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For large-angle long-term true polar wander (TPW) there are currently two types of nonlinear methods which give approximated solutions: those assuming that the rotational axis coincides with the axis of maximum moment of inertia (MoI), which simplifies the Liouville equation, and those based on the quasi-fluid approximation, which approximates the Love number. Recent studies show that both can have a significant bias for certain models. Therefore, we still lack an (semi)analytical method which can give exact solutions for large-angle TPW for a model based on Maxwell rheology. This paper provides a method which analytically solves the MoI equation and adopts an extended iterative procedure introduced in Hu et al. (2017) to obtain a time-dependent solution. The new method can be used to simulate the effect of a remnant bulge or models in different hydrostatic states. We show the effect of the viscosity of the lithosphere on long-term, large-angle TPW. We also simulate models without hydrostatic equilibrium and show that the choice of the initial stress-free shape for the elastic (or highly viscous) lithosphere of a given model is as important as its thickness for obtaining a correct TPW behavior. The initial shape of the lithosphere can be an alternative explanation to mantle convection for the difference between the observed and model predicted flattening. Finally, it is concluded that based on the quasi-fluid approximation, TPW speed on Earth and Mars is underestimated, while the speed of the rotational axis approaching the end position on Venus is overestimated.
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Journal of Geophysical Research: Planets
A Full-Maxwell Approach for Large-Angle Polar
Wander of Viscoelastic Bodies
H. Hu1, W. van der Wal1, and L. L. A. Vermeersen1
1Department of Aerospace Engineering, Delft University of Technology, Delft, Netherlands
Abstract For large-angle long-term true polar wander (TPW) there are currently two types of nonlinear
methods which give approximated solutions: those assuming that the rotational axis coincides with the
axis of maximum moment of inertia (MoI), which simplifies the Liouville equation, and those based on the
quasi-fluid approximation, which approximates the Love number. Recent studies show that both can have
a significant bias for certain models. Therefore, we still lack an (semi)analytical method which can give
exact solutions for large-angle TPW for a model based on Maxwell rheology. This paper provides a method
which analytically solves the MoI equation and adopts an extended iterative procedure introduced in
Hu et al. (2017) to obtain a time-dependent solution. The new method can be used to simulate the effect
of a remnant bulge or models in different hydrostatic states. We show the effect of the viscosity of the
lithosphere on long-term, large-angle TPW. We also simulate models without hydrostatic equilibrium
and show that the choice of the initial stress-free shape for the elastic (or highly viscous) lithosphere of a
given model is as important as its thickness for obtaining a correct TPW behavior. The initial shape of the
lithosphere can be an alternative explanation to mantle convection for the difference between the observed
and model predicted flattening. Finally, it is concluded that based on the quasi-fluid approximation, TPW
speed on Earth and Mars is underestimated, while the speed of the rotational axis approaching the end
position on Venus is overestimated.
Plain Language Summary The North and South Poles of the Earth are slowly moving. This is
because a large mass can form on the Earth’s surface on geologic time scales, which changes how the
Earth rotates. Examples of large masses are ice sheets and large mountains. This phenomenon of moving
poles takes place on many planets or moons and is referred to as polar wander. Polar wander can be the
explanation for why we find surface features at certain locations. For instance, we often observe mountains
near the equator of a planet or moon. This is usually the consequence of polar wander: these mountains
could have formed anywhere on the planetary body, but due to the polar wander, they eventually end up
on the equator. This is the case for the Tharsis plateau on Mars. The mathematical description of polar
wander on planetary bodies is difficult, and only approximated solutions have been obtained in the past.
Our study establishes a new method which can give an accurate prediction of how the polar wander
proceeds through time. We show that previous studies have incorrectly estimated the speed of polar
wander on Mars and Venus. With the new method, polar wander on a wide range of planetary bodies can
be simulated. Predictions of our method can be compared to observations of surface features to get a
better understanding of the interior structure of planets and moons.
1. Introduction
Concerning the study of large-angle true polar wander (TPW) on a viscoelastic body such as terrestrial planets
like the Earth, Mars, and Venus, there are currently two types of nonlinear approaches to obtain a time-
dependent solution. One of them is from Nakada (2007) which applies an iterative scheme but simplifies the
Liouville equation by ignoring the time derivative of the MoI term. This approximation is equivalent to assum-
ing that the rotation axis coincides with the axis of the maximum moment of inertia during the process of
TPW. The validity of this assumption was discussed in detail in Cambiotti et al. (2011) who showed that even
for the Earth this assumption is not always appropriate. Another approach which is more commonly applied
in recent studies was formulated originally by Sabadini and Peltier (1981) and further developed by Sabadini
et al. (1982), Spada et al. (1992), and Ricard et al. (1993) which is based on the quasi-fluid approximation.
RESEARCH ARTICLE
10.1002/2017JE005365
Key Points:
• A semianalytical method for
large-angle true polar wander that
can deal with a complete scheme
of multilayer Maxwell rheology
is presented
• We find that using the quasi-fluid
approximation leads to a large error
for TPW on the Earth, Mars, and Venus
• We show the effect of the permanent
shape of an elastic lithosphere and
a model that is not in hydrostatic
equilibrium on TPW
Correspondence to:
H. Hu,
h.hu-1@tudelft.nl
Citation:
Hu, H., van der Wal, W., &
Vermeersen, L. L. (2017). A full-Maxwell
approach for large-angle polar wander
of viscoelastic bodies. Journal of
Geophysical Research: Planets,122.
https://doi.org/10.1002/2017JE005365
Received 9 JUN 2017
Accepted 10 NOV 2017
Accepted article online 22 NOV 2017
©2017. The Authors.
This is an open access article under the
terms of the Creative Commons
Attribution-NonCommercial-NoDerivs
License, which permits use and
distribution in any medium, provided
the original work is properly cited, the
use is non-commercial and no
modifications or adaptations are made.
HU ET AL. A FULL-MAXWELL APPROACH FOR TPW 1
Journal of Geophysical Research: Planets 10.1002/2017JE005365
Mathematically, this approximation is the first-order Taylor expansion of the Love number in the Laplace
domain. The consequence of adopting the quasi-fluid approximation is that the elastic response of a Maxwell
model is missing, and it also simplifies the individual viscous relaxation of different modes. Hu et al. (2017)
tested the validity of the quasi-fluid approximation and showed that it can lead to a large error in the tran-
sient behavior of TPW for a model whose strong modes have very different relaxation times. This could lead
to erroneous conclusions when the model-predicted TPW speed is compared with the speed that is observed
or inferred from surface features (e.g., on Mars, Bouley et al., 2016). The method developed by Hu et al. (2017)
is a numerical approach which requires that the change in the inertia tensor is calculated either by convo-
lution or by a finite-element package. Both the numerical convolution and the finite-element package are
not suitable for studies of models containing layers with very different viscosities since the large contrast in
viscosity results in a large increase in computational time for numerical methods. The increase in the com-
putational time is caused by the fact that the total integration time has to be long to account for the long
relaxation time while the integration step size must be small to accurately simulate the layers with short
relaxation time.
Another issue which is intensively studied in recent years is the effect of an elastic or highly viscous layer on
TPW (Cambiotti et al., 2010; Chan et al., 2014; Harada, 2012; Harada & Xiao, 2015; Mitrovica et al., 2005; Moore
et al., 2017; Willemann, 1984). The existence of such a layer can create a delayed readjustment of the equa-
torial bulge (often called remnant bulge) which significantly changes the behavior of TPW as discussed by
Willemann (1984) and Mitrovica et al. (2005). Mitrovica et al. (2005) show the importance of a correct choice
for the initial hydrostatic state when the TPW is estimated. They used the fluid tidal Love number which cor-
responds to the observed flattening instead of the model predicted flattening. Recent studies often assume
that this extra flattening comes from mantle convection (Cambiotti et al., 2010; Mitrovica et al., 2005). Alter-
natively, if we do not assume that the model is in hydrostatic equilibrium, this difference can also come from
the elastic lithosphere which has its background shape corresponding to a faster rotational speed. As far
as we know, this issue has not been discussed yet. Recent studies concerning the time-dependent solution
of long-term large-angle TPW with an elastic or highly viscous lithosphere (Chan et al., 2014; Harada, 2012;
Moore et al., 2017) are all based on the method developed by Ricard et al. (1993) and adopt the quasi-fluid
approximation.
Compared with the linear approach (e.g., Wu & Peltier, 1984), which can only simulate TPW for small-angle
changes, the method from Ricard et al. (1993) enables the study of issues such as the coupling of the rotational
perturbation in the Xand Ydirections. This means that in the body-fixed frame, a mass distribution imbal-
ance in the X-Zplane would cause a rotational perturbation not only in the X-Zplane but also in the Y-Z
plane. This coupling effect increases as the rotational speed of the object decreases and can turn TPW into
a mega-wobble for some objects like Venus which rotates very slowly (Spada et al., 1996). The phenomenon
of the mega-wobble is caused by the increase of the contribution from the mass anomaly itself compared to
that of the equatorial bulge readjustment. When the contribution from the equatorial bulge readjustment is
dominant, the periodic behavior, often called Chandler wobble, damps out quickly and its secular effect on
the long-term TPW can be ignored. However, when the rotational speed decreases which causes the equa-
torial bulge to decrease, the change in the inertia tensor becomes dominated by the mass anomaly itself.
As a result, the rotational behavior resembles the free nutation of a rigid body (Lambeck, 2005). This cou-
pling effect, or periodic behavior, is almost always neglected in the linear scheme for the study of the Earth
(e.g., Cambiotti et al., 2010; Wu & Peltier, 1984).
To conclude, a semianalytical approach which can accurately calculate TPW of a Maxwell model in different
hydrostatic states is missing. It is the main purpose of this paper to develop such a method and show how
more accurate solutions are obtained. We also show if the difference in results between the methods has a
significant impact on planetary studies (e.g., observation and modeling) in the following cases: TPW for slowly
rotating objects: mega-wobble of Venus; effect of a remnant bulge caused by an elastic or highly viscous
lithosphere on large-angle TPW; and TPW on a body that is not in hydrostatic equilibrium.
In section 2, the influence of the quasi-fluid approximation is discussed in more detail and our new method
will be presented. Sections 3–5 will cover the above listed issues. Sections 3 and 4 contain a case study of
Venus and Mars, respectively, which compares the results obtained in previous studies and that from our
new method.
HU ET AL. A FULL-MAXWELL APPROACH FOR TPW 2
Journal of Geophysical Research: Planets 10.1002/2017JE005365
2. Method
The governing equation for the rotation of a rigid body in the body-fixed rotating reference frame is the
well-known Euler’s equation. As the body becomes deformable, it is often referred to as the Liouville equation.
For a torque-free case, it reads (Sabadini et al., 2016)
d
dt(I𝝎)+𝝎×I𝝎=0(1)
where Iis the inertia tensor and 𝝎is the rotational vector whose magnitude is the rotating speed. Both val-
ues are defined in a body-fixed coordinate system. When the moment of inertia of the body is perturbed by a
geophysical process which causes mass redistribution, the rotational axis shifts, and consequently, the equa-
torial bulge readjusts. Analytically, given a rotational vector as 𝝎=𝜔1,𝜔
2,𝜔
3T̄𝜔1𝜔2𝜔3T, where Ω
is the angular speed of the rotation and ̄𝜔1𝜔2𝜔3Tis a unit vector which represents the direction of the
rotation, the total moment of inertia attributable to such a process is given by (Ricard et al., 1993)
Ii,j(t)=I𝛿ij +kT(t)a5
3G𝜔i(t)𝜔j(t)− 1
3Ω(t)2𝛿ij
+𝛿(t)+kL(t)Ci,j(t)
(2)
where Iis the principle moment of inertia of the spherical body in hydrostatic equilibrium, Gis the gravita-
tional constant, and ais the radius of the planet. kT(t)and kL(t)are the degree 2 potential tidal Love number
and load Love number, respectively. Love numbers are obtained by the normal mode method and based on
the Maxwell rheology (Farrell, 1972). The asterisk denotes convolution in the time domain. Ci,jrepresents the
change in the moments and products of inertia without considering the deformation and this is the triggering
load for the TPW. The most difficult part of solving equations (1) and (2) is the convolution of the tidal Love
number and the centrifugal potential, in particular, the part kT(t)∗𝜔i(t)𝜔j(t). In the following subsection, we
first show how this problem is tackled by adopting the quasi-fluid approximation and the influence of this
approximation on calculation of the inertia tensor. Following this subsection a new approach is presented to
calculate the MoI equation analytically. Section 2.3 demonstrates how to use the developed algorithm and
provides initial results from our method.
2.1. Conventional Approach Based on the Quasi-Fluid Approximation
The tidal Love number in the Laplace domain for a given harmonic degree is expressed as (Peltier, 1974)
kT(s)=kT
e+
m
i=1
kT
i
ssi
(3)
where kT
eis the elastic Love number, kT
iare the residues of each mode, and siare the inverse relaxation times.
This form of the Love number contains all the information about how a multilayered Maxwell body deforms:
an instantaneous elastic response characterized by kT
efollowed by viscous relaxation of separate modes char-
acterized by their different inverse relaxation time siand mode strength kisi. We call this form of the Love
number the full-Maxwell rheology scheme, as opposed to the quasi-fluid approximation introduced below.
In order to solve equation (2), Ricard et al. (1993) took the quasi-fluid approximation which approximates the
tidal Love number with its first-order Taylor expansion:
kT(s)≈kT
e
m
i=1kT
i
si
+kT
is
s2
i
=kT
f(1T1s)
(4)
where kT
fis the fluid Love number which is the sum of the mode strength:
kT
f=ke
m
i=1
kT
i
si
(5)
The time constant T1is
T1=1
kT
f
m
i=1
kT
i
s2
i
(6)
Thus, by taking the quasi-fluid approximation, all information for a viscoelastic layered model is combined into
one constant T1. Because of the kis2
iterm, this constant is dominated by the slowest modes. Thus, applying
HU ET AL. A FULL-MAXWELL APPROACH FOR TPW 3
Journal of Geophysical Research: Planets 10.1002/2017JE005365
Figure 1. O-X<mathdollar>YZis the body-fixed frame where the Zaxis
is the original rotational axis. O-XYZ’ is the bulge-fixed frame where
the Z’ axis is the instantaneous rotational axis and the X’ axis lies within
the Z-Z’ plane.
the quasi-fluid approximation on a physical model which contains modes
with both very long and short relaxation times will result in a large bias. It
can also be seen that with this approximation, the elastic response as well as
the viscous relaxation characterized by the function 1∕(ssi)in the Laplace
domain (esitin the time domain) are missing (as will be shown in equation
(7)); therefore, the ongoing deformation does not agree with the complete
Maxwell rheology scheme. The convolution of equation (4) with a linear load
function F(t)=a+bt, where a,bare constants, results in a time domain
response of the form
R(t)=kT
fF(t)−F(t)kT
fT1(7)
where the derivative F(t)=band R(t)is the response function. This response
demonstrates the effect of the quasi-fluid approximation: for a near-constant
load (F(t)≈0), the response reaches its fluid limit kT
fF(t)immediately with-
out the time-dependent viscous behavior. This means that when the speed of
TPW is very slow compared to the characteristic relaxation speed of the body,
the results based on the quasi-fluid approximation approach those which are
obtained from the fluid limit method which diagonalizes equation (2), such as
in Matsuyama and Nimmo (2007). For other loads which change linearly in
time, the instantaneous fluid limit response kT
fF(t)is shifted by a value which
is proportional to the speed of the change of the load, as given by the sec-
ond term of the equation (7). For the complete Maxwell rheology scheme, the
original form of the Love number (equation (3) needs to be convoluted with the loading function, resulting
in damping of this part with a function of eAt , where Ais a positive constant. As a result, compared to the
original Maxwell rheology, the response of a Heaviside or fast changing load based on the quasi-fluid approx-
imation will likely result in a very different TPW path. For example, the change in the inertia tensor due to an
impact crater which appears instantly and is preserved afterward is a Heaviside load to the planet.
Next, we quantitatively show that adopting quasi-fluid approximation can either underestimate or overesti-
mate the equatorial readjustment for certain components of the inertia tensor in the normal polar wander
case and in the mega-wobble case. Substituting equation (4) into (2) gives the change in the moment of inertia
as (Ricard et al., 1993)
ΔIi,j(t)= kT
fa5
3G𝜔i(t)𝜔j(t)− 1
3Ω(t)2𝛿ij
kT
fa5
3GT1̇𝜔i(t)𝜔j(t)+𝜔i(t)̇𝜔j(t)− 2
3𝜔l(t)𝜔l(t)𝛿ij+Eij
(8)
where Ei,j(t)and ̇
Ei,j(t)are obtained by convolving Ci,j(t)and ̇
Ci,j(t)with 𝛿i,j+kL(t). We compare the change in
the moment of inertia calculated by equations (8) and (2) to show the influence of adopting the quasi-fluid
approximation. The latter is obtained by numerical calculation of the convolution. As a representative of
terrestrial planets, we use a SG6 Earth model (which represents a multilayered model with interior density,
rigidity, and viscosity change) and let the rotational axis move in two ways:
1. The rotational axis drifts with a constant speed along the X-Zplane in the body-fixed coordinates for 90.
This is to simulate the normal polar wander case for fast-rotating planets such as the Earth and Mars.
2. The rotational axis initially stays at 30colatitude and 0longitude in the X-Zplane and moves longitudinally
with a constant speed along the 30colatitude circle for 720. This is to simulate the mega wobble for very
slowly rotating objects such as Venus.
In the first case, the drift speed of the rotational axis is chosen to be fast enough to view the effect of the
quasi-fluid approximation. We define a bulge-fixed frame whose Z’ axis coincideswith the instantaneous rota-
tional axis and whose X’ axis lies within the Z-Z’ plane as shown in Figure 1. So a pure rotation (around Y’ axis)
can transform the body-fixed frame into the bulge-fixed frame.
Figure 2 gives the change in the six components of the MoI tensor in the bulge-fixed frame. In this figure, the
differences in the I11,I22,andI33 are small, but the most important component I13 calculated with the quasi-
fluid approximation is significantly larger than the accurate value. The magnitude of the I13 in the bulge-fixed
HU ET AL. A FULL-MAXWELL APPROACH FOR TPW 4
Journal of Geophysical Research: Planets 10.1002/2017JE005365
0 200 400
-7.15
-7.14
-7.13
-7.12
-7.11
-7.1 x 1034 I11
ka
kg m2
0 200 400
-7.1423
-7.1422
-7.1421
-7.142
-7.1419 x 1034 I22
ka
0 200 400
1.424
1.425
1.426
1.427
1.428
1.429
1.43 x 1035 I33
ka
Quasi
Numerical
-fluid
0 200 400
-2
0
2
4
6x 1033 I13
ka
kg m2
0 200 400
-1
-0.5
0
0.5
1
I23
ka
0 200 400
-1
-0.5
0
0.5
1
I12
ka
Figure 2. Normal polar wander case: change in the six components of the MoI tensor when the rotational axis initially
stays at 30colatitude and 0longitude in the X-Zplane and precesses with a constant speed along the 30colatitude
circle for 720in 500 ka. Red lines are the accurate results and blue lines are from the quasi-fluid approximation.
0 200 400 600
-7.15
-7.14
-7.13
-7.12
-7.11
-7.1 x 1034 I11
ka
kg m2
0 200 400 600
-7.15
-7.1
-7.05
-7
-6.95
-6.9 x 1034 I22
ka
0 200 400 600
1.4
1.41
1.42
1.43
1.44 x 1035 I33
ka
Quasi
Numerical
-fluid
0 200 400 600
0
1
2
3
4
5
6x 1032 I13
ka
kg m2
0 200 400 600
-4
-3
-2
-1
0
1x 1034 I23
ka
0 200 400
-6
-4
-2
0
2x 1032 I12
ka
Figure 3. Mega-wobble case: change in the six components of the MoI tensor. The rotational axis initially stays at 30
colatitude and 0longitude in the X-Zplane and precesses longitudinally with a constant speed along the 30colatitude
circle for 720in 600 ka. Red lines are the accurate results, and blue lines are from the quasi-fluid approximation.
HU ET AL. A FULL-MAXWELL APPROACH FOR TPW 5
Journal of Geophysical Research: Planets 10.1002/2017JE005365
frame determines how fast the equatorial bulge readjusts. A larger value of I13 suggests a slower readjustment
(if the readjustment is complete, then the rotational axis coincides with the new principle axis and I13 would
be zero). So adopting the quasi-fluid approximation in the normal polar case causes a large underestimation
for the speed of the equatorial bulge readjustment.
For the second case where the rotational axis tends to wobble around a fixed point, in the bulge-fixed frame,
the I23 implies the readjustment in the direction along the track of the rotational axis (called along-track direc-
tion in the following) and I13 gives the readjustment in the direction which is perpendicular to the plane that
contains the along-track direction and the rotational axis (called normal direction in the following). The results
are shown in Figure 3. For components I11,I22 , and I33, the quasi-fluid approximation misses the small oscilla-
tions but the differences are still very small between the two methods; thus, for these three components the
error introduced by the approximation can be ignored. However, the magnitude of the along-track compo-
nent I23, is largely underestimated just like the I13 component in Figure 2 (I13 is the along-track component
for the normal polar wander case). On the other hand, the normal directional component I13 in Figure 3 is
underestimated by the quasi-fluid approximation which suggests an overestimation of the equatorial bulge
readjustment in this direction.
The key information obtained in this subsection is that by adopting the quasi-fluid approximation, the speed
for equatorial readjustment can be, depending on the model and load, largely underestimated in the along-
track direction but overestimated for the mega wobble case in the normal direction. These are the main rea-
sons for the difference of the TPW path calculated by different methods which will be discussed in sections 3
and 4.
2.2. A New Approach
In order to eliminate the convolution in the part kT(t)∗𝜔i(t)𝜔j(t)while staying consistent with the fundamen-
tal rheology of the system, we adopt the strategy of approximating the load term 𝜔i(t)𝜔j(t)instead. Within
the considered time period Tn, at time t=Tp,p=0,1,,n, values of 𝜔i(t),i=1,2,3are known, then we have
𝜔i(Tp)=Wi,p. Assuming that 𝜔i(t),i=1,2,3changes linearly between each time step, 𝜔i(t)can be written as a
piecewise linear function:
𝜔i(t)=
n
p=1
𝜔i,p(9)
where
𝜔i,p=Wi,p1+Wi,pWi,p1
TpTp1tTp1HtTp1HTpt(10)
and H(t)is the Heaviside step function. With this form, k(t)∗𝜔i(t)𝜔j(t)and its derivative can be expressed
analytically by applying the Laplace transformation
kT(t)∗𝜔i(t)𝜔j(t)=1[[kT(t)∗𝜔i(t)𝜔j(t)]] (11)
where and 1stand for the Laplace and inverse Laplace transformation, respectively.The explicit expression
of equation (11) can be found in Appendix A as equations (A1a)– (A1c). Substituting equations (A1a) (A1c)
into (2), the inertia tensor and its derivative at time t=Tncan be expressed analytically as
Ii,j(Tn)=I𝛿i,j+a5
3G(Ai,j(Tn)+Bi,j)− 1
3
3
k=1
(Ak,k(Tn)+Bk,k)+Ei,j(Tn)(12a)
̇
Ii,j(Tn)= a5
3G(Ci,j(Tn)+Di,j)− 1
3
3
k=1
(Ck,k(Tn)+Dk,k)+̇
Ei,j(Tn)(12b)
where expressions for Ai,j(t),Bi,j(t),Ci,j(t)and Di,j(t)can be found in Appendix A. When t=Tpwith p=0,1, ...n
and 𝜔i(Tp)=Wi,pwith p=0,1,2...n1are given, equations (12a) and (12b) express the moments and products
of inertia and its derivative as a function of Wi,n. Then Wi,ncan be solved by equation (1). To this end, it helps to
see the problem as a global optimization problem: we seek the value of Wi,nin the neighborhood of Wi,n1so
that the value of d
dt(I𝝎)+𝝎×I𝝎is minimized. As a result, the method which is introduced in Hu et al. (2017)
as algorithm 2 (p. 10) can be applied. This method applies the linearized form of the Liouville equation and
an iteration procedure to obtain Wi,n. It will be briefly explained in the following and outlined in Appendix B.
HU ET AL. A FULL-MAXWELL APPROACH FOR TPW 6
Journal of Geophysical Research: Planets 10.1002/2017JE005365
This algorithm is validated in (Hu et al., 2017) by both the linear (Wu & Peltier, 1984) and the nonlinear method
(Ricard et al., 1993) as shown in Figure D1 in Appendix D. We define the perturbed rotational vector as
𝝎(m1,m2,1+m3)T(13)
where m1,m2, and m3are small real numbers. The Liouville equation can be linearized to obtain the form
(Hu et al., 2017)
m1(t)= ΔI13(t)
CA+CΔ
.
I23 (t)
Ω(CA)(CB)(14a)
m2(t)= ΔI23(t)
CBCΔ
.
I13 (t)
Ω(CA)(CB)(14b)
m3(t)=−
ΔI33
C(14c)
In equations (14a)–(14c), the terms of the inertia tensor, A,B,Cand I13,I23 ,̇
I13,̇
I23 are not in the body-fixed
frame but in the bulge-fixed frame. The transformation matrix from the body-fixed frame to the bulge-fixed
frame by a pure rotation can be obtained as
Q=
𝜔3+𝜔2
2
1+𝜔3
𝜔1𝜔2
1+𝜔3
𝜔1
𝜔1𝜔2
1+𝜔3
1𝜔2
2
1+𝜔3
𝜔2
𝜔1𝜔2𝜔3
(15)
A coordinate transformation is required before we can substitute the value of the inertia tensor calculated
from equations (12a) and (12b). The detailed procedure of algorithm 2 in Hu et al. (2017) is given in Appendix B.
In general, the only assumptions we make in the entire calculation in this study are two linear approximations:
the changes in the rotational vector and the inertia tensor (see equation (10) in Hu et al. (2017)) are small in
each step and can be treated as linear. These assumptions are valid when the step sizes (Δtp=TpTp1,p=
2,3,,n) are small enough. Since we do not approximate Love numbers, our method gives the TPW path
for a viscoelastic body which is consistent with the complete scheme of Maxwell rheology. We will label our
method in the following as full-Maxwell method.
2.3. Initial Setting and Validation
One of the major factors that controls the TPW behavior is the shape of the equatorial bulge (and the tidal
bulge which is discussed in Hu et al., 2017). When the interior model and rotational speed is given, this shape
is controlled by the hydrostatic state of the model. Due to the limitation of the method, previous studies
based on either linear (Sabadini et al., 1982; Wu & Peltier, 1984) or nonlinear (Chan et al., 2014; Moore et al.,
2017; Ricard et al., 1993) approaches can only simulate TPW on a model which is assumed to be in hydrostatic
equilibrium. However, as will be shown in section 5, the choice of the hydrostatic state can have a significant
impact on the TPW behavior. With our method, we can choose the hydrostatic state of the model at which
the TPW starts.
To simulate the TPW of a body at a certain hydrostatic state, we need to apply a centrifugal force to the model
for a certain length of time. If the rotational vector at the start of the simulation is given by 𝝎0(0,0,1)T,
applying centrifugal force to the model for a duration of This expressed in our scheme as
T0=0(16a)
T1=Th(16b)
W1,0,W2,0,W3,0T=W1,1,W2,1,W3,1T=𝝎0(16c)
The triggering load Ei,j(t)needs to be applied at t=Thto start the TPW. To simulate a model in hydrostatic
equilibrium, Thneeds to be large enough so that all modes of the model are sufficiently relaxed. In order to
achieve this, we can choose a Thso that the slowest mode is relaxed more than 99.999%. Assuming s1is the
slowest mode,
1es1Th>0.99999 (17)
HU ET AL. A FULL-MAXWELL APPROACH FOR TPW 7
Journal of Geophysical Research: Planets 10.1002/2017JE005365
0 2000 4000 6000 8000 10000 12000 14000
-60
-50
-40
-30
-20
-10
0
Time (ka)
Colatitude (degree)
T =2e3 ka
N, T =2e3 ka
T =1e6 ka
Quasi-fluid
Figure 4. Polar wander in the X-Zplane for the SG6 Earth model
triggered by a point mass of 2×1019 kg attached at the surface at
30colatitude.
translates into
Th>11.513
s1
(18)
This choice is similar to that in Hu et al. (2017), which sets Thso that
kT(Th)>99.95%kT
f, but much more strict. If the model contains a very slow
and strong mode, we can obtain a very large value of Thand this will cause an
extremely long calculation time for a pure numerical method (Hu et al., 2017).
By expressing the inertia tensor analytically, a very large Thcan be dealt with.
Eventually, there is no limit for the choice of Thas well as the initial loading
𝝎0as long as numerical errors (e.g., truncation error) are avoided. As a result,
TPW for a body in a different hydrostatic state can be obtained.This makes our
method suitable to study the effect of a remnant bulge or TPW on a model
without hydrostatic equilibrium as will be shown and discussed in detail in
sections 4 and 5.
The algorithm, as shown in Appendix B, was developed in Hu et al. (2017). The
main idea is to decouple the two governing equations. The MoI equation was
solved by either direct convolution or from a finite element method and the
result is fed back into the linearized Liouville equation and solved by an iterative procedure. Such algorithm
has been validated by both comparing to the linear (Wu & Peltier, 1984) and nonlinear method (Ricard et al.,
1993) in Figures 6 and 9 of Hu et al. (2017). The difference between the method in this paper and that of Hu
et al. (2017) is that here the MoI equation is solved analytically with the assumption that the rotational vector
changes linearly during each time step of TPW. If the step size is set to be much smaller than the relaxation
time of the dominant modes, which is the same requirement for calculating TPW, the analytical solution of the
MoI equation with the linear assumption will be sufficiently equivalent to the result from direct convolution.
Therefore, the TPW solution generated in this paper can approach that from Hu et al. (2017) for a small enough
step size. We first demonstrate the result of TPW calculated with Th=2,000 ka, which is the choice in Hu et al.
(2017) for a six-layer incompressible Earth model SG6 (Table 2 in the same paper), and a much higher value
(Th=106ka) for which the model can be considered in hydrostatic equilibrium (equation (18) holds). Weplace
a stationary (kL(t)=0) mass anomaly, which means that the mass anomaly does not “sink” into the body, at
the surface at 30colatitude. Here we do not consider the effect of the remnant bulge (which will be discussed
in detail in section 4), so we ignore the slowest mode generated by the lithosphere (its viscosity is set to 1031
Pa s to calculate the Love numbers). We also include the result obtained by the quasi-fluid approximation
according to Ricard et al. (1993).
As can be seen in Figure 4, the result obtained forTh=2,000 k a by the full-Maxwell method is very close (within
0.05% difference) to the result of the numerical method from Hu et al. (2017), for a step size of 5 ka. It is also
clear that choosing Th=2,000 ka still shows a TPW path that is different from that of a body which can be
considered to be in hydrostatic equilibrium (Th=106ka). This suggests the sensitivity of the TPW to a small
deviation from its hydrostatic equilibrium. So choosing Thlarge enough is the first guarantee that the correct
TPW path for models with hydrostatic equilibrium is obtained.
The computational cost of our method depends on three factors: (1) the complexity of the layered model, or
more precisely, the number of the modes in the Love numbers; (2) the number of iteration necessary to obtain
a convergent result in each step; and (3) the number of time steps. The computational time increases roughly
linearly with these factors. For the results shown in Figure 3, which is for a SG6 Earth model that contains
12 modes, the number of time steps is 2,600 (each step is 5 ka) and every step requires 50 (first step) to 18
(last step) iterations. The program is written in MATLAB and the total computational time on our desktop com-
puter is about 3 min. Compared with the numerical computation (Hu et al., 2017) which is about several hours,
the speed from the semianalytical approach is much faster.
Based on the analysis in section 2.1, the largely underestimated TPW speed by adopting the quasi-fluid
approximation is caused by the underestimation of the speed of the equatorial readjustment for the case of
Earth and Mars. For the mega-wobble case in Venus, the situation can be quite different as will be discussed
in the following section.
HU ET AL. A FULL-MAXWELL APPROACH FOR TPW 8
Journal of Geophysical Research: Planets 10.1002/2017JE005365
3. Mega-Wobble: TPW on Venus
The linearized Liouville equation (equations (14a)– (14c)) is obtained by two fundamental assumptions. First,
during the TPW, the components of the inertia tensor satisfy I33(C)>I11(A)and I33 (C)>I22(B). Second, the peri-
odic terms which represent the Chandler wobble can be ignored. While these assumptions are true for cases
like Mars and the Earth, it can be invalid for some slowly rotating objects such as Venus. Since the rotational
speed of Venus is so slow that its equatorial bulge is also extremely small, the difference of the two principle
moment of inertia: CA, which can be calculated by
CA=kT
fa5Ω2
3G(19)
for Venus is less than 1.5×105of that for Earth. For magnitudes of a mass anomaly of 105or 106of the
total mass of the planet considered in Spada et al. (1996), depending on the depth and position, the moment
around the rotational axis may not be the largest of the diagonal components anymore (C>A,Bare not
satisfied). Furthermore, the period of the wobble, which can be estimated as 2𝜋
ΩAB
(CA)(CB)(see equation (26))
when C>A,B, is about 4 months (depending on the interior model) on Earth or Mars but can be, depending
on the interior model, over 10 Ma on Venus. Because of such low frequency, the periodic terms will have a
secular effect for TPW on Venus and cannot be ignored as for the Earth and Mars. Therefore, in order to study
TPW on Venus, it is necessary to first derive a new set of linearized Liouville equations suitable for a body with
a very long wobble period. The linearized Liouville equation for a triaxial body reads (Sabadini et al., 2016)
̇
m1=−
CB
AΩm2+Ω
AΔI23 Δ̇
I13
A(20a)
̇
m2=CA
BΩm1Ω
BΔI13 Δ̇
I23
B(20b)
̇
m3=−
Δ̇
I33
C(20c)
We first deal with the cases of C>A,C>B, and C<A,C<B. By assuming that the change in the moment of inertia
is linear, equations (20a)– (20c) can be solved analytically. The result contains the non-periodic terms in
equations (14a)–(14c) and periodic terms
̄
m1(t)=B
A(CA)3(CB2sin (CA)(CB)
AB ΩΔt(ACI23(t)+CΔ̇
I13(t)
cos (CA)(CB)
AB ΩΔtΔI13(t)
CA+CΔ̇
I23(t)
Ω(CA)(CB)(21)
̄
m2(t)=A
B(CA)(CB)3Ω2sin (CA)(CB)
AB ΩΔt(CBI13(t)+CΔ̇
I23(t)
cos (CA)(CB)
AB ΩΔtΔI23(t)
CB+CΔ̇
I13(t)
Ω(CA)(CB)(22)
When the period of the wobble becomes very long and the step size Δtis small enough, the magnitude of
(CA)(CB)
AB ΩΔtin the trigonometric functions is very small, and we can apply sin(𝜃)≈𝜃and cos(𝜃)≈1𝜃22.
Applying these approximations, combining equations (14a)– (14c), (21), and (22) and ignoring the derivative
terms of the inertia tensor gives
m1(t)= (CB2ΔI13(tt2+2BΩΔI23 (tt
2AB (23a)
m2(t)= (CA2ΔI23(tt22AΩΔI13 (tt
2AB (23b)
m3(t)=−
ΔI33
C(23c)
HU ET AL. A FULL-MAXWELL APPROACH FOR TPW 9
Journal of Geophysical Research: Planets 10.1002/2017JE005365
Tab le 1
Venus Model
Outer radius (km) Density (kg m3) Shear modulus (Pa) Viscosity (Pa s)
6,052 2,900 0.36 ×1011
6,002 3,350 0.68 ×1011 0.6×1021
5,500 3,725 0.93 ×1011 1.6×1021
5,200 4,900 2.07 ×1011 6.4×1021
3,250 10,560 0 0
The derivation for other situations such as B<C<Aor B>C>Aare shown in Appendix C. Note that in
equations (23a)–(23c), as well as in equations (C5a), (C5b), (C6), and (C6b), we do not have the CAor CB
terms in the denominator; thus, these expressions have no singularity problem for C=Bor C=A. When TPW on
a very slowly rotating object like Venus is calculated, equations (23a)– (23c) instead of equations (14a)–(14c)
should be applied. Basically, equations (14a)– (14c) and equations (23a)(23c) give two extreme situations
for calculating the rotational perturbation. When the step size of the calculation Δtcan be set to much larger
than the Chandler period, equations (14a)– (14c) should be used to give the secular behavior. When the step
size Δtis chosen to be much smaller than the period of the Chandler wobble, equations (23a)– (23c) can give
the periodic “short”-term behavior which leads to the mega-wobble on Venus or, when the step size is set to
days, the Chandler wobble on Earth or Mars.
Next, we apply our method to a model of Venus and test some results obtained in Spada et al. (1996) who
apply the quasi-fluid approximation. We create a five-layer Venus model which approximates the density and
rigidity profile used in Armann and Tackley (2012), and the viscosities are chosen similar to those of Earth.
The interior properties are shown in Table 1. The effect of a remnant bulge is not included here and will be
discussed in the next section.
Figure 5. A point mass of 5×1018 kg is attached at the surface at 45colatitude. (top row)The displacement of
the rotational axis in the along-track and normal direction. (bottom row) The movement of the mass anomaly in the
bulge-fixed frame where the rotational axis is always pointing upward at the center. Blue lines are obtained by applying
the quasi-fluid approximation, and red lines are from the full-Maxwell method. The black dots are the original locations
of the mass anomalies in the bulge-fixed frame.
HU ET AL. A FULL-MAXWELL APPROACH FOR TPW 10
Journal of Geophysical Research: Planets 10.1002/2017JE005365
For this model, the period 2𝜋
ΩAB
(CA)(CB)is about 11.45 Ma. The step size needs to be much smaller than this
value; therefore, a value of less than 5,000 years is sufficient to obtain an accurate solution. One way to test if
the chosen step size is indeed small enough is to recalculate the TPW with half of the step size. If the chosen
step size is indeed sufficiently small, the result will not change significantly.We first simulate the TPW on Venus
driven by a negative mass anomaly of magnitude 5×1018 kg (which is about the maximum value simulated
by Spada et al. (1996)) which is attached at the surface at 45colatitude. We decompose the displacement of
the rotational axis in the direction which lies in the plane that contains the mass anomaly (normal direction)
and the along-track direction. The result is shown in Figure 5. As we can see, for the along-track displace-
ment which describes the long-term wobble of the rotational axis around the mass anomaly, the results from
the two methods differ less than 1%. However, for the displacement in the normal direction, representing
the movement of the rotational axis toward the negative mass anomaly, the result obtained based on the
quasi-fluid approximation overestimates the speed. The agreement in the along-track direction displacement
and a disagreement in the normal direction displacement between the two methods can be explained by the
small contribution of the equatorial bulge readjustment to the rotational perturbation for Venus. As pointed
out by Spada et al. (1996), for Venus, the rotational behavior is largely dominated by the long-term wobble
which resembles the free nutation of a rigid body. The long-term wobble is mainly caused by the mass
anomaly itself, while the equatorial readjustment contributes very little. Based on the discussion of Figure 3,
the component I23 also contributes to the along-track speed of the wobble. This component is largely over-
estimated by the quasi-fluid approximation, but since its magnitude is much smaller than the contribution
from the mass anomaly itself (less than 1% in this case), the difference between the two methods can be
ignored. However, the damping of the oscillatory motion, or the displacement of the rotational axis in the
normal direction, is solely controlled by the viscous relaxation of the body, specifically the component I13 in
Figure 3. Since this component is underestimated by the quasi-fluid approximation, the rotational behavior
on Venus obtained by adopting the approximation results in too much damping.
In the study of terrestrial planets whose tidal bulge can be ignored, like Earth, Mars, Venus, and Mercury, we
may see certain geographic features (e.g., the supercontinent Pangaea on the Earth or the Tharsis plateau
on Mars) which have the potential to cause (or have caused) polar wander. Based on an interior model and
TPW history, we can estimate the age of the feature or the history of its relocation by its (estimated) former and
current latitude since we know that positive mass anomaly tends to relocate toward equatorand negative one
toward the pole. The latitudinal information in many situations is much more important than the longitudinal
one if the body is not tidally locked. As a result, correctly estimating the TPW speed in the latitudinal (normal)
direction is crucial for a better understanding of the planet reorientation. In the following the same Venus
model is used and several different magnitudes of mass anomaly are tested. We compare the difference in
speed as a function of colatitude of the mass anomaly in the normal direction between the two methods.
Magnitudes of mass anomaly 1×1016,1×1017 ,1×1018, and 5×1018 kg are chosen, which are about 105to
106times of the total mass of Venus, similar to the values chosen in Spada et al. (1996). The results are shown
in Figure 6. Generally, methods based on the quasi-fluid approximation overestimate the normal-directional
speed by a factor of 3 to 5.
Spada et al. (1996) state that for the same mass anomaly, the instantaneous velocity of rotational pole on
Venus is about 30 times larger than that of Earth and Mars. But they compared the complete rotational
behavior of Venus whose largest part is the wobble with the secular rotational behavior of Earth and Mars in
which the Chandler wobble has been filtered out. A more proper comparison would be either between the
speed of Chandler wobble on Earth or Mars with the mega-wobble on Venus, or between the secular speed
of rotational variation on Earth or Mars and the normal-directional speed of the rotational axis on Venus.
In the latter case, for an Earth, Mars, and Venus model with the same average viscosity (ranging from 1020 to
1022 Pa s), it can be shown that TPW on Earth and Mars is 10 to 15 times larger than the normal-directional
speed of Venus’ rotational axis for the mass anomalies considered in this study. This means that for the same
scale of mass anomaly, it will take much longer on Venus than on Earth or Mars before it can reach the pole
or equator. The knowledge of Venusian viscosity is very limited, if the observed normal-directional change of
Venus’ rotational axis is out of this range (1/10 to 1/15 times of the Earth’s TPW), then the average viscosity of
Venus must be lower or higher than that of the Earth.
HU ET AL. A FULL-MAXWELL APPROACH FOR TPW 11
Journal of Geophysical Research: Planets 10.1002/2017JE005365
Figure 6. Normal directional speed of the rotational axis toward the mass anomaly as a function of the colatitude.
The results we obtained in this section (e.g., the speed in Figures 5 and 6) are, of course, dependent on the
interior model of Venus. With different viscosities (1019 Pa s to 1022 Pa s), the TPW speed can be very different
but adopting the quasi-fluid approximation always largely overestimates the normal-directional speed.
4. Effect of a Remnant Bulge on TPW and a Study of Mars
For a viscoelastic model which can be sufficiently relaxed, a positive mass anomaly with any magnitude will
end up at the equator, while a negative mass anomaly will eventually reach the poles. However, most of the
observed geophysical features which are thought to have triggereda reorientation, such as the Tharsis plateau
on Mars (Bouley et al., 2016) or Sputnik Planitia on Pluto (Keane et al., 2016), are not located exactly at the
equator. A common explanation is that certain parts of the planet, usually the lithosphere which is considered
to be elastic or to have a very high viscosity, have not yet relaxed, preventing the mass anomaly from being
relocated further. The effect of such elastic or highly viscous lithosphere on TPW has been studied for the
linear scheme (e.g., Cambiotti et al., 2010; Mitrovica et al., 2005), and the nonlinear scheme with the quasi-fluid
approximation (e.g., Chan et al., 2014; Harada, 2012; Moore et al., 2017). Here we demonstrate the effect of a
remnant bulge on the large-angle TPW with the full-Maxwell method.
First, the origin of the remnant bulge is shown using the normal mode method. This has also been discussed
by Moore et al. (2017). The remnant bulge, either formed by an elastic layer or a highly viscous layer, appears
because of a certain mode(s) which has a much longer (or infinite) relaxation time compared to other domi-
nant relaxation modes of the model. We demonstrate this with a simple two-/three-layer Earth models with
and without a lithosphere of varying viscosity. In Table 2, the physical properties of the models are shown.
Tab le 2
Properties of the Two-/Three-Layer Earth Models (M1– M5)
Layer Outer radius (km) Density (kg m3) Shear modulus (Pa) Viscosity (Pa s)
Lithosphere 6,371 4,448 1.7×1011 1×1021,24,26,29,
Mantle 6,361 4,448 1.7×1011 1×1021
Core 3,480 10,977 0 0
HU ET AL. A FULL-MAXWELL APPROACH FOR TPW 12
Journal of Geophysical Research: Planets 10.1002/2017JE005365
Tab le 3
Potential Tidal Love Number of Models M1– M5
M1 (1021 Pa s) M2 (1024 Pa s) M3 (1026 Pa s) M4 (1029 Pa s) M5 (Pa s)
Modes sikisisikisisikisisikisisikisi
ke0.3449 0.3449 0.3449 0.3449 0.3449
T10.5341e1 0.4731e20.5343e3 0.4204e20.5343e6 0.4200e2
T20.7886e1 0.3045e20.2764e1 0.2035e20.2713e1 0.2027e20.2712e1 0.2027e2
C00.4086 0.2376 0.4208 0.2337 0.4193 0.2351 0.4193 0.2352 0.4193 0.2352
M02.225 0.4591 2.2344 0.4552 2.2343 0.4553 2.2343 0.4553 2.2343 0.4553
kf1.0416 1.0416 1.0416 1.0416 1.0374
Model M1 has a lithosphere viscosity of 1021 Pa s which is the same as the mantle; thus, this is actually a
two-layer model without lithosphere. The lithosphere of models M2– M5 have viscosities of 1024,1026 ,1029,
and Pa s, respectively.
The degree 2 potential tidal Love numbers of models M1– M5 are shown in Table 3, where the inverse relax-
ation time sihas unit 1/ka. Following Sabadini et al. (2016), we can see that the two-layer model M1 only
contains two relaxation modes, C0corresponding to the core-mantle boundary and M0corresponding to the
surface. When a viscoelastic layer with the same density is added to the model (M2– M5), two additional tran-
sition modes, ̄
T1and ̄
T2, are triggered if the Maxwell time on either side of the boundary is different. In the case
of model M2, these two additional modes have relaxation times not too different from the dominant modes
(C0and M0in this case). Consequently, the delayed relaxation of both ̄
T1and ̄
T2is not large enough to cause a
remnant bulge. However, in the case of M3 and M4, as the viscosity of the lithosphere increases, the relaxation
time for one of the ̄
Tmodes increases with the same order as the viscosity. It is this ̄
Tmode that determines if
the remnant bulge is present. As the viscosity of the lithosphere increases further and eventually approaches
infinity, as is the case in model M5, the ̄
T1mode disappears and its mode strength k1s1becomes absent in
the fluid Love number kf. It can be seen that the difference in the fluid Love number between M5 and M4 is
almost the same as the mode strength of ̄
T1in M4: kM4
fkM5
f≈(k1s1)M4. The remnant bulge is dealt with in
previous studies by either those dealing with an elastic lithosphere (Cambiotti et al., 2010; Chan et al., 2014;
Harada, 2012; Mitrovica et al., 2005) or viscoelastic lithosphere (Cambiotti et al., 2010; Moore et al., 2017) by
isolating this part of the Love number and formulating the influence of it separately.
0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000
-45
-40
-35
-30
-25
-20
-15
-10
-5
0
Time (ka)
Colatitude (degree)
LT=30km
LT=20km
LT=10km
Figure 7. Polar wander in the X-Zplane for the Earth models with
different lithosphere thickness triggered by a point mass of 2×1019 kg
attached at the surface at 20colatitude. Solid lines show results
obtained with the full-Maxwell method from this paper, the line dots
are calculated with the method by Chan et al. (2014) which is based
on the quasi-fluid approximation.
One of the advantages of the full-Maxwell method is that we can choose any
value for the initial loading time Th. This enables us to simulate the influence
of the remnant bulge without any extra formulation. To include such a bulge
caused by a very slow relaxation mode, we only need to set the initial loading
time Thto a value large enough so that this slow mode is fully relaxed to the
centrifugal force, according to the condition in equation(18). For instance, for
the model M4 in Table 3, we can set Th=1×109ka which guarantees that the
̄
T1mode is relaxed. In practice, we do not need to simulate the case with a fully
elastic layer. Instead, we can always set the viscosity of the layer high enough
to guarantee that its relaxation within the considered time can be ignored. In
that case, the lithosphere is effectively elastic.
Now we demonstrate the effect of a remnant bulge with the full-Maxwell
method and compare the results with those obtained by applying the
quasi-fluid approximation. A more realistic SG6 model is used, in contrast with
those shown in Figure 4 where the slowest mode is ignored. Three cases with
different thickness of the lithosphere are considered and the viscosity of this
layer is set to 1031 Pa s so that within the considered time span (10 Ma), the
relaxation of the slowest mode can be ignored. We compare our results with
those obtained by using the method of Chan et al. (2014) which is based on
the quasi-fluid approximation. The results are shown in Figure 7. While pre-
dicting the same end position of TPW, the quasi-fluid approximation gives a
much slower transient response.
HU ET AL. A FULL-MAXWELL APPROACH FOR TPW 13
Journal of Geophysical Research: Planets 10.1002/2017JE005365
00.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
x 104
-60
-50
-40
-30
-20
-10
0
Time (ka)
Colatitude (degree)
LT=20km
LT=10km
LT=0km
Figure 8. Apointmassof2×1019 kg attached at the surface at 30
colatitude and removed after 10 Ma. Lines are for models with
lithosphere viscosity of 1031 Pa s and line crosses are for models with
lithosphere viscosity of 1027 Pa s.
Next we show the effect of a viscoelastic lithosphere whose viscosity has a
relaxation time which is comparable with the considered time span. This issue
is important because whether or not the lithosphere can relax during the con-
sidered period can affect the TPW behavior both in the short term (e.g., the
speed) and the long term. The comparison between the effect of an effectively
elastic and a viscoelastic lithosphere on small-range TPW (less than 2) has
been done by Cambiotti et al. (2010) using a linear scheme. Here we show the
effect when large-angle TPW is considered. For this issue Moore et al. (2017)
extend the theory in Chan et al. (2014) and consider the slowest mode(s) sep-
arately while applying the quasi-fluid approximation to the rest of the modes.
Apart from the bias introduced by the quasi-fluid approximation as shown in
Figure 7, another problem of this approach is that for a complex multilayered
model like the SG6 Earth model with its lithosphere viscosity smaller than a
certain value (e.g., 1028 Pa s for SG6 model), the slowest mode might not be
the ̄
T1mode from the lithosphere but one of the Mmodes (M2,M3,) which
is generated by a density difference of the inner layers. Moreover, there might
not be a large difference in the relaxation time between two modes (such as
cases M3 and M4 in Table 3); as a result, it is not clear which modes need to
be modeled separately. Our approach does not have this limitation. Similar to
the case of an effectively elastic lithosphere, we only need to choose a large
enough value for Th. Here the SG6 model with lithospheric thicknesses of 0, 10, and 20 km and with viscosi-
ties of 1027 and 1031 Pa s is used. A mass anomaly of 2×1019 kg is placed at 30colatitude for 10 Ma, then
removed. Within the considered time span of 20 Ma, the lithosphere with viscosity of 1031 Pa s can be con-
sidered as effectively elastic while that with a viscosity of 1027 Pa s is partially relaxed. The result is shown in
Figure 8. We see that the behavior of the TPW is very sensitive to the thickness of the lithosphere. A thicker
lithosphere gives stronger resistance against TPW as well as a faster rebounding of the rotational axis when
the triggering load is removed. Due to the partial relaxation of the ̄
T1mode, which allows the equatorial bulge
Figure 9. TPW on Mars triggered by a mass anomaly of 3.5×1019 kg which is attached at the surface at 45latitude
for four different values of mantle viscosity. This magnitude is about the same as for Q=1of the normalized load
parameter Qdefined in Willemann (1984).
HU ET AL. A FULL-MAXWELL APPROACH FOR TPW 14
Journal of Geophysical Research: Planets 10.1002/2017JE005365
to adjust faster for the high viscosity lithosphere, the TPW for models with a low viscosity lithosphere is larger
and the rotational axis cannot go back to its original position after the mass anomaly is removed.
We also apply our method to a model of Mars to compare the speed of TPW on Mars calculated by the two
methods. We establish a five-layer model whose density and rigidity approximates the model in Zharkov
and Gudkova (2005) which contains a 50 km lithosphere. The viscosity of the model is divided into an effec-
tively elastic lithosphere and a uniform mantle of viscosities 1019 Pa s, 1020 Pa s, 1021 Pa s, and 1022 Pa s
which covers the value used in most recent studies (Breuer & Spohn, 2006; Hauck & Phillips, 2002). We load the
model with a surface mass anomaly of magnitude 3.5×1019 kg (which is about the magnitude for Q=1for
the normalized load parameter Qdefined in Willemann (1984) and used by Chan et al. (2014) and Matsuyama
and Nimmo (2007)) placed at 45latitude. The results are shown in Figure 9. Similar to the Earth model, the
quasi-fluid approximation has a large underestimation of the speed for most of the duration of the TPW.
The instantaneous speed from the full-Maxwell method is, for all four viscosities, about 4.6 times as large as
those obtained based on the quasi-fluid approximation. The reason for this is, as mentioned before, the under-
estimation of the equatorial bulge readjustment when the approximation is adopted. When the rotational
axis approaches its end position, the speed of TPW obtained from the full-Maxwell method drops faster than
for the quasi-fluid approximation which results in the end position being reached later. Generally, the method
based on the quasi-fluid approximation underestimates the time it takes for a mass anomaly to reach its end
position by about half, compared to the full-Maxwell method.
5. TPW on a Model Without Hydrostatic Equilibrium
In practice, a physical model (consisting of layers with given density, rigidity, and viscosity) can be derived from
a geochemical model and the density profile matches the total mass and/or gravitational data. However, it can
happen that the predicted tidal fluid Love number based on the physical model does not match the observed
value for the present-day rotational speed Ω. By assuming that the model is in hydrostatic equilibrium, the
fluid Love number can be estimated from the observed difference in the polar and equatorial moments of
inertia CA(Mound et al., 2003):
kobs
f=3G
a5Ω2(CA)obs (24)
For the Earth, this issue appears to have been studied first by Mitrovica et al. (2005) who introduced the 𝛽
correction term to the tidal fluid Love number when the present-day TPW speed is estimated. Usually, it is con-
sidered that this extra nonhydrostatic contribution stems from mantle convection. Here we simulate another
possible cause for this contribution and its effect on TPW. Before the TPW starts, the lithosphere is not in
hydrostatic equilibrium, or more specifically, the permanent shape of the elastic lithosphere does not match
the present-day rotational speed. The stress-free flattening of the elastic layer can be either larger or smaller
because the rotational speed during the formation of the planet (or moon) was either faster or slower than the
present-day value. We demonstrate here the influence on the TPW of a lithosphere with the same thickness
but in different hydrostatic state.
Since the influence of each relaxation mode siis formulated separately in terms ai,j,p,q(t)and ci,j,p,q(t)in
equations (A1a)–(A1c), we can set both the initial loading potential characterized by 𝜔i(t)and the load-
ing period differently for each relaxation mode to simulate the model being in a different hydrostatic state.
For each relaxation mode si, we set the initial loading period as Tihand the rotational speed as 𝝎i0. In contrast
to equations (16a)–(16c), we now have
T0,i=0(25a)
T1,i=Tih(25b)
W1,0,W2,0,W3,0i=W1,1,W2,1,W3,1i=𝝎i0(25c)
In the following, we show that the choice of the hydrostatic state is as important as the choice of the model.
We demonstrate this concept using SG6 models with a thin (10 km) and thick (20 km) lithosphere. As shown
in Figure 8, when the models are in hydrostatic equilibrium, different lithospheric thicknesses result in sig-
nificantly different TPW behavior. This SG6 model is put into two categories which results in four scenarios
in total.
HU ET AL. A FULL-MAXWELL APPROACH FOR TPW 15
Journal of Geophysical Research: Planets 10.1002/2017JE005365
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
x 104
-45
-40
-35
-30
-25
-20
-15
-10
-5
0
Time (ka)
Colatitude (degree)
I. a
I. b
II. a
II. b
Figure 10. TPW for a point mass of 2×1019 kg attached at the surface
at 30colatitude and removed after 10 Ma. Red and blue lines are
models with an effectively elastic (viscosity 1031 Pa s) lithosphere with
thickness of 10 km and 20 km, respectively, which are in hydrostatic
equilibrium. Green and black lines represent models with a lithosphere
thickness of 10 km and 20 km, respectively, and their lithospheres
initially contribute the same to the moment of inertia as the models
in hydrostatic equilibrium.
Category I: Scenario Ia. A thin-lithosphere model in hydrostatic equilibrium
with the present-day rotational speed.Ib. A thick-lithosphere model whose
slowest mode has relaxed so that its mode strength is equal to that of the thin
lithosphere model:
k1
s1thick
×1es1Th1=k1
s1thin
(26)
Since other modes are much faster than the slowest mode, this Th1is set for
all the modes. In this way, all other modes except the slowest one are fully
relaxed. The partial relaxation of the slowest mode of the thick-lithosphere
model has the same strength as the slowest mode of the thin-lithosphere
model.
Scenario Ib is created to simulate that the rotational speed during the forma-
tion of the lithosphere was slower than present-day.
Category II: Scenario IIa. A thick-lithosphere model in hydrostatic equilibrium
with the present-day rotational speed.IIb. A thin-lithosphere model whose
slowest mode has fully relaxed for a faster rotational speed of
𝝎Thin
10=
k1s1Thick
k1s1Thin 𝝎0(27)
where 𝝎0is the present-day rotational speed. k1s1Thick and k1s1Thin are the mode strengths of the
slowest modes of the thick and thin lithosphere model, respectively.
Scenario IIb corresponds to the situation that the rotational speed during the formation of the lithosphere is
faster than the present-day value. With equations (26) and (27), we configure the scenarios of models without
hydrostatic equilibrium, Ib and IIb, such that the influence of the slowest mode has the same contribution to
the inertia tensor of the entire model at the start of the simulation in each category. We test the model with
an effectively elastic lithosphere (viscosity 1031 Pa s) with the same loading as those in Figure 8. The results
are shown in Figure 10.
As we can see in Figure 10, although the models in each category have different thicknesses of the lithosphere
and hydrostatic states, the performance of TPW is almost identical. This can be understood as follows: when
the lithosphere is thin enough (e.g., less than 100 km for the Earth’s case), the influence of changing the prop-
erties of this layer (thickness, rigidity, and viscosity) is very small on modes other than ̄
T1and ̄
T2generated
from the nonlithosphere part of the model. Its largest effect is, as shown in Figure 4, a remnant bulge which
resists the readjustment of the equatorial bulge. Once we choose a proper hydrostatic state so that the same
contribution to the inertia tensor is guaranteed from the lithosphere of a different model, the path of TPW
will also be almost the same. As for the issue concerning the difference between the observed and model
predicted flattening, it can be seen from equation (27) that if the elastic layer generates a mode with strength
k1s1and the rotational speed during its formation is Ω0, and Ωcat present-day, then this elastic layer would
cause an extra flattening represented in the observed fluid Love number as
Δkobs
f=−
k1
s1Ω0
Ωc2
1(28)
For the Earth model, for instance, considering that 6 Ma ago the rotational speed of Earth was about 15%
faster (Zahnle & Walker, 1987), the rotational speed during the formation of the lithosphere would be even
higher considering that tidal dissipation started when the Earth-Moon system formed about 4.5 Ga ago.
An elastic lithosphere of 80 km with a rotational speed during formation that is 30% faster than the present-
day speed would cause the observed fluid Love number to increase by about 0.01.
We conclude that for the study of TPW for a model with a lithosphere, it is necessary to know both the initial
stress-free shape of the lithosphere as well as its thickness before a correct TPW behavior can be predicted.
HU ET AL. A FULL-MAXWELL APPROACH FOR TPW 16
Journal of Geophysical Research: Planets 10.1002/2017JE005365
This can be significant for the study of TPW on Mars since it has less tectonic activity compared to Earth
to release the stress in its lithosphere, which results in a higher chance that the Mars is not in hydrostatic
equilibrium.
6. Conclusion
We have established a new semianalytical method for calculating large-angle true polar wander (TPW) which
is consistent with the complete scheme of Maxwell rheology, meaning that both fast and slow modes are cor-
rectly taken into account, in contrast with previous studies which adopt the quasi-fluid approximation that
approximate the Love number. We extend the scheme of the linearized Liouville equation in Hu et al. (2017)
which can also be used to simulate the mega-wobble on Venus. Theinfluence of the delayed relaxation of elas-
tic or highly viscous layers as well as models in different hydrostatic states (e.g., not in hydrostatic equilibrium)
has been studied. With the help of our model, the following conclusions can be drawn:
1. The quasi-fluid approximation can introduce a large error in the transient response for a time-dependent
solution. The TPW speed on Earth and Mars based on the quasi-fluid approximation can be underestimated,
while the speed of the rotational axis approaching the end position on Venus is overestimated.
2. Depending on the considered period, the TPW for a model with a viscoelastic lithosphere can have a
larger displacement and a weaker restoring response compared to a model with an (effectively) elastic
lithosphere. Due to the sensitivity of TPW to the shape of the lithosphere, the viscosity of the lithosphere
needs to be taken into account when the studied period is not much shorter than the relaxation time of
the lithosphere.
3. The permanent shape of the elastic lithosphere can have a large effect on the TPW and this can be
an additional explanation for the difference between the observed and model-predicted flattening of a
rotating body.
Thus, studies involving TPW can benefit from our method to achieve better accuracy in the pole path,
especially while taking into account a viscous lithosphere or nonhydrostatic background.Our method can also
be extended to tidally deformed models. In this paper we applied algorithm 2 of Hu et al. (2017) to calculate
the rotational perturbation. In the same paper, a new iterative procedure was also presented to combine
both the rotational and tidal perturbation (algorithm 3). Combining this algorithm with the analytical solution
for the change in the moment of inertia presented in the present paper can give a semianalytical solution for
large-angle reorientations of rotating tidally deformed bodies.
Appendix A: Expressions for Ai,j(t),Bi,j(t),Ci,j(t),andDi,j(t)
kT(t)∗𝜔i(t)𝜔j(t)t=Tn=Ai,jTn+Bi,j(A1a)
dkT(t)∗𝜔i(t)𝜔j(t)
dtt=Tn=Ci,jTn+Di,j(A1b)
where
Ai,j(t)=
n
p=1
m
q=1
ai,j,p,q(t)(A2)
Ci,j(t)=
n
p=1
m
q=1
ci,j,p,q(t)(A3)
and
Bi,j=kT
eWi,nWj,n(A4a)
Di,j=kT
eWi,n1Wj,n+Wi,nWj,n12Wj,n
Tn1Tn
(A4b)
HU ET AL. A FULL-MAXWELL APPROACH FOR TPW 17
Journal of Geophysical Research: Planets 10.1002/2017JE005365
where for t>Tp
ai,j,p,q(t)= kq
Tp1Tp2s3
qesq(tTp1)Wi,nWj,n1TpTp1sq2+2Wj,n
+Wi,n1Wj,n1Tp1TpsqTp1Tpsq+2+2+Wj,nTpTp1sq2
esq(tTp)Wi,n1Wj,nTp1Tpsq2+2Wj,n1
+Wi,nTp1Tpsq2Tp1TpsqWj,n+Wj,n1+2Wj,n
(A5)
ci,j,p,q(t)= kqesqTp1
Tp1Tp2s2
qetsqWi,nWj,n1TpTp1sq2+2Wj,n
+Wi,n1Wj,n1Tp1TpsqTp1Tpsq+2+2+Wj,nTpTp1sq2
esq(Tp1Tp+t)(Wi,n1Wj,nTp1Tpsq2+2Wj,n1
+Wi,nTp1Tpsq2Tp1TpsqWj,n+Wj,n1+2Wj,n
(A6)
and for tTp
ai,j,p,q(t)= kq
Tp1Tp2s3
qWi,nWj,n1sqsqtTp1tTp+TpTp1esq(tTp1)
Tp1Tp+2t2esq(tTp1)+2
+Wj,n2esq(tTp1)1sqtTp1sqtTp1+2
+Wi,n1Wj,n12esq(tTp1)1+sqTp1TpTp1Tpsq+2esq(tTp1)
+tTpsqTpt2
+Wj,nsqsqtTp1tTpTp1Tp+2t+TpTp1sq2esq(tTp1)+2Wj,n
(A7)
ci,j,p,q(t)= kqesqTp1
Tp1Tp2s2
qetsqWi,nWj,n1TpTp1sq2+2Wj,n
+Wi,n1Wj,n1Tp1TpsqTp1Tpsq+2+2)+Wj,nTpTp1sq2
eTp1sq(Wi,nWj,n1sqTp1+Tp2t2+2Wj,nsqtTp1+1
+Wi,n12Wj,n1sqtTp+1+Wj,nsqTp1+Tp2t2
(A8)
Appendix B: Iterative Algorithm for Calculating TPW
1. Assume that step istarts at time tiwith the vector of the rotation being 𝝎i
i(𝜔i
1,𝜔
i
2,𝜔
i
3)and ends at
time ti+1with the vector of the rotation being 𝝎i+1. For the first iteration we assume that the vector of the
rotation does not change: 𝝎i+1=𝝎i.
2. Obtain ΔIand its derivative Δ̇
Ifrom equations (12a) and (12b). With Qas defined in equation (15) being the
coordinate transformation matrix from the body-fixed coordinates to the local coordinates where the Zaxis
aligns with the direction of the rotation, the inertia tensors in the transformed coordinates are obtained by
ΔI1=QTΔIQ and Δ̇
I1=QTΔ̇
IQ.
3. Substitute ΔI1and Δ̇
I1into equations (14a) –(14c) and obtain 𝝎i(m1,m2,1+m3)T. Normalize this vector
as 𝝎i+1̄
𝝎, where ̄
𝝎is the direction of the perturbed rotational axis in the local coordinate system and
transform back into the body-fixed frame to obtain 𝝎i+1i+1Q̄
𝝎.
4. Substitute 𝝎i+1into step 2 until the result converges.
Appendix C: Governing Equation for the Case B<C<Aand B>C>A
For the case B>C>A, we assume that the changes in the inertia tensor are linear:
ΔI13(t)=a1+b1t(C1a)
ΔI23(t)=a2+b2t(C1b)
HU ET AL. A FULL-MAXWELL APPROACH FOR TPW 18
Journal of Geophysical Research: Planets 10.1002/2017JE005365
Then equations (20a)– (20c) can be solved analytically which results in
m1(t)=B
A(CA)3(BC2sinh (CA)(BC)
AB ΩΔt(CAI23(t)−CΔ̇
I13(t)
cosh (CA)(BC)
AB ΩΔtΔI13(t)
CA+CΔ̇
I23(t)
Ω(CA)(BC)
+ΔI13(t)
CA+CΔ̇
I23(t)
Ω(CA)(BC)
(C2)
m2(t)=A
B(CA)(BC)3Ω2sinh (CA)(BC)
AB ΩΔt(CBI13(t)+CΔ̇
I23(t)
+cosh (CA)(BC)
AB ΩΔtΔI23(t)
BCCΔ̇
I13(t)
Ω(CA)(BC)
ΔI23(t)
BCCΔ̇
I13(t)
Ω(CA)(BC)
(C3)
where sinh(u)and cosh(u)are hyperbolic functions
sinh(u)= eueu
2(C4a)
cosh(u)= eu+eu
2(C4b)
When Δtis small enough, we can apply approximation sinh(𝜃)≈𝜃and cosh(𝜃)≈1𝜃22. Ignoring the
derivative terms of the inertia tensor simplifies equations (C2) and (C3) into
m
1(t)= (BC2ΔI13(tt2+2BΩΔI23 (tt
2AB (C5a)
m
2(t)= −(CA2ΔI23(tt22AΩΔI13 (tt
2AB (C5b)
The same procedure can be applied to the case A>C>B, which results in
m′′
1(t)= (CB2ΔI13(tt2+2BΩΔI23 (tt
2AB (C6a)
m′′
2(t)= −(AC2ΔI23(tt22AΩΔI13 (tt
2AB (C6b)
Figure D1. (left) The polar wander path in the x-zplane of the two-layer (blue) and SG6 (red) Earth models triggered by
a mass anomaly of 2×1019 kg attached at 45colatitude in the x-zplane. Lines show the results with the semianalytical
method of Wu and Peltier (1984) (Figure D1, left). Circles represent our method. (right) The polar wander path in the x-z
plane of the two-layer Earth model triggered by a mass anomaly of 2×1019 kg attached at 30colatitude in the x-z
plane. Lines show the results with the semianalytical method of Ricard et al. (1993) (Figure D1, left). Circles represent
our methods.
HU ET AL. A FULL-MAXWELL APPROACH FOR TPW 19
Journal of Geophysical Research: Planets 10.1002/2017JE005365
Appendix D: Validation of the Iterative Algorithm
Figure D1 contains the validation done in Hu et al. (2017). The iterative procedure shown in Appendix B has
been compared for short-term small-angle TPW, where linear method (Wu & Peltier, 1984) is accurate. For
large-angle TPW we compare results against the nonlinear method from Ricard et al. (1993). That method can
only simulate TPW for models without large relaxation time contrast in the modes (Hu et al., 2017). It can be
seen that our iterative procedure matches the results obtained with the methods from Wu and Peltier (1984)
and Ricard et al. (1993) very well.
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Acknowled gments
We thank two anonymous reviewers
for their review and constructive
suggestions. This research has
been financially supported by the
GO program of the Netherlands
Organization for Scientific Research
(NWO), project ALW-GO-12/11. All data
used to produce the figures and the
codes for the algorithm have been
uploaded to 4TU Center for Research
Data (https://data.4tu.nl/) under the
name of this paper.
HU ET AL. A FULL-MAXWELL APPROACH FOR TPW 20
... Such an approximation is commonly referred to as the "quasi-fluid" approach (Lefftz et al., 1991;Ricard et al., 1993). Recently, Hu et al. (2017bHu et al. ( , 2019 analyzed the quasi-fluid approximation, and by comparing the traditional solutions to their full-Maxwell approach they showed that the prediction of the TPW speed on Earth and Mars were underestimated in previous literature. ...
... Hereafter I label this approximation as the ω‖MIA approximation, but it is typically referred to as the "quasi-fluid" approximation as well, because its derivation was linked to the Taylor expansion discussed in the previous paragraph. Hu et al. (2017aHu et al. ( , 2017b develop a new approach, in which they iteratively solve the linearized LE to obtain a large angle TPW. Hu et al. (2017b) argue that the validity of the ω‖MIA approximation "was discussed in detail in Cambiotti et al. (2011) who showed that even for the Earth this assumption is not always appropriate". ...
... Hu et al. (2017aHu et al. ( , 2017b develop a new approach, in which they iteratively solve the linearized LE to obtain a large angle TPW. Hu et al. (2017b) argue that the validity of the ω‖MIA approximation "was discussed in detail in Cambiotti et al. (2011) who showed that even for the Earth this assumption is not always appropriate". However, such a statement may be misleading, because Cambiotti et al. (2011) actually employ the ω‖MIA approximation, see their Equation 7. ...
Article
Full-text available
Almost three decades ago, the problem of long‐term polar wander on a dynamic planet was formulated and simplified within the framework of normal mode theory. The underlying simplifications have been debated ever since, recently in a series of papers by Hu et al. (2017a, https://doi.org/10.1002/2016je005114, 2017b, https://doi.org/10.1002/2017je005365, 2019, https://doi.org/10.1016/j.icarus.2018.08.013), who clarify the role of neglecting short‐term relaxation modes of the body. However, the authors still do not solve the governing equations in full, because they make approximations to the Liouville equation (LE). In this paper, I use a time domain approach and, for previously studied test loads, both the planet's relaxation and the LE are solved in full. In order to analyze the existing LE approximations, I compute the energy balance of true polar wander (TPW). For fast rotating bodies such as Earth, the rotation axis becomes aligned with the main inertia axis (ω‖MIA) once free oscillations are damped. The ω‖MIA assumption is re‐derived theoretically–contrary to previous beliefs, I demonstrate that it is not necessarily linked to the quasi‐fluid limit of the viscoelastic response to loading and rotation, but that it is an expression of neglecting the Coriolis and Euler forces from the equation of motion. It is thus important to distinguish between simplifying the LE and simplifying the planet's response to forcing. For slow rotators such as Venus, the full LE together with energy analysis indicate that previous estimates of TPW rate need to be revisited. The numerical code LIOUSHELL is released on GitHub.
... The mathematical description of TPW includes the Liouville equation which 15 represents the conservation of the angular momentum in a co-rotating frame and the moment of inertia equation which describes the change in the inertia tensor due to the deformation of the body and the mass redistribution. Due to the viscoelasticity of the body, computing the deformation involves convolution of the tidal Love number with the centrifugal and tidal potential, which increases 20 the difficulty to solve these two equations analytically. ...
... In this case, the body acts like an inviscid body. On the other hand, as pointed out in [15], as the change in the load decreases, deformation based on the quasi-fluid approximation approaches the fluid limit solution which itself is close to the complete solution. ...
... This is computationally ulated with an extremely high viscosity (effectively elastic) which results in an extremely long computational time. [15] provided an analytical solution for calculating the inertia tensor without approximating the Love number and studied the effect of the lithosphere, referred to as a remnant bulge, on planetary bodies 65 without a tidal bulge. Adding the tidal potential to the model and combining this analytical expression for the inertia tensor with the algorithm developed in [3] generates a semi-analytical method which can deal with tidally deformed bodies with a lithosphere. ...
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In past studies, the reorientation or true polar wander (TPW) of visco-elastic bodies has been studied with approximated solutions. Two types of methods are commonly adopted: those based on the quasi-fluid approximation e.g. Ricard et al., 1993 and those based on the fluid limit approximation which only consider the final orientation (e.g. Matsuyama and Nimmo, 2007). Recently, Hu et al., (2017b) established a method which provides a dynamic solution for calculating the reorientation of tidally deformed bodies. However, they did not provide the links between the complete solution and the fluid limit solution. This paper provides a semi-analytical method for calculating the reorientation of tidally deformed bodies and shows the relation between the complete and the approximated solutions. Furthermore, we provide a criterion, the fluid limit process number F, to test for a given model and estimated TPW speed if the quasi-fluid approximation or fluid limit solution is valid. This number is a quantitative description of how close the body stays in hydrostatic equilibrium during a reorientation process. We use this number to obtain the largest allowed TPW speed of Mars as a function of viscosity, for which the approximated solutions may be used.
Article
A sea level model which is normally used to calculate sea level changes during glacial-interglacial cycles is modified to solve the approximate nonlinear Liouvelle equation, in order to calculate ‘large’ true polar wander (TPW) that may be induced by ice sheet loading. The purpose is to understand when the TPW will be too large to be solved properly by the linear model, and to properly calculate the TPW induced by snowball Earth events. It is found that the relative error for TPW calculated with a linear model will be >10% when the TPW exceeds ∼2° for ice sheets that develop near the poles, but remains <10% when the TPW exceeds 20° for ice sheets centered at between 45° and 60° latitude. To ensure the relative error for TPW speed to be <10%, the TPW should not exceed 1° and 10° for ice sheets near the poles and 50° latitude, respectively. Because the hypothesized ice sheets during the Neoproterozoic snowball Earth events were located near 55°S for one of the continental configuration, and not much lower than 45°S for the other, the TPW calculated with the linear model in a previous study is overall correct even though its magnitude could be >10° for certain viscosity profiles of the Earth.
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Widespread evidence points to the existence of an ancient Martian ocean1,2,3,4,5,6,7,8. Most compelling are the putative ancient shorelines in the northern plains2,7. However, these shorelines fail to follow an equipotential surface, and this has been used to challenge the notion that they formed via an early ocean9 and hence to question the existence of such an ocean. The shorelines’ deviation from a constant elevation can be explained by true polar wander occurring after the formation of Tharsis10, a volcanic province that dominates the gravity and topography of Mars. However, surface loading from the oceans can drive polar wander only if Tharsis formed far from the equator10, and most evidence indicates that Tharsis formed near the equator11,12,13,14,15, meaning that there is no current explanation for the shorelines’ deviation from an equipotential that is consistent with our geophysical understanding of Mars. Here we show that variations in shoreline topography can be explained by deformation caused by the emplacement of Tharsis. We find that the shorelines must have formed before and during the emplacement of Tharsis, instead of afterwards, as previously assumed. Our results imply that oceans on Mars formed early, concurrent with the valley networks15, and point to a close relationship between the evolution of oceans on Mars and the initiation and decline of Tharsis volcanism, with broad implications for the geology, hydrological cycle and climate of early Mars.
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Existing approaches for simulating the true polar wander (TPW) of a viscoelastic body can be divided into three categories: (i) a linear dynamic approach which uses the linearized Liouville equation (e.g., Wu and Peltier (1984) and Mitrovica et al. (2005)); (ii) a nonlinear dynamic approach which is based on the quasi-fluid approximation (e.g., Sabadini and Peltier (1981), Ricard et al. (1993), and Cambiotti et al. (2011)); and (iii) a long-term limit approach which only considers the fluid limit of a reorientation (e.g., Matsuyama and Nimmo (2007)). Several limitations of these approaches have not been studied: the range for which the linear approach is accurate, the validity of the quasi-fluid approximation, and the dynamic solution for TPW of a tidally deformed rotating body. We establish a numerical procedure which is able to determine the large-angle reorientation of a viscoelastic celestial body that can be both centrifugally and tidally deformed. We show that the linear approach leads to significant errors for loadings near the poles or the equator. Second, we show that slow relaxation modes can have a significant effect on large-angle TPW of Earth or other planets. Finally, we show that reorientation of a tidally deformed body driven by a positive mass anomaly near the poles has a preference for rotating around the tidal axis instead of toward it. At a tidally deformed body which does not have a remnant bulge, positive mass anomalies are more likely to be found near the equator and the plane perpendicular to the tidal axis, while negative mass anomalies tend to be near the great circle that contains the tidal and rotational axes.
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The Tharsis region is the largest volcanic complex on Mars and in the Solar System. Young lava flows cover its surface (from the Amazonian period, less than 3 billion years ago) but its growth started during the Noachian era (more than 3.7 billion years ago). Its position has induced a reorientation of the planet with respect to its spin axis (true polar wander, TPW), which is responsible for the present equatorial position of the volcanic province. It has been suggested that the Tharsis load on the lithosphere influenced the orientation of the Noachian/Early Hesperian (more than 3.5 billion years ago) valley networks and therefore that most of the topography of Tharsis was completed before fluvial incision. Here we calculate the rotational figure of Mars (that is, its equilibrium shape) and its surface topography before Tharsis formed, when the spin axis of the planet was controlled by the difference in elevation between the northern and southern hemispheres (hemispheric dichotomy). We show that the observed directions of valley networks are also consistent with topographic gradients in this configuration and thus do not require the presence of the Tharsis load. Furthermore, the distribution of the valleys along a small circle tilted with respect to the equator is found to correspond to a southern-hemisphere latitudinal band in the pre-TPW geographical frame. Preferential accumulation of ice or water in a south tropical band is predicted by climate model simulations of early Mars applied to the pre-TPW topography. A late growth of Tharsis, contemporaneous with valley incision, has several implications for the early geological history of Mars, including the existence of glacial environments near the locations of the pre-TPW poles of rotation, and a possible link between volcanic outgassing from Tharsis and the stability of liquid water at the surface of Mars.
Book
This volume opens up new perspectives on the physics of the Earth’s interior and planetary bodies for graduate students and researchers working in the fields of geophysics, planetary sciences and geodesy. It looks at our planet in an integrated fashion, linking the physics of its interior to geophysical and geodetic techniques that record, over a broad spectrum of spatial wavelengths and time scales, the ongoing modifications in the shape and gravity field of the planet. Basic issues related to the rheological properties of the Earth and to its slow deformation are considered, in both mathematical and physical terms, within the framework of an analytical relaxation theory. Fundamentals of this theory are developed in the first two Chapters. Chapters 3-9 deal with a wide range of applications, ranging from changes in the Earth’s rotation to post-seismic deformation and from sea-level variations induced by post-glacial rebound to tidal deformation of icy moons of the Solar System. This Second Edition improves substantially our formalism implementing compressibility in viscoelastic relaxation. Chapter 5 now contains new developments in the physics of the gravitational effects of large earthquakes at subduction zones, made possible by new gravity data from space missions. The new Chapter 9 of this Second Edition on deformation and stresses of icy moons enlarges the applications of the book to Planetology, dealing with the additional complications in the theory of viscoelastic relaxation introduced by the shallow low-viscosity zones and inviscid water layers of the moons of Jupiter and Saturn.
Article
We extend previous work to derive a non-linear rotational stability theory governing true polar wander (TPW) of terrestrial planets with viscoelastic lithospheres. We demonstrate, analytically and using numerical examples, that our expressions are consistent with previous results in the limiting cases of low and infinite (i.e., purely elastic) viscosity lithospheres. To illustrate the stability theory, we compute TPW on Mars driven by a simple, prescribed mass loading. Our calculations demonstrate that on short time scales relative to the relaxation time of the viscoelastic lithosphere, the rotation axis follows a constrained path that reflects stabilization by remnant strength in the lithosphere, but that on long times scales this stabilization disappears and the load ultimately reaches the equator. Earlier work based on the assumption of a permanent remnant bulge in the case of a purely elastic lithosphere has suggested that Martian TPW would not persist for any significant time after a load is emplaced, and thus an equilibrium stability theory is sufficient to model long-term (order 1 Myr or longer) polar motion of the planet. Our results suggest, in contrast, that TPW on Mars can continue over time scales on the order of the relaxation time of the lithosphere after load emplacement; for sufficiently high lithospheric viscosities, this time scale may be comparable to the age of the planet.
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Pluto is an astoundingly diverse, geologically dynamic world. The dominant feature is Sputnik Planitia-a tear-drop-shaped topographic depression approximately 1,000 kilometres in diameter possibly representing an ancient impact basin. The interior of Sputnik Planitia is characterized by a smooth, craterless plain three to four kilometres beneath the surrounding rugged uplands, and represents the surface of a massive unit of actively convecting volatile ices (N2, CH4 and CO) several kilometres thick. This large feature is very near the Pluto-Charon tidal axis. Here we report that the location of Sputnik Planitia is the natural consequence of the sequestration of volatile ices within the basin and the resulting reorientation (true polar wander) of Pluto. Loading of volatile ices within a basin the size of Sputnik Planitia can substantially alter Pluto's inertia tensor, resulting in a reorientation of the dwarf planet of around 60 degrees with respect to the rotational and tidal axes. The combination of this reorientation, loading and global expansion due to the freezing of a possible subsurface ocean generates stresses within the planet's lithosphere, resulting in a global network of extensional faults that closely replicate the observed fault networks on Pluto. Sputnik Planitia probably formed northwest of its present location, and was loaded with volatiles over million-year timescales as a result of volatile transport cycles on Pluto. Pluto's past, present and future orientation is controlled by feedbacks between volatile sublimation and condensation, changing insolation conditions and Pluto's interior structure.
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By means of a simple parameter sensitivity analysis, we demonstrate the effect of a low-viscosity layer inserted inside a mantle of a hypothetical Earth on the timescale of large-scale and long-term true polar wander. Here the timescale in our parameter study means the characteristic scale of viscoelastic readjustment of the rotational bulge in the framework of the quasi-fluid approximation for the long-term reorientation of the Earth. Based on this assumption, we calculate the characteristic timescale and associated viscoelastic tidal Love number with the effect of this layer in order to see the dependences on the viscosity, depth, and thickness of the inserted layer. We also compute the characteristic timescale without this layer for the sake of comparison. Our results indicate that the timescale strongly depends on the existence of this layer: positive dependences on its viscosity and depth and a negative dependence on its thickness. We conclude that the low-viscosity layer has a strong impact on the characteristic timescale, especially if this layer exists at the top of the mantle. Although a few previous studies on the small-scale and short-term true polar wander have also suggested a possible effect of inserting a low-viscosity layer, our study implies that the sensitivity to the low-viscosity layer over a long timespan is not necessarily the same as that over a short timespan.
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We present a coupled thermal-magmatic model for the evolution of Mars' mantle and crust that may be consistent with estimates of the average crustal thickness and crustal growth rate. By coupling a simple parameterized model of mantle convection to a batch-melting model for peridotite, we can investigate potential conditions and evolutionary paths of the crust and mantle in a coupled thermal-magmatic system. On the basis of recent geophysical and geochemical studies, we constrain our models to have average crustal thicknesses between 50 and 100 km that were mostly formed by 4 Ga. Our nominal model is an attempt to satisfy these constraints with a relatively simple set of conditions. Key elements of this model are the inclusion of the energetics of melting, a wet (weak) mantle rheology, self-consistent fractionation of heat-producing elements to the crust, and a near-chondritic abundance of those elements. The latent heat of melting mantle material is a small (percent level) contributor to the total planetary energy budget over 4.5 Gyr but is crucial for constraining the thermal and magmatic history of Mars. Our nominal model predicts an average crustal thickness of ~62 km that was 73% emplaced by 4 Ga. However, if Mars had a primary crust enriched in heat-producing elements, consistent with SNC meteorite geochemistry, then our models predict a considerably diminished amount of post 4 Ga crustal emplacement relative to the nominal model. The importance of a wet mantle in satisfying the basic constraints of Mars' thermal and crustal evolution suggests (independently from traditional geomorphology or meteorite geochemistry arguments) that early Mars had a wet environment. Extraction of water from the mantle of a one-plate planet such as Mars is found to be extremely inefficient, such that 90-95% of all water present in the mantle after the initial degassing event should still reside there currently. Yet extraction of even 5% of a modestly wet mantle (~36 ppm water) would result in a significant amount (6.4 m equivalent global layer) of water available to influence the early surface and climate evolution of the planet.
Article
[1] True polar wander (TPW), a reorientation of the rotation axis relative to the solid body, is driven by mass redistribution on the surface or within the planet and is stabilized by two aspects of the planet's viscoelastic response: the delayed viscous readjustment of the rotational bulge and the elastic stresses in the lithosphere. The latter, following Willemann [1984], is known as remnant bulge stabilization. In the absence of a remnant bulge, the rotation of a terrestrial planet is said to be inherently unstable. Theoretical treatments have been developed to treat the final (equilibrium) state in this case and the time-dependent TPW toward this state, including non-linear approaches that assume slow changes in the inertia tensor. Moreover,remnant bulge stabilization has been incorporated into both equilibrium and linearized, time-dependent treatments of rotational stability. We extend the work of Ricard et al. [1993] to derive a non-linear, time-dependent theory of TPW that incorporates stabilization by both the remnant bulge and viscous readjustment of the rotational bulge. We illustrate the theory using idealized surface loading scenarios applied to models of both Earth and Mars. We demonstrate that the inclusion of remnant bulge stabilization reduces both the amplitude and timescale of TPW relative to calculations in which this stabilization is omitted. Furthermore, given current estimates of mantle viscosity for both planets, our calculations indicate that departures from the equilibrium orientation of the rotation axis in response to forcings with timescale of 1 Myr or greater are significant for Earth but negligible for Mars.
Article
Numerical convection models of the thermochemical evolution of Venus are compared to present-day topography and geoid and recent resurfacing history. The models include melting, magmatism, decaying heat-producing elements, core cooling, realistic temperature-dependent viscosity and either stagnant lid or episodic lithospheric overturn. In stagnant lid convection the dominant mode of heat loss is magmatic heat pipe, which requires massive magmatism and produces very thick crust, inconsistent with observations. Partitioning of heat-producing elements into the crust helps but does not help enough. Episodic lid overturn interspersed by periods of quiescence effectively loses Venus's heat while giving lower rates of volcanism and a thinner crust. Calculations predict 5-8 overturn events over Venus's history, each lasting ˜150 Myr, initiating in one place and then spreading globally. During quiescent periods convection keeps the lithosphere thin. Magmatism keeps the mantle temperature ˜constant over Venus's history. Crustal recycling occurs by entrainment in stagnant lid convection, and by lid overturn in episodic mode. Venus-like amplitudes of topography and geoid can be produced in either stagnant or episodic modes, with a viscosity profile that is Earth-like but shifted to higher values. The basalt density inversion below the olivine-perovskite transition causes compositional stratification around 730 km breakdown of this layering increases episodicity but far less than episodic lid overturn. The classical stagnant lid mode with interior temperature ˜rheological temperature scale lower than TCMB is not reached because mantle temperature is controlled by magmatism while the core cools slowly from a superheated start. Core heat flow decreases with time, possibly shutting off the dynamo, particularly in episodic cases.