. The second-degree term causes oblateness or prolateness. The subscript f on the lefthand side of eqs. 5 implies the forcing term. •_

3. The equation of equilibrium 2. The centrifugal force The centrifugal force acts in a direction normal to the axis of rotation at all points of the body. On an element of mass the force density is: 2 CA~ r Sifl ~ where p is the density, r the radial distance and w the angular rate of rotation. The force vector f may be written as: 2r sin2 0 + i 2rsinO cos 0)p (2) f= (i,w 0t~ In terms of the associated Legendre functions we may -

The vector equation of equilibrium (Smylie and Mansinha, 1971, eq. 11) is usually written in the y~ notation introduced by Alterman et al. (1959). The y~(i = 1, 6) are the radial coefficients of: radial displacement, radial stress, transverse displacement, transverse stress, change in the gravitational potential and the radial gravitational flux density, respectively. The displacement in our case is given by: u(r,O) = ir[{yi}8(r)P8(cos 0) + {Yi }~(r)1i(cos0)] 0 dp~(cos0) .,

+i

0 {y3}2(r)

write: r

~ dP°V1 Iow2rI! ~ (3) L ~ 3 dO ~ An arbitrary vector can be decomposed into spheroidal and toroidal terms (Bullard and Gellman, 1954). Here flacks a toroidal component. An arbitrary spheroidal vector s may be written as:

I

=

s=

I~

~

—

n m~n [iru~

+ i0~~j vm

~p02)p +

..~

(r)P,r(cos 0) eim~

r~d~),~m(~~os do 0) eim~

4, u~(r)~sin0 0) ~~imq) dØ

(6)

Because m = 0 there is no dependence on ~ and also there is no component in the i~direction. Hereafter the superscript ofy1 andy3 can be dropped; m = 0 is implied. The equations of equilibrium for n = 0 and n = 2 are different. Hence, the subscript can be dropped for convenience; the context will make the subscript clear. For n = 2 the spheroidal equations of equilibrium for an elastic earth form a system of six simultaneous first-order differential equations (Smylie and Mansinha, 1971, eq. 32). dy1 2X 1 n(n+1)X 4p

dr dy2

+i

dO

dr.

+ 2.t)r Y1 + ~ + 22Y2 + (A + 2~z)r~’~ r(X4p0g0 (3X+2pfl 3” (A + 2p)r)’2

L rn(n

Specifications of the radial coefficients ufl” and v~”~

—

+

,-

+

r

4~z(A + 2,.z)r2]

1)

2n(n

+

1) (3X

0g0 1) r Y4 P0Y6

‘~(A + 2p)

—

are sufficient to uniquely determine the spheroidal vector. From eqs. 3 and 4 the centrifugal force f is

n(n +

+ 2p)1

p

+

—

m —

U~f

~f~

202

L. MANSINHA AND P.-Y. SHEN

dy3

1

1

cb’62

1

~

Ipogo

dy4

(3X+2~)1

TQ~= [Tr

—

2ji (A

+

2 + 2n

—

—

7)

3’2

A

2p)r2i ~1

(A + 2ji)r

1)X + 2(n2 + n

—

1 2 ~—~Y1 dy

Po

m _Vnf

~

6

~Y2~7Y3

4Pogo r y

2 3

1

l)p]y3

(X+2ji)r2 2i~ [(2n

+

Similarly, by setting 4u(r) = 0 in eq. 7 one obtains, for ndy= 2 and a fluid earth:

1+~P0g0y3—P0Y6—U~f

0p0g0y1 —Y2

dy5

PoY5 —fl)2f

(10)

dy5 =

4irGp~y1~Y6

dy6

n(n

—~-=—4irGp0

+

4irGp0y1 ~Y6

-~-

n(n 2+ 1) Y5~Y6 2

1)

~

—i— r y3+~j~y5__Y6 dy6 24irGp0 6 2

r

y3+

Here X(r) and p(r) are the Lamé parameters; p 0(r) and g0(r) the undisturbed density and gravity, and Grepresent is the gravitational constant. The corresponding equations for n = 0 for an elastic earth are:

The9fhomogeneous form eq. the 10, subject obtainedofby setand V9f to zero, hasof been some ting U discussion in literature (Smylie and Mansinha, 1971; Israel et al., 1973). After some algebraic manipulation

dy 1

2X

1

one obtains:

ri

dr(X+2p)rY1~X+2’2 I 4p0g0

dy2

r

=

—POY6 dy5 =

dy6

—

~ (3A+2p) +

(A

Y2[~

1

+ ~p)~2jJ’1

(x + 2p)r y2

2

1 Yij—Ys

The equations for a fluid earth, as well as for the

dy1 2 1 —~-=.~—y~+-~-Y2

Y2 = 0

1 =

I

dy5 Yl—POY6 —U~~-

dy6

4~=47rGp 0~1~Y6

n(n

+ l)g0

~

+ ‘Y6]

(12)

Y4°

4p0g0

r -

bution everywhere in the fluid it is not preferred. Instead one choosesy2 = 0. Physical arguments support-

ing this choice have been adduced by Crossley and Gubbins (1975). The homogeneous equations then become:

4irGp0y1 +y6

fluid outer core of an elastic earth are obtained by setting p = 0. We then have, for n = 0:

dy2

(11)

Because the expression in parentheses implies chemical homogeneity and an adiabatic temperature distri(8)

U~f

g0p0]0

dp0 d

(9)

4ii-Gp0

r

g0

y5+y6

i6irGp0

g0r

n(n+1)] [41rGpo2]~ 2 jy~ L g + r 0

ROTATIONAL DEFORMATION OF THE EARTH

203

Sincey

1 andy3 can be expressed in terms ofy5 andy6, we have to solve only the last two equations

2y /d 1 ~y—

dy1 ~ 2-i-— ~y1

)

dy1

6p0 +—

~

0

(17)

of(l2). Similarly, the inhomogeneous equations are: Y5

g0

+

We intend to show that eq. 17 is equivalent to the last equation of the set (12). In addition to eqs. 12 we use the following relations:

PV9f

p0g0

-~-=——g ~o 2

0 Y2 =

0+4irGp0

~-[4Y~ ~?Y6 +—~~(U~f +4v~f)]

(18)

3

(19) =

0

(13)

dy5

4ITGp0 dr = g0

4irGr 0 ~

+y~+

I l6irGp0 g0r

dy6

From eqs. 12 we have:

61

1 Yl =Y~

~—V2f g0

[4lrGpo +~]Y6 g0 r

Using eqs. 12 and eq. 18, one can write:

dy1

4irG r [u2f + 4v9f] Only the last two simultaneou~differential equation

Y6 (20)

2y d 1

1 dy6

/ 4

=

4lrGp0\

2 2g g02 )Y6 + r~YS0

deed be solved for the liquid part of the core or for a liquid-earth model. For n = 2, eqs. 7 and 13 can be

~2 g0 dr — Substituting eqs. 19 and 20 in eq. 17 and after some

solved by including the centrifugal potential and its derivative in y~and Y6~The differential equations then assume the form of eqs. 12.

algebraic manipulation we obtain:

4. Equivalence of Clairaut’s equation From Jeffreys (1959, p. 147, eq. 15) we have Clairaut’s equation: 2e 6e~ 6P0(de e) + 0 ‘d \c~2 r~i r dr r (14) where e(r) is the ellipticity; Po is the undisturbed density; and ~ is the mean density and is given by: 3

fr p~x~2dx

(15)

=

(16IrGPO g0r ‘~

6 \ +—~-)~5

(2÷4~Gp0\ ~r g0 )Y6

This is the same as the last equation in the set (12). The correspondence between Clairaut’s equation and the inhomogeneous equations (13) for a fluid earth can be demonstrated by an equivalent procedure. 5. Initial and boundary conditions Solutions of eqs. 7 and 8 for the elastic earth and eqs. 9 and 13 for the fluid earth must satisfy the condition of regularityboundary at the center of the earth well as the appropriate conditions at theasexternal surface and internal surfaces of discontinuity. For the elastic earth and n = 0 the regular solutions

The ellipticity e is: 3 Y1(r) e(r)=—~--—-——

dy6 dr

(16)

Substituting eq. 16 in eq. 14 we have Clairaut’s equation in terms ofy 1:

neaEr0 are: ~1

3X

y2 =A

+

Ar+... ...

I

I (21)

204

L. MANSINHA AND P.-Y. SI-lEN

2+...

y5=B+...

y5B+2irGp0Ar

y 60

y6=O and forn

Forn=2wehave:

=

2,wehave:

y1=~r+Br~+... P

2+...

y

y2=2A—(k+jffl)r A

6=(—C+w 3

1

Y3

=

~—~r +-~-{k + (5?~.+ 7p)B}r

Y4

=

A

+ —p--3X

1

{

k + (8X

+

0)r+...

stants be determined conditions. In the to above expressionsby A, boundary B, and C are free con-The

...

elastic constants A, p and the function k are assumed

7p)B }r2 +

to be of r when r is sufficiently At independent internal surfaces of discontinuity in thesmall. solid

2irGp 0

~

=

+

(24)

2p

{C

2

—

earth all thebetween Yi are assumed to be continuous. boundaries the liquid outer core and At thethe

2irGp0 ~ + B)r~+ ~ (-~

A r

+

solid earthy y 5 andy4 are continuous. y1, y2, Y3 and 6 are discontinuous. The discontinuity in)1 at the solid—liquid boundary was first introduced in Smylie

}

3+ =

4irGp~ 2k

—

Y6

A

Here

(A + 2p)B} r

7

and Mansinha Buty6 was out assumed to be continuous. It has(1971). now been pointed by several

2 2p

k = 2irGp0A

+

— -~-

w

(22)

0

In the fluid earth the initial solution for n

Yl

=

Ar +

=

authors (see Crossley and Gubbins, 1975) that Y6 is discontinuous.

0 is

6. Results and discussion Two earth models were used to compute the rotational deformation of the earth. The two models have

...

_f

Y2 =A [3?~.

2]+

..

irGp~r

(23)

TABLE I

Deformation of earth model HB1 with solid inner core and the corresponding liquid earth due to a change of rotation by fi

=

7.292 1 . iO~rad/sec; the radial expansion is in km and the ellipticity has been multiplied by 106 Depth (km)

Model HB1 with solid inner core

Liquid HB1

radial

radial

ellipticity

expansion 0 1,000 2,000 2,800

0.73 0.72 0.64 0.54

2,878 2,878

ellipticity

expansion

ellipticity

from Bullen

and Haddon (1973)

1,052 1,298 1.615

1.02 1.08 0.89

3,329 3,036 2,731

3,353 3,056 2,750 2,569

0.53 0.53

1,947 1,980 1,726

0.62 0.61 0.61

2,551 2,545 2,545

3,000 4,000

0.51 0.36

1,722 1,692

0.59 0.41

2,538 2,494

2,556 2,511

5,000 5,121 5,121

0.21 0.19 0.19

1,675 1,674 —13

0.26 0.22 0.22

2,469 2,467 2,467

2,486

— —

— —

-

ROTATIONAL DEFORMATION OF THE EARTH

205

different radii for the inner core, in addition to slightly-varying elastic properties. One model is HB1 of Bullen and Haddon (1967a, b) modified to include a solid inner core of radius 1,250km (Stacey, 1969, p. 281). The other is model Bi of Bullen and Haddon (l967c) with inner core radius of 1,660 km. Liquidearth models were derived by simply setting p = 0. In our solutions we have assumed that the earth was initially spherical and slowly accelerated to its present angular velocity, 12. For any other angular velocity to, the results should be multiplied by w2/f12.

2.c

,

I -

I I

E

-

The radial expansion and ellipticities for model

INNER CORE

HB 1 are presented in Table I. For comparison the internal ellipticities computed by Bullen and Haddon (1973) are also listed. Their results were obtained by assuming an observed surface ellipticity of 0.3353 1o_2. This is slightly higher than the hydrostatic value. Consequently the internal ellipticities of Bullen and

0

-

2

.

C 0f.

Haddon not exactly hydrostaticfor values. Fig. 1 (1973) shows are the also zeroth-degree deformation a real earth (HB 1) as well as the corresponding liquid earth. Note that the radial expansion is greater m case of the liquid earth. Also the maximum expansion is not at the surface but at 5,984 km for HB1 and at 5,721 km for the liquid earth. The ellipticities for model HBI and the corresponding liquid earth are shown in Figs. 2 and 3, respectively.

~ 3 KU RADIUS IN 1O ellipticities of a spherical-earth Fig. 2. Surface and internal model HB1 caused by rotation. Note the prolateness of the

solid inner core. The rest of the earth is oblate.

For the real earth, there are sharp discontinuities in the ellipticity at the inner-core—outer-core and the outer-core—mantle boundaries. The maximum ellipticity is at the bottom of the mantle. The inner core is slightly prolate. For the liquid earth the ellipticity remains close to 0.0025 in the core and then increases uniformly to the surface value. The only significant difference exhibited by model

BI is in the ellipticity of the inner core for the real~I.o

earth model. It is partly oblate and partly prolate

(Fig. 4). ~

~

4

RADIUS IN io~ KM Fig. 1. Radial expansion of a spherical earth due to change in angular velocity of fi = 7.292 11 - i(Y-~rad/sec. The earth model used is Hill modified to include a solid inner core of radius 1,250 km. The liquid HB1 was obtained by setting =

0.

a

I

‘

0

I

2

3

5

6

RADIUS IN 10 KM Fig. 3. Ellipticity ofa liquid HB1 earth model due to rotation.

206

L. MANSINHA AND P.-Y. SHEN

2.0

N

Bullard, E.C. and Qellman, H., 1954. Homogeneous dynamos

I

~

and terrestrial i~iagnetism.Philos. Trans. R. Soc. London, Ser. A, 247: 213—277. Bullen, K.E. and Haddon, R.A.W., 1967a. Earth oscillations and the Earth’s interior. Nature, 213: 574—576. Bullen, K.E. and Haddon, R.A.W., 1967b. Derivation of an Earth model from free oscillation data. Proc. U.S. Nati. Acad. Sci., 58: 846—852. Bullen, K.E. and Haddon, R.A.W., 1967c. Earth models based

\ \

I

INNER CORE ~ 1.0

2

.

I Id

-I I

0

I

2

-

_______

0

I

I

I

I

I

I

2

3

4

6

6

RADIUS IN iO~KM Fig. 4. Surface and internal ellipticities of a spherical-earth model Bi. The solid inner core has a radius of 1,660 km. The inner córe is partly prolate and partly oblate.

Acknowledgment We thank the National Research Council of Canada support of this project.

for financial

on compressibility theory. Phys. Earth Planet. Inter., 1: 1—13. B~illen,K.E. and Haddon, R.A.W., 1973. The ellipticities of surfaces of equal density inside the earth. Phys. Earth Planet. Inter., 7: 199—202. Chandrasekhar, S., 1963. Ellipsoidal Figures of Equilibrium. Yale University Press, New Haven, Conn., 252 pp. Crossley, D.J. and Gubbins, D., 1975. Static deformation of the Earth’s liquid core. Geophys. Res. Lett., 2: 1—4. Fischer, I., 1975. The figure of the Earth — changes in concepts. Geophys. Surv., 2: 3—54. Israel, M., Ben-Menaliem, A. and Singh, S.J., 1973. Residual deformation of real Earth models with application to the Chandler wobble. Geophys. J.R. Astron. Soc., 32: 219—247. Jeffreys, H., 1959. The Earth. Cambridge University Press, London, 392 pp. Jeffreys, H., 1963. On the hydrostatic theory of the figure of the Earth. Geophys. J.R. Astron. Soc., 8: 196—202. King-Hele, D., 1967. The shape of the Earth. Sci. Am., 217: 67—76. O’Keefe, J.A., 1966. The equilibrium shape of the Earth in the light of recent discoveries in space science. Lect. Appi. Math., Space Math. Part II 6:119—154. Smylie, D.E. and Mansinha, L., 1971. The elasticity theory of dislocations in real Earth models and changes in the

rotation of the Earth. Geophys. J.R. Astron. Soc., 23:

References Alterman, Z., Jarosch, H. and Pekeris, C.L., 1959. Oscillations of the Earth. Proc. R. Soc. London, Ser. A, 252: 80—95.

329—354. Stacey, F.D., 1969. Physics of the Earth. Wiley, New York, N.Y. 324 pp. Stoneley, R., 1924. The shrinkage of the Earth’s crust through diminishing rotations. Mon. Not. R. Astron. Soc. Geophys., Suppl., 1: 149—155.