Geophys. J. Int. (1997) 131,607-617
Effects of a mushy transition zone at the inner core boundary on the
Slichter modes
Z. R.Peng*
Departfnent of Earth Sciences, Memorial Uniaersity of Nelyfoundland, St. JohnL NF, Canada, A1B 3x5. E-mail: [email protected]
Accepted 1997 June 26. Received 1997 June 2; in original form 1996 September 19
SUMMARY
This article investigates the effects of a mushy inner core boundary o n the eigenperiods
of the Slichter modes for a simple, but realistic, earth model (rotating, spherical
configuration, elastic inner core and mantle, neutrally stratified, inviscid, compressible
liquid core). It is found that the influence of the mushy boundary layer is substantial
compared with some other effects, such as those from elasticity of the mantle, nonneutral stratification of the liquid outer core and ellipticity of the Earth and centrifugal
potential. The results obtained here may set a lower bound o n the eigenperiods of the
Slichter modes for a realistic earth model. For example, for a PREM model, the lower
bound of the central period of the Slichter modes should be about 5.3 hr.
Key words: eigenperiods, mushy zone, Slichter modes.
1 INTRODUCTION
The Slichter mode, which was first proposed by Slichter
following the great 1960 Chilean earthquake (Slichter 1961),
is the degree-1 long-period free oscillation of the Earth which
involves translational vibration of the solid inner core in the
surrounding liquid of the outer core. Due to the Earth’s
rotation, a single mode is split into three different polarizations
(a triplet): one axial and two equatorial (one retrograde and
one prograde with respect to the sense of the Earth’s rotation).
Collectively, these are called the Slichter ‘modes’. It is known
that the eigenperiods of the Slichter modes critically depend
on the density contrast across the inner core boundary (ICB),
a key parameter for evaluating the gravitational energy released
from segregation of core material upon its cooling and solidification (Loper 1978; Masters 1979; Morse 1986). This is
probably the most important source of energy to power the
geodynamo action in the Earth‘s fluid core (Verhoogen 1961;
Braginskii 1963; Gubbins 1977; McElhinny & Senanayake
1980; Anderson & Young 1988). Because conventional shortperiod seismology has so far not very confidently determined
the density contrast across the ICB, the Slichter modes have
become ideal candidates for inferring this parameter.
Because of the above-mentioned importance in the geodynamo theory, a few attempts have been made to detect the
Slichter modes, but so far these have been unsuccessful. For
example, after carefully analysing the Earth tides data from
the South Pole, Jackson & Slichter (1974) were unable to
detect evidence of its existence in a record of nearly 1 yr
* Now at: Department of Physics, University of Toronto, Toronto,
ON, Canada, M5S 1A7.
0 1997 RAS
duration (1970 October to 1971 September) there. Smylie
(1992) announced that he had detected the Slichter triplet
based on his theoretical prediction and analysis of the stacked
superconducting gravimeter data. However, the latter discovery
is still highly controversial (Crossley, Rochester & Peng 1992;
Rochester & Peng 1993).
The detection of the Slichter modes requires the development
of a satisfactory theory based on a realistic earth model,
without which one cannot be sure of correctly identifying
the Slichter periods in the spectra of disturbances. A number
of researchers have devoted their efforts to the theoretical
modelling of the oscillation and related problems (Slichter
1961; Busse 1974; Crossley 1975a; Smith 1976; Dahlen & Sailor
1979; Rochester & Peng 1990; Smylie 1992; Smylie & Qin
1992; Crossley et al. 1992; Rochester & Peng 1993; Wu &
Rochester 1994; Buffett & Goertz 1995). Because the present
paper will deal with a specific problem in the modelling of the
Slichter modes, namely the effects of a mushy transition zone
below the ICB, we will not carry out a detailed review of
the literature here. Instead, we list in Table 1 some results for
benchmark cases. It is clear that as practical earth models
come closer to each other, the respective eigenperiods of the
Slichter modes fall into a narrower range. Note that SPLM3
is no longer considered a reliable earth model.
Conventional short-period seismology infers the density
contract across ICB mainly from the amplitude ratio
PKiKPIPcP, where PKiKP is a P wave reflected at the ICB,
and PcP is a P wave reflected at the core-mantle boundary
(CMB). The poor determination of this density contrast is
mainly due to insufficient detection of the PKiKP wave
(Doornbos 1974; Shearer & Masters 1990), and also due to
the strongly scattered nature of the PcP wave. While the
607
608
2. R. Peng
Table 1. Selected list of central period of the Slichter modes for
benchmark cases. As the earth models come closer to each other, the
eigenperiods of the Slichter modes also fall into a narrower range.
Note that SPLM3 is no longer considered a reliable earth model, and
Smylie ( 1992) is still highly controversial.
Authors
Bliss(>(1971)
Crossley (1975a)
Dahlen & Sailor [ 1979)
Smylie (1992)
Crossley. Ruc1icst~c.rki Pctig (1992)
Rochester, W i i k Pmg (1992)
Rochcst,c:r k Pcng (1993)
Earth Models
SPL\IS
1066X
1066.A
J06G.4
Core1 1
PREhl
as
Eigenperiod (hr)
6-40
7.72
1.53
2.70
1.53
5 80
5.30
ys(r) is the radial part of the additional gravitationd potential
field, and y6(r) is the gravitational flux
(4)
where i,
p and po are the Lame parameter, rigidity and density
of the solid parts of the Earth, respectively. The validity of
descriptions (1)-(4) is based on basic definitions
uncertainties in interpreting PcP observations are introduced
by possible complex structure at the CMB (Buchbinder, Wright
& Poupinet 1973; Souriau & Souriau 1989), the weakness of
the signals reflected at the ICB may indicate uncertainty in
the ICB itself it may not be a sharp discontinuity; instead it
may well be a transition zone with low rigidity (Loper &
Roberts 1981; Cummins & Johnson 1988; Shearer & Masters
1990). The thickness of the possible transition zone is also
poorly determined. In their study based on synthetic seismograms using PKiKP and P K J K P data, Cummins & Johnson
( 1988) proposed that the thickness of the transition zone must
be 5 km or less, which broadly agrees with the result of
Phinney (1970). The latter author obtained a thickness of
1.5 km or less using a different approach which retains a
special form of the transition zone where analytic solutions
exist.
Mochizuki (1991) studied qualitatively the effects of a
transition zone on short-period free oscillation of the Earth.
He concluded that the transition-zone effects are expected to
become important for the inner core boundary for two reasons:
(1) the discontinuity of elastic parameters is large; (2) the
existence of a transition zone is probable for the boundary
between fluid and solid iron. In the present study, we will
incorporate a mushy transition zone with a finite thickness
into our earth model, and examine quantitatively how it will
affect the eigenperiods of the Slichter modes.
where ii is the unit normal pointing out of the liquid core
(therefore ii = -f at the ICB, and fi = f at the CMB), u is the
displacement field, V, is the additional gravitational potential
due to disturbance, and Y: is the standard spherical harmonic
of degree n and order m.
The matrix A in eq. ( I ) consists of coefficients of y,s of the
governing equations, and they are functions of radius Y,
frequency w , azimuthal order Iy1 and/or degree n. The non-zero
elements of A are
4 . 3
=
+
+
n(n 1)A
(1- 2p)r '
~
2 GOVERNING EQUATIONS O F THE
I N N E R AND OUTER CORES
The governing system of differential equations for an elastic
sphere or anelastic spherical shell with only the first-order
Coriolis self-coupling (to the spheroidal field) taken into
account is most conveniently expressed in AJP notations
(Alterman, Jarosch & Pekeris 1959):
dY
=Ay,
dr
-
-
1
A3.3 = -
1
where
1
A4,1
=
A2,3 >
yl(r) and y3(r) are the radial parts (functions of Y only) of the
normal and transverse displacement fields, yz(r) and y4(r) are
the radial parts of the normal and shear stress fields, defined
0 1997 RAS, GJI 131, 607-617
Mushy transition zone
609
1) + 2p(n2 + n -
+-2mp0wR
n(n + I ) ’
r = 1 -v’kk- c*c
+ ivl; x 1,
B
n..
~
2R
V=-.
0
A6,5
=
where 1 is a unit tensor. The stability parameter /Iwas first
used by Pekeris & Accad (1972), and it satisfies the following
relation:
+
n(n 1)
r2 ’
~
These expressions of matrix A are analogous to those given
by Crossley & Rochester (1980), except they also included the
Coriolis coupling to the toroidal field.
Assuming an equilibrium configuration in which the Earth
is at rest in a steadily rotating reference frame, free oscillations
of the liquid core without a dissipative mechanism are governed
by a set of six equations (three scalar equations and three
components of a vector equation) which express the conservation of entropy (equation of state), mass, momentum
and gravitational flux, respectively:
-w2u
1
P1
+ 2 i w ~ l x; u = - -Vpl
+ v V, + -go,
Po
Po
where eq. (9) is a vector equation that is equivalent to three
scalar equations. Here, tl is the compressional-wave speed, p1
is the Eulerian flow pressure, po and go are the density of the
liquid core and the gravitational acceleration, respectively, at
equilibrium, p1 is the Eulerian density increment, w is the
vibration frequency, and i< is the unit vector aligned with the
Earth’s rotation axis.
Wu & Rochester (1990) showed that the governing system
(7), (8), (9) and (10) can be reduced to a coupled pair of
second-order linear partial differential equations based on
two scalar potential fields x (termed the reduced pressure)
and V,:
“*I
r e v x+ w2(1 - v2)(x+ h)-B
d(1-
In a neutrally stratified liquid core ( p = 0), the pair of equations
(11) and (12) becomes
V2 Vl
+ 4nG Po(Xtl2+ Vl) = O
3 REPLACING THE INNER CORE
B O U N D A R Y BY A T H I N M U S H Y
TRANSITION ZONE
For the purpose of investigating the influence of the mushy
layer at the ICB on the Slichter modes, it may be useful to look
at a variation of zone thickness, for example from 1 to 5 km,
with the upper bound at the PREM (Dziewonski & Anderson
1981) inner core boundary (1221.5 km). Also, for the sake of
simplicity, the earth model is taken to be spherical in configuration, with an elastic inner core and mantle, and a
neutrally stratified, inviscid, compressible liquid core.
First we turn to Loper & Fearn (1983) for guidance in
constructing a model of the transition zone at the ICB.
Currently, the earth models deduced from seismology all place
a sharp ICB immediately beneath the liquid outer core.
However, studies of the anelasticity and P-wave speed of the
inner core also indicate a non-homogeneous region several
hundred kilometres thick immediately below the ICB. Therefore,
from a seismological point of view, the transition zone at the
ICB is most likely of a form in which the outer core contains
no solid but the inner core may contain liquid in a dendritic
mushy zone (Loper & Fearn 1983). This mushy zone should
have a significant mass fraction of solid which forms a mechanically rigid matrix capable of sustaining shear waves. The
model should also allow for liquid inclusions to occur in the
interstices of the solid matrix. If the shear modulus of the inner
core is ps,the shear modulus pt of the transition zone can be
estimated using Loper & Fearn’s (1983) formula (eq. 64):
Pt = ( 1 -f YP.3 >
(21)
where f is the fluid fraction parameter defined as
f
0 1997 RAS, GJI 131, 607-617
= VJV,
(22)
610
2. R. Peng
V' is the volume of the fluid in the transition zone, V is the
total volume, and q5 ( 2 1) is an integer parameter.
In principle, the density (po) and the elastic parameters
(2 and p) should change smoothly through the transition zone
and continue across its upper and lower boundaries. While
the requirement for continuity of the Lam& parameter iand
the density po can be implemented easily, such a requirement
for the shear modulus p will pose some difficulty. When
p=0, two of the six ordinary differential equations of the
transition zone, listed in eq. ( l ) ,become infinite, so a special
treatment is needed at this point. To avoid this difficulty, we
allow for a small discontinuity in shear modulus at the upper
boundary of the transition zone (PREM's ICB). For example,
one can take ps = 0 just outside the ICB, at a h/20 or so,
where a is the radius of the ICB and h is the thickness of the
transition zone. A fitted polynomial using this point just above
the ICB and another point at the lower boundary of the
transition zone will give a small non-zero psjust below the ICB.
The procedure for modifying the PREM inner core model
to have a mushy transition zone of thickness h can be outlined
as follows.
Table 3. Rigidity pt (10" kg m 5 ~ ' ) of the transition zone.
h: 1-5 km; ,f: 0.0-0.5; 4 = 1. r l = u - 11, the lower boundary of the
transition zone; r2 = u - h:2, the mid-point in the transition zone:
r3 = a , the upper boundary of the transition zone. (cr = 1221.5 km, the
radius of the ICB).
J=O.(I
r1
X
7-2
r3
1.23898 1.23909 1.23920 1.23931 1.23942
1.27159 1.27165 1.27170 1.27176 1.27181
1.30420 1.30420 1.30420 1.30420 1.30420
r\h (km)
1
1.41093
1.37759
1.34425
2
1.41121
1.37773
1.34425
1
1.25416
1.22453
1.19489
2
1.25441
1.22465
1.19489
1
1.09739
1.07146
1.04553
.f =0.3
2
3
4
0
1.09761 1.09782 1.09803 1.09824
1.07157 1.07167 1.07178 1.07189
1.04553 1.04553 1.04553 1.04553
T:I
4
1.56862
1.53111
1.49361
5
1.56892
1.53127
1.49361
1
3
1.41148
1.37787
1.31425
f =0.2
4
1.41175
1.37800
1.34425
1.41203
1.37814
1.34425
3
1.25463
1.22477
1.19489
4
1.25489
1.22489
1.19489
5
1.25314
1.22501
1.19489
2
f'=o.
TI
7-2
'r:3
r\h (km)
TI
l.2
I.3
r\h (km)
r1
r2
r3
5
f =0.4
r\h (km)
T1
7-2
+
PREM's ICB).
r\h (km)
1
2
3
4
5
r1
12.7642 12.7647 12.7652 12.7657 12.7663
po
~2
12.4635 12.4637 12.4640 12.4643 12.4645
rg
12.1628 12.1628 12.1628 12.1628 12.1682
3
1.56801 1.56831
1.53081 1.53096
1.49361 1.49361
r2
(1) Find values of the density, rigidity and Lam&parameter
at the lower boundary of the transition zone a - h using the
profiles of these properties of the inner core.
(2) Find values of the density and LamC's parameter just
above the upper boundary of the transition zone (PREM's
ICB) using the profiles of these properties of the liquid core
(rigidity vanishes at this point).
(3) Use a cubic spline technique to determine polynomials
for these properties in the transition zone (note that to obtain
a small non-zero ps at the upper boundary of the transition
zone, we choose the second point for interpolating ps to be
a h/20).
(4) With ps computed at any point in the transition zone
by the corresponding fitted polynomial, eq. (21) can then
be used to obtain the value of pt at that point for any chosen
f and 4.
Table2. Density po (lo3kgm-3) and Lame parameter I
(10" kg m - ' s-*) of the transition zone. h: 1-5 km (for any chosen .f
and #). tl= a - h, the lower boundary of the transition zone;
rz = a - h/2, the mid-point of the transition zone; rJ = a, the upper
boundary of the transition zone, where a = 1221.5 km (the radius of
1
1.56770
1.53066
1.49361
rl
+
Table 2 lists the values of the density and Lame parameter
at the lower and upper boundaries of the transition zone, as
well as at its mid-point. Since the density and Lame parameter
are functions of radius r (hence h) only, we tabulate them
separately from the rigidity. The latter is not only a function
of r, but also a function of f and 4. We have chosen the
thickness h to vary from 1 to 5 km and 4 = 1. Table 3
displays the values of pt as f varies from 0.0 to 0.5, h varies
from 1 to 5 km and 4 = 1. Fig. 1 shows, for a transition zone
(km)
r\h
T3
1
0.94062
0.91839
0.89617
2
0.94080
0.91849
0.89617
3
0.94099
0.91858
0.89617
4
0.94117
0.91867
0.89617
5
0.94135
0.91876
0.89617
f =0.5
r\h (km)
7.1
T2
7-3
1
2
3
0.78385 0.78400 0.78416
U.7ti599 U.76541 U.76548
0.74681 0.74681 0.74681
4
5
0.78431 0.78446
0.76556: 0.76563
0.74681 0.74681
of 5 km thickness, changes of the rigidity at the upper bound
(r = a ) , at the mid-point ( r = a - h / 2 ) and at the lower
bound (r = a - h) of the transition zone versus changes of the
fluid fraction f from 0 to 0.5. The rigidity at f = 0.0 represents
the PREM's values. The changes for the zone thickness of 3
and 1 km have a very similar pattern to that of 5 km. It is
obvious that, owing to the intrusion of fluid in the transition
zone, the rigidity of the zone is reduced. Softening of the
transition zone then in turn reduces the effective elastic
restoring force a t the inner core boundary, hence the Slichter
eigenperiods should be slightly lengthened.
LOAD LOVE NUMBERS AT THE UPPER
BOUND O F THE TRANSITION ZONE
4
System (1) can be integrated numerically by a Runge-Kutta
method through the inner core to obtain solutions at the ICB,
and through the mantle to obtain solutions at the surface of
the Earth. For the inner core in this study, due to the presence
of a mushy zone at the top, we integrate first the inner core
0 1997 RAS, G J l 131, 607-617
Mushy transition zone
1.7
I
I
I
I
I
I
61 1
I
1.5
1.3
1.1
0.B
-(a)
-
Lower bound, r = a
-
h
Zone thickness h = 5 km
I
0.7
1.7
1.5
1.3
r -
I
I
I
(b)
-
I
-
1.1
0.9
-
Middle of the zone, r = a - h/2
-
0.7
1.7
1.5
-
1.3
-
1.1
-
0.9 - ( c )
0.7
I
I
I
I
I
-
-
-
-
-
-
Upper bound, r = a
Zone thickness h = 5 km
I
I
I
I
7
Figure 1. Changes of the rigidity (a) at the lower bound (r = u - h), (b) at the middle point ( r = u - h/2), and (c) at the upper bound (r = a) of the
transition zone versus changes of the fluid fraction f from 0 to 0.5. The rigidities at f = 0.0 represent the PREM values.
governing differential equations from the centre of the Earth
to the lower boundary of the transition zone, and then continue
across the transition zone to reach the upper boundary-the
ICB. In the transition zone, the density, rigidity and Lamb
parameter of the inner core are replaced by the above modified
ones. At the lower boundary of the transition zone, we need
not do any extra work since this is a boundary where the
density and elastic parameters change smoothly. Finally, a set
of internal load Love numbers can be found at the upper
boundary of the transition layer, where the general solid/liquid
continuity conditions apply.
Since the geocentre is a regular singular point of the sixthorder differential system ( l ) , only three of the six fundamental
solutions are regular there, and care must be taken in selecting
the starting values of these solutions at the geocentre. To start
integration, Pekeris ( 1966) suggested using a power-series
expansion near the geocentre, whereas Takeuchi & Saito (1971)
used analytic solutions in a homogeneous sphere of small
radius. Crossley ( 1975b) introduced a variable transformation
to reduce the singularity from second order to first order, and
showed how to select solutions that are finite at the geocentre
(also see Haardeng-Pedersen 1975). Wu & Rochester (1994)
give detailed starting values at or near the geocentre for
Crossley's method or Pekeris' method, respectively. For the
sixth-order differential system here, either method yields the
same load Love numbers.
0 1997 RAS, GJI 131, 607-617
Now, for convenience, we define the dimensionless radius
r
R'
x=-
where R is the radius of the Earth. By defining the dimensionless AJP variables as jis,the dimensioned yis are expressed
in terms of these dimensionless variables according to the
following relation:
(24)
where Sij= 1 if i = j , and S i j = 0 otherwise. Then the fundamental solutions at the geocentre can be represented as the
following, which are equivalent to those of Crossley (1975b)
except for the presence of R here, or that of Wu & Rochester
(1994):
jj 1--
~ ~ n - +A'x"+'+
l
j 2 = BXn-2
...
+ (B'x" + ." ,
j 3 = C x " - l + C ' x " + l + ...,
+ ... ,
j 5= EX" + ~ E ' x "+
+ ~... ,
+
j4
=D X " - ~ [D'x~
jj6 = ,,Fx"-' + y y F ' x n + l
+ .,.,
612
Z. R . Peng
where
boundary, the solutions
y(u-)
(32)
(33)
where p(0) and po(0) are the rigidity and density of the inner
core at the geocentre, a is the radius of the ICB, and p o ( a + )
the density of the liquid core just above the ICB. Note that
hereafter for convenience in writing, we will drop the overbar
on the dimensionless yis. Whenever the dimensioned yis arise,
they will be referred to explicitly.
Of the 12 coefficients listed in eqs (25)-(30), only three are
independent. We choose C, E and A‘ as the independent
coefficients which will enable the case n=O (purely radial
deformation) to be handled easily-no separate treatment is
needed. All the other coefficients are expressed in terms of
these three as follows:
= C-y ‘ ” ( a - ) +
EJ”2’(a-)+
A ’-y ‘ 3 y L )
must satisfy the necessary boundary conditions, namely the
continuity of normal displacement 8 * u , incremental gravitational potential Vl, normal stress 8 . t , and gravitational flux
8 * [ V V, - 4zGpOu]; here T is the additional stress due to the
disturbance of deformation.
By the definitions (5) and (6), the first two boundary
conditions mean that y , and j 5 are continuous across the ICB.
Note that for simplicity, the notation for the j i s adopted here
for calculating the load Love numbers will not explicitly
include their dependence on frequency and on azimuthal order
number rn, nor their dependence on degree n.
Before proceeding further, it is necessary to look at the
expression of the radial component of the normal stress
(denoted T ~ , )at the liquid side of the ICB. First, the reduced
pressure x may be represented by the superposition of a radial
function and the standard spherical harmonics
A=nC,
(49)
B = 2n(n - l)[C,
Referring to eqs (7) and (13) of Section 2, the normal stress in
the liquid core can be written as
D = 2(n - l)[C,
F
(48)
= n(E - y C ) ,
z, = rlv-u = -po(u’go
+ x + v,)
m
c
= -Po
+ (&
2mwR + (3 -
D‘ = A‘ + nC’,
1
[(n
2(2n 3)
+
~
(50)
At the point just above the ICB, the normal stress expressed
in AJP notations is
(39)
B‘ = 420‘ - qIC’,
E‘=
[-go(r)Yi(r)+X~(r)+Y,(r)lY::.
n=/ml
(40)
+ 3)A’- n(n + l)C’],
F’= ( n + 2)E’- A’,
(41)
(42)
(43)
c y2(a+)Y,”
m
1v-u=
n=lml
As defined in eq. (24), all AJP field variables of the inner core
in the boundary conditions can also be expressed in terms of
the corresponding dimensionless quantities. Therefore, with
dimensioned quantities replaced by dimensionless ones, the
condition of a continuous normal stress across the ICB,
y z ( a + )= y z ( a - ) , can then be simplified as
(44)
(45)
q2 = ( n
+ 3)-4 - 4 ,
P;
where a,, and Po are the compressional- and shear-wave
velocities at the centre of the Earth, respectively. If one chooses
to start the integration at a point away from the geocentre,
Wu & Rochester (1994) demonstrated that it is adequate to
keep only two terms in the above power expansion, provided
that the starting point is very close to, for example a few
kilometres from, the centre.
The general solutions in terms of these three independent
constants have the form
-y = Cy‘”
-
where the fully dimensioned quantity x r ( a + )is the radial part
of the potential x just above the ICB.
For an inviscid liquid core, the shear stress vanishes at the
ICB, i.e. at
+ Ey‘”
- + A ’-Y ‘ ~ ’ ,
(47)
where -y(k) are the fundamental solutions. At the inner core
y,(a-)=O.
(53)
Finally, the continuity of the gravitational flux across the ICB
gives
(54)
Note that (dy5/dr),=a+ of the liquid core remains a fully
dimensioned quantity.
Eqs (52), (53) and (54) are seen to depend on xy and dy,/dr
of the liquid side of the ICB only. Guided by this dependence,
the internal load Love numbers can be defined in such a
0 1997 RAS, G J I 131, 607-617
Mushy transition zone
way that
(55)
where the load Love numbers h, and h, are the solutions y ,
and y5 when
(57)
x%+ 1 = 0
a priori. But again, one can take advantage of the natural
character of the boundary conditions on x and V,, and relax
the latter requirement by adding necessary surface integrals
containing these boundary conditions to the resulting Galerkin
equations.
Suppose the two equations (19) and (20) are written in a
compact form:
Cx=O,
(66)
where C is constructed from the linear operators in these two
equations, i.e.
L = 0-1
and kl and k5 are those solutions when
(67)
L2IT
and
9
_x=(xV,IT.
(68)
At the outer core boundaries,
With the above conditions, the coefficients C, E and A' can be
obtained, and hence the internal load Love numbers:
The internal load Love numbers at the CMB can be obtained
with the same procedure as that at the ICB (Rochester & Peng
1993), subject to the boundary conditions at the free surface
of the Earth
=0 9
y,(R - 1 = 0
2
(61)
(62)
and similar boundary conditions as eqs (53), (52) and (54) at
the CMB, but with ~ : ( a + and
) go@) replaced by ~ r ( b - and
)
go@). The radial displacement and additional gravitational
fields at the CMB are now expressed in terms of the mantle
load Love numbers h,(b), h,(b), k,(b) and k,(b) at the CMB,
and the quantities ~ : ( b - )and ( r dy,/dr),=,- just below the
CMB:
5
N U M E R I C A L RESULTS: EIGENPERIODS
We now use a Galerkin method to solve the pair of liquidcore-governing equations (19) and (20), subject to appropriate
continuity conditions at the outer core boundaries. This
weighted residual procedure allows the trial functions to satisfy
the governing partial differential equations in a weighted mean
sense. The difference between the Galerkin method and other
residual methods is that the former takes the trial functions
themselves as the weighting functions (Zienkiewicz & Morgan
1982). The resulting expressions from substituting the trial
functions into the governing equations are orthogonal to the
respective set of trial functions. In principle, one can require
the trial functions, x and V, in the case of the two-potential
description, to satisfy the associated boundary conditions
0 1997 RAS, GJI 131, 607-617
x
are required to satisfy continuity conditions of normal -displacement, normal stress,
incremental gravitational potential and gravitational flux.
These conditions can be represented as
Kx=_0,
y,(R-
613
(69)
Table 4. Eigenperiods (hr) of the Slichter modes for the PREM model
with a mushy ICB. j,the fluid fraction; h, the thickness of the transition
zone. L = 5, N = 5 and Q = 1.
f\h (km)
0.0
0.1
0.2
0.3
0.4
0.5
1
5.42694
5.42695
5.42696
5.42697
5.42698
5.42698
non-rotating (PREM: 5.42047)
2
3
4
5
5.43342 5.43992 5.44643 5.45296
5.43344 5.43994 5.44647 5.45301
5.43346 5.43997 5.44650 5.45305
5.43347 5.44000 5.44654 5.45310
5.43349 5.44003 5.44658 5.45315
5.43351 5.44006 5.44662 5.45319
rotating, m=O (PREM: 5.30868)
f\h (km)
1
2
3
4
5
0.0
5.31475 5.32081 5.32688 5.33297 5.33907
0.1
5.31475 5.32082 5.32691 5.33301 5.33912
0.2
5.31476 5.32084 5.32693 5.33304 5.33916
0.3
5.31477 5.32086 5.32696 5.33307 5.33920
0.4
5.31478 5.32088 5.32698 5.33311 5.33925
0.5
5.31479 5.32089 5.32701 5.33314 5.33929
rotating, m=l (PREM: 4.76424)
f\h (km)
1
2
3
4
5
0.0
4.76915 4.77405 4.77897 4.78391 4.78885
0.1
4.76915 4.77407 4.77900 4.78393 4.78888
0.2
4.76916 4.77408 4.77902 4.78396 4.78891
0.3
4.76917 4.77410 4.77904 4.78399 4.78895
0.4
4.76918 4.77411 4.77906 4.78402 4.78899
0.5
4.76918 4.77413 4.77908 4.78405 4.78902
rotating, m=-1 (PREM: 5.97827)
f\h (km)
1
2
3
4
5
0.0
5.98594 5.99359 6.00126 6.00895 6.01666
0.1
5.98595 5.99361 6.00129 6.00900 6.01672
0.2
5.98596 5.99363 6.00133 6.00904 6.01677
0.3
5.98598 5.99366 6.00136 6.00908 6.01683
0.4
5.98598 5.99368 6.00139 6.00913 6.01688
0.5
5.98600 5.99370 6.00143 6.00917 6.01694
614
2. R. Peng
where, in the associated boundary conditions, K are also linear
operators:
K=(KlK2)T.
(70)
Then the compact form of the Galerkin equations incorporating
the natural boundary conditions is
where Y* are independent weighting functions,
Y
= (YlY2)T.
(72)
We construct the trial functions for
)I
and V, as follows:
(73)
(74)
where
k = p + 2(L- l ) ( n - l),
n = 1,2, ... , N ,
forl<p<L-l,
t=p-(L+l),
z=a,
t = p - ( L - l),
z = b, for L < p
< 2L-
2.
Note that the choice of scaling factor z as above ensures that
r/z is always less than 1, which then prevents the value of the
radial trial function from being too large.
5.342
I
Upon using the divergence theorem and choosing the
weighting functions accordingly, the Galerkin equation set 71 1
can then be written as
C;!?=o,
GI,,, and G,,,, are the i.jth components of the left-hand side
of equation set (71).
The Slichter eigenperiods for a suite of transition-zone
models are listed in Table 4. To obtain these results, we have
set the parameter 4 of eq. (21) to be 1. We also examined the
cases where 4 equals 2 and 3, and the results show that the
eigenperiods are affected very little, only about 0.003 per cent
difference from the eigenperiods in Table 4. In other words,
the Slichter modes seem to be insensitive to this parameter.
The effect of adding the mushy transition zone at the ICB is
to lengthen the eigenperiods of the Slichter modes, as would
be expected, since the effective density jump is slightly reduced
and the rigidity of the ICB is also reduced. The results shown
in Table 4 show that, with the thickness of the mushy zone up
to 5 km and the fluid content up to 50 per cent, the eigenperiods
of the Slichter modes are altered slightly, with 0.604, 0.577,
0.567 and 0.647 per cent increments for non-rotating, rotatingaxial, prograde and retrograde modes, respectively. This outcome reflects the nature of the Slichter modes, with their main
I
I
(a)
I
I
-
-
5.338
5.330
75)
where
Rotating, m=O (PREM 5.30868 hr)
Fluid fraction f = 10%
5.324
5.318
5.312
5.342
I
5.338
(b)
5.330
I
I
I
5.336
5.330
-
I
-
Rotating, m=O (PREY 5.30868 hr)
Fluid fraction f = 30%
I
- (c)
-
Rotating, m=O (PREY 5.30888hr)
Fluid fraction f = 50%
I
-
-
-
5.324
5.318
5.312
'
0.0
I
1.0
I
2.0
I
3.0
I
4.0
I
5.0
8.0
Thickness of the transition zone (km)
Figure 2. Change of the central (m= 0) period of the Slichter modes caused by variation of the thickness of the transition zone. The fluid fraction
is fixed at (a) 10 per cent, (b) 30 per cent, and (c) 50 per cent.
0 1997 RAS, GJI 131, 607-617
Mushy transition zone
4.795
4.789
4.763
I
E
(a)
I
I
I
Rotating, m = l (PREM 4.76424 hr)
Fluid fraction f = 10%
I
615
I
-
4.777
4.771
4.708
-
4.783
-
4.777
4.771
-
(b) Rotating, m = l (PREM 4.76424 hr)
I
Fluid fraction f = 30%
-
-
-
-
I
4.789 -(c)
4.703
4.777
4.771
4.765
-
1
t
I-
I
0.0
-
R o t a t i , m = l (PREM 4.76424 hr)
-
Fluid fraction f = 50%
-
I
1.0
I
2.0
I
I
3.0
-
I
5.0
4.0
1
6.0
Thickness of the transition zone (krn)
Figure Change o :he retrograde (m = 1) period of the Slichter modes caused by variations in the thickness of the transition zone. The fluid
fraction is fixed at (a) 10 per cent, (b) 30 per cent, and (c) 50 per cent.
restoring force being gravitation (the elasticity of the inner
core plays only a minor role in sustaining the oscillation). On
the other hand, the smallness of the eigenperiod increment is
also due to the limited thickness of the transition layer. It is
evident from Table 4 that the eigenperiod change with respect
to the change of the thickness of the mushy zone is much
larger than that with respect to the change of the fluid fraction.
This latter fact implies that the thicker the transition zone, the
more it absorbs vibration energy. The trend of the change is
better illustrated in Figs 2-4 for the Slichter triplet (rotating,
m = 0, m = 1 and m = - l), where we plotted the eigenperiods
of the vibration versus the zone thickness (which varies from
1 to 5 km). For each member of the triplet, we only chose to
illustrate the variation relation of periods and zone thickness
for three fixed fluid fractions: 10, 30 and 50 per cent.
Note that although the density profile of the transition zone
has been slightly altered from that of the PREM inner core,
which may result in a small change in the mass of the inner
core, it can be shown that the effect of this change on the
eigenperiods of Slichter modes is negligible. For the purpose
of illustration, we employ a simpler earth model that possesses
a non-rotating, spherical rigid mantle and inner core, and a
homogeneous, inviscid, incompressible liquid outer core. Peng
(1990) showed, in this case, that the vibration frequency
0 1997 RAS, G J I 131, 607-617
where
(79)
E=fi(-) b3 + 2a3
2p,,
b3 - a 3
’
and a and b are the radii of the ICB and CMB, respectively.
It is clear that the frequency of the oscillation is only related
to the density contrast at the inner core boundary and the
locations of the inner and outer core boundaries. In other
words, the latter are the most important parameters for the
determination of the eigenperiods of the Slichter modes. Note
that, although the result in eq. (78) is derived for a special
case, the conclusion reached there should be applicable to the
present study due to the fact that the mass change of the inner
core, caused by introducing a transition zone, is very small.
Therefore, it seems safe to assume that the effect of the change
in mass of the inner core on the eigenperiods of the Slichter
modes is negligible here.
6 CONCLUSIONS
This is the first time that the effect of a mushy inner core
boundary on the Slichter modes has been investigated. We
have modelled the transition zone with different thicknesses
(1-5 km) and different liquid contents (0-50 per cent), which
616
2. R. Peng
6.020
,
8.012
-
8.004
-
5.996
-
5.988
-
I
6.004
5.996
5.988
I
(a) Rotating, m=-1 (PREM 5.97827 hr)
Fluid fraction f = 10%
I
6.020
6.012 -(c)
-
5.988
-
5.980
-
-
-
I
I
I
I
-
i
I
I
Rotating, m=-1 (PREM 5.97827 hr)
Fluid fraction f = 50%
I
I
-
-
-
-
-
-
-
I
I
-
I
5.998
I
-
1
-
-
-
(b) Rotating, m=-1 (PREM 5.97827 hr)
Fluid fraction f = 30%
8.004
I
-
I
6.012
I
I
I
I
I
Figure 4. Change of the prograde (m = - 1) period of the Slichter modes caused by variations in the thickness of the transition zone. The fluid
fraction is fixed at (a) 10 per cent, (b) 30 per cent, and (c) 50 per cent.
include most possible states of the zone. With the presence of
the mushy zone, the rigidity of the inner core boundary and
the density jump across it are reduced. These effects in turn
reduce the effective gravitational restoring force, which results
in an increase in the vibration periods. The results obtained
here suggest that the influence of the mushy inner core
boundary on the Slichter triplet is relatively small. The
increments in eigenperiods are about 0.577, 0.467 and 0.647
per cent for a realistic earth model such as PREM, with a
mushy zone 5 km in thickness and 50 per cent in fluid content.
However, though small, the effects of the mushy inner core
boundary on the Slichter modes are comparable to, or larger
than, some effects previously studied, such as elasticity of the
mantle (maximum 0.126 per cent; Peng 1995), non-neutral
stratification of the liquid core (maximum 0.126 per cent, Peng
1995), and the ellipticity and centrifugal potential (maximum
0.90 per cent, Dahlen & Sailor 1979; maximum 0.097 per cent,
Wu & Rochester 1994; maximum 0.134 per cent, Peng &
Rochester 1997). Therefore, the possibility of a mushy inner
core boundary could significantly influence mode identification
and confirmation. It seems reasonable to say that the central
period of 5.3 hr is the Lower bound of the Slichter modes for
PREM, in so far as a softer inner core boundary is a sound
and practical theory.
ACKNOWLEDGMENTS
I am grateful to Professor M. G. Rochester of Memorial
University of Newfoundland for his guidance and comments
during this study. The research was supported partly by the
School of Graduate Studies of Memorial University of
Newfoundland, and partly by a NSERC research grant
(A-1 182) held by Professor Rochester. The computation was
performed using facilities of the Computing Services of
Memorial University of Newfoundland. I also thank reviewers
of the manuscript for their valuable comments and suggestions.
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