Earth's geoid anomaly, mantle convection and stress field under the lithosphere

50

Physics of the Earth and Planetary Interiors, 54 (1989) 50—54 Elsevier Science Publishers By., Amsterdam — Printed in The Netherlands

Earth’s geoid anomaly, mantle convection and stress field under the lithosphere Fu Rong-shan Department of Earth and Space Sciences, University of Science and Technology of China, Hefei, Anhui (People’s Republic of China) (Received November 13, 1987; revision accepted February 22, 1988)

Fu R.-S., 1989. Earth’s geoid anomaly, mantle convection and stress field under the lithosphere. Phys. Earth Planet. Inter., 54: 50—54. The contributions of the core—mantle boundary and the lithosphere—mantle boundary to the Earth’s geoid anomaly are computed using different thermal convection models that are constrained to fit the Earth’s geoid anomaly. The results show that the contributions of the lower boundary for two models decrease as the degree increases and the effects of the lower boundary on the stress field under the lithosphere can be ignored. These results indicate that using Runcorn’s correlation equations of the Earth’s geoid and mantle convection to calculate the stress field under the lithosphere is a reasonable approximation.

I. Introduction The pioneer work to establish the relationship between the Earth’s geoid anomaly and the mantle thermal convection was developed by Runcorn (1964). In a later study, he obtained the famous correlation equations of the Earth’s geoid anomalies and mantle convection (Runcorn, 1967). In addition, Runcorn calculated the stress field under the lithosphere with low degree coefficients of the geoid, ignoring the contributions of the lower boundary. Following Runcorn’s work, Liu and coworkers (1976, 1977, 1978, 1980a, b) computed the stress field of many selected areas with degree 12—25 harmonic coefficients of the geoid while Huang and Fu (1982 a, b) investigated Asia and the global stress field with degree 2—30 harmonic coefficients of the geoid anomaly. All of these results provide information about the stress state under the lithosphere as well as the mantle convection patterns, which can be used to investigate the driving mechanism of the plate motions. A global stress map on the surface of the Earth was computed by Fu and Haung (1983) by adopting 0031-9201/89/$03.50

© 1989 Elsevier Science Publishers B.V.

Runcorn’s equation as the boundary condition on the bottom of the lithosphere. But two important questions remain: is Runcorn’s simplification correct and how much does the lower boundary contribute to the geoid anomaly? Obviously, before establishing a mantle convection model which is directly associated with the Earth’s geoid anomaly, it is not easy to answer these questions. Fu (1986 a, b) developed a physical—mathematical model of mantle convection, with which the mantle convection patterns can be calculated using the harmonic coefficients of the Earth’s geoid anomaly. This model provides a possible way for us to answer the above problem. The purpose of this paper is to investigate the contributions of lower and upper boundaries to the Earth’s geoid anomaly for different boundary conditions.

2. Correlations between Earth’s geoid and mantle convection Assuming that the Earth’s mantle is a spherical shell of homogeneous, isoviscous, incompressible,

51

Newtonian viscous fluid, the motion of the fluid must follow the Navier—Stokes equation. Ignoring the Coriolis and inertial forces, the equation can be written as

3. Earth’s geoid anomaly and physical—mathematical models of mantle convection

v vp + pg (1) where ~tsis viscosity, u is velocity, p is pressure, g is the gravitational acceleration and p is density difference caused by temperature variations in the convective fluid. The low-degree anomaly of the external gravity potential of the Earth has two contributions: the density distribution in the mantle and the boundary deformations caused by the

associated with the Earth’s geoid anomaly was developed by Fu (1986a, b). The basic equations governing written as thermal convection in the mantle can be

—

,i

=

—

mantle convection. Based on this idea, Runcorn obtained the famous correlation equation between the mantle convection and the Earth’s geoid anomaly: 2W 2 dW~\ / d dr2 ,

—

=

~(ba)

fl

+

=

2~ / ddr2 ~

~

~)

(2) where J’V~ is potential function of velocity, a and b are the upper and lower boundary radii respectively, M is the mass of the Earth, a’ is the radius of the Earth, C~m and S~ are harmonic coefficients of the Earth’s geoid anomaly and P~m(cos0) is the associated Legendre polynomial. Since the factor of the second term on the left 1 is << 1 with increasing side of n, eqn. (2) (b/a)’~~ degree Runcorn gives the stress field under the lithosphere as follows

=

2

(~) ~

2n + 1 n +1

1 sin9

4~ra x(—mC,~sin m4+mS,~cos mq) x P~m(cos8)

n=2 m=O

a /~

—

a~r -~j--

=

—

.

+

ax

+

agx1O

2U + v

V

k V 2~

(4)

1 dU.

ax,

0

=

where a is the coefficient of volume expansion, ~p is the perturbation of pressure, 0 is the perturbation of temperature T, U~is the velocity vector, x

son’s equation V2V=4~Gp

x (c,~cos mq + S~sin mq)P~m(cos8)

a’

=

p

r=~

Mg (a’ ~ 2n + ~ ~ 4~fLa~a) n+1 m~O

~

au,

1 is the position vector, is kinematic viscosity, k is thermal conductivity. The gravitational potential V must satisfy Pois-

2 dJ4’~ +

The thermal convection model of the mantle

(5)

where G is the universal gravitational constant and p is density. Following Chandrasekhar’s work (1961), eqns. (4) can be expressed in terms of a velocity potential function W. Assuming that no external forces act on the boundary and taking the toroidal cornponent of velocity Z 0, the velocity potential W can be written as (Fu, 1986a) =

W

~ ~

~B

1W1~1(c~ COS m4 + S,~sin m4i) P~”~(cos 0) (6) where B 1 are constants to be determined by boundary conditions and W are composed of the half-order Bassel function fl

m

X

w;= [Jfl+(aJr)+J(fl+)(aJr)]/V~

(7)

(3) where a1 are calculated by the equation

aN

=

~ Mg a’ n+1 2n + I x (C~cos4~a m~(a) + S~sin n+1 m~)dP~~ n=2m=O x (cos 0)/do

where 6=Cn(n+1).n=~ Cn is the Rayleigh number of the fluid. (8) a The boundary conditions are constant temperature and zero vertical velocity on the lower and

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upper. boundaries, with the horizontal velocity equal to zero for a rigid boundary and shear stress equal to zero for a free boundary. In order to constrain the fit to the Earth’s geoid anomaly, a frictional boundary condition is introduced. It implies that there are both horizontal velocity and shear stress boundary conditions on a boundary. In this case the shear stress on the bottom of the lithosphere can be written as aN

TABLE I Computation models Model Boundary condition

Rayleigh number

I

lOX io~

II

L—M frictional C-M free Free surface

III

C-M frictional L—M frictional

4. Computation results The velocity field of thermal convection which (9)

aE

=

is associated with the Earth’s geoid anomaly is controlled by the Rayleigh number as well as boundary conditions (Fu, 1986 a, b). Table I lists

d d2W sin 8 d4 dr2 IIr=a ~u

—

the Rayleigh numbers and the boundary conditions of the computation models which are used in our calculations. In our computations, the radius of core—mantle boundary (C—M) is b 3480 km, the radius of lithosphere—mantle boundary (L—M)

It is then easy to obtain the ratio of the lower boundary and the upper boundary contributions to the Earth’s geoid anomaly lb \

R=

=

d2W~ 2 dJ4~,\~

n+1/

~~) ~

dr~

)

d

+

is a

~r=b

dr2

/ Ir=a

j

~~-\\

¶

\

~

//~5~

I /

\

~

40~ ~

~

~

~

/

¶

\\\~~

200

\ N.

\

‘~

~

0

I.~-20

/ /

-40°

0~Dso~D

-60° /

/

\

~

~

-

—

\ \ \

/

~ \

~

I

/

/

~///

/ —.

~T

I

a

/

I~o

/

~

.

ID

~

~

I

00fi0.

~. ~

/~D

~__

o~

~

tC\\~~

-.

~\

/

\

—

C -/

0 0

0

~//

~

‘

—

._

1

~

/

\

/ /

C

I

~

-. —

~ —

~—

\— CC —

.~ ~

\

,0,0s0

0 \D~ —

— —

~

-

~80°

0°

20°

40°

60°

80°

100°

120°

140° 160°

180°

200° 220°

240’

260° 280° 300° 320°

LONGTITUDE

Fig. 1. Stress field of sublithosphere. D = divergent plate boundary, C

=

1

/ C

0 /

1

-

/ -

~

/

//

0

~

/~7~

\

\ \\ ~ /

/

~ ,//

.

/ /

/

~

~

-

/-

-

~_//////~

I

I

/

~

\

C

0

///////~ / / / /

N

C

I /

\ ~C~flb\C\\

C~C\

—

1

.

/ 0

~

.~\

-

.-

~

0

6280 km, the average viscosity of the mantle

=

J

r ~

=

is ~.t 1022 poise and the harmonic coefficients of the Earth’s geoid anomaly are as obtained by Lerch et al. (1979). The computed stress field2—8 under the lithosphere with modelshear I and degree is illustrated

(10)

(d2w~+ 2 dJ3’~\

80’r

1.6 X i0~

CM rigid

2W I d d~‘~~r=a

~

5.0 X iO~

convergent plate boundary.

340° 360°

53

in Fig. 1, in which the symbols C and D are to be taken as the convergent and divergent boundaries respectively. The computed shear stress agrees with the orientations of Runcorn’s (1967) mapping of the stress and is also similar to the plate motions. The orientations of the stress field are compatible with the underlying velocity field (Fu, 1986b). The maximum value of the shear stress is ~ dyn crn2, and is comparable with Runcorn’s estima—

ilOflS.

The contribution ratios of lower boundary to upper boundary R as a function of n are listed in Table II and Fig. 2 for each model considered. It is very clear from these results that the lower

_____________________________

0_iD_i

~

io~ io6 (a)

~ io~ ~ 1O~

________

1

5

________________

10

,,

:o’

~ io~

~ io~

_________________________

15

0.73100E—07

0.54850E+0O

0.24690E—04

3 4 5 6 7 8

io

Earth’s geoid

anomaly for models I and III is much smaller than the upper boundary contribution, and it decreases with increasing degree. This is particularly true for model I, where R is 0.05 when n 2, decreasing to order i0~when n 15. Boundary model III is more similar to Runcorn’s suggestion (both rigid boundary), with decrease in the ratio R for increasing n approximated by (b/a)”~’.Finally, model II is very different from the other two models and the lower boundary contribution to the Earth’s geoid anomaly cannot be ignored, =

=

7.

Models I and III are not directly equivalent to Runcorn’s boundary conditions, but are fairly similar, especially model III. The computed results show that the shear stress field under the lithosphere based on the mantle thermal convection model, which is constrained to fit the Earth’s low-degree geoid anomaly, agrees with the stress orientations of Runcorn’s field. This implies that

4”N~

-

Model III O.24830E+O0 0.88840E—O1 O.41930E—01 0.21000E01 O.l073OE—01 O.53900E—02 0.28O40E—02 0.15120E—O2 O.92710E—03 0.8853OE—O3 0.52340E03 O.50700E —04 0.10210E—O4

5. Discussion and conclusion

~

MODEL 1ff

io-

Model II 0.1O53OE+O1 0.1l230E+0l O.79650E+O0 0.46020E+00 0.19340E+OO 0.20110E—02 0.14530E+00 0.25130E+O0 0.33190E+O0 O.39690E+OO 0.44620E+00 O.48770E +00 0.5214OE+O0

except for n

)b)

0

11 12 13 14

Model I 0.48680E—01 O.16600E—01 0.72210E—O2 O.33910E—02 0.16660E—02 O.85180E—03 O.49660E—03 O.44750E—03 0.2776OE—03 O.24730E—04 0.47500E05 0.1 I000E —05 O.27700E—O6

2

=

15

MODEL 11

~,

Degree

boundary contribution to the

MODEL

:J

TABLE II Ratio of contributions of boundaries to geoid anomaly (lower boundary/upper boundary)

~

10 2 1o~

~

c)

____________________________ ii

10

15

Fig. 2. Contribution of boundary to geoid anomaly: (a) Model I, (b) Model II, (c) Model III.

Runcorn’s simplification is probably right. Boundary model I is more plausible than Runcorn’s boundary model because even the Earth’s lithosphere is a mechanical element inde-

54

pendent of the underlying mantle and hence can be modelled using a cooling plate spreading from oceanic ridges. Thus, there is not a sharp boundary between the lithosphere and underlying mantle, and both horizontal velocity and shear stress must exist on this boundary. On the other hand, the computation speed of the stress field under the lithosphere using Runcorn’s equation is much laster than in our calculations, especially for large n, so Runcorn’s stress equation is still useful for obtaining the stress orientations under the lithosphere. However, the simplified model of the mantle either in Runcorn’s paper or in our study includes an assumption that the mantle is an isoviscous fluid. Obviously, this approximation is quite sketchy. Generally, a modest increase in viscosity with depth is now widely accepted (Wu and Peltier, 1982, 1983, 1984; Yuen et al., 1982; Peltier, 1985). The studies of numerical simulation with depth-dependent viscosity (Gurnis and Davies, 1986; Fang and Fu, 1987) show that the stream lines concentrate on the upper boundary of a convective fluid in this case and the convection is slower in the lower part of the fluid than in the upper part. This implies that the upper part in a depth-dependent viscosity fluid plays a more important role than in an isoviscous fluid, so that the upper boundary contributions to the Earth’s geoid anomaly in a modest depth-dependent viscosity mantle are probably larger than in an isoviscous mantle. It must be noted that the results from both the thermal convection model (this paper) and Runcorn’s equation depend not only on the thermal dynamical parameters but also on the boundary conditions. More detailed information on the thermal state and the mechanical behaviour of the lithosphere—mantle boundary is required, in order to obtain more realistic models of the stress state under the lithosphere,

Acknowledgement This work was supported by the National Joint Seismological Science Foundation of China No. (86) 008.

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phys. Res., 69: 4389—4394. Runcorn, S.K., 1967. Flow in the mantle inferred from low degree harmonics of the geopotential. Geophys. J. R. Astron. Soc., 14: 375—384. Wu, P. and Peltier, W.R., 1982. Viscous gravitational relaxation. Geophys. J. R. Astron. Soc., 70: 435—486. Wu, P. and Peltier, W.R., 1983. Glacial isostatic adjustment and the free air gravity anomaly as a constraint on deep mantle viscosity. Geophys. J. R. Astron. Soc., 74: 377—450. Wu, P. and Peltier, W.R., 1984. Pleistocene deglaciation and the Earth’s rotation: a new analysis. Geophys. J. R. Astron. Soc., 76: 753—791. Yuen, D.A., Sabadini, R. and Boschi, E., 1982. Viscosity of the lower mantle as inferred from rotational data. J. Geophys. Res., 87: 10745—10762.