Geophys. J . Int. (1996) 127, F5-F9
FAST-TRACK PAPER
Significance of the fundamental mantle rotational relaxation mode in
polar wander simulations
L. L. A. Vermeersen* and R. Sabadini
Dipartimento di Scienze della Terra. Sezione Geojisica, Universitci degli Studi di Milano, Via L. Cicognara 7,20129 Milano, Italy
Accepted 1996 September 5. Received 1996 August 5; in original form 1996 June 5
SUMMARY
In some studies on glacially induced true polar wander (TPW), the tidal-effective
relaxation of the fundamental mantle mode (MO) is lacking. We show that this is
caused by the deletion of the Chandler wobble in an early stage of the theory
development to facilitate the retrieval of the rotational relaxation modes. We derive an
analytical approximation formula for the MO rotational relaxation mode (including
the Chandler wobble), which can be of practical value for TPW simulations with
realistically stratified earth models. However, we point out that the contribution of the
MO rotational relaxation mode has, to a high approximation, the same effect on secular
TPW as the contribution from an elastic term in models that do not have the MO
rotational relaxation mode. The two model approaches lead to the same polar wander
results whenever the Chandler wobble is filtered from models in which the MO
rotational relaxation mode is retained.
Key words: Chandler wobble, mantle viscosity, TPW (true polar wander).
INTRODUCTION
A number of authors have reported during the past decade
that studies on glacially induced TPW allow in general for
multiple solutions for the lower-mantle viscosity if the TPW
is known and all the other rheological, elastic and constitutional parameters are fixed [Fig. 20 of Yuen et al. (1986),
Fig. 5 of Spada et al. (1992), and, most recently, Fig. 1 of
Milne & Mitrovica ( 1996)l. These multiple-branch solutions
are also commonly found in other geophysical signatures
related to glacially induced solid-earth deformations such as
postglacial rebound, free-air gravity anomalies, and changes in
the non-tidal acceleration of the Earth. In a recent study
(Peltier & Jiang 1996, Fig. 7), such multi-branch solutions for
the lower-mantle viscosity are conspicuously absent; only
solutions in which the lower mantle is slightly more viscous
than the upper mantle are allowed.
Although the theories described by Sabadini, Yuen & Boschi
(1982) and Wu & Peltier (1984) appear to have a number of
differences, Sabadini, Yuen & Boschi (1984) have shown that
the formulations are equivalent to some extent. The proof of
the equivalence of eqs (16) and (17) of Sabadini et al. (1984)
for the secular rotation term is an important result in this
respect. Another result mentioned in Sabadini et al. (1984) is
*Now at: Department of Geodetic Science, University Stuttgart,
Keplerstrasse 11, 70174 Stuttgart, Germany.
01996 RAS
that each of the load relaxation modes has a corresponding
rotational relaxation mode. This correspondence remains an
important issue, since in Wu & Peltier’s (1984) theoretical
development one of the corresponding modes, the MO
rotational relaxation mode, is lacking (see also Table 1 of
Sabadini et al. 1984).
Peltier & Jiang (1996) extend the analysis of Wu & Peltier
(1984) to consider a basic state in which the two principal
equatorial moments of inertia are distinct. The extension is,
however, unimportant in predictions of TPW (Peltier & Jiang
1996). More importantly, the extended theory retains the
Chandler wobble filtering procedure of Wu & Peltier (1984)
and this raises the possibility that the different behaviour of
the TPW curves in Peltier & Jiang (1996) is due to the neglect
of the MO rotational relaxation mode. We show explicitly that
the absence of the MO rotational relaxation mode cannot be
the cause of this difference in TPW-rate behaviour. As support
for this result, note that the TPW predictions of Milne &
Mitrovica (1996; Fig. l), which agree with earlier analyses (e.g.
Spada et al. 1992), are based on theory which is equivalent to
Wu & Peltier’s (1984) approach.
The present paper is structured as follows. First, we show
that Wu & Peltier’s (1984) approach to deleting the Chandler
wobble acts to remove the MO rotational mode. We then
proceed by deriving a new analytical approximation formula
for the MO rotational relaxation mode, which incorporates
the Chandler wobble frequency for a stratified earth as the
F5
F6
L. L. A. Vermeersen and R. Sabadini
imaginary part. It will be shown, using numerical tests, that
this approximation formula is extremely accurate. Finally, we
show that the model approaches used in Sabadini et al. (1982)
and Wu & Peltier (1984) lead to the same secular TPW results.
OMISSION O F THE MO ROTATIONAL
RELAXATION M O D E
We start with eq. (64) of Wu & Peltier (1984), reading in our
notation
with %(s) the complex-valued Laplace-transformed polar wander
(s is the Laplace variable, which in general is complex-valued),
i=
6, the Chandler wobble frequency for a rigid earth,
fDL the complex-valued Lapiace-transformed forcing, and M
the total number of load relaxation modes j with (negative)
inverse times s j . These sj are real-valued. The auxiliary variable
x j is defined by
fl,
in which kj' is the tidal-effective Love number associated with
modej, and k; the fluid tidal-effective Love number. Note that
the terms kj' have the same dimension as s j , while the fluid
Love number k; is dimensionless, so that the terms xi have
the same dimension as sj [note that our parameters kj' are the
same as the parameters t j used in Wu & Peltier (1984)l.
Inverse Laplace transformation of ( 1) requires finding the
complex-valued roots of the denominator on the right-hand
side of (1). At this stage, Wu & Peltier (1984) make the point
that the unity term in the denominator of (1) may be neglected.
This is only correct, however, if the imaginary parts of the
roots have the same magnitude or are much smaller than the
magnitude of the real parts. This is indeed the case for M - 1
roots, but it is not true for one root which has a much larger
imaginary part than real part. This root turns out to be the
rotational root that gives the relaxation of the fundamental
mantle mode as the real part and the Chandler wobble as the
imaginary part.
The omission of this rotational root in Wu & Peltier (1984)
is more apparent when we rewrite the term inside the square
brackets of ( 1 ) as
j=1
It is clear that if the first term of the numerator on the
right-hand side of (3) were deleted, the numerator would be
reduced from an expression of order M to an expression of
order M - 1. This implies that one of the M load relaxation
modes would have no rotational counterpart. Neglecting
the first term of the numerator of (3) on the right-hand side
of the equation is correct for the M - 1 roots for which the
imaginary part is orders of magnitude smaller than the real
part, as outlined by Wu & Peltier (1984). For these roots, the
approximation
(4)
k=l
j#k
is valid, being a purely real expression resulting in M - 1 real
roots. These real roots constitute the M - 1 rotational inverse
relaxation times associated with all modes except the MO mode.
For a root with a large imaginary value, comparable in
strength with the variables x j given by (2), an argument that
the first term of the numerator on the right-hand sight of (3)
is negligible with respect to the second term of this numerator
is no longer valid. In fact, such a complex-valued root leads
to a real part, being the MO rotational relaxation mode, which
has the same order of magnitude as the other (real) M - 1 roots.
This complex-valued mode, with the Chandler wobble frequency for a stratified earth as the imaginary part, has thus to
be derived from the complete complex-valued equation
M
M
M
We will show in the next section that a highly accurate
analytical approximation formula can be derived from (5)
for the MO rotational relaxation mode. This approximation
facilitates polar wander simulations on earth models with a
large number of layers. Indeed, it is well known that complex
root-finding procedures are numerically more difficult to apply
and are less reliable than real-valued root-finding procedures.
When an analytical formula can be obtained for the only
complex root that has a non-negligible imaginary part, one
can use root-finding procedures for real numbers using (4)
instead of complex numbers using (5) in TPW calculations.
Before deriving this formula, an example is given to illustrate
the foregoing remarks. Table 1 gives the values for the densities
and rigidities for an earth model consisting of an elastic
lithosphere, viscoelastic shallow mantle, transition zone and
lower mantle, and inviscid fluid core. The mantle has a uniform
viscosity of loz1Pa s. In Table 2, the nine inverse load relaxation times sj and the nine rotational relaxation roots aj are
given for the five-layer earth model of Table 1. The complex
rotational relaxation roots aj are determined by applying a
complex root-finding procedure to (5). It is clear from Table 2
that eight roots have imaginary values that are negligible in
strength, and that one root has a large imaginary value. This
large imaginary value represents the Chandler wobble. The
real value of this root is the MO rotational relaxation mode.
This mode is not negligible at all; on the contrary, it is often
the strongest relaxation mode, as is illustrated by the last two
columns of Table 2. In Table 4 of Wu & Peltier (1984), where
nine load relaxation and eight rotational relaxation roots are
given, it is this MO rotational relaxation mode that is lacking.
The small imaginary values of the other rotational relaxation
roots in Table 2 indicate that these modes are also characterized
by wobbles. One can easily prove from (5) that the imaginary
Table 1. Density and rigidity values for the five-layer model.
radial distance (km)
density (kg m-3)
rigidity (Pa)
6371-6250
6250-5951
5951-5701
5701-3480
3480-0
3184
3434
3857
4878
10 932
6.02 x lo1'
7.27 x 10''
1.06 x 10"
2.19 x 10"
0
0 1996 RAS, GJI 127, F5-F9
Fundamental mantle rotational relaxation mode
F7
Table 2. Inverse load relaxation times sj, inverse rotational relaxation times aj, load Love
numbers kj" and tidal-effective Love numbers kj' for the nine relaxation modes of the five-layer
model of Table 1, with a uniform mantle viscosity of 102' Pas. The Lam6 parameter A is infinite.
The labelling of the modes agrees with the labelling of Table 4 of Wu & Peltier (1984). Note
that the MO mode is the strongest mode for both load relaxation and tidal-effective relaxation.
mode j
M2
M1
LO
co
MO
Tl
T2
T3
T4
-sj
(kyr-')
2.90 x
1.29 x
1.09 x lo-'
4.50 x lo-'
2.02
2.48
2.84
3.56
4.00
--a,
(kyr-l)
7.32 x
-9.64 x 10-loi
1.06 x
- 1.73 x 10-6i
1.19 x lo-' - 1.89 x 10-6i
1.02 - 1.50 x 10-4i
1.95 - 5.10 x 1 0 3 i
2.38-2.19 x 10-5i
2.62 - 2.45 x l O W 5 i
3.47 - 2.60 x lO-'i
3.89 - 2.94 x 10-5i
parts of these modes are not equal to zero, thus excluding
the possibility that these imaginary values are the result of
numerical inaccuracies. This is interesting from a physical
point of view, although the amplitudes of these wobbles are
too small to be of any immediate significance.
Filtering the Chandler wobble out after the rotational relaxation roots have been found is not the same as deleting the
mode in which the Chandler wobble will appear before (5) is
solved. One might thus think that the omission of the MO
rotational relaxation mode has major consequences for the
TPW simulations. However, eq. (79) of Wu & Peltier (1984)
contains the extra term Dlf( t ) inside the square brackets, which
is only created when the number of rotational relaxation modes
is one less than the number of load relaxation modes. It will be
shown that this term contains approximately the same contribution as is found from the relaxation of the MO rotational
relaxation mode after the Chandler wobble is filtered out. This
equivalence will be pointed out and discussed after the analytical
formula for the MO rotational relaxation mode is derived.
ANALYTICAL F O R M U L A FOR T H E MO
ROTATIONAL RELAXATION MODE
As shown in the last section, the numerator on the right-hand
side of (3) has M - 1 solutions s = aj ( j = 1, ... ,M - 1) for
which the imaginary part can be neglected. These M - 1
solutions can be found by applying a real-valued root-finder
procedure to (4). The root that contains the Chandler wobble
as the imaginary part and the MO rotational relaxation mode
as the real part must be solved from (5). If we split this root
s = aMOinto its real and imaginary parts as aMO= aR ia,, then
it is clear from, for example, Table 2 that JaII>> laRl and
(a,(>> lskl for all M load relaxation modes k.
With this, the first term of ( 5 ) can be approximated by
+
j= 1
k=l
while the second term of (5) can be approximated by
M
1
k=l
n
-1.45 x
-7.01 x
- 1.69 x
- 1.41 x
-3.32 x
-7.82 x
-2.80 x
-9.95 x
- 1.15 x
7.17 x
7.72 x
7.84 x
8.08 x
4.21 x
8.13 x
3.73 x
1.13 x
1.79 x
lo-'
lo-'
lo-'
lo-'
lo-'
lo-'
lo-'
lo-'
lo-'
lo-'
M
k= 1
and
(9)
k=l
k=l
j#k
Eq. (8) yields the expression for the imaginary part of the root
as
while (10) in (9) leads to
and this can be reduced to
M
From (10) and (12) we thus have as the Mth complex-valued
root of eq. (5)
The real part of this root gives the rotational inverse relaxation
time of the fundamental mantle mode MO, while the imaginary
part gives the Chandler wobble frequency cro of the stratified
model [compare also with Wu & Peltier's (1984) eq. (68)]:
M
.._
00%
c
xk,
k= 1
or, with (2):
For the five-layer model of Table 1, the complex root as given
by (13) has the value
aMO
= - 1.9467033513-t5096.9417% kyr-' ,
M
xk
kj' (kyr-')
kj" (kyr-')
(s-sj)
while from a complex root-finder applied to (5) with quadrupole precision (COMPLEX* 16) the root is determined as having
the value
j#k
k=l
k=l
j#k
The sum of (6) and (7) has terms which all contain either iM
or i'-'. Irrespective of the value of M , the terms must thus
obey the relations
0 1996 RAS, G J I 127, F5-F9
aMO =
+
- 1.9467033558 5096.942041 kyr-'.
The analytical formula (13) gives thus an extremely accurate
approximation of aMO.
F8
L. L. A. Vermeersen and R. Sabadini
UNIFICATION O F THE TWO APPROACHES
Formulas (13) and (15) are not only useful in the model
approach following the method of Sabadini et al. (1982), but
also prove helpful in establishing the equivalence with the
model approach of Wu & Peltier (1984) for secular TPW.
If we consider eq. (79) of Wu & Peltier (1984), then the
formulation of Sabadini et al. (1982) alters this equation by the
following four points (J. X. Mitrovica, personal communication).
(1) The term D, of Wu & Peltier’s eq. (79) becomes zero.
This term D, is an elastic term which arises from the first term
(being 1) on the right-hand side of Wu & Peltier’s eq. (75).
The term D, is a direct consequence of the fact that there is
one rotational relaxation mode less than the number of load
relaxation modes. This term of unity would be absent if there
were N rotational relaxation modes corresponding with the N
load relaxation modes inside the square brackets of the last
line in Wu & Peltier’s eq. (74).
(2) The term D 2becomes - ia,D, in Wu & Peltier’s eq. (79),
but at the same time the term D2is divided by the extra root
-aMO of our eq. (13) in Wu & Peltier’s eq. (80). As the
imaginary part of eq. (13) is orders of magnitude larger than
the real part, the effect on Wu & Peltier’s eq. (80) is that D, is
to a high approximation divided by -h0,so that the total
effect on Wu & Peltier’s eq. (79) is that the original term D 2
remains unchanged (note that our roots ai have the opposite
sign of Wu & Peltier’s corresponding roots A i ) .
(3) The terms Ei in Wu 8c Peltier’s eq. (79) become - iaoEi,
but at the same time the terms Eiin Wu & Peltier’s eq. (80) are
to a high approximation divided by the extra term - ia,, which
is again a consequence of the fact that the extra rotational
I
I
relaxation mode has an imaginary term in our eq. (13) being
orders of magnitude larger than the real part. The net effect is
thus, just as in point (2), that the terms Eiremain to a high
approximation unchanged in Wu & Peltier’s eq. (79).
(4) The final change in Wu & Peltier’s eq. (79) concerns the
addition of the extra term E,f* exp(a,,t). This term causes
the wobble. It turns out (J. X. Mitrovica, personal communication) that if this extra term is averaged over time, i.e. when
the wobble is filtered out, then the contribution that remains
is numerically equal to D, to a high approximation.
The net effect of points (1)-(4) is thus that the elastic term D 1
in Wu 8c Peltier’s (1984) theory contains the signal that is
contributed by Sabadini et al.’s (1982) MO rotational relaxation
mode when the Chandler wobble is filtered out. Together with
points (2) and (3) one thus can conclude that, to a high
approximation, the theoretical developments of Sabadini et al.
(1982) and Wu & Peltier (1984) lead to the same results for
secular TPW simulations.
A corollary of this result is that the discrepancy in
the glacially induced TPW results between Peltier & Jiang
(1996) and other authors, which was briefly discussed in the
Introduction, is due to the application of the theory.
TPW I N D U C E D BY PLEISTOCENE
GLACIAL CYCLES
Fig. 1 shows the TPW rate as a function of lower-mantle
viscosity for three cases: a five-layer earth model with
parameters given in Table 1, a 31-layer model and a 56-layer
model. All three models have volume-averaged properties of
PREM (Dziewonski & Anderson 1981) and a value for the
I
I
I
I
LOG,,( Lower mantle viscosity)
Figure 1. TPW velocity as a function of the lower-mantle viscosity. The horizontal band marks the error bounds of the present-day TPW datum
given by McCarthy & Luzum (1996).
0 1996 RAS, GJI 127, F S F 9
Fundamental mantle rotational relaxation mode
upper-mantle viscosity of 10” Pas. The ice model that has
been used is based on ICE-3G of Tushingham & Peltier (1991).
The ICE-3G model gives the deglaciation of the last Pleistocene
cycle by a set of decrements, starting at 18 kyr before present
and ending at 5 kyr before present. After 5 kyr before present
the ice loads remain constant until present. We have used
ICE-3G for times between 18 and 5 kyr before present, and
extended it with seven glacial sawtooth pre-cycles and a linear
glaciation phase, which ends at 18 kyr before present. Each
cycle is connected to its previous and following one, and the
end of the seventh cycle is the beginning of the linear glaciation
phase which ends at 18 kyr before present. The seven precycles each consist of a 90kyr linear growth phase and a
10 kyr linear decay phase. The minimum amount of ice is the
same as in ICE-3G at 5 kyr before present, while the maximum
amount of ice is the same as in ICE-3G at 18 kyr before
present. The TPW rates are determined 5 kyr after I C E 3 G
ends, i.e. the rates are determined at present.
The most recent observed value of the present-day TPW
comes from McCarthy & Luzum (1996). They determine a
mean TPW velocity during the period from 1899 to 1994 of
0.925 0.022 deg Myr-l. The area between the error bounds
of this TPW velocity datum is shaded in Fig. 1.
The form of the curves of Fig. 1, which exhibit a clear nonmonotonic behaviour, are consistent with independent calculations by Yuen et al. (1986), Spada et a/. (1992) and Milne &
Mitrovica (1996). They are, in contrast, in disagreement with
the recent predictions of Peltier & Jiang (1996). Fig. 1 illustrates several interesting points which we will discuss more
fully in a companion article (Vermeersen, Fournier & Sabadini
1996). As an example, the number of layers used to discretize
the viscoelastic earth model can influence the accuracy of the
predictions. However, the fact that the 31- and 56-layer models
virtually cover each other confirms the results on ‘saturated
limit’ behaviour for stratified models in Vermeersen & Sabadini
(1996b). Also, the curves of Fig. 1 show TPW rates which are
significantly higher than those of Spada et al. (1992). The
reason for this is that Spada et al. (1992) adopt a five-layer
fixed boundary contrast model rather than the volume-averaged models we use here (see also Vermeersen & Sabadini
1996b). In the case of lower-mantle viscosities lower than
lo2’ Pa s, our predictions are significantly higher than those
of Milne & Mitrovica (1996). This can be attributed to the
fact that we have employed incompressible rheologies to
produce Fig. 1, whereas Milne & Mitrovica (1996) have used
compressible rheologies (Vermeersen & Sabadini 1996a;
J. X. Mitrovica, personal communication).
Apart from these differences in modelling, there are a number
of reasons that make it difficult to establish a unique relationship between the viscosity of the lower mantle and TPW. For
example, uncertainties in lithospheric thickness and uppermantle viscosity will corrupt any such inference and, furthermore, it is highly questionable whether glacial processes are
the sole driver of the present-day TPW. For example, tectonic
processes such as subduction (e.g. Ricard, Sabadini & Spada
1992) and mountain building (e.g. Vermeersen et al. 1994) have
been shown to be potentially effective contributors to the
present-day TPW.
CONCLUDING REMARK
A first step towards unifying the approaches by Sabadini et al.
(1982) and Wu & Peltier (1984) was taken in Sabadini et a/.
0 1996 RAS, GJI 127, F5-F9
F9
(1984). The present paper completes this unification, in the
sense that the theories are shown to be equivalent to a high
approximation whenever the Chandler wobble is filtered out.
The complex-valued approximation formula (13) for the
strength of the MO rotational relaxation mode can be helpful
in models that follow the procedures outlined in Sabadini et at.
(1982), either with the Chandler wobble included or with the
Chandler wobble filtered out.
ACKNOWLEDGMENTS
This work was financially supported by the European Space
Agency by contract PERS/mp/4178 and by the Italian Space
Agency by grant AS1 95-RS-153. Alexandre Fournier is
acknowledged for assistance in producing the figure. We are
grateful to Jerry Mitrovica for many discussions, benchmark
comparisons, and a thorough review of the original manuscript.
Detlef Wolf is thanked for thoughtful remarks.
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