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Excitation of polar motion by earthquake displacement field
Chin.Astron.Astrophys.5 (1982) 60-63 22 (1981) 383-388 -
Act.Astron.Sin.
EXCITATION SONG
OF POLAR
Guo-xuan,
MOTION
Pergamon Press. Printed in Great Britain 0275-1062/82/010060-04$07.50/O
BY EARTHQUAKE
ZHAO Ming, ZI’iE~:G Da-wei
DISPLACEMENT
Shanghai
FIELD
Observatory
Received 1980 October 23
ABSTRACT. We discuss the excitation of polar motion by earthquake displacement field. Instead of the usual static equilibrium equations in the literature, we use an improved set as given in /I/, which guarantee continuity at the core-mantle boundary. We take the parameter values of three earthquakes from !2/. To obviate the singularity at r=O, we use asymptotic solutions by power series within a small sphere around the centre. Outisde this sphere, the equations are numerically integrated by the Runge-Kutta algorithm. Our equations !l! gave polar shifts some 3 times larger than Dahlen's equations /2/.
1.
INTRODUCTION
When considering the excitation of polar motion by earthquakes, it is important to calculate the additional tiisplacementfield caused by earthquake dislocation. Since this was pointed out by Smylie and Nansinha in 1967, IS/, there has been much discussion. But a number of inconsistencies remain, and the crux of the matter concerns the static equations for the liquid core of the Earth. It was pointed out by Longman 131, that, for an Earth model with a liquid core, the static problem under the effect of a surface load has a solution only if the liquid core satisfies the Adams-Williamson condition. namely zpo+p~&&=O, (1' ., where X is the elastic constant of the liquid core, p0 is the density distribution, o0 is the self-gravity acceleration and 'dot' denotes differentiationwith respect to r. If we take the Mu Barth model /4/, then the Adams-Williamson condition is obviously not satisfied in the liquid core. Thus, the corresponding static equations do not haveasolution. This is manifested in the incomplete satisfaction of the continuity conditions at the coremantle boundry, the number of boundry conditions being more than the number of adjustable constants of the differential equations. This has provoked a series of attempts at solving this difficulty /2,5,6!, which have not proved to be completely successful. Ln Ref./l/, we have proposed an improved set of equations for the Earth's core. We believe the failure of having a self-consistant solution when the Adams-Williamson condition does not hold, lies in the assumption of hydrostatic pressure when deriving the equations. Wemodified this assumption and showed that our improved core equations are consistent with the present dynamic equationsfor the core and that, even when the Adams-Williamson condition does not hold our improved equations still have self-consistent solutions that satisfy all the boundary or continuity conditions. In this paper, we apply these equations to the question of excitation of the Chandler motion by some specific earthquake events. 2.
BASIC E(jUATIONS
We adopt the spherical coordinates (r,e,$) and develop all the physical quantitiesbyspherical harmonics S.(e,'p)
A = 4 =
X(r)S,(@, ~1% 5x~)s+o, up>,
I
where ur,ue,u+ are the displacement components along the coordinate directions, A is the rate of change of volume and J,is the perturbing self-gravitationalpotential.
Excitation by Earthquake Dispfacement Field.
61
For the mantle, the equations of elastic equilibrium are expressed in terms of the variables, f61, Y>=
u,
y1= 1x+ Zpb, Y3=== v,
i
(3)
Y4= P(3 -tv/r - V/r),
Y5"Q, YC=== 15-4%&&, I They are convenient in numerical calculations and have been generally adopted in geophysical problems. The equations are then 21 3r= -(A +2&Y'+ $1 = [-4@&7+
n(n-f1)X (1+ 2p)r ySt
*+
4~;;~2;;']+
(1. ::,,, yt n(n+ l)&gOl- 2p(3R+ 2p)n(n+ 1)] ++ I I+ 28 -I + I! + Y4 -, 91= r + P 28(32+ 2~1 Y$ $+ = I@@ 2+ 2p yz +
n(n+ 11 Y4-p0YO r (4)
17 (1:2&
e [I(Zn'+2n -l)+(n'+
PaYI ti- 1)2~].___&_ 23 - & (1+2p) 9 r -7
3s= 4nC?%Yl+Ycr 4aG#z(n + l)Y,+ n(n+ 1) 31= YS--3Y69 S. I' where G is the gravitational constant and X ,n the elastic constants. For a liquid core which does not satisfy the Adams-Williamsoncondition, our improved static equations /li are p+~~+[!EG/_G$+=o, &Jr+ po o,+ f& _&!+V,]=O, I 2Bu, -tJ-[4&4U‘f n(n+ 1)(-u‘+ 3VI dr' d =.-.-r-d 3, -lli+!!!) r r dr ( dr ( +'[5U,+3rS, I
r3,) 1
-22n(n+ 1)V,l];
The equations of elastic equilibrium for the mantle (4) are of order 6 in 6 variables, their general solution contains 6 adjustable constants. The equilibrium equations for the core (5) are also of order 6, and a general solution also contains 6 adjustable constants. Boundaq Conditions. At the surface, normal stress and sheer stress are zero and the --7: gravitationX~X&iSZl is continuous. At the Farth's centre, all.quantites must be finite. Six Conditions of Continuity at the Core-Mantle Boundary. Normal displacement continuous normal stress continuous, zero sheer stress in mantle, zero additional stress in core, gravitational potential and its first derivative continuous. Self-consistent solutions can be found for the static equationsfor the whole Earth thus defined. 3. METHOD OF CALC~AT~ON After transforming all the differential equations by means of propagating matrices /7! into
62
Excitation by Earthquake Displacement Field
an initial-valueoroblem,thesewere integrated numerically by the Runge-Kutta algorithm, Here we should emphasize a point the equations (5) for the core show that their coefficients are singular atr =O. Hence, for the boundary condition atr =0, we can only propose the conditions of finiteness. Such conditions are adequate when solving the equations analytically, but they give rise to dif!icultieswhen we solve the equations numerically. To obviate these difficulties, we take a small sphere about the Earth's centre, and use a power series approximation fat the solution within the sphere. Outside the sphere, right to the Earth's surface, numerical integration can be made throughout, since there is no longer any singularity. Because we now use series approximation near the centre, half of the adjust -able constants of a general solution can be taken to be 0 by the finiteness conditions. This solution is then connected to the numerical solution outside the sphere bythecontinuity requirement across the surface of the sphere. The radius of the sphere can be chosen arbitrarily, but preferably not too large so that one or two terms of the series expansion will adequately represent the solution inside. 4.
RESULTS AND DISCUSSION
We made calculation for three large earthquakes. Different authors have given rather different values for the earthquake parameters. We took those from Ref. /2/ foreasy comparison. For two of the events, the values of)f, were calculated from Fig.4 of Ref. /2/, as theywerenot givendirectly . The parameters we adopted are
6
MI
Earthquake
(dyne_cm)
--
1960
QI”
(km) 25
5x 10”
a
6
170*
35.5’
-~~ 285.5”
Chile
:p,-,:,
/
5X10”
/
5G? i
29.0’
j
Z13.0°
/
135’
/
170’
!
1
270’
/
270’
Our calculated results are
Discussions on the effect of earthquakes on polar motion have been going on for many years and are still continuing /a/. They centre mainly on two questions.1) Eow big are the changes in the products of inertia caused by the displacement fieldsofindividual earthoualres? 2: Do these changes produce sufficient excitation to maintain the Chandler wobble? The latter question has been discussed by us in Ref. .'9.'. The primary difficulty encountered when attempting toanswer the first question is how to specify the static equations for an Earth with a liquid core. This involves the AdamsWilliamson condition. Generally speaking, this condition is not satisfied in the liquid core, and the equations available at present have no self-consistent solutions when this condition is not satisfied. Hence it is necessary to modify the static equations. We believe that it is reasonable to insist that continuity conditions should be satisfied across the coremantle boundary. Whatever the deformation of the core, cavities should not occur, nor should the core and the mantle interpenetrate. This was the starting point of our proposedequations in Ill. For the excitation by earthquakes, our results are larger than those of Dahlen. The polar motion data available at the present are not precise enough to decide which set is closer to reality. However, it seems that the previous equations may imply the appearance of cavities on the core-mantle boundary, whereas in our derivation, the continuity between the core and mantle is guaranteed. Under the constraint of continuity, a deformation of the mantle brings
Excitation by Farthquake Displacement Field
63
about a deformation of the core, and the latter reacts and causes a greater deformation of the mantle. This is why the earthquake displacement field and the changes in the products of inertia of the Earth found are larger than estimated by previous workers. ACKNOWLEDGEMENT. This paper forms a part of the topic 'Relationshipbetween ChandlerWobble and Earthquakes' proposed by Professor YE Shu-hua, and we thank Prof. YE for her guidence and help,
REFERENCES SONG Guo-xuam, ZHAO Ming, Celiang yu Di,iquwuli Jikan. (Studies on Geodsy and Geowhysicsf 1980 No. 4 [21 I31
Dahbn.
141
Shhen. PO-yu.
151
&nyIie.
ISI 171
Cbinnery,
r8i
(91
F. A., Ge~~hysiool
Longman.
Gilbert, Man&ha.
I.
M., Journal Mansinba,
D. E., Mansinha, M. A., F.,
JOWW
SW., 25(1971), 157, and ibid., 32(1973), 20% Bea., 68(1QG3). 485. I., Geophw. Jmtrnol of Roy. Aair. SW., 46(1976). 467. I., &id.,
ibid., 42(1975),
Backus.
of Eoy B&r.
of Geophys.
Z’$(lQ?lf.
329.
461.
G. E.. Geophysics. 51(1366),
I., EJmylie. D. E., Chapman,
C. E.,
326.
Ceophys.
Jormnl
ZHAO Ming, SONG Guo-xuan, I:exueTongbao
Chin.Astron.Astrophys.5 (1982) 63-67 Act.Astron.Sin.22 129311 389-394
of Boy.
ostr.
a?~., 5@(1@:9).
Pergamon Press. Printed in Great Britain o275-1062182faloo63-04$07.5010
ON THE SOLUTIONS OF THE EARTH ROTATION OBSERVATIONS OF TIME AND LATITUDE
Analysis
1.
(1980! 738.
PAR~ETERS
FROM CLASSICAL
Centre for Classical Observations, Pxoject MERIT. (Shanghai Observatory)
Received 1981 May 30
ABSTRACT. We compare the Earth's rotation parameters claculated from various input sets and conclude 1) A reference system comparable to the BIF sustem can be set up using just a few high-precision, evenly-distributedinstruments. 2) Chinese instruments for time and latitude determination play an important role in the setting up and maintenance of a global reference system. 3f There seem to be no systematic differences, of an annual or a semi-annual character between the observations by the classical methods and the newer techniques. The difference BIH(l.979) - BIH(l968)is probably what has remained of the station errors when the BIH (1968) system was set up. 4) It is possible that some unknown common source of error may exist over a large geographical region, hence, to set up a good reference system, the observing insrtuments should be distributed as evenly as possible over the globe.
1. INTRODUCTION The main purpose of the project MERIT is to compare the various observing techniques and reduction methods, especially between the classical and the newer techniques, comparing their precision, thier differences. In order to make it easier to interpret the eventual results of this comparison, a careful analysis of the various means by themselves seems desirable. In this paper, we shall analyse the classical observations of time and latitude, estimate their precision and attempt to identify the source of some of the errors. Our method is as follows. We shall first use the largest possible set of global data and the best tested procedures to establish a set of 'best' values of the rotational parameters, called the Regular Solution. We then introduce some arbitrary modifications in the input
Act.Astron.Sin.
EXCITATION SONG
OF POLAR
Guo-xuan,
MOTION
Pergamon Press. Printed in Great Britain 0275-1062/82/010060-04$07.50/O
BY EARTHQUAKE
ZHAO Ming, ZI’iE~:G Da-wei
DISPLACEMENT
Shanghai
FIELD
Observatory
Received 1980 October 23
ABSTRACT. We discuss the excitation of polar motion by earthquake displacement field. Instead of the usual static equilibrium equations in the literature, we use an improved set as given in /I/, which guarantee continuity at the core-mantle boundary. We take the parameter values of three earthquakes from !2/. To obviate the singularity at r=O, we use asymptotic solutions by power series within a small sphere around the centre. Outisde this sphere, the equations are numerically integrated by the Runge-Kutta algorithm. Our equations !l! gave polar shifts some 3 times larger than Dahlen's equations /2/.
1.
INTRODUCTION
When considering the excitation of polar motion by earthquakes, it is important to calculate the additional tiisplacementfield caused by earthquake dislocation. Since this was pointed out by Smylie and Nansinha in 1967, IS/, there has been much discussion. But a number of inconsistencies remain, and the crux of the matter concerns the static equations for the liquid core of the Earth. It was pointed out by Longman 131, that, for an Earth model with a liquid core, the static problem under the effect of a surface load has a solution only if the liquid core satisfies the Adams-Williamson condition. namely zpo+p~&&=O, (1' ., where X is the elastic constant of the liquid core, p0 is the density distribution, o0 is the self-gravity acceleration and 'dot' denotes differentiationwith respect to r. If we take the Mu Barth model /4/, then the Adams-Williamson condition is obviously not satisfied in the liquid core. Thus, the corresponding static equations do not haveasolution. This is manifested in the incomplete satisfaction of the continuity conditions at the coremantle boundry, the number of boundry conditions being more than the number of adjustable constants of the differential equations. This has provoked a series of attempts at solving this difficulty /2,5,6!, which have not proved to be completely successful. Ln Ref./l/, we have proposed an improved set of equations for the Earth's core. We believe the failure of having a self-consistant solution when the Adams-Williamson condition does not hold, lies in the assumption of hydrostatic pressure when deriving the equations. Wemodified this assumption and showed that our improved core equations are consistent with the present dynamic equationsfor the core and that, even when the Adams-Williamson condition does not hold our improved equations still have self-consistent solutions that satisfy all the boundary or continuity conditions. In this paper, we apply these equations to the question of excitation of the Chandler motion by some specific earthquake events. 2.
BASIC E(jUATIONS
We adopt the spherical coordinates (r,e,$) and develop all the physical quantitiesbyspherical harmonics S.(e,'p)
A = 4 =
X(r)S,(@, ~1% 5x~)s+o, up>,
I
where ur,ue,u+ are the displacement components along the coordinate directions, A is the rate of change of volume and J,is the perturbing self-gravitationalpotential.
Excitation by Earthquake Dispfacement Field.
61
For the mantle, the equations of elastic equilibrium are expressed in terms of the variables, f61, Y>=
u,
y1= 1x+ Zpb, Y3=== v,
i
(3)
Y4= P(3 -tv/r - V/r),
Y5"Q, YC=== 15-4%&&, I They are convenient in numerical calculations and have been generally adopted in geophysical problems. The equations are then 21 3r= -(A +2&Y'+ $1 = [-4@&7+
n(n-f1)X (1+ 2p)r ySt
*+
4~;;~2;;']+
(1. ::,,, yt n(n+ l)&gOl- 2p(3R+ 2p)n(n+ 1)] ++ I I+ 28 -I + I! + Y4 -, 91= r + P 28(32+ 2~1 Y$ $+ = I@@ 2+ 2p yz +
n(n+ 11 Y4-p0YO r (4)
17 (1:2&
e [I(Zn'+2n -l)+(n'+
PaYI ti- 1)2~].___&_ 23 - & (1+2p) 9 r -7
3s= 4nC?%Yl+Ycr 4aG#z(n + l)Y,+ n(n+ 1) 31= YS--3Y69 S. I' where G is the gravitational constant and X ,n the elastic constants. For a liquid core which does not satisfy the Adams-Williamsoncondition, our improved static equations /li are p+~~+[!EG/_G$+=o, &Jr+ po o,+ f& _&!+V,]=O, I 2Bu, -tJ-[4&4U‘f n(n+ 1)(-u‘+ 3VI dr' d =.-.-r-d 3, -lli+!!!) r r dr ( dr ( +'[5U,+3rS, I
r3,) 1
-22n(n+ 1)V,l];
The equations of elastic equilibrium for the mantle (4) are of order 6 in 6 variables, their general solution contains 6 adjustable constants. The equilibrium equations for the core (5) are also of order 6, and a general solution also contains 6 adjustable constants. Boundaq Conditions. At the surface, normal stress and sheer stress are zero and the --7: gravitationX~X&iSZl is continuous. At the Farth's centre, all.quantites must be finite. Six Conditions of Continuity at the Core-Mantle Boundary. Normal displacement continuous normal stress continuous, zero sheer stress in mantle, zero additional stress in core, gravitational potential and its first derivative continuous. Self-consistent solutions can be found for the static equationsfor the whole Earth thus defined. 3. METHOD OF CALC~AT~ON After transforming all the differential equations by means of propagating matrices /7! into
62
Excitation by Earthquake Displacement Field
an initial-valueoroblem,thesewere integrated numerically by the Runge-Kutta algorithm, Here we should emphasize a point the equations (5) for the core show that their coefficients are singular atr =O. Hence, for the boundary condition atr =0, we can only propose the conditions of finiteness. Such conditions are adequate when solving the equations analytically, but they give rise to dif!icultieswhen we solve the equations numerically. To obviate these difficulties, we take a small sphere about the Earth's centre, and use a power series approximation fat the solution within the sphere. Outside the sphere, right to the Earth's surface, numerical integration can be made throughout, since there is no longer any singularity. Because we now use series approximation near the centre, half of the adjust -able constants of a general solution can be taken to be 0 by the finiteness conditions. This solution is then connected to the numerical solution outside the sphere bythecontinuity requirement across the surface of the sphere. The radius of the sphere can be chosen arbitrarily, but preferably not too large so that one or two terms of the series expansion will adequately represent the solution inside. 4.
RESULTS AND DISCUSSION
We made calculation for three large earthquakes. Different authors have given rather different values for the earthquake parameters. We took those from Ref. /2/ foreasy comparison. For two of the events, the values of)f, were calculated from Fig.4 of Ref. /2/, as theywerenot givendirectly . The parameters we adopted are
6
MI
Earthquake
(dyne_cm)
--
1960
QI”
(km) 25
5x 10”
a
6
170*
35.5’
-~~ 285.5”
Chile
:p,-,:,
/
5X10”
/
5G? i
29.0’
j
Z13.0°
/
135’
/
170’
!
1
270’
/
270’
Our calculated results are
Discussions on the effect of earthquakes on polar motion have been going on for many years and are still continuing /a/. They centre mainly on two questions.1) Eow big are the changes in the products of inertia caused by the displacement fieldsofindividual earthoualres? 2: Do these changes produce sufficient excitation to maintain the Chandler wobble? The latter question has been discussed by us in Ref. .'9.'. The primary difficulty encountered when attempting toanswer the first question is how to specify the static equations for an Earth with a liquid core. This involves the AdamsWilliamson condition. Generally speaking, this condition is not satisfied in the liquid core, and the equations available at present have no self-consistent solutions when this condition is not satisfied. Hence it is necessary to modify the static equations. We believe that it is reasonable to insist that continuity conditions should be satisfied across the coremantle boundary. Whatever the deformation of the core, cavities should not occur, nor should the core and the mantle interpenetrate. This was the starting point of our proposedequations in Ill. For the excitation by earthquakes, our results are larger than those of Dahlen. The polar motion data available at the present are not precise enough to decide which set is closer to reality. However, it seems that the previous equations may imply the appearance of cavities on the core-mantle boundary, whereas in our derivation, the continuity between the core and mantle is guaranteed. Under the constraint of continuity, a deformation of the mantle brings
Excitation by Farthquake Displacement Field
63
about a deformation of the core, and the latter reacts and causes a greater deformation of the mantle. This is why the earthquake displacement field and the changes in the products of inertia of the Earth found are larger than estimated by previous workers. ACKNOWLEDGEMENT. This paper forms a part of the topic 'Relationshipbetween ChandlerWobble and Earthquakes' proposed by Professor YE Shu-hua, and we thank Prof. YE for her guidence and help,
REFERENCES SONG Guo-xuam, ZHAO Ming, Celiang yu Di,iquwuli Jikan. (Studies on Geodsy and Geowhysicsf 1980 No. 4 [21 I31
Dahbn.
141
Shhen. PO-yu.
151
&nyIie.
ISI 171
Cbinnery,
r8i
(91
F. A., Ge~~hysiool
Longman.
Gilbert, Man&ha.
I.
M., Journal Mansinba,
D. E., Mansinha, M. A., F.,
JOWW
SW., 25(1971), 157, and ibid., 32(1973), 20% Bea., 68(1QG3). 485. I., Geophw. Jmtrnol of Roy. Aair. SW., 46(1976). 467. I., &id.,
ibid., 42(1975),
Backus.
of Eoy B&r.
of Geophys.
Z’$(lQ?lf.
329.
461.
G. E.. Geophysics. 51(1366),
I., EJmylie. D. E., Chapman,
C. E.,
326.
Ceophys.
Jormnl
ZHAO Ming, SONG Guo-xuan, I:exueTongbao
Chin.Astron.Astrophys.5 (1982) 63-67 Act.Astron.Sin.22 129311 389-394
of Boy.
ostr.
a?~., 5@(1@:9).
Pergamon Press. Printed in Great Britain o275-1062182faloo63-04$07.5010
ON THE SOLUTIONS OF THE EARTH ROTATION OBSERVATIONS OF TIME AND LATITUDE
Analysis
1.
(1980! 738.
PAR~ETERS
FROM CLASSICAL
Centre for Classical Observations, Pxoject MERIT. (Shanghai Observatory)
Received 1981 May 30
ABSTRACT. We compare the Earth's rotation parameters claculated from various input sets and conclude 1) A reference system comparable to the BIF sustem can be set up using just a few high-precision, evenly-distributedinstruments. 2) Chinese instruments for time and latitude determination play an important role in the setting up and maintenance of a global reference system. 3f There seem to be no systematic differences, of an annual or a semi-annual character between the observations by the classical methods and the newer techniques. The difference BIH(l.979) - BIH(l968)is probably what has remained of the station errors when the BIH (1968) system was set up. 4) It is possible that some unknown common source of error may exist over a large geographical region, hence, to set up a good reference system, the observing insrtuments should be distributed as evenly as possible over the globe.
1. INTRODUCTION The main purpose of the project MERIT is to compare the various observing techniques and reduction methods, especially between the classical and the newer techniques, comparing their precision, thier differences. In order to make it easier to interpret the eventual results of this comparison, a careful analysis of the various means by themselves seems desirable. In this paper, we shall analyse the classical observations of time and latitude, estimate their precision and attempt to identify the source of some of the errors. Our method is as follows. We shall first use the largest possible set of global data and the best tested procedures to establish a set of 'best' values of the rotational parameters, called the Regular Solution. We then introduce some arbitrary modifications in the input