On the excitation, detection and damping of core modes

Physics of the Earth and Planetary Interiors, 68 (1991) 97—116

97

Elsevier Science Publishers B.V., Amsterdam

On the excitation, detection and damping of core modes D.J. Crossley

~,

J. Hinderer and H. Legros

Institut de Physique du Globe, Strasbourg, France

(Received 28 June 1990; revision accepted 11 January 1991)

ABSTRACT Crossley, D.J., Hinderer, J. and Legros, H., 1991. On the excitation, detection and damping of core modes. Phys. Earth Planet. Inter.. 68: 97—116. We continue work on the possibility that internal oscillations in the Earth’s core might be detectable by a gravity meter at the surface. As a possible excitation mechanism we use earthquakes and for the damping we consider the contribution of anelasticity in both solid and liquid parts of the Earth. The calculation of the eigenfunctions is restricted in this paper to a non-rotating Earth model. We first review the spectrum of modes for degrees 1 up to 20 using a stably stratified fluid core based on the Preliminary Reference Earth Model (PREM) and compute the earthquake excitation and Q values using standard seismic theory. We then show that the effects of boundary rigidity and both Boussinesq and subseismic approximations in the core have little effect on the eigenfrequencies of the core modes. The rigid boundary core solutions are then extended into both the mantle and inner core by making use of a generalised internal load Love number approach. This method yields good approximations to the elastic eigenfunctions throughout the Earth, the agreement with full theory improving as the degree 1 increases. Thus we have confidence that this method can be extended successfully to models with rotation.

1. Introduction

1975). The strength of the coupling between any

Permanent address: Department of Geological Sciences, Mc-

two harmonics is approximately proportional to the ratio T/12, where T is the period of the motion in hours. Because the core modes are likely to have periods in the range 6 h to ~, this coupling results in a major departure from seismic normal mode theory. In the absence of rotation, the seismic modes are degenerate with respect to m so one has simply to sum the contributions of the eigenfunctions of different m values to obtain the surface displacement associated with a multiplet of degree 1. With the addition of rotation to the model, this summation continues to be true for those seismic normal modes with periods shorter than a few hundred seconds (due to the smallness of the splitting), whereas the eigensolutions of the long period normal modes are computed using first or second order perturbation theory. Perturbation theory is no longer valid for the

Gill University, Montreal, Canada.

core spectrum where a large (possibly infinite)

This paper can be regarded as a continuation of the excitation problem for long period oscillations of the Earth’s core given in Crossley (1989). In that paper, denoted here by [1], it was shown that the excitation of the modes, in a non-rotating Earth with an artificially stabilised fluid core, was generally below the detection level even for the new generation of superconducting gravimeters. The calculation in [1] however was restricted to only a few sample motions of low spherical harmonic degree. With the inclusion of the Coriolis force in the equations of motion it is now well understood that all the harmonic terms of identical azimuthal order m, but different angular degrees 1, are coupled together (e.g. Crossley, 1975b; Dahlen and Smith,

0031-9201/91/$03.50

© 1991

—

Elsevier Science Publishers B.V.

98

number of terms of different I values will have to be summed to compute the displacement fields throughout the Earth. Any study of the excitation and detection of core modes must therefore inelude the important role of the Coriolis coupling before we can be confident that the global solulions will be useful for predicting surface gravity perturbations. Since 1975 much effort has been expended trying to understand the theoretical spectrum of a rotating, thick shell of inviscid fluid, the usual model for the Earth’s liquid outer core. Unfortunately, the problem remains complex even when assumptions are made for a spherical geometry, rigid boundaries and various physical simplifications for the behaviour of the core fluid. The primary simplifications are the subseimic approximation introduced by Smylie and Rochester (1981) and the Boussinesq approximation (Crossley and Rochester, 1980). Even the mathematical structure of the core spectrum is still debatable, with recent work by Valette (1989, 1990) indicating that the spectrum is dense and possibly even continuous, contradicting previous assumptions on the discrete nature of the spectrum (Friedlander, 1985a). In the face of such slow progress, we have decided to take a pragmatic, seismic view of the core spectrum, as in [1]. In other words, we start with a spherical harmonic expansion of the displacement field, presumably valid in the nonrotating limit, and gradually add the Earth’s rotation by taking more and more terms into the expansion of the displacement field. By concentrating on the gravimetric effect at the Earth’s surface, in the presence of a realistic excitation such as a large earthquake, we try to assess the effect of rotation only approximately, while acknowledging that the core displacement field may not itself be particularly well modelled. In particular, if we can demonstrate that the gravity effect of the higher degree I motions decreases sufficiently rapidly as I increases, then we can claim that a truncated core expansion yields useful gravimetric information. In the next section, we give a brief review of the theory involved and extend previous calculations [1] of the eigenfrequencies and excitation to large

Lxi. CROSSLEY ET AL.

values of I. Then we investigate the effect of anelasticity throughout the Earth as a damping mechanism for the modes. The next step is to simplify the problem, first by introducing rigid boundaries to the fluid core and then allowing approximations to the dynamics of the core fluid. The main result of this paper is to introduce internal load Love numbers to extend the eigenfunctions both to the shell where the source excitation exists and to the surface where a gravimeter may be situated. The theory is but one aspect of the generalised Love number approach to geodynamics discussed by Hinderer and Legros (1989) and is based on the thesis work of Legros (1987). Smylie et al. (1989) have also applied this method to a similar calculation on the solution of modes in a rotating fluid core obeying the subseismic approximation.

2. Review of the theory

The essential equations were introduced in [11 and here we add a few additional notes. The time domain problem is [2’ ( r) + 2Q x 8, + 8,21 S ( r, t) = f( r, t) (1) in which 2’ is the linearised elastic-gravitational operator for small adiabatic deformations, S is the Lagrangian displacement, S~the constant rotational velocity of the Earth and f is the forcing term (r position, 1 time). The formal solution of eqn. (1) can be written L

M

5(r

t)

~

=

‘

=

—

m=

o

[s

(2)

(r t)]

1= I m I

where the real component of displacement, associated with a spherical harmonic Y~m(O,q), is (1 cos w,, ,t) —

s,,,m.,(r, t)

=

,,,

Re[u,,mi(r)fnm,] n.m.!

+ ~

(

\

c

1 sin

mtun,m.l~r,Jn.m.1J

C~2nmit 2

(3) In eqn. (2) the radial quantum numbers (n ~ 0) are seismic modes, negative n values denote inter-

99

ON THE EXCITATION. DETECTION AND DAMPING OF CORE MODES

nal gravity modes in the fluid core and the upper limits M and L for the azimuth and degree of the spherical harmonic are considered to be finite integers. The quantities anm/, the eigenfrequency

As far as the detection is concerned, the response of a gravimeter is determined by the perturbation in gravity at the Earth’s surface z~g= ~ (g~+ g 2 + g,)

associated with a complex free mode unmi(r, t) unmi(r) exp(ic~.)nm,t), and fn,m,i’ the scalar ex-

n,m,I

=

citation, are all assumed to refer to a single cornponent of the motion (n, m, 1). In the non-rotating case, Im[unmi(r) fnmi] = 0 and eqn. (3) reduces to the result of Gilbert (1970). The total complex free-mode displacement u = u(r) satisfies the frequency domain equation 2u = 0 (4) 1t’u + 2iw(l X u — w and if it is expanded as a sum of spheroidal and torsional vectors

()

g1 =Y~(4~TGP0 — ~go) g2

dy5 g3 = — dr = — 4~rGp0y1— in which g0 is the unperturbed gravity and y1, y3, y 5, y6 are the usual radial coefficients (Alterman et al., 1959), here referred to the forced solutions (eqn. (3)); all quantities are evaluated at the Earth’s

surface r u(r)=

~

~ [nSim+nTtm] n.m.!

U,,mi=

n.m.!

(5)

and substituted into eqn. (4), we find the two distinct solutions Um

( r)

=

S(~, + T~ +

um(r)

=

. . .

±I

+

S~,flI ±2

EVEN

+

T~I +

Ti’~1+Sjmi±i +T +...ODD

m ImI+2 ±Sirni+3

= d. As is evident in eqn. (3), the spatial part of the forced solution is simply a free mode eigenfunction scaled by the excitation fnm/~ Further, L~g is equivalent to the gravity effect one might compute from the gravimetric ~ factor in a

tidal problem, the equivalence through the expressions

demonstrated

(6) seismic

(7) g3

+

tidal (I+1)k

4~rGp0y1= —y6

~

2 —g0y1

~

—

*-*

—

1

d

The EVEN chain (symmetric about the equator)

includes the equatorial Slichter modes and the ODD chain (antisymmetric about the equator) includes the axial Slichter mode and the wobble modes. The chains are truncated by choosing the upper limit L in eqn. (2). The excitation in eqn. (3) is specified by the complex scalar quantity fn.m.1

=

f

un*mi(r) -f(r)p0

dv

(8)

g1

—

4~Gp0y1 =

—

(missing) g2

2h1

~

—~—

I”,,

(usually neglected) (10)

TABLE I Evaluation of free-mode eigenfunctions u(r)

where p0 is the unperturbed density. In eqn. (8), f(r) is the radial part of f(r, 1) and the time

Region

Description

dependence is assumed to be a unit step function. As discussed in [1], f(r) can be further decomposed into spheroidal and torsional vector compo-

0 ~r~ d

y,... y6 throughout the Earth

(4)

a©r~ b

~1 ~ throughout the outer core (to determine the eigenfrequencies

(14)—(19)

in text

nents depending on the geometry of the source

function. It is completely adequate for the current problem to consider point-source earthquake functions and so eqn. (8) reduces to combinations of the radial components of u(r) and their derivalives evaluated at the radius r0 of the source.

Equation

and eigenfunctions) r = r0

y1, y3 at the earthquake focus (to determine the scalar excitation)

r= d

~1’

~ at the Earth’s surface

(to find the gravity effect)

(8) (9)

100

as in Hinderer and Legros (1989, eqn. (3.4)). The disturbing potential l’~in 2.10) is here entirely due to the perturbation in gravity associated with the deformation itself and k 1 and h, are the equivalent Love numbers. The essential point of the above review is to emphasise that the free-mode eigenfunctions have to be evaluated at various regions within the Earth, as summarised in Table 1.

3. Numerical details All the results in this paper share the following computational similarities, including the important fact that we consider only the non-rotating limit ~ = 0 in the equations from the previous section.

Di.

CROSSLEY

ET AL.

other models of the core. This is clear from the fact that ~g decreases as TB increases [1] (Table 3). 2 in the It turns out that the compromises. modification of fluid core entails some AsNin previous work, we choose to retain c and the mass of the Earth at the original model values and allow the moment of inertia of the outer core to be unconstrained when choosing the modified density profile. It is worth noting that near the fluid core boundaries it is difficult to constrain accurately N2 due to the lack of control on the density gradient at these boundaries. 3.2. Numerical integration The equations of motion (4) are first transformed to the standard form (Alterman et al., 1959) and then integrated using a 4th order

3.1. Earth model with a stable core

Runge-Kutta integration scheme with variable step

We choose PREM (Dziewonski and Anderson, 1981) as the basis model with the following modifications. The crustal layer is taken to be continental to avoid the necessity of introducing another solid/liquid interface in the integration, Since we are primarily concerned with the fluid core, and the mantle acts rather passively as far as

size to ensure numerical accuracy. The accuracy of the integration was verified by changing the error tolerance and the number of steps at which the eigenfunctions are evaluated throughout the model. No particular tricks were needed to find the eigenfunctions beyond the standard seismic approach (basically variations of the original scheme of Alsop (1963)) for zeroing the character-

the core motions are concerned, this is not a

istic determinant at the outer boundary of the

serious approximation. Further, the stability profile in the core is modified to give a value of the Brunt-Väisälä buoyancy period T 8 = 6 h throughout, i.e. from the inner core boundary (ICB) to the core mantle boundary (CMB). This period is defined as 2 TB = -~

integration region and starting low-degree solutions at the Earth’s centre using the polynomial expansions of Crossley (1975a). Due to the lack of

2 (11) 2 = — g~dp N 0 — 2 c where dpPo dr 0/dr is the density gradient and c is the local P-wave acoustic velocity. This value continues to be the upper limit set by seismic observations (Crossley, 1984) and is no doubt unrealistic in practice. However since it represents the most stable core possible, we can be virtually certain that the surface gravity effects we compute here will be the upper limits of those possible with

simpler than for the seismic normal modes.

significant displacement in the inner core, it was found to be more efficient to begin integration of modes with 1> 5 at a finite initial radius (10 km) in the inner core. In this respect the calculation of high degree modes with high radial number is

The PREM coursedispersion, frequency and dependent due tomodel P and isS of velocity this was extended to the core-mode period range. . 3.3. Normalisation

All eigenfunctions were normalised to give unity for the inner product (U*,

u.)

=

Ju*(r) v

‘

.

u (r)p0 d~=

(12)

101

ON THE EXCITATION. DETECTION AND DAMPING OF CORE MODES

where p is the domain of the solution. The scheme for doing this outlined in [1] was found to be unnecessarily complicated. It is sufficient to first normalise with y 1 = I at r = d. compute the inner product (12) and then re-adjust the arbitrary constants accordingly. 3.4. Earthquake source function As in [11, we take the 1960 Chilean earthquake as a standard source with the parameters: seismic moment 1025 Nm, depth 25 km, dip 10° and of a dip-slip type.

4. Elastic solutions for high degree To date example solutions for the core modes have been presented only for low degree harmonics. We now extend the calculations of the eigenfrequencies up to I = 20, retaining M = 2 and n = — 1 ... — 6. These solutions, for a non-rotating Earth model, are obtained in a manner identical to seismic normal mode calculations except that a modified core stability profile is used, see above, No other approximations are made.

The results, shown in Fig. 1, demonstrate that the higher degree eigenperiods accumulate towards the limiting period TB with the more cornplicated radial motions (larger — n values) having the longer periods. Instantly it is clear that the spectrum for a rotating Earth with all these modes coupled will be complex and qualitatively ‘dense’ (this does not necessarily imply a dense spectrum in the mathematical sense). One feature of the eigenfunctions as 1 —~ 20 is the difficulty of maintaining sufficient numerical accuracy in the core to ensure the boundary conditions at r = d are adequately met. This point will be emphasised again later when comparisons are made with the Love number approach. Using the eigenfunctions u(r) computed from eqn. (4), the excitation f was computed from eqn. (5), inserted into eqn. (3) and the total surface gravity effect was calculated from eqn. (6). The results are shown in Fig. 2a for low-degree modes, including some seismic overtones as a comparison, and in Fig. 2b for all modes up to I = 20. It is evident that the excitation of all core modes, cxcept the special modes /=1, lies below the threshold of detectability of even superconducting gravimeters. Considering the difficulties which

COMPARISON OF ELASTIC AND RIGID BOUNDARY EIGENPERIODS

1O-~

degree 1

Fig. 1. Eigenperiods of the spheroidal modes of a stable core without rotation. Continuous curves: for complete Earth model; D: for a core with rigid boundaries (but otherwise the full fluid equations). The asymptote for n —1 is the buoyancy period of 6 h.

102

Di. CROSSLEY ET AL.

currently exist in recording and processing long

in the log domain indicates that ~g is an exponen-

gravimetric records at the 1—10 ngal level, it is unlikely that we can expect any improvement in this threshold in the foreseeable future.

tially decreasing function of degree I. This implies that ~ g computed from expressions such as eqns. (6) and (7) for a rotating Earth might converge

Therefore we have to turn to the question of

rapidly as

whether the effect of rotation is likely to change these predictions. The linear decrease of ~g with 1 EARTHQUAKE EXCITATION

I—

Note that we are unable to add directly a linear combination of the non-rotating eigenfunctions to

Low Degree Harmonics

—

(a) a 1=m1 3

-

1~m=2

A

4

M=1O

5

25

N.m

43 5

~‘

2

j

-

‘P

‘~i °

—2

2

detection

~

.4

f~i~It

4-

>.

CC,

-1

• —3-

+

~—e A

—6 A

—4

—

—

—

—

—

—

—

—

_.__

EARTHQUAKE EXCITATION

—

—

—

—

—

—

Higher Harmonics

4 —

m

m = i £m=2

=

(b)

~

a

2-

detection —

0

~.:

—

.4.)

-

limit

-

--

o-j-

a

a —2

~

—

6j

I

g~5~

a

‘~‘—6-7-

...~.

—e 0

1

2

3

4

5

6

I 7

I 6

I

4

9

10

11

degree 1 Fig. 2. (a) Magnitude of surface gravity for / = I and 1 2 for various seismic overtones and core modes. The detection limit has been taken as I ngal. (b) As (a) for / up to 10. The m = 1 and m = 2 solutions are separated.

103

ON THE EXCITATION. DETECTION AND DAMPING OF CORE MODES

form the displacement field because the proporlion of each harmonic is unknown. Instead we have to choose a level of truncation (M, L) and solve for all harmonics simultaneously. In the face of formidable numerical problems with this approach (see e.g. Johnson and Smylie, 1977), we will shortly adopt some reasonable approximations that simplify the model.

tional effects). Using recent estimates of the core viscosity, Poirier (1988) concludes that E — 4 X 10 ~s implying a viscous boundary layer less than 1 m thick. Hence the spin-down time of an oscillation at a 12 h period is of the order of iO~years. Comparable results were found for electromagnetic dissipation (Crossley and Smylie, 1975). Our initial thought was that shear dissipation in the mantle would damp the motions in the core, even though that mechanism would be weak due to the small amplitude of the motion in the mantle. We decided therefore to calculate the anelastic dissipation for core modes based on standard seismic methods. The appropriate expression for Q can be found in Sailor and Dziewonski (1978) 1 a = = —i ( r) M1 ( r ) q~( r) = +K(r)K 2 dr, (13) 1(r)q4(T)J r

5. Damping by seismic anelasticity Previous estimates of the damping of core modes have yielded very high Q values. S. Friedlander (unpublished preprint, 1987) concludes from an analysis of boundary layer viscous dissipation that all core oscillations decay at the 2)where E is spin-down scale(the—ratio O(E~” the Eckmantime number of viscous to rota-

j

~

TABLE 2 Partitioning of /

‘I

m

Q~1

Q,~’

Q

Inner core

Outer core

Shell

Inner core

Outer core

Shell

I seismic overtones 5 1 0.50x104 4 t 0.14x105

(days)

0.11 X106 0.15x105

0.67x107 0.40>10’

0.10x10’ 0.30x10’

0.00 0.00

0.69x105. 0.28xI0’

91 355

0.19 0.93

=

3 2 m

Q for low degree harmonics

=

1

1

0.24x105

0.68x105

0.74X105

0.26x106

0.00

0.11 xlO’

831

3.25

1

0.19x106

0.87x106

0.42xI05

0.10x108

0.00

0.24x10’

400

3.65

I Slichter mode 1 0.17x103

0.13x104

0.37x105

0.50x385

0.00

0.80x106

5213

424.28

0.00

0.82x106 0.31 x 106

7552 7838

1482.38 2714.74

0.17x106

7924

3998.68

317 434 370 96 311 513

0.56 0.93 1.23 0.38 1.69 6.11

8966 8196 8053 7709 7699 7227

1115.52 1689.14 2380.86 2988.43 3701.29 4150.78

m =1 core modes

—1 —2 —3 m

=

5 4

3 2 1 0 m

=

—1 —2

—3 —4 —5 —6

1 1

0.29x104 0.92x i0~

0.10x103 0.11 x 10’

0.35x105 0.14x l0”

0.53x106 0.86x iO5

1

0.49x105

0.12x103

0.69x109

0.14xI0~7

2 seismic overtones 2 0.16x106 2 0.22x107 2 0.81 x106 2 0.22x104 2 0.14x106 2 0.53x107

0.34x105 0.13x105 0.38x10’ 0.12x10’ 0.26x106 0.30x105

0.20x105 0.68.<10~ 0.53x106 0.30x10’ 0.19x10’

0.58x104 0.17x104 0.90x103 0.10x10’ 0.36><10~ 0.56x105

0.00 0.00 0.00 0.00 0.00 0.00

0.10x302 0.10x222 0.10x172 0.10x123 0.10x3L’ 0.10,<192

2 core modes 2 0.33x107 2 0.15x107 2 0.89 x iO~ 2 0.43x108 2 0.35x105 2 0.20x108

O.10x103 0.11 x103 0.12 x iO~ 0.12x103 0.12x103 0.13x103

0.18x108 0.98x109 0.51 x I0~ 0.30x109 0.20x109 0.14x109

0.23x105 0.12x105 0.66 x 10—6 0.39x106 0.26x106 0.18x106

0.00 0.00 0.00 0.00 0.00 0.00

0.13x10’ 0.78x106 0.36 x 10~ 0.24x106 0.14x106 0.11x106

0.00 0.00

104

Di. CROSSLEY

where is the overall Q~for mode j, p.(r), K(r) are the elastic shear modulus and incom1 pressibility of the model, q~(r),q~(r)are the Q values for each region of the model and M~(r), K 1(r) are Fréchet kernels for the shear and com-

ET AL.

times for some low-degree harmonics. It is evident that the smallest Q values are associated with compressibility in the fluid core, an effect which completely dominates shear dissipation in the

mantle. Thus the most important PREM Q value is the poorly constrained 52828 for QK throughout the of the Earth. There is veryoflittle information bulk on the frequency dependence Q~(by cornparison with that for shear dissipation) though it seems unlikely that such high seismic values will be significantly reduced by going to tidal periods. In this sense we consider the constant Q assumption reasonable. In Fig. 3 the damping times are given for all core modes up to I 20. It can be seen that the

pressional energies given in Backus and Gilbert (1967, Appendix B). Note thecompensate appearance for of the 2 in eqn. (13) to the factor 1/win normalisation between eqn. (12) and difference that used by Backus and Gilbert. A damping time can be associated with each Q value, T QT/ir, where T is the period of the motion. In order to apply eqn. (13) in practise, we need to assume that the PREM seismic model can be legitimately extended to core periods (i.e. the tidal band, nominally 12—24 h). This requires an cxtrapolation of the constant Q model on which PREM is based to a period range outside its =

=

times are all significantly longer than a year and that the decay time increases with radial number n and decreases with degree 1. Interestingly, these relations have the same sense as the decay though boundary layer viscous dissipation discussed by Friedlander (1987). Even though these Q values are very high, it is significant that seismic anelas-

proven validity. However, considering the results to follow, this extrapolation can be considered verified a posteriori. Table 2 shows the distribution of Q throughout the Earth and damping

DAMPING TIME BY SEISMIC ANELASTICITY 500C -

4D00~-s a >. a

6a>. a -a

—

~30OD--2

—

a,

c

2000

0

0

-

2-

1

degree I -

-

-

0

0

0

-

I

0

2~

~

a ~0

0

-

00

--

C

1000

seismic modes

0

I 2

I 4

~

I 5

B- --!-

I B

I:

---‘~

I 10

:]~--l-:I I 12

I 14

-I--! I 16

I lB

I 20

22

degree 1 Fig. 3. The anelastic decay (e-folding) time for the modes of Fig. 1. Apart from ,S 1 all modes will persist for several years according to this linear theory.

105

ON THE EXCITATION. DETECTION AND DAMPING OF CORE MODES

ticity is several orders of magnitude more important than viscous or magnetic dissipation.

Similarly for the mantle, 1+1

y

5 ( r)

=

y5 (b

)

(~)

r I+ 1 y6(r)=—————y5(r)

6. Simpllficatlons to the model

(21) (r~b)

6.1. Rigid core boundaries The first approximation is to allow the inner and outer core boundaries to be rigid, as in CrossIcy and Rochester (1980). This essentially confines the calculations to the fluid core. We choose to express the 4th order system of equations in the form (without any approximation as yet)

2

dy1 =

—

~YI

+

1(1+1) r y-,

r ~0

1

[—(g0y1 (~2 — N2)y

92

+

=

d

1

-

—

r

2~5)j N —(92

(14) +y5)

d y5 —=4i~Gp0y1+y6 dr 2 1) ~ dy6 1(1 + 1) 1(1 + = —4irGp~ r y3 + r

=

92

(15)

g0 /

~,16) —

r)’6 2

Again we normalise using eqn. (12) but v is now simply the domain of the outer core. An example of the eigenfunctions obtained by this procedure is shown in Fig. 4a, showing a comparison with the full elastic solution. The zero displacements in the mantle have two obvious consequencies. The first is that because the Q values are determined almost exclusively from the outer core displacements, the damping assessed by eqn. (13) remains unaltered in the case of rigid core boundaries. The second remark, evident from of Table 1, is that the excitation/detection portion the calculation is meaningless without mantle displacements and we are forced to use a Love number approach to regenerate . A suite agreement plotted in

the mantle portion of the solutions. . . . of rigid boundary eigenfrequencies is is generally compared Fig. I for L =excellent 20. It can be seenwith the

(17)

the full elastic theory, numerically the rigid boundary eigenfrequencies are only 0.07% smaller

(18)

than those computed for the whole Earth.

(19)

6.2. The Boussinesq approximation

g

=

0y1

—

-~—y2— p0

where 92 is the radial coefficient of the reduced potential x = p 1/p0 — V1 (see also Smylie and Rochester, 1981) in which Pi’ V1 are the perturbations in Eulerian pressure and gravitational potential respectively. With rigid boundaries, there are two free constants to be integrated from the ICB (r = a):

The transition from the full core equations (14)—(19) to the Boussinesq equations has been given in detail in Crossley and Rochester (1980). Suffice to state here that the Boussinesq approximation combines the following set of assumptions

92(a) = a1, y5(a) = a2 and two conditions to be satisfied at the CMB (r = b): y1(b) = 0 and y4(b) + (1 + 1)y5(b)/b = 0. Naturally we take V1 to be harmonic in the inner core and the gravity per-

K

turbation and its radial derivative can be determined once an eigensolution is found,

V

y~( r)

=

a2 a

y (r)

=

-~y(r) r

(20) (r~a)

~: the fluid is incompressible ‘u = 0: the fluid motion is solenoidal p1. V~—s 0: the density and gravity perturbations are vanishingly small N2 is a parameterised stratification no longer derived from eqn. (11)

v

—~

theseinapproximations is to theThe finaleffect two of terms brackets [] from theremove righthand side (14) of eqns. (14) and (15). It is then seen that eqn. expresses the solenoidal condition

106

Di. CROSSLEY ET AL.

for u and that the gravity equations (16)—(17) are decoupled from the motion. This simplifies the dynamics considerably as the equations are now 0.3882

8.405

hr

y

0.3715

only second order throughout the fluid. Moreover, no model variables other than the parameterised 2 enter the equations. form of N

y 2 2.9039

0.0016

y4 0.0085

3.8895

(a)

-

I

;;~

0

9.405

0

796

hr

I 795

1593

2389 3186 3982 radius 1km)

I

I

4778

5575

Y

0.3882

~ 2 2.9039

V5 0.0016

y

0.3715

0.0085

3.8895

I 2389 3185 3982 radius 1km)

_____________

1593

I 4778

5575

5371

(b) 6371

Fig. 4. (a) Eigenfunctions of the full solution (smooth curves) for the simple mode — 1S~.The maximum values of the various radial functions are shown above the figure. The symbols show the equivalent calculation for a core with rigid boundaries — y1 (13), .Y2 (o, dotted line), y3 (si), y4 (dotted line). y~(X, dashed line), Y6 dashed line). (b) As (a) but the symbols are for a Boussinesq approximation to the core. The usual normal stress y~is plotted, not the reduced potential j~5,eqn.(19).

((u’,

ON

107

THE EXCITATION. DETECTION AND DAMPING OF CORE MODES

2 y

1

9.401

hr

0.3885

y2 2.9050

y5 0.0015

y3 0.3715

y4 0.0000

y6 3.8911

-

—

0

4.

a, -

~. -

,‘

-~

(c)

~

0

4)

0

C

-

a

-,

0

--

U

00

a, >

-

4) a

4

U

0

•0

-

0~

a, C-

ft —

,0

—

~

I 795

0

I 2389

~

1593 -

I 3186

I 3982

I 4778

I 5575

5371

radius )km)

Fig. 4(c). As (a) but the symbols are for a subseismic approximation to the core. The smooth (reference) curves refer to the rigid boundary solutions shown in symbols in Fig. 4a.

Friedlander (1985b) found that the Boussinesq approximation retained the essential mathematical structure of the subseismic equation of Smylie and Rochester (1981) and predicted only small differences in the eigenfrequencies. We here test this conjecture more thoroughly than before by recalculating the same suite of modes as before. The

although we note that to do so leads to some inconsistency in the dynamics as the Boussinesq

result demonstrates very good agreement with the full elastic solutions, the average difference (Boussinesq elastic) eigenperiods is 2.4%. Naturally

full equations and the Boussinesq approximation. We have taken one version of the subseismic equations (Smylie et al., 1984, equations (2.43)—(2.44))

the Boussinesq eigenfrequencies also agree well with the full core solutions computed with the assumption of rigid boundaries and exactly with equation (21) of Crossley and Rochester (1980) in the case C 0.

and converted them to spherical harmonics. The result is dy1 p0g0 2 1(1+ 1) d7 ~ + r (23)

=

Figure 4b shows that the Boussinesq eigenfunc-

and y3 compare well with the elastic Thus we see the Boussinesq solution reproduces the displacement and stress fields but ignores the gravity perturbations. It is possible to reconstruct y5 and y6 from u using equations tions ~1’ solutions.

2~

=

6.3. The subseismic approximation

The subseismic approximation lies between the

=

(—~~—

=

(w

d92 r

-~—

—

-~)

2)y

2 —

N

1

(24)

.~2

(16)—(17) which are the radial counterparts of the Poisson equation

v

approximation assumes V~is vanishingly small.

—4irGp

1

=

4irGv ‘(p,~u)

(22)

with the other four eqns. (16)—(19) exactly as before. Clearly, like the Boussinesq assumption, the gravity equations are decoupled from the mo-

lion and it is clear that the only distinguishing difference is the term p0g0/sc in eqn. (23) which vanishes in the Boussinesq approximation. Unlike

the latter assumption however, in the subseismic approximation the gravity perturbation is pre-

108

Di.

cisely described by eqn. (22) and so we should be able to recover a good approximation to the full solution. Again using the modes shown in Fig. 1, we found that the subseismic eigenperiods are almost identical to their Boussinesq counterparts, i.e. the mean percentage difference from the elastic case is also 2.4. The subseismic eigenfunctions are computed in two steps. We first solve eqns. (23)—(24) for the displacements and the reduced potential and then (optionally) substitute into eqn. (22) to find the gravity variables. This latter calculation is simply done using eqns. (16) and (17) by allowing y5(a) to be a free constant and requiring the harmonic condition y4(b) + (1 + 1)y5(b)/b = 0 at r = b. The solution converges quickly from virtually any reasonable initial guess. The results, shown in Fig. 4c, are very close to the full solutions for all the variables, verifying the utility of the subseismic approximation. Interestingly, if we attempt to use eqn. (22) with the Boussinesq displacement field, we find that V~is about an order of magnitude larger than the full solution. We shall return to this point later, As a final comment on the approximations, we record here the various CPU times for a typical integration (i.e. 1 mode at I trial frequency). The times are 22.69 s (elastic), 8.07 s (rigid), 2.69 ~ (subseismic) and Boussinesq 0.88 s. The Boussinesq gains over the subseismic because no Earth model interpolation is required.

7. The internal load Love numbers demonstrated that the integration of theHaving equations can be restricted to the fluid core,

CROSSLEY ET AL.

we need to take into account the elasticity of the inner core and mantle. This will allow a continuation of the eigenfunctions from the core into the mantle (in particular) and thereby allow the cxcitation and detection to be estimated (Table 1). 7.1. Computational procedure The details of the rigid boundary integrations are summarised in Table 3. It can be seen that the core integration yields 3 variables Y 2, Y5 and Y~, each of which is defined at the ICB and the CMB, the other variables are not continued into the solid regions. The internal load Love numbers are based on the observation that, for example at the CMB, there are a source terms (such as a normal pressure 1’2) that load the mantle from below. The mantle responds by deforming in a manner such that (for example) the radial surface displacement can be assumed proportional to the source ~“2~ The constant of proportionality is a load Love number (when suitably scaled). To compute the mantle deformation, we integrate the standard 6th order system of equations in the inner core and mantle at a frequency we can choose. The normal procedure is to take w = 0 to provide a static solution. This would be reasonable for the core modes as they are long period deformations. However PREM is also a frequency dependent model and we cannot extrapolate it to zero frequency. To settle the question we tested the stability of the mantle propagator as a function of period. The results are shown in Fig. 5 for two values of 1 from periods 1—1000 h. The upper (dashed) of each pair of curves shows the mantle solution as a function only of held the inertial 2u, the PREM velocities being at their term val~ ues for 1 h. The lower curves show the behaviour

TABLE 3 Solutions for a core with rigid boundaries. Quasi-static propagator matrices are calculated for the inner core and shell

Inner core

Outer core

y,(O)

y,(a)

0

Y2@)

Y2(°)

Y~’

Shell 0 Y~’

y,(b)

y,(d)

y 2(b)

y~(O)

Q(a, 0)

y~(O)

y3(a) y4(a) y5(a)

0 Y5”+~”

,,b~yb

Y6(O)

y6(a)

Y6”+4””

44~b~yb

)~4(0)

y3(b)

0

0 y5(b) 6) y~(

P(d, b)

y2(d) y3(d) y4(d)

y5(d) y 6(d)

ON

THE

EXCITATION.

L)ETECTION

AND

109

DAMPING OF CORE MODES

MANTLE PROPAGATOR AS A FUNCTION OF PERIOD 33.6

-

32.1

1=2

-

30.7-

-

S.---

1=10

56.8 -

56.5

-

56.1 55.8

29.2

-

-

27.8

-

-

E

~25.3a

-55.1

*

C

-

54.8 ~

23.4

-

54.4

21.9

-

54.1

-

53.8

24.8

-

20.5

-

19.0 0

I 1 log period

I 2 )hrs)

I 3

Fig. 5. The ‘sum’ refers simply to the sum of all elements of the propagator P(d, b) (an arbitrary measure of the overall numerical stabiblity of the matrix). The scale on the left refers to / = 1, on the right to / = 10. Dashed lines are the inertial effect, solid lines include inertial + velocity dispersion effects.

with both the inertial term and the velocity dispersion present. For reasons fairly obvious from the figure, we chose 12 h (the vertical dotted line) as the reference period, Suppose in the mantle we define P(d, b) as the

of five unknown constants /3., (/3~= 0) and any inner core solution written using three constants a,. Before applying these solutions (containing 8 unknown constants) we note that there are 10 boundary conditions, namely continuity of Y2’ .Y

propagator matrix from radius r

and y6 at both the ICB and the CMB; the vanishing of the stresses y2, y4 and the harmonic condition for y5, y6 at r d and finally the vanishing of y4 at r a. The system is therefore overde-

=

b to r

=

d

(Gil-

bert and Backus, 1966), and similarly Q(a, 0) for the inner core. Then y(d) = P(d, h) y(b) and y(a) = Q(a, 0) y(O) (Table 3). We let the elements of P be p,, and of Q be q,~ The solution in the mantle is composed of five linearly independent solutions (the solution for y4 is null); that for the inner core consists of only three independent solutions. By choosing each of the five mantle variables to be unity at r = b, the rest set to zero, and integrating to r = d, we construct the P matrix column by column. For the inner core, a similar procedure is used to find the Q matrix at r = a, but we note that when starting at r = 0 the linearly independent solutions are various combinations of the y variables, as discussed in Crossley (1975a). Having found the propagators, we now assume they can be applied to the rigid boundary solulions. Any mantle solution can be written in terms

5

=

=

termined unless we drop two of the conditions. A better procedure is to recognise that we can solve for an additional potential in the core of the form (±1

~(r)

=

A(s)

/

+

B(~)

(a ~ r ~ b)

(25)

r where A, B are arbitrary constants. The need for such a potential arises from the deformation in the solid regions. Remember that we have already solved for y5, y6 in the outer core with a particular eigenfrequency, assuming the mantle and inner core are rigid. When elasticity is allowed, the deformation induces a gravity perturbation in the solid regions which is extended harmonically into the outer core from the ICB and CMB; hence the

110

D.i.

CROSSLEY ET AL.

form of eqn. (25). It is clear the 2 additional constants in eqn. (25) now equalise the total num-

The constants now enable various boundary values to be found, the most useful being

ber of constants and boundary conditions in the problem, as well as tying together the solutions for the inner core and mantle. We note that the additional potential 4~(r)in

y

the outer core falls off very rapidly away from the boundaries for large values of I. This has the effect of again isolating the inner core solution from the

mantle solution and could cause numerical instability in the set of equations to follow eqn. (26). From the above analysis, we need to solve the following set of equations for the constants a,, /3. A, B: 3, + P22/32 +

Y2 (d) ~V

(

4 d)

=

P21/ +P25/35 + P26/36

=

P41 /3i + P42

0

/3~+ p43/33

+

~

d

~

)

d “ P53 + 1+1 P63 )

(

d

/3~

(27)

=

Before proceeding to define the Love numbers, it is worth remarking that we could iterate the whole

$~ /3~

tar the mantle and inner core static integrations only depend on the harmonic degree and so can be used for every core oscillation of that degree

\

(no matter what eigenfrequency). We therefore

d

7i~jP66) /36

=

q21a1 + q22a2 + q23a3

j’~(a)

=

q41a1 + q42a2 + q43a3

y5(a)

= =

q31a1 + q32a2 + q33a3

the core, we felt it unnecessary to implement this additional step. The above procedure does not have to be repeated for every core mode calculation. In particu-

+ (Pss + 7T~’jP65) /35 + ( P56 +

=

Considering the already good approximation we have to the eigenfrequencies and eigenfunctions in

(p51 + ~-~~1-P6i)/3l + (P52 +

(

=

zero displacement assumed for rigid boundaries.

d

=

y1 (b) ~ b)

values calculated from eqn. (27) rather than the

+ P45/35 + P~/36= 0

y5(d) + ~—~-j-y6(d)

.Y3 (a)

problem by re-calculating the eigenfrequencies and eigenfunctions in the outer the the radial displacement at the ICBcore andassuming CMB have

~/3 =

36

1(d) =p11/31 + P12/32 + p1~/33+p~/35+ P16/ y 3i +P62/3 3 3S+ P66/36 2 ±P631 3±P65/ y6(d) =P6i/ 1(a) = q11a1 + q12a2 ±q13a3

0

=

=

va 12

=

0

q51a1 + q52a2 + q53a3 Y~+A+(~)B

condense the above information in a set of internal load Love numbers with the notation given in Table 4. The variables eqn. (27) are then defined according to Table 5. The actual Love number calculation procedes 2 (see Table 5). as in the following example for hb TABLE 4

y 6(a)

=

q61a1 + q62a2 + q63a3

=

ya

— ~

6

y2(b)

=

y5(b)

=

$2

=

=

/36

a

Notation for load love numbers. The harmonic degree should be added in parentheses, e.g. (hr), as required

(

B a a ~b)

Notation h7

=

y5b

y 6(b)

A+

=

a ~ 1 ±A(~) + B 1+1 a

‘6~’—

+

/

—h-—A(~)

-~B

9~

/,~.

k’

Variable radial displacement gravity perturbation transverse displacement

derivative of gravity perturbation

Index values: g, effect: 0 = surface, 1 = CMB, 2

(26)

=

ICB (s = source radius)

.9”, source: 0 = surface, 1 = CMB, 2 = ICB (s = source radius) .~, type: 2 = pressure, 5 = gravity, 6 = gravity gradient

ON

THE

EXCITATION,

AND DAMPING OF

DETECTION

111

CORE MODES

TABLE 5

First we set all source terms (except Y~= 1) in

Load Love number expressions”

eqn. (26) in turn to zero and solve for the constants a,, /3,, A and B. Then y 1(d) is= found from eqn. (27) and finally we compute h’~ p 0g0y1(d). In Table 5, the Love numbers for the constants A, B in the additional potential 4 have been omitted

7.. +

y,(d)

=

h~—~— + h~f-~+ p0g0

h106....~...+ h~2

g0

g0

p0g0

g0

+

as they will be rarely needed. We continue the

g0 y3(d)

=

p0g0 +

y5(d)

+

/~2_~_

=

+ ~

g0

~

+

g0

=

policy of Hinderer in d adopting standard values forand p0, Legros g0 and(1989) radius in the scaling of the Love numbers (Table 5) rather than

~~6_~_d

~

5Y 1’+ k~Y 1’d+ ~ 5 6

+

k~5Y~’ + k~6y:d

using values specific to a particular Earth model. thefirst Loveset numbers shown in Typical Figs. 6a results and 6b.forThe of Loveare numbers

+ k~

g 0

y6(d)

r g0

+ ~025

p0g0

g0 +

~i2~’2

+

+ ~22~

i~sY~’

+

26y” 06

shows the surface radial displacement due to pressure and gravity sources at the CMB and ICB. The ments atsetthe(Fig. second CMB6b) andshows ICB due the toradial sources displaceat the

respective boundaries themselves. These should be

+ ~

y 1(b)

=

h~2_~_ + h~~-~+~ p0g0

g0

g0

+ h~2_~_+ ~ p0g0

g0

+ ~ g0

y3(b)

=

+

~

+ /~6~ + ~

p0g0

g0

g0

g0

To compute the surface gravity, we are re-

+~ g0

1--+ h~—6—+ h~2—7--+ h~~-~-h’~—~— + h~—

y,(a)

p0g0

g0

g0

p0g0

g0

=

/I2~

+

p0g0

/~s....i + g0

/i6~.~

+

+

g0

more shell integration from r

= b. We decided instead to introduce Love numbers for the excita-

255

p0g0

g0

tion as y(~) well by S(,b, defining a propagator at the source radius b) y(b). By analogy with eqn.

+ /~6~ g0

=

1-+ ~ =

+

h22.2~.. +

h~25Y~

0g0

g°

-,

p0g0

g0

26

=

g0

p0g0

y1(r0) y3(r0)

=

s11/31 +

=

~31/31

+

~12/32 ~32/32

+ ~i3/33+ s15/35 + ~16/36 + s33/33 ±s35/35 + ~36/36

(28)

ji2....~... + /~5__i_+ ~

g0

g0

+ /22.......

p0g0

+

g0

g0

O

(27) we then have

p h~2_~_ + h~— +h0

y3(r0)

minded from Table 1 that the shell eigenfunctions y 1, y3 are required at the focal radius r = r0 as well as y1, y6 at the surface. The latter values are given directly the Lovecould numbers (Tableby5) but the focalthrough displacements be found one

+ h26_~t~_ 2 g0

y3(a)

puted by Smylie et al. (1989) for the case of the subseismic equations in a rotating core.

7.2. Results for the displacements

+ 25Y1

p0g0

compared to the equivalent Love numbers com-

All Love numbers are normalized by taking the nominal values Po = 5.5 cm ~, g0 = 9.81 m s~ and d = 6371 km.

and the additional Love numbers can be found in Tables 4 and 5. Sample results for the radial displacements at

the ICB and CMB are shown in Figs. 7a and 7b. The alternating effect caused by the odd/even values of / reflects the normatisation used for the

eigenfunctions which is dependent on the azimuth m. The agreement at the ICB between elastic and

112

D.i.

rigid boundary solutions is remarkably good, though worse at the CMB, especially for the lowdegree modes. The discrepancies at the larger val-

ues of 1 are not due to any breakdown of the Love number approach but in numerical difficulties with the elastic solutions (as indicated earlier); the

INTERNAL LOAD LOVE NUMBERS 0.01

0.05

a, a, a, a, a, a, a, a, a, a, a, a,

—0.07

IS a, a,

a, a, a, a, 5,

0.02

-

h12 O

—0.20-

—0.09

-0.45

-

—0.12

10

(5

(a)

h15

a, -

a 5

(5

0

-

.

5, (55, 5, 5, 5, (55,

a,

O.05

—0.32

-O

CROSSLEY ET AL.

15

20

0.0000

O~15

I

I

I

5

10

15

20

I 15

20

0.0059 A A A -A -A A A -A A -A £ A £ -A A A -

A

—0.0001

-

-0.0002

-

+

0.0046

h 22

-

h 25

0.0033

O —0.0003

-

—0.0004

-

0 0.0020 0.0008

+

A

—O

ooo~

I 5

I 10 degree 1

I 15

—0 OO0~ 20 ‘ 0

I 5

I 10 degree I

INTERNAL LOAD LOVE NUMBERS 0.061

0.05

b

0

—0.12

O.O4~-

-

-0.47

-

—0.65

-

h12

a

—0.30-

h16

0.035-

0.021

-

0.008

-

5’

a,

—0.82 0

I 5

I 10

I 15

0.1258

20

—0.005 0

I 15

20

0.0083-

O.0762~

h22

A

.

A

0.0038

A A.A..ALAA

I 10 degree

2

+

A

I 5

h26

0.0060

2 -

O.0265~

O

20

+

0.1010

0 ~017

I 15

0.0106 A

0.0513

I 10

5

I 15 1

0.0015—0.0005 20 0

-

~-+.

+

I 5

I 10 degree

1

Fig. 6. (a) Love numbers as a function of degree. The ‘h’ denotes a Yi component and, the ‘0’ that the effect is measured at the surface. The upper curves are for a source at the CMB, the lower for a source at the ICB. See Table 4 for further details. (b) As (a) but showing Yi effect due to ~‘2, Y 6 sources at the same boundary; upper two curves CMB, lower two ICB.

113

ON THE EXCITATION, DETECTION AND DAMPING OF CORE MODES

VERTICAL

DISPLACEMENT AT ICB

10. C (a) 9.0 0 )-~

8.0

-

C

o

a,

E a, -

U

a -

.26.0-

--

a,

~ 5.0-

~

4w-t-~-

~.

w401

0~

a,

0

.230S ~•

2.01.0

5’

I

0

I

2

4

I

6

8

I

I

10 12 degree 1

I

I

14

I

16

18

I

20

22

VERTICAL DISPLACEMENT AT CMB 8.0

(b)

2

7.0

a4) C a,

--

C

.

cv

-

-

U

54

‘.1

-4 54

a

a,

0

-

54

o

is

~40 0

a, C-

g

I

1

o

‘~

S --..

3.0-

-~

t

-1,.’

-:~ ~

~: 54

2.0

-

a

1 ~ 0

2

4

I

I

I

I

I

I

I

I

6

8

10

12

14

16

18

20

degree

22

1

FIg. 7. (a) Radial coefficients at ICB comparing the elastic solutions (U) to the Love number solutions in the rigid boundary case (dotted lines). (b) As (a) but at CMB. The discrepancy at the low degree end is probably due to the limitations of the theory, at the

high end due to numerical inaccuracies for the elastic integrations.

surface boundary conditions have to be applied

7.3. Results for the surface gravity

far from where the energy in the oscillation is

concentrated (i.e. in the outer core). In practise the

We are now in a position to recalculate the

Love number approach should be regarded as the more stable method,

surface gravity using the foregoing theory. To evaluate the performance of the various approxi-

114

DJ. CROSSLEY ETAL.

mations we have taken as a control the calculation of the surface gravity assuming we start with the elastic values of the sources (Yl’ etc.) in equations (26). The resulting value of ~xg(eqn. (9)) is shown

The other columns of Table 6 show sample calculations for some of the other approximations, all using the appropriate load Love numbers for the inner core and mantle. The Boussinesq solu-

in the second column of Table 6 where it is seen

tion is obtained in the same manner as the full

that the agreement with the full elastic theory is

solution except we do not solve for the Love

close. This establishes (a) that the numerical

numbers associated with Y~ and Y

scheme is sound and (b) that the period-depend-

6 at the core boundaries as these are identically zero. Effec-

ency of the mantle solutions is indeed small (Fig. 5).

tively, the surface gravity is then dependent on only the pressure term at the ICB and CMB. It

TABLE 6 Surface gravity under various assumptions Mode ‘1

/

m

~g(pgal) Elastic

Elastic

Rigid

Boussinesq

Subseismic

Subseismic

(— gravity)

0.14 —0.38x101 0.27x10~

(+0.63x10~ gravity) 0.25X101 0.10x10~

—0.12x104 —0.63x105 —0.33x105 —0.20x105 —0.13x105 —0.93X106

—0.13X104 —0.72x105 —0.38x105 —0.23x105 —0.15x105 —0.11x105

—0.32x104 —0.16x104 —0.93 x105 —0.50X105 —0.37x105 —0.24x105

(original) —2 —1 —3

1 1

1 1

2 —0.23x102 —0.29x10 —0.37x102

(Love) —0.57x102 —0.54x103 —0.12x10~

—0.97x10~ —0.33x1O~ —0.15x101

0.12 —0.43x101 0.23x101

—1 —2 —3 —4 —5 —6

2 2 2 2 2 2

2 2 2 2 2 2

—0.32x105 —0.15x105 —0.95x106 —0.49x106 —0.38x106 —0.23x106

—0.32x105 —0.15x105 —0.95x106 —0.49x106 —0.38x106 —0.23x106

—0.13x104 —0.74x105 —0.39x io~ —0.24X105 —0.15x105 —0.iixi0~

—1 —2 —3 —4 —5 —6

3 3 3 3 3 3

1 1 1 1 1 1

0.89x104 0.70X104 0.44x104 0.28x104 0.19x104 0.14x104

0.21x103 0.17x103 0.10x103 0.66x104 0.45x104 0.32x104

0.18x103 0.14x103 O.85x104 0.54x104 0.37x104 0.26x104

0.20x103 0.16x103 0.97x104 0.62X104 0.42x104 0.30x104

0.39x103 0.30x103 O.19x103 0.12x103 0.83x104 0.58x104

—1 —2 —3 —4 —5 —6

4 4 4 4 4 4

2 2 2 2 2 2

—0.25x106 —0.28x106 —0.20x106 —0.13x106 —0.95x107 —0.69x107

—0.43x106 —0.48x106 —0.34x106 —0.23x106 —0.16x106 —0.12x106

—0.28X106 —0.31x106 —0.22x106 —0.15X106 —0.i0xio~ —0.75x107

—0.32x106 —0.35x106 —0.25x106 —0.17X106 —0.12x106 —0.87x107

—0.67x106 —0.73x106 —0.52x106 —0.35x106 —0.25x106 —0.18x106

—1 —2 —3 —4 —5 —6

5 5 5 5 5 5

1 1 1 1 1 1

O.34x105 0.48x105 0.39x i0~ 0.29x105 0.21x105 0.16x105

0.49x105 0.71x105 0.58x iø~ 0.42x105 0.31x105 0.23x105

—1 —2 —3 —4 —5 —6

6 6 6 6 6 6

2 2 2 2 2 2

—0.84x108 —0.15x107 —O.14x107 —0.11 xlO7 —0.82x108 —0.64x108

—0.11x107 —0.20x107 —0.18x107 —0.15x107 —0.11x107 —0.86x108

0.89x10~4 0.70x104 0.44x10~ 0.28x104 0.19x104 0.14x104 —0.25x106 —0.28x106 —0.20x106 —0.13x106 —0.95x107 —0.69x107 0.34x105 0.48x105 0.39x iO~ 0.29x105 0.21x105 0.16x105 —0.84x108 —0.15x107 —0.14x107 —0.lIxlO7 —0.82x108 —0.64x108

0.31x105 0.44x105 0.35 x io~~ 0.26x105 0.19x105 0.14x105 —0.65x108 —0.11x107 —0.lOxIO7 —0.82x108 —0.62x108 —0.48x108

0.35x105 0.50x105 0.41 x 10~ 0.30x105 0.22x105 0.16x105 —0.74x108 —O.13x107 —0.12X107 —0.94x108 —0.72x108 —0.55x108

0.71x105 0.10x104 0.82 x i0~ 0.59x105 0.43x105 0.32x105 —0.15X107 —0.27X107 —0.25x107 —0.19x107 —0.15x107 —O.iiXiO’

ON

THE

EXCITATION,

DETECTION

AND

DAMPING

OF CORE

115

MODES

can be seen that the /

= 1 solutions are the wrong sign but otherwise the gravity values are of the same magnitude as the rigid (boundary) solutions, The subseismic surface values are computed both without and with the additional gravity variables

This last value deserves special comment as it

in the core. Figure 8 summarises and extends Table 6 to higher degrees. The ordinate is the ratio R of the gravity computed via a Love number approximation to that using the full elastic solution. The

tion to the displacement field (as implied by our

differences are averged over the appropriate over-

to accurately determine the gravity contribution at

tone numbers. We note again that the synthesised

the core boundaries. In other words, it seems

elastic (Love) values almost perfectly match the full solution, but the other approximations start off by over-estimating the surface gravity for the

better to leave out the gravity contribution and use a simple Boussinesq approach. Perhaps the subseismic calculation would benefit from a fur-

low-degrees, say / 2—5, and approach a constant ratio as / increases. Beyond / 10 the elastic solutions degrade due to loss of integration accu-

ther iteration, as suggested earlier. As can be inferred from the above ratios, the Boussinesq gravity (Y2 pressure contribution) is about 69% of

racy (fixable by re-normalising the standard equa-

that computed from the full core dynamics (pres-

tions, but we did not pursue this further). The averaged ratio R = ~ g/L~g( elastic, original) for each column of Table 6 for / = 2 to 10 is elastic (Love) R = 0.999 rigid boundaries R = 1.796 Boussinesq R = 1.242 subseismic (—gravity) R = 1.418 subseismic (+ gravity) R = 2.995

sure ±density perturbations in the core).

=

=

SURFACE GRAVITY

:

indicates that the inclusion of the gravity variables in the subseismic solution has degraded the agreement with the full solution. The apparent reason for this lies in the sensitivity of the Poisson equaearlier attempt to calculate the gravity with the Boussinesq displacements). It seems as though the

accord demonstrated in Fig. 4c between the rigid and subseismic eigenfunctions is not good enough

8. Conclusions We have now completed a survey of the dynamics of the core modes for the non-rotating case, including for the first time a discussion of

LOVE NUMBER VS ELASTIC SOLUTIONS

5.0 4.5

-

11 degree

1

Fig. 8. The ratio of surface gravity computed using various approximations to the full elastic case. A perfect agreement would be 1 (dotted line). The symbols refer to elastic Love (0), rigid (0), Boussinesq (s~)and subseismic without gravity (c)).

116

Di.

the level of earthquake excitation and damping for higher degree harmonics. In doing so we have established the weak excitation but long persistence of the oscillations (for an artificially stabilised core model) and demonstrated that both the Boussinesq and subseismic approximations

yield excellent approximations to the eigensolutions. In addition the viability of the Love number approach for dealing with elasticity in the inner core and mantle is now confirmed, providing one is willing to accept a factor of 4 error for the surface gravity for / 2 and a factor less than 2 =

overall for degrees 2—10 (the / = I case is special

and will be treated in more depth in a forthcoming paper). With the above as a basis, the core modes can now be integrated for a rotating Earth model using the Boussinesq or subseismic approximations in a core surrounded by rigid boundaries. The surface gravity effect from an earthquake can then be assessed through the use of the Love numbers presented here.

CROSSLEY FT AL.

quakes. In: D.E Smylie and R. Hide (Editors), Structure and Dynamics of Earth’s Deep Interior. Geophys. Monogr. Am. Geophys. Union, 46 (1): 41—50. Crossley, D., 1984. Oscillatory flow in the liquid core. Phys. Earth Planet. Inter., 36: 1—16. Crossley, Di., 1975a. The free-oscillation equations at the centre of the Earth. Geophys. J., 41: 152—163. Crossley, D.J., 1975b. Core undertones with rotation. Geophys. J,, 42: 477—488. Crossley, D.J. and Rochester, MG., 1980. Simple core undertones. Geophys. J., 60: 129—161. Crossley, D.J. and Smylie, D.E., 1975. Electromagnetic and viscous damping of core oscillations. Geophys. J., 42: 10111033. Dahlen, F. A. and M. L. Smith, 1975. The influence of rotation on the free oscillations of the Earth. Phil. Trans. R. Soc. London Ser. A, 279: 583—629. Dziewonski, AM. and Anderson, DL., 1981. Preliminary reference Earth model. Phys. Earth Planet. Inter., 25: 297—356. Friedlander, S., 1985b. Stability of the subseismic wave equation for the Earth’s fluid core. GAFD, 31: 151—167. Friedlander, S., 1985a. Internal oscillations in the Earth’s fluid core. Geophys. J., 80: 345—361. Gilbert, F., 1970. Excitation of the normal modes of the Earth by earthquake sources Geophys. J., 22: 223—236. Gilbert, F. and Backus, G., 1966. Propagator matrices in elastic wave and vibrational problems. Geophysics, 31: 326—332. Hinderer, J. and Legros, H., 1989. Elasto-gravitational defor-

Acknowledgments This study has been supported by CNRS—INSU and is DBT contribution number 259. One of us (D. Crossley) wishes to acknowledge funding through a Canadian NSERC Operating Grant and to thank the Institute de Physique du Globe for support and hospitality during a very enjoyable

sabbatical visit. We also thank Bernard Valette for valuable discussions and Michael Rochester for pertinent comments.

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