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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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Journal of Geophysical Research: Planets
A numerical method for reorientation of rotating tidally
deformed viscoelastic bodies
H. Hu1, W. van der Wal1, and L. L. A. Vermeersen1
1Department of Aerospace Engineering, Delft University of Technology, Delft, Netherlands
Abstract 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.
1. Introduction
True polar wander (TPW), the nonperiodical secular part of the displacement of the rotation axis with respect
to surface topography or internal signatures, has been proposed to be the cause of many geographic fea-
tures on various planets and moons (e.g., Mars [Schultz and Lutz, 1988], Venus [Malcuit, 2014], Enceladus
[Nimmo and Pappalardo, 2006], and Europa [Ojakangas and Stevenson, 1989]). The theoretical study of TPW
can be dated back to Gold [1955] who introduced the modern concept and general mechanism of TPW.
After the development of the normal mode method [Farrell, 1972], the Liouville equation could be solved
semianalytically [Sabadini and Peltier, 1981] for a viscoelastic multilayer model to arrive at the dynamic solu-
tion of TPW. Early studies focused on the speed of present-day TPW and small angular change, so a linear
approach, which applies the linearized form of the Liouville equation [Munk and MacDonald, 1960], was
adopted to calculate TPW when the rotational axis is not too far away from the initial position [Nakiboglu
and Lambeck, 1980; Sabadini and Peltier, 1981; Wu and Peltier, 1984]. In order to deal with the long-term rota-
tional variation of Earth which may include large angular TPW, nonlinear methods have been developed, but
they adopt the quasi-fluid approximation which assumes that the variation of the driving force for TPW is
much slower compared to the characteristic viscous relaxation. Mathematically, the quasi-fluid approxima-
tion is a first-order approximation in the Taylor expansion of the tidal Love number [Spada et al., 1992; Ricard
et al., 1993; Cambiotti et al., 2011]. Thus, these semianalytical solutions haveseveral limitations. Specifically, the
approximations which have been adopted in the development of the methods have not been quantitatively
tested. They will be discussed in the following.
First, although some later studies solve the Liouville equation in different ways such as with a finite difference
method [Nakada, 2002; Mitrovica et al., 2005], the linearized form of the Liouville equation is still used and
there is a limit for the allowed range of TPW in order for the error to remain small. In order to show the limit
of the linear method, Sabadini and Peltier [1981], within the frame of the quasi-fluid approximation, carried
out a comparison between the nonlinear scheme and the linear scheme, arriving at the conclusion that the
linear method is valid for TPW of about 10from the initial position of the rotation pole. The linearized form of
RESEARCH ARTICLE
10.1002/2016JE005114
Key Points:
• A numerical method for large-angle
true polar wander is presented
• The applicability of linear rotation
theory and the quasi-fluid
approximation is shown
• A dynamic solution for the
reorientation of tidally deformed
bodies is obtained
Correspondence to:
H. Hu,
h.hu-1@tudelft.nl
Citation:
Hu, H., W. van der Wal, and
L. L. A. Vermeersen (2017), A numerical
method for reorientation of rotating
tidally deformed viscoelastic bodies,
J. Geophys. Res. Planets,122,
doi:10.1002/2016JE005114.
Received 24 JUN 2016
Accepted 3 JAN 2017
Accepted article online 9 JAN 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 NUMERICAL METHOD FOR REORIENTATION 1
Journal of Geophysical Research: Planets 10.1002/2016JE005114
Figure 1. The bulge-fixed coordinate system in a tidally deformed
rotation body. The Xaxis is the tidal axis and points toward the central
body. The Zaxis is the rotation axis, and the Yaxis completes a
right-handed coordinate system. The colored arrows show three
reorientations around the axes, which will be labeled as X,Y,andZ
reorientation.
the Liouville equation is derived in the
body-fixed frame where the rotational
axis coincides with the vertical axis in
the beginning [Munk and MacDonald,
1960]. Since the loading (the inertia ten-
sor representing the geophysical pro-
cess on the solid model) is also defined
in the body-fixed fame, the linear the-
ory actually also assumes that the rela-
tive location of the loading with respect
to the rotational axis does not change
during TPW. This assumption can lead
to a large error for certain locations of
the loading. For instance, when a point
mass is located near the poles or equa-
tor, the effect of a change in colatitude
of the point mass is relatively large. As a
result, the linear methods should have a
much smaller applicable range for load-
ings near the pole or equator. Currently,
no study gives the expected error as a
function of the angle of TPW and the
position of the load when the linearized
form of the Liouville equation is applied.
Second, the nonlinear approach is currently the only general way to calculate large-angle TPW. As a result,
the effect of the quasi-fluid approximation, which has been the fundamental assumption of many previous
studies [Spada et al., 1992, 1996; Ricard et al., 1993; Harada, 2012; Chan et al., 2014], has not been tested. So
it is not clear what the effect is of taking the quasi-fluid approximation and ignoring the effects of the slow
modes (such as the M1 and M2 modes for Earth) on the path of TPW.
Third, a rotating tidally deformed body can be very difficult to deal with by current linear or nonlinear rota-
tion theory. As shown in Figure 1, there are three different reorientations of a tidally deformed body, while
there is only one type of the reorientation when only a centrifugal force is applied. As a result, the complete
description of the reorientation of a tidally deformed rotating body consists of the polar wander of both the
rotational and tidal axes. We are not aware of other methods which solve the Liouville equation to give a
time-dependent solution for the reorientation of a rotating tidally deformed viscoelastic body. Most studies
concerning TPW of a tidally deformed body only focus on the fluid limit of the viscoelastic response which
gives the final position of the rotational or tidal axis [Willemann, 1984; Matsuyama and Nimmo, 2007]. In prac-
tice, it is difficult to know if the TPW or reorientation has already finished and the rotational or tidal axis are
in their final position. This limits the application of methods which only calculate the final position and not
the full reorientation path. More importantly, since these methods do not provide dynamic solutions, we
do not have a clear insight on how the reorientation is accomplished. Studies which concern the direction
of polar wander of tidally deformed bodies driven by either a positive mass anomaly such as ice caps on
Triton [Rubincam, 2003] or a negative mass anomaly such as a diapirism-induced low-density area on
Enceladus [Matsuyama and Nimmo, 2007; Nimmo and Pappalardo, 2006] suggest that the polar motion is
directly targeting its end position. However, these suggestions are not tested in these papers because a theory
for combined centrifugally and tidally induced TPW is lacking.
Considering all above mentioned problems and the difficulty of solving the Liouville equation analyti-
cally, we create a numerical model to tackle these problems. Another advantage of adopting a numerical
approach is that the normal mode method, which is the foundation of all above mentioned dynamic rota-
tion methods, can only be applied for a radially symmetric model, while many planets and moons can have
considerable lateral heterogeneity, for example, Mars [˘
Srámek and Zhong, 2012] or Enceladus [Nimmo and
Pappalardo, 2006].
HU ET AL. A NUMERICAL METHOD FOR REORIENTATION 2
Journal of Geophysical Research: Planets 10.1002/2016JE005114
The purpose of this paper is to establish a general numerical method which can determine the secular part of
the rotational variation of a centrifugally and tidally deformed viscoelastic body. With the help of this method
we aim to answer the following questions:
1. What is the effect of assuming that the load is stationary relative to the rotational axis in the linear method?
2. What is the effect of the quasi-fluid approximation for the TPW path of a planetary model?
3. How is the reorientation of a tidally deformed body driven by a certain mass anomaly accomplished?
It is important to note that we only consider pure viscoelastic bodies without a remnant bulge in this study. For
some planets such as Earth, during the early stages of their formation, the outer layer cools down in an ellip-
soidal shape and becomes fixed. The existence of such a bulge can have a significant effect on the behavior
of TPW. For the case of Earth and some other celestial bodies, this issue has been intensively studied dur-
ing the past decades [Willemann, 1984; Mitrovica et al., 2005; Matsuyama and Nimmo, 2007; Cambiotti et al.,
2010; Mitrovica and Wahr, 2011; Chan et al., 2014]. The existence of a remnant bulge would have two effects.
First, during the TPW, since the stress in the outer layer cannot relax as the rest of the viscoelastic parts, the
remaining stress in this layer would prevent the equatorial bulge to fully adjust into the vertical position to
the rotational axis. Because of this, a positive anomaly, for instance, will not reach the equator as for the case
of a pure viscoelastic body, as is demonstrated in Mitrovica et al. [2005, Figure 14]. Second, when the TPW is
finished or during the TPW, if the mass anomaly which causes the TPW is removed from the body, the stress
in the outer layer would try to restore the shape of the body back into its initial form before the TPW starts, so
the rotational axis would go back to its initial position. This is different from the case of a purely viscoelastic
body in which the rotational axis is expected to retain its final position when the mass anomaly is removed.
As a result, the study of TPW on models with such an elastic layer is significant. However, the numerical pro-
cedures and the validation of such models are beyond the scope and purpose of this paper. So in this paper,
only models without a remnant bulge are considered.
The content is organized as follows: section 2 shows how the change in the inertia tensor can be obtained by
a finite element modeling (FEM). Section 3 presents a numerical method for solving the Liouville equation.
After validating our numerical results with previous semianalytical methods, we test the above mentioned
assumptions. Finally, section 4 presents a method to calculate the reorientation of a tidally deformed rotating
viscoelastic body and shows the cases of a body driven by a positive and negative mass anomaly respectively.
This paper only focuses on the laterally homogeneous case.
2. Finite Element Approach for Calculating the Change in the Moment of Inertia
The Liouville equation gives the general dynamics of a rotational body that can deform. When no external
torque is applied, it reads [Sabadini and Vermeersen, 2004]
d
dt(I𝝎)+𝝎×I𝝎=0(1)
where Iis the inertia tensor and 𝝎is the rotational vector. Both values are defined in a body-fixed coordinate
system. In order to solve this equation, information about the change in the inertia tensor must be given.
When the moments of inertia are perturbed by a geophysical process for a centrifugally deformed body with-
out tidal deformation, the rotational axis shifts, and the resulting change in the centrifugal force also deforms
the body. Analytically, given a rotational vector as 𝝎(𝜔1,𝜔
2,𝜔
3)T, where Ωis the angular speed of the
rotation and (𝜔1,𝜔
2,𝜔
3)Tis a unit vector which represents the direction of the rotation, the total moment of
inertia attributable to such process is given by (similar to equation 2 in Ricard et al. [1993])
Iij(t)=I𝛿ij +kT(t)a5
3GΩ2𝜔i(t)𝜔j(t)− 1
3𝛿ij+[𝛿(t)+kL(t)] ∗ Cij(t)(2)
where Iis the principle moment of inertia of the unloaded laterally homogeneous spherical body and Gand
aare the gravitational constant and the radius of the planet, respectively. kT(t)and kL(t)are the degree 2
tidal Love number and load Love number, respectively. The denotes convolution in the time domain. Cij
represents the change in the moments and products of inertia without considering the dynamic deformation.
These values trigger the polar wander. Thesecond and third terms in equation (2) represent the changes which
derive from the perturbed centrifugal force and from the mass redistribution induced by the original load,
HU ET AL. A NUMERICAL METHOD FOR REORIENTATION 3
Journal of Geophysical Research: Planets 10.1002/2016JE005114
Figure 2. Deformation of a boundary layer whose radius changes from rto
r+ur. The densities inside and outside of the layer are 𝜌iand 𝜌i+1,
respectively.
respectively. The use of Love numbers
limits the simulation to the case of
a laterally homogeneous model, and
thus, we also seek a numerical method
as an alternative for equation (2) to cal-
culate the change in the inertia tensor.
This will be the foundation for deal-
ing with heterogeneous cases in our
second paper.
In order to obtain the change in the
inertia tensor, we need to know the
deformation of the body for which
we can use FEM. This part is mainly
based on the method of Wu [2004].
One of the problems of applying FEM
to calculate the viscoelastic response
of a 3-D celestial body is the effect of
gravitation which is dependent on the
deformation itself. In Wu [2004], the
deformation is first determined by assuming the perturbed potential is zero, and the result from FEM is used
to calculate the perturbed potential. The potential is applied to the model again and iteration continues until
the result converges. We develop a finite element (FE) model with the commercial package Abaqus version
6.13 in which the average grid size for the Earth model is chosen to be around 400 km and linear brick ele-
ments are used. With the information of the deformation the change in the inertia tensor is also calculated
numerically after the result from FEM converges. In the FE model, the Poisson ratio of the planet model can be
set to that of a compressible material, but the effects of a change in density on gravity and inertia are not taken
into account. Hence, our method does not include the full effect of compressibility but only material com-
pressibility (similar to, e.g., Wang et al. [2008]). Since we ignore the density changes, when the deformation is
small, only the radial displacement for each layer is required for calculating the change in the moment of iner-
tia, which is shown in the following method. As we can see in Figure 2, the deformation changes the shapes
of the boundaries which switches the density of certain parts: for the shaded area in Figure 2, the density of
the green parts changes from 𝜌i+1to 𝜌iand the density of blue parts changes from 𝜌ito 𝜌i+1. As a result, for a
model which contains Nlayers, at the pth internal boundary, the change in the inertia tensor is calculated as
ΔIij,p=ΔV
(𝜌p+1𝜌p)(rkrk𝛿ij rirj)dV
S
(𝜌p+1𝜌p)(rkrk𝛿ij rirj)urdS,p=0,1,2,,N1
and at the surface
ΔIij,N=S
(𝜌N)(rkrk𝛿ij rirj)urdS(3)
Here ΔVis the perturbed volume which contains the above mentioned density switch and Sis the complete
interface or the surface. The complete change of the inertial tensor is given by the sum of the changes at all
interfaces and at the surface:
ΔIij =
N
p=0
ΔIij,p(4)
Tab le 1. Properties of the Two-Layer Earth Model
Layer Outer Radius (km) Density (kg m3) Shear Modulus (Pa) Viscosity (Pa s)
Mantle 6,371 4,448 1.7364 ×1011 1×1021
Core 3,480 10,977 0 0
HU ET AL. A NUMERICAL METHOD FOR REORIENTATION 4
Journal of Geophysical Research: Planets 10.1002/2016JE005114
Figure 3. Change in the moment of inertia for a two-layer Earth model with the rotation axis linearly drifting from 0 to
45colatitude in the x-zplane in 5000 years.
We validate our FE model by calculating the change in the inertia tensor of a two-layer Earth model (Table 1)
which is forced by a varying centrifugal force.
We apply the centrifugal force to an initially unloaded model and let the rotational axis move toward the
equator with a constant speed of 45in 5000 years. The change in the moment of inertia for this case can be
calculated by equation (2) with Cij(t)=0. The comparison between the semianalytical and FEM results is given
in Figure 3. For the nonzero components I11,I22 ,I33,and I13 , the numerical results show very good agreement
with the semianalytical results. FEM results of I12 and I23 are also nonzero while they should be theoretically
zero. However, the numerical results of these two values are about 4 orders of magnitude smaller than the
other four components in the inertia tensor. Thus, these values result in a numerical error which is around
0.1% for our configuration of a mesh with an average grid size of 400 km for the Earth model.
As will be shown in the next section in the linearized Liouville equation and algorithm 2, the accuracy
of the TPW is controlled by four terms which are combinations of components of the inertia tensor and
the angular speed in a coordinate system whose zaxis coincides with the rotational axis: ΔI13 (t)
CA,ΔI23(t)
CBand
CΔ
.
I13(t)
Ω(CA)(CB),CΔ
.
I23(t)
Ω(CA)(CB). In order to show that the change in the moment of inertia obtained from FEM is accu-
rate enough for calculating TPW, we compare both the analytical and numerical values of these four terms
for a given TPW history. As shown in Figure 4, the theoretical nonzero terms ΔI13(t)
CAand CΔ
.
I13(t)
Ω(CA)(CB)show very
good agreement. For a grid size of 300 km, the ΔI13(t)
CAand CΔ
.
I13(t)
Ω(CA)(CB)terms have less than 0.5% error level. We
see again that two theoretical zero components are at least 4 orders of magnitude smaller than the nonzero
components. It was found that in order to get results close to the analytic result which is based on Maxwell
rheology, in Abaqus, a “Viscoelastic” option needs to be used. The viscoelastic setting in Abaqus uses the
Prony series which is a general scheme that encompasses a simple Maxwell rheology. We show this issue in
Appendix A.
HU ET AL. A NUMERICAL METHOD FOR REORIENTATION 5
Journal of Geophysical Research: Planets 10.1002/2016JE005114
Figure 4. Values of four terms for a SG6 Earth model with the rotation axis linearly drifting from 0 to 45 degree
colatitude in the x-z plane in 10 thousand years after the centrifugal force is applied to the spherical model for 20
thousand years.
3. Numerical Solutions of Liouville Equation
With the information about the change in the inertia tensor obtained either by equation (2) or FEM, the
Liouville equation can be solved numerically. Cases with small angular change, to which linear theory can
be directly applied, and large angular change will be dealt with separately. We validate our numerical meth-
ods by comparing the results with semianalytical linear [Wu and Peltier, 1984] and nonlinear [Ricard et al.,
1993] methods with the same assumptions. After that, we test the validity of the assumptions made in these
methods.
3.1. Small-Angle Polar Wander
Considering that we want to deal with lateral heterogeneity in paper II and tidally deformed bodies, we first
need to derive a more general form of the linearized Liouville equation. The procedure is similar to that given
on page 104 of Sabadini and Vermeersen [2004].
In equation (1), when assuming that changes in Iare small, the perturbed inertia tensor can be written as
I=
AI11 ΔI12 ΔI13
ΔI21 BI22 ΔI23
ΔI31 ΔI32 CI33
(5)
Here A,B, and Cdenote the moments of inertia of the rotational body for the equatorial principal axes and
polar principal axis. We do not assume A=Bas in Sabadini and Vermeersen [2004]. We define the perturbed
vector of the rotation as
𝝎(m1,m2,1+m3)T(6)
HU ET AL. A NUMERICAL METHOD FOR REORIENTATION 6
Journal of Geophysical Research: Planets 10.1002/2016JE005114
where Ωis the angular speed of the rotation and mi,i=1,2,3are small values with m1and m2representing
the TPW and m3the change in the length of the day (LOD). By omitting the products of the small quantities
miand ΔIij,wehave
I𝝎
AΩm1ΔI13
BΩm2ΔI23
CΩ+Cm3Ω+ΔI33 Ω
(7)
𝝎×(I𝝎)≈
Bm2Ω2+Cm2Ω2−ΔI23 Ω2
Am1Ω2Cm1Ω2I13 Ω2
0
(8)
Substituting equations (7) and (8) into equation (1) we have
.
m1=−
CB
AΩm2+Ω
AΔI23 Δ
.
I13
A(9a)
.
m2=CA
BΩm1Ω
BΔI13 Δ
.
I23
B(9b)
.
m3=−
Δ
.
I33
C(9c)
Note that now we cannot define the Eulerian free precession frequency as 𝜎r=CA
AΩto further combine
these equations. Equation (9) gives the rotational dynamics of a triaxial body for a small perturbation which
has been previously studied by Hinderer et al. [1982] and Matsuyama et al. [2010] for the case of Earth.
Matsuyama et al. [2010] made the assumptions that the time scale of the loading is much longer than both
the period of rotation and the Euler wobble periods. Based on these assumptions, the derivatives on both
side of equation (9) are ignored. The same procedure is also used in Sabadini and Vermeersen [2004]. These
assumptions might be true for Earth but not for some slow rotating bodies like Venus. In order to establish
a more general method, we cannot directly ignore these derivative terms. Instead, we take advantage of the
fact that numerically, the TPW is calculated stepwise and deal with equation (9) as follows: In each step of the
numerical integration, we assume that the size of the step is small enough so that the change in the inertia
tensor can be treated as linear, which gives
ΔI13(t)=a1+b1t(10a)
ΔI23(t)=a2+b2t(10b)
After substituting (10) into (9), m1,m2can be solved analytically. The results contain both secular terms and
periodic terms, which represent the TPW and the Chandler wobble, respectively. We ignore the periodical
terms and obtain
m1(t)= ΔI13(t)
CA+CΔ
.
I23 (t)
Ω(CA)(CB)(11a)
m2(t)= ΔI23(t)
CBCΔ
.
I13 (t)
Ω(CA)(CB)(11b)
m3(t)=−
ΔI33
C(11c)
Besides the Aand Bterms, the equations also contain the derivatives of the elements of the inertia tensor.
When only the centrifugal force is considered for a laterally homogeneous model, (CA)∕C=(CB)∕C
represents the flattening of the model and the magnitude of (CA)∕Cis proportional to the square of the
rotational rate Ω2. As a result, the magnitudes of the second terms on the right side become significantly
larger for slowly rotating bodies such as Venus. When the magnitude of the second terms on the right side
becomes comparable to that of the first terms, it results in the phenomenon of so-called mega wobble [Spada
et al., 1996; Sabadini and Vermeersen, 2004] as shown on the right Figure 5a. For most of the bodies in the solar
system including Earth, long-term TPW acts as in Figure 5b. In this case the part which contains the derivatives
HU ET AL. A NUMERICAL METHOD FOR REORIENTATION 7
Journal of Geophysical Research: Planets 10.1002/2016JE005114
Figure 5. Two types of polar wander path. The green arrow is the initial position of the rotational axis and the red dot is
the point mass load.
of the changes in the inertia tensor is small and the path of TPW driven by a point mass is almost along the
great circle of the body. So if we place a point mass load in the x-zplane, the TPW can be almost completely
described by the value m1.
Since we study the TPW of bodies in the hydrostatic state,the centrifugal force needs to be applied for a certain
length of time T0until the model can be considered to have reached its equilibrium. For laterally homoge-
neous models, the choice of T0can be derived from the time history of the tidal Love number. We choose T0
to be the time at which the tidal Love number reaches more than 99.95% of the fluid tidal Love number:
kT(T0)>99.95%kT
f(12)
Here kT
fis the fluid tidal love number. For the two-layer Earth model of Table 1 it follows that T0=15 ka.
In the FE model, we apply a centrifugal force at its original rotational axis for T0before we start to apply the
algorithm to calculate the path of TPW. If equation (2) is used, then we have
𝝎(t)=(0,0,Ω) for 0tT0(13)
where Ωis the angular velocity of the body. For a centrifugally deformed body triggered by a mass anomaly
with inertia tensor ΔIL, the algorithm for calculating the small-angle polar wander and LOD m=(m1,m2,m3)
is given as follows.
Algorithm 1
1. Assume that the step istarts at time tiwith the rotational axis being located at 𝝎i
i(mi
1,mi
2,1+mi
3)and
ends at time ti+1with the rotational axis at 𝝎i+1. For the first iteration we assume that the rotation axis does
not change: 𝝎i+1=𝝎i.
2. For a laterally homogeneous model we use equation (2) for calculating the change in the inertia tensor. In
equation (2), set Cij(t)=ΔIL_ij and let
𝝎(t)=𝝎i+tit
titi+1
𝝎i+1for titti+1(14)
then the result of equation (2) can directly give the total change in the inertia tensor ΔIand its derivative Δ̇
I.
For a laterally heterogeneous model we use FEM to obtain the inertia tensor. We change the centrifugal
potential from its initial direction along 𝝎iat tilinearly to its new direction of 𝝎i+1at ti+1in the FEM and
calculate the change in the inertia tensor ΔIDand its derivative Δ̇
IDdue to centrifugal deformation and
HU ET AL. A NUMERICAL METHOD FOR REORIENTATION 8
Journal of Geophysical Research: Planets 10.1002/2016JE005114
Tab le 2. Properties of the Viscoelastic Earth Model SG6
Layer Outer Radius (km) Density (kg m3) Shear Modulus (Pa) Viscosity (Pa s)
Lithosphere 6,371 4,120 0.73 ×1011
Upper mantle 6,271 4,120 0.95 ×1011 0.6×1021
Transition zone 5,950 4,220 1.10 ×1011 0.6×1021
Shallow lower mantle 5,700 4,508 2.00 ×1011 1.6×1021
Deeper lower mantle 5,040 4,508 2.00 ×1011 3×1021
Core 3,480 10,925 0 0
surface load by equation (4). The total change in the inertia tensor is the sum of that due to the deformation
and the tensor of the initial load: ΔIIDIL,Δ̇
I
̇
IḊ
IL.
3. Substitute ΔIand Δ̇
Iinto equation (11) and obtain the updated 𝝎i+1. This value is fed back into step 2 until
the result converges.
The small-angle numerical results are compared with results from the linear semianalytical method of Wu and
Peltier [1984]. In that paper, the Chandler wobble is filtered out by assuming that all rotational modes have
a much longer relaxation time than the Chandler wobble. By taking s≪𝜎
0, where sand 𝜎0are the Laplace
frequency and Eulerian free precession frequency, respectively, the imaginary, harmonic part of the funda-
mental mantle mode (M0) which contributes most to the Chandler wobble, is omitted. It has been proven
that the elastic term of equation (79) in Wu and Peltier [1984] is a highly accurate approximation of the effect
on TPW of the real part of the M0 mode [Vermeersen and Sabadini, 1996; Peltier and Jiang, 1996]. So for small
angular motion, since the method stated in Wu and Peltier [1984] contains the effects of all modes, we expect
that it gives an accurate prediction of the TPW on a layered viscoelastic model against which our method
can be benchmarked. We test our method both with the two-layer Earth model (Table 1) and the six-layer
Earth model SG6 as defined in Table 2. We calculate the corresponding Love number of this model by set-
ting the viscosity of the lithosphere to an extremely high value but exclude the slowest mode generated
by this layer. This scenario corresponds to the situation where the elastic layer exists before the centrifugal
potential is applied to the spherical Earth, as demonstrated in Figure 14 of [Mitrovica et al., 2005] as case
B. Of course this situation does not correspond to the real Earth (cases C and D in the same figure) which
has a remnant bulge. However, as mentioned in the introduction, the purpose here is method development
and validation, and we leave the effect of a remnant bulge in our method to future work. The models are
driven by a constant point mass of 2×1019 kg which is attached to the surface at 45colatitude. We assume
Figure 6. 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 semianalytical method of Wu and Peltier [1984], and circles
represent our numerical ones.
that the point mass is stationary at
the surface, so in equation (2) kL=0.For
the SG6 model, the initial time T0for
which the centrifugal force needs to be
applied is chosen to be 4 million years
according to equation (12). The numer-
ical results for both models agree very
well with the prediction of Wu and Peltier
[1984], see Figure 6. From this figure, we
can also see the effect of the delayed vis-
cous adjustment of the rotational bulge.
This stabilization is larger for layers with
higher viscosity, and this is why TPW of
SG6 model driven by the same anomaly
is slower.
Usually 7–8 iterations in each step are
necessary for the use of equation (2),
and 9–10 for FEM with equation (4)
are required to achieve an accuracy of
0.1%. The required number of iterations is
reduced for smaller step sizes.
HU ET AL. A NUMERICAL METHOD FOR REORIENTATION 9
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The agreement of our numerical method and the method from Wu and Peltier [1984] shows the validity of the
assumption s≪𝜎
0. However, as mentioned in section 1, there is one problem with the method in [Wu and
Peltier, 1984]: the loading itself is assumed to be stationary relative to the rotational axis and is not updated
by the polar motion. For the case of TPW due to a mass anomaly which is shown in Figure 6, during the polar
wander, the mass anomaly is calculated in the body-fixed coordinates, which means it is treated as being
always located at 45colatitude. However, when the rotation axis drifts away from the mass anomaly by 1,in
that instantaneous moment, the mass anomaly is actually placed at 46colatitude. Of course, when only very
small angle TPW is considered, the difference can be small but the exact effect has not been studied. We will
show the magnitude of the error in the next section with a new method for calculating large-angle TPW.
3.2. Large-Angle Polar Wander
The limitation of the method in Wu and Peltier [1984] and the numerical method presented in the previous
section is that they are based on the Liouville equation which is linearized at the position where the zaxis of
the coordinate system is the rotational axis and the equatorial bulge is perpendicular to this axis. As a result,
this method cannot deal with large-angle TPW when the rotation axis drifts away from this position. However,
if we assume that during the process of polar wander the equatorial readjustment is fast enough (or the
polar wander is slow enough) so that the equatorial bulge is always nearly (but not necessarily exactly)
perpendicular to the rotational axis, then we can define a new reference frame in which the new zaxis coin-
cides with the current rotational axis and we can linearize the Liouville equation in the new coordinate system.
Physically, the process of TPW is the process of the rotational axis moving toward the axis of the maximum
moment of inertia while the axis of the maximum moment of inertia is being pushed further away by the
viscoelastic relaxation of the body and the displacement of the mass anomaly. What we assume is that the
angle between these two axes is small enough so that the linearization of the Liouville equation is valid. This
assumption is fundamentally different from assuming that during the TPW the rotational axis and the princi-
ple axis of the moment of inertia coincide [Jurdy, 1978; Rouby et al., 2010; Steinberger and O’Connell, 1997]. The
validity of this assumption is studied and discussed in detail in the study of Cambiotti et al. [2011] which devel-
ops, within the frame of a nonlinear approach for TPW induced by internal mass anomalies, a linear scheme of
the Liouville equation in the system of the principal moments of inertia reference frame of the mass anomaly.
Apparently, this assumption can be violated by a situation where the TPW is triggered by a very large mass
anomaly which corresponds to an inertia tensor that is comparable in magnitude to the inertia tensor of the
rotational body itself. In this case, the angle between the largest moment of inertia (the sum of the inertia
tensors of both rotating body and the mass anomaly) and the rotational axis would be too large to apply the
linearized Liouville equation. One advantage of our numerical method is that during the calculation, we can
constantly monitor the validity of this assumption as will be shown by the end of this section. Generally, we
can do a coordinate transformation in each step and apply the method we used for small angular change in
the new coordinate system so that the local angular change in each step remains small enough.
We define the vector of the rotation as 𝝎(𝜔1,𝜔
2,𝜔
3)T, where (𝜔1,𝜔
2,𝜔
3)is a unit vector. For an arbitrary
𝝎, the TPW which starts from this vector needs to be calculated in the frame whose Zaxis coincides with 𝝎.So
we need to transform the original body-fixed coordinates into this new frame . The coordinate transformation
matrix of a rotation from the vector (0,0,1)to the unit vector 𝝎can be obtained from a general rotation matrix
[Arvo, 1992] in which the third column of the matrix is (𝜔1,𝜔
2,𝜔
3)T.
Q=
𝜔3+𝜔2
2
1+𝜔3
𝜔1𝜔2
1+𝜔3
𝜔1
𝜔1𝜔2
1+𝜔3
1𝜔2
2
1+𝜔3
𝜔2
𝜔1𝜔2𝜔3
(15)
For a centrifugally deformed body triggered by a mass anomaly which corresponds to the inertia tensor ΔIL,
the algorithm for calculating the large-angle TPW is as follows:
Algorithm 2
1. Assume that the 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 𝝎i+1. For the first iteration, we assume that the vector of the rotation
does not change: 𝝎i+1=𝝎i.
HU ET AL. A NUMERICAL METHOD FOR REORIENTATION 10
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2. Obtain ΔIand its derivative Δ̇
Ifrom FEM or using equation (2) in the same way as step 2 in algorithm 1. 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 equation (11) and obtain 𝝎
i(m1,m2,1+m3)T. We normalize this vector as
𝝎
i+1̄
𝝎where ̄
𝝎is the direction of the perturbed rotational axis in the local coordinate system and
needs to be transformed back into the body-fixed frame to obtain 𝝎i+1
i+1Q̄
𝝎where Ωi+1is the same
as in the previous equation.
4. Substitute 𝝎i+1into step 2 until the result converges.
There are two major differences between algorithms 1 and 2. First, in algorithm 2, the rotational perturbation
is calculated in a transformed coordinate system instead of the original body-fixed frame in each step. Second,
the initial load ΔILis also updated in each step in response to the change of the rotational axis. As we can see
in step 2 in algorithm 2, since ΔIcontains both the change in the moment of inertia due to deformation ΔID
and the initial load ΔIL,wehaveQTΔIQ =QTΔIDQ+QTΔILQ.SoQTΔILQinstead of ΔILis used as the input
for the driving factor of the TPW. In this way we lift the assumption of a stationary load as in Wu and Peltier
[1984] and algorithm 1.
For validation purposes, we test if algorithm 2 can produce the same result as algorithm 1 for a small angle
when we disable the updating of ΔIL. This means in step 2 of algorithm 2, we only do a coordinate transform
for the inertia tensor due to deformation but keep the one for the loading the same, so the total change for
the inertia tensor is calculated as ΔI1=QTΔIDQILinstead of ΔI1=QTIDIL)Q. Then the condition
is the same as in Wu and Peltier [1984] and algorithm 1. To show the effect of the assumption of a stationary
load, we also calculate the result with the original algorithm 2 (ΔILis updated). The comparison of the semi-
analytical result from Wu, our numerical result without updating the loading and the numerical result with
updated loading is shown in Figure 7. When the loading is not updated in each step, the numerical and the
normal mode results show perfect agreement. This validates algorithm 2 as well as the assumption that the
equatorial readjustment in this case is fast enough to catch up the polar wander. On the other hand, when
the loading is updated in each step, as we can see in Figure 7, the normal mode result overestimates the TPW
by about 2.5% for 2of TPW. This is understandable, as the mass anomaly has its largest loading effect when it
is at 45(or 135) colatitude. When the positive mass anomaly is attached at the surface at 60colatitude, as the
TPW proceeds, the mass anomaly moves toward the equator and the loading effect decreases. As a result, the
speed of TPW slows down. Thus, with the method of Wu and Peltier [1984], depending on whether or not
the TPW is displacing the mass anomaly toward or away from 45latitude, the result can be either underes-
timated or overestimated, respectively. The bias becomes much larger if the mass anomaly is close to 0, 90,
and 180colatitude: If we place the same mass anomaly at 10colatitude, after a polar wander of 2the error
can be up to 12 % with the stationary loading assumption. Consequently, the applicable range of the linear
method becomes even smaller when the loading is close to poles or the equator. The comparison between
the result from Wu and Peltier [1984] and the updated linear method (algorithm 2) is similar to that in the
Figure 3 of Sabadini and Peltier [1981] which compares the TPW path on a homogeneous viscoelastic sphere
from both a linear and nonlinear scheme. One apparent difference is the lack of an elastic response in the
results of Sabadini and Peltier [1981]. Figure 8 shows how large the TPW can be as a function of colatitude in
order to keep the error below 1.5%.
As can be derived from Figure 8, for the situation when the initial load is close to the pole or equator, the
applicable range of the linear theory is quite limited. As a consequence, results obtained from linear rotation
theory may need to be reconsidered for studies such as TPW on Earth driven by ice loss from Greenland or
Antarctica, since these areas are close to the poles.
Next we test the behavior of our numerical method for large-angle polar wander and compare the result with
the method of Ricard et al. [1993]. Ricard et al. assume that the Earth model has no internal nonadiabatic
HU ET AL. A NUMERICAL METHOD FOR REORIENTATION 11
Journal of Geophysical Research: Planets 10.1002/2016JE005114
Figure 7. 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 60colatitude. The line, red circles, and blue crosses represent the semianalytical result from Wu and Peltier
[1984], our numerical result without updating loading, and the numerical result with updated loading, respectively.
density gradients (no M1 or M2 modes) and with s<< si, the tidal Love number is approximated as
kT(s)=kT
e+
M
i=1
kT
i
ssi
kT
e
M
i=1
kT
i
si
=kT
f(1T1s)
(16)
where kT
eis the elastic Love number, kT
iare the residues of each mode, and siare the inverse relaxation times.
The time constant T1is
T1=1
kT
f
M
i=1
kT
i
s2
i
(17)
This assumption, which is called the quasi-fluid approximation, is actually the first-order approximation of the
tidal Love number. It assumes that the relaxation time of every mode is much shorter than the time span for
long-term polar wander.With this approximation, the nonlinear equation (1) can be simplified into a first-order
Figure 8. The allowed range of polar wander in order to obtain less than 1.5% error as a function of the colatitude of the
loading.
HU ET AL. A NUMERICAL METHOD FOR REORIENTATION 12
Journal of Geophysical Research: Planets 10.1002/2016JE005114
Figure 9. The polar wander in x-zplane of the two Earth models triggered by a point mass of 2×1019 kg placed at 30
colatitude in the x-zplane. Lines show the semianalytical results from Ricard et al. [1993], and symbols represent our
numerical results (only half of the data points for the two-layer model and one tenth of the SG6 data points are shown
in the figures).
differential equation for 𝜔(t)and solved numerically. We first validate the accuracy of our numerical method
by calculating TPW for the two-layer Earth model (Table 1). This is because a two-layer Earth model, which
is a viscoelastic layer over a fluid core, does not contain the relatively slow modes (M1 and M2 modes). So
the quasi-fluid approximation is reliable in this case and the TPW calculated by the method in Ricard et al.
[1993] can be expected to be accurate. The comparison of our numerical and the semianalytical results for a
two-layer model is shown in Figure 9a. As we can see, the two methods have very good agreement. Except for
the first few points, where the difference is due to the elastic response that is missing in Ricard et al.’s results,
the differences are below 0.5%.
Then we use both methods to calculate TPW for the six-layer SG6 model. This model approximates the real
Earth better and also contains the slow M1 and M2 modes, which allows us to test the quasi-fluid approxi-
mation. The results are shown in Figure 9b. It is clear that for the SG6 model, the two methods show large
differences and the polar wander given by Ricard et al. [1993] is much slower due to the lack of the relaxation
from the M1 and M2 modes. To further validate our results and rule out the cause of numerical error, we also
compare the semianalytical results from both Wu and Peltier [1984] and Ricard et al. [1993] for short-angle
changes. As shown in Figure 10, we see again that in the two-layer model, despite the lack of the elastic
response which gives the initial jump in Ricard et al.’s results, both results stay almost parallel. However, for
the SG6 model, Ricard et al.’s results, which lack the contribution from M1 and M2 modes, lag behind from the
beginning.
Figure 10. Polar wander in the x-zplane of the two Earth models
triggered by a point mass of 2×1019 kg placed at 45colatitude in the
x-zplane, for two different semianalytical methods.
After long enough time results from
Ricard et al. [1993] will converge to the
same end position as the numerical
one, but the large difference in transient
behavior suggests that the quasi-fluid
approximation is not a good choice for
obtaining a time-dependent solution.
The numerical method we developed in
this section is very general since the only
assumption we take is that the equato-
rial readjustment is fast enough that the
equatorial bulge is almost always perpen-
dicular to the rotational axis. This means
that the largest principle axis for the
moment of inertia must nearly coincide
with the rotational axis, so the inertia ten-
sor in the coordinate system where the
Zaxis is the direction of the rotation is
close to a diagonal matrix. We can check
if this condition is satisfied during the
HU ET AL. A NUMERICAL METHOD FOR REORIENTATION 13
Journal of Geophysical Research: Planets 10.1002/2016JE005114
numerical calculation by comparing the diagonal elements in the inertial tensor with the nondiagonal ones:
in the transformed coordinate system, the condition ΔI
ij,ijΔI
ii must be satisfied. Tests show that for
TPW on two-layer Earth model with magnitudes of the mass anomalies below 2×1022 kg (this amount is
about 100 times that of the ice sheets melted during the last deglaciation [Ricard et al., 1993]), this condition
is satisfied. Only when the model is driven by an even larger mass anomaly, this condition fails by a significant
amount and the linear and nonlinear methods do not agree any more like in Figure 9a.
4. Reorientation of a Rotating Tidally Deformed Viscoelastic Body
As mentioned in section 1, Willemann [1984] and Matsuyama and Nimmo [2007] presented a solution which
only calculates the final position of the reorientation. We are not aware of a general dynamic solution for the
reorientation of a tidally deformed body. Two major difficulties prevent applying the existing rotation theory,
linear or nonlinear, to a tidally deformed body. First, the principle inertia moments Aand Bare not equal in this
case. Second, it is difficult to combine the effects of the centrifugal and tidal potential so that the deformed
body and load can achieve the minimal potential state throughout the reorientation process. In the previous
section we have already solved the first problem by deriving a more general linearized form of the Liouville
equation (equation (11)). The main focus for the development of the method in this section is on how the tidal
potential is treated and how the centrifugal and tidal potential can be combined.
When the reorientation of a tidally deformed body is studied, it is necessary that not only the rotational axis is
considered but also the direction of the tidal axis which is the vector pointing to the central body. In this paper,
we only consider the situation in which the rotational body is tidally locked in a circular orbit so the body is
co-rotating with its central body and the direction of the tidal axis is always perpendicular to the rotational
axis (the obliquity or axial tilt is zero).
For an incompressible model, the effective centrifugal potential is [Murray and Dermott, 2000]
Φc=1
3Ω2r2P0
2(cos𝜃)(18)
with 𝜃being the colatitude and P0
2the associated Legendre function of degree 2, order 0. The tidal potential
due to the central body at the same point can be written as [Murray and Dermott, 2000]
Φt=−
GM
a3r2P0
2(cos𝜓)(19)
Here G,M, and aare the gravitational constant, the mass of the central body, and the radius of the orbit,
respectively. 𝜓is the angle between the radius vector and the direction of the tidal bulge. Generally, if we
define the equivalent angular speed of the tidal potential as
Ω=3GM
a3(20)
then the form of the tidal potential becomes the same as the centrifugal potential except for the negative
sign. When the rotational body is tidally locked in a circular orbit, then the rotational period is the same as the
orbital period which is T=2𝜋a3GM, from which it is easy to see that the magnitude of the tidal potential
is always 3 times the magnitude of the centrifugal potential.
Because of the negative sign, the effect of applying a tidal potential to a certain object is the same as applying
the centrifugal potential of the same magnitude but with opposite direction of the force. As a result, a positive
mass anomaly driven by a centrifugal potential acts exactly like a negative mass anomaly driven by a tidal
potential and vice versa. A centrifugal force alwayst ries torelocate a positive mass anomaly to the equator and
a negative mass anomaly to the poles to minimize the total potential, while a tidal potential tries to relocate
the positive mass anomaly to the subhost point (the closest point on the body to the central body) or its
antipodal and a negative mass anomaly to the great circle which is perpendicular to the direction of the tidal
bulge. In order to calculate the change of the inertia tensor due to both the rotational and tidal potential,
in FEM we need to add the tidal force to the model and apply equation (4). If equation (2) is used instead
of the FE model, then we need to add an extra term for the contribution of the tidal potential. Since the
effect of the tidal potential is exactly the same as the centrifugal potential except the direction, similar to the
rotational vector, we can define a tidal vector which describes the strength and direction of the tidal force
HU ET AL. A NUMERICAL METHOD FOR REORIENTATION 14
Journal of Geophysical Research: Planets 10.1002/2016JE005114
Figure 11. Bulge-fixed coordinate system where the Xand Zaxes are aligned with the direction of the tidal axis and the
rotational axis, respectively. m1,m2,andm3correspond to the perturbation of the rotational axis. m
1,m
2,andm
3
correspond to the perturbation of the tidal axis.
as X
(𝜔
1,𝜔
2,𝜔
3)T, where Ωis equivalent angular speed and (𝜔1,𝜔
2,𝜔
3)Tis a unit vector. We have the
perturbed inertia tensor due to tidal potential as
ΔIT_ij(t)=−
kT(t)a5
3GΩ2𝜔
i(t)𝜔
j(t)− 1
3𝛿ij (21)
This term needs to be added to the right side of the equation (2). Notice the negative sign because in the
case of the tidal potential, the direction of the force to the body is opposite to the centrifugal potential, so the
perturbed inertia tensor is also negative.
In previous sections, the assumption was stated that the centrifugal potential is applied along the Zaxis. Since
the centrifugal force flattens the body, we have C>A,B. On the other hand, if we treat the tidal potential in
the same way and apply it along the Zaxis, the tidal force would elongate the body and we have C<A,B.
For equation (11) to be valid, the conditions CAand CBare required. So we can use equation (11) to
calculate the perturbation of both rotational and tidal axes. We define a moving bulge-fixed coordinate sys-
tem in which the Zaxis aligns with the direction of rotation and the Xaxis with the tidal bulge, as shown in
Figure 11. In order to use equation (11), the direction of the tidal potential needs to be along the third axis
of the coordinates (just as the rotation axis is along the Zaxis of (X,Y,Z) frame). So the “polar” wander of the
tidal axis is calculated in the coordinate system (X,Y,Z) where Xand Zalign with the negative Z and X axes,
see Figure 11. The transformation matrix from (X,Y,Z)to(X,Y,Z) is written as
S=
001
01 0
10 0
(22)
If the inertia tensor for both the triggering load and the deformation and its derivative is obtained in the
(X,Y,Z)frame as ΔIand Δ̇
I, these values are to be used to calculate the rotational axis change. Then the
corresponding values for the tidal axis change are
ΔIT=−STΔIS (23)
Δ̇
IT=−STΔ̇
IS (24)
ΔITand Δ̇
ITare substituted into equation (11) to determine the local change of the tidal axis. When both
the centrifugal and tidal potential are applied to the body perturbed by a certain load, as can be seen from
Figure 11, the centrifugal force tries to relocate the rotational axis from Z=(0,0,Ω)Tto
Z(m1,m2,1+m3)T(25)
HU ET AL. A NUMERICAL METHOD FOR REORIENTATION 15
Journal of Geophysical Research: Planets 10.1002/2016JE005114
and the tidal force tries to push the tidal axis from X=(Ω
,0,0)to
X
1+m
3,m
2,m
1T(26)
Here Ωis the equivalent angular speed defined by equation (20). The problem here is that these two con-
ditions cannot be simultaneously satisfied since the new Xand Zaxes should also be perpendicular to each
other because the feature of the tidal axis being orthogonal to the rotation axis does not change due to the
reorientation. Considering that the X-Zplane is the one which constrains the directions of both rotational
and tidal axes, this plane has to satisfy both conditions. So we have to make a compromise of the conditions:
we let one of the axes, either Xor Z, to be relocated at the exact desired location of equation (26) or (25) but
the other one only lies within the new X-Zplane defined by vector Xand Z.Let ̄
Xand ̄
Zbe the normalized
vectors of Xand Z, then we can calculate the new Yaxis which is perpendicular to the X-Zplane as
̄
Y=̄
Z×̄
X(27)
Here ×is the cross product. If we put the Xaxis at x, then the new Zaxis which lies in the X-Zplane is
determined as
̄
Z′′ =̄
X×̄
Y(28)
and the coordinate transformation matrix from the frame (X,Y,Z) to the frame (x0,y0,z
0)is given by
V=[
̄
X,̄
Y,̄
Z′′](29)
Physically, the relocation of Xand Zaxesminimizes the total potential of the body and the initial mass anomaly
related to the tidal and centrifugal force, respectively. Our method first achieves the minimization of the
potential corresponding to one of the forces, then the other. When the relocation of Xaxis converges, then
the change of the Xaxis becomes nearly 0, namely, ̄
X≈(1,0,0). As a result, from equations (28) and (29), we
have ̄
Z′′ ̄
Z. So eventually, the minimal potential state associated with both the centrifugal and tidal force is
found. This is similar to multiple-objective optimization [Miettinen, 1999]. Of course we can also first put the
Zaxis at Zand then we calculate the new Xaxis as ̄
X′′ =̄
Y×̄
Zand obtain the coordinate transform matrix
V=[
̄
X′′,̄
Y,̄
Z]. Tests show that in either way, the results converge to the same final position of both rotational
and tidal axes in each step.
For a tidally deformed body triggered by a certain mass anomaly which corresponds to inertia tensor ΔIL, the
complete algorithm for calculating the reorientation is given as follows:
Algorithm 3
1. Assume that the step i, from time tito ti+1, starts with the direction of the rotational axis given by
𝝎i
r
i
r(𝜔i
1,𝜔
i
2,𝜔
i
3)Tand the direction of the tidal axis by 𝝎i
t
i
t(𝜔i
4,𝜔
i
5,𝜔
i
6)Tin which (𝜔i
1,𝜔
i
2,𝜔
i
3)Tand
(𝜔i
4,𝜔
i
5,𝜔
i
6)Tare unit column vectors which satisfy 𝜔i
1𝜔i
4+𝜔i
2𝜔i
5+𝜔i
3𝜔i
6=0.Ωi
tis the equivalent angular
speed of the tidal potential calculated from equation (20). For the first iteration, we assume that the rotation
and tidal axes in this step do not change: 𝝎i+1
r=𝝎i
rand 𝝎i+1
t=𝝎i
t.
2. Apply both the centrifugal and tidal potential to the model in the same way as stated in step 2 in algorithm 1
to either FE model (equation (4)) or use equation (2) and add the term from equation (21). Obtain the total
change in the inertia tensor and its derivative as ΔIand Δ̇
I. The coordinate transformation matrix from the
body-fixed to the bulge-fixed coordinate system is given by
U=𝝎i+1
t,𝝎i+1
r×𝝎i+1
t,𝝎i+1
r(30)
The local values of the inertia tensor for the centrifugal part are obtained by ΔI1=UTΔIU and
Δ̇
I1=UTΔ̇
IU. The corresponding inertia tensors for calculating tidal perturbation are ΔI2=−STΔI1Sand
Δ̇
I2=−STΔ̇
I1S
3. Apply equation (11) to ΔI1and Δ̇
I1and obtain the perturbation for the rotational axis as Ω1(m1,m2,m3).
Apply equation (11) to ΔI2and Δ̇
I2and obtain the perturbation for the tidal axis as Ω2(m
1,m
2,m
3). Then
we have the perturbed Zand Xaxis as Z
1(m1,m2,1+m3)Tand X
2(1+m
3,m
2,m
1)T.We
normalize these vectors as Z
i+1
r̄
Zand X
i+1
t̄
X. The local coordinate transformation matrix from
HU ET AL. A NUMERICAL METHOD FOR REORIENTATION 16
Journal of Geophysical Research: Planets 10.1002/2016JE005114
Tab le 3. Properties of Triton
Layer Outer Radius (km) Density (kg m3) Shear Modulus (Pa) Viscosity (Pa s)
Ice I 1352 937 3.6×1091×1021
Ice II 1100 1193 6.2×1091×1021
Mantle 950 3500 65 ×1091×1019
Core 600 5844.8 0 0
the bulge-fixed frame at time tito the new frame at time ti+1is obtained as V=[
̄
X,̄
Z×̄
X,̄
X×(̄
Z×̄
X)].The
updated direction of the rotational and tidal axes in the original body-fixed coordinates are obtained as
𝝎i+1
r
i+1
rUV[0,0,1]T(31a)
𝝎i+1
t
i+1
tUV[1,0,0]T(31b)
4. Substitute 𝝎i+1
rand 𝝎i+1
tin step 2 until the results converge.
In order to show how the reorientation of a tidally deformed rotating body is accomplished, we choose a
model of Triton, the largest moon of Neptune. Triton is an icy moon and tidally locked. It has zero obliquity and
small orbital eccentricity which is about 1.6×105and can be ignored, so it fits the situation of our assumption.
The interior structure is chosen according to the empirical model presented in Spohn et al. [2014] which fits
observations of Triton’s mass and moment of inertia. Depending on the amount of internal heating, there can
be an ocean between the high pressure and low pressure ice [Spohn et al., 2014]. For simplicity, the effect of a
possible ocean is ignored. The physical properties of the model are shown in Table 3.
To trigger the reorientation, a surface mass anomaly with a magnitude of 3.6×1017 kg, either positive or
negative, is chosen. This amount is approximately the accumulation of nitrogen snow during 10,000 years
which is sublimated from the equatorial area, then moves to the polar areas where it is deposited [Rubincam,
2003]. We simulate the polar wander of Triton for two cases: (1) a positive mass anomaly at high latitude (20
colatitude) and (2) a negative mass anomaly at low latitude (60colatitude). In each case, the mass anomaly is
placed at three different longitudes: 15, 45, and 75. Due to the symmetry, only situations in one quadrant
are considered. The initial loading time T0and the time span for the TPW are both chosen to be 4 million years
at which, from normal mode method, the time history of the tidal Love number reaches about 99.75% of its
fluid Love number, following equation (12). With this range of time, the reorientation should be close to its
equilibrium position. In order to better describe the three types of the reorientation as shown in Figure 1, in
the bulge-fixed frame we define the reorientation of the tidal deformed rotating body around the Z,X, and
Yaxes as the Zreorientation, Xreorientation, and Yreorientation, respectively. The results for the cases of
positive mass anomalies are shown in Figures 12 and 13.
Figure 12. Reorientation caused by positive mass anomalies placed at 20colatitude, 15, 45, and 75 longitudes.
(a) The traces (lines) of the positive mass anomalies (filled circles) in the bulge-fixed frame where the sub-Neptune point
is at 0longitude. (b) The traces of the north pole of Triton (lines) and sub-Neptune point (dashed lines) in the
body-fixed frame.
HU ET AL. A NUMERICAL METHOD FOR REORIENTATION 17
Journal of Geophysical Research: Planets 10.1002/2016JE005114
Figure 13. Reorientation caused by positive mass anomalies placed at 20colatitude, 15, 45, and 75longitudes.
The time history for the colatitudes (solid lines) and longitudes (dashed lines) of the north pole. (b) The time history for
the colatitudes (solid lines) and longitudes (dashed lines) of the sub-Neptune point.
Figure 12a gives the paths of the mass anomalies in the bulge-fixed coordinates. We can see that when the
positive mass anomaly is placed at high colatitude, the reorientation first pushes the mass anomalies toward
the equator with an Xreorientation. When the mass anomaly is close to the equator, a Zreorientation fol-
lows and eventually the mass anomaly tries to reach the sub-Neptune point. Due to the Zreorientation, in
the body-fixed coordinates as shown in Figure 12b, the pole does not drift away from the mass anomalies in a
straight line; instead, it moves closely along the great circle which is perpendicular to the tidal axis. This is dif-
ferent compared to the case of polar wander on a centrifugally deformed body such as Earth. FromFigures 13a
and 13b, by comparing the speed of the pole in colatitude direction and that of the subhost point in longitu-
dinal direction, we can see that the Xreorientation is much faster than the Zreorientation. This indicates that
it is much easier to reorient the tidal bulge around the tidal axis than the rotational axis. So the direction of the
polar wander due to unbalanced ice caps on Triton is more likely to go around the tidal axis instead of going
toward it. As a result, the suggestion that the direction of the polar wander for the viscoelastic case of Triton
would be toward the sub-Neptune point when the reorientation starts [Rubincam, 2003] does not seem to
be correct.
Cases of negative anomalies are shown in Figures 14 and 15. These cases apply to situations like the south
polar terrain of Enceladus, in which Nimmo and Pappalardo [2006] suggests that the diapirism of the lower
density material creates a negative mass anomaly which is relocated to the south pole due to the reorientation
of Enceladus. In contrast with the case of positive mass anomaly, we see in Figure 14a that the reorientation
has a slight preference to first push the mass anomaly to the great circle where both tidal and rotational
axes are located or to the 0 and 180 degree longitude circle. This indicates that the Zreorientation is slightly
faster than the Yreorientation, so the direction of both rotational and tidal axes are still not directly targeting
their end positions. Figures 15a and 15b shows that except for the case where the negative mass anomaly is
Figure 14. Reorientation cased by negative mass anomalies placed at 60colatitude, 15, 45, and 75 longitudes. The
traces (lines) of the negative mass anomalies (filled circles) in the bulge-fixed frame where the sub-Neptune point is at
0longitude. The traces of the north pole (lines) in the body-fixed frame.
HU ET AL. A NUMERICAL METHOD FOR REORIENTATION 18
Journal of Geophysical Research: Planets 10.1002/2016JE005114
Figure 15. Reorientation cased by negative mass anomalies placed at 60colatitude, 15, 45, and 75longitudes.
The time history for the colatitudes (solid lines) and longitudes (dashed lines) of the north pole and the time history for
the colatitudes (solid lines) and longitudes (dashed lines) of the sub-Neptune point.
close to the ±90 longitude (case with the green color), there is no drastic speed change. This is quite different
compared to cases with positive anomalies which always begin with a relatively fast Xreorientation.
From both cases of positive and negative anomalies, we see that for a tidally deformed rotating body, the
Xreorientation is much faster than the Zreorientation, while the Zreorientation is slightly faster than the Y
reorientation. Such preference for the direction of the reorientation can be explained in the following way.
Since the tidal potential is larger than the centrifugal potential, diagonal elements of the inertia tensor in the
bulge-fixed frame satisfies A<B<Cand CB<BA. The rotation of the same small angle around X,Y, and
Zaxes changes the diagonal inertia tensor Iinto QT
iIQi, where Qi,i=1,2,3, are the transformation matrices
for rotations around X,Y, and Zaxes. These transformations produce nondiagonal elements with magnitude
ΔI23,ΔI13 , and ΔI12, respectively, and it is easy to prove that they have the relation: ΔI23 <ΔI12 <ΔI13. These
cross products represent the resistance of the bulge, either rotational or tidal, against the polar (tidal) wan-
der, so the reorientation around the Xaxis is the fastest while that around the Yaxis is the slowest. From the
track of both positive and negative mass anomaly in various positions shown in Figures 12 and 14, we can
also conclude that except for the six dead zones where both centrifugal and tidal force are either very small
or in equilibrium (the areas around the poles, the subhost point and its antipode, and the two points facing
the orbit), positive mass anomalies are more likely to be found around the equator and the great circle per-
pendicular to the tidal axis while negative mass anomalies tend to be around the 0 and 180longitude great
circle in the bulge-fixed frame.
It is also worth to mention that our method can be extended to situations where the obliquity or orbit eccen-
tricity is nonzero. In these cases, we need to change the transformation matrix Sgiven by equation (22), which
would become time dependent and needs to be updated according to the position of the body in the orbit
and the relative location of the rotational and tidal axes in each step of the numerical calculation.
5. Conclusions
Numerical methods for calculating both small- and large-angle reorientation of a centrifugally and tidally
deformed viscoelastic body are established. The methods are validated by comparing with existing nor-
mal mode methods which were developed for both small-angle and large-angle TPW. With the help of the
developed numerical methods, the following conclusions can be drawn:
1. Linear rotation theory leads to a bias which can be very large when the initial position of the mass anomaly
causing the true polar wander (TPW) is close to the poles or equator. This significantly limits the applicable
range of the linear method if loads are close to poles or equator.
2. The time-dependent result of TPW obtained by taking the first-order approximation of the tidal love num-
ber, namely, the quasi-fluid approximation, gives large errors for the transient behavior and only when the
model is close to its final orientation, results taking quasi-fluid approximation give reliable prediction. This
makes quasi-fluid approximation not a good choice for studying transient viscoelastic readjustment of Earth
or other planets which contain significant slow relaxation modes.
3. A tidally deformed body has a preference of the reorientation around the tidal axis over that around the
rotational axis. The rotational axis driven by a positive mass anomaly near the poles tends to first rotate
around the tidal axis instead of toward it. For tidally locked bodies which do not have a remnant bulge,
HU ET AL. A NUMERICAL METHOD FOR REORIENTATION 19
Journal of Geophysical Research: Planets 10.1002/2016JE005114
positive mass anomalies are more likely to be found around the equator and the great circle perpendicular
to the tidal axis, while negative mass anomalies tend to be near the great circle that contains the tidal and
rotational axes.
Appendix A
Since the results from the FEM are validated by comparing with the analytical results obtained from a nor-
mal mode method which is based on Maxwell rheology, the material properties need to be defined in FEM
such that the viscoelastic response of the material is equivalent to that of a Maxwell material. In the Abaqus
FEM package, the viscoelastic property of the material is defined in the following way: (1) the initial elasticity
is defined separately by giving the Young’s modulus in the option “Elasticity.” (2) the normalized viscoelastic
behavior can be defined either with the “Creep” option which uses power law strain-hardening or a “Vis-
coelastic” option which uses the Prony series which is a general scheme that encompasses a simple Maxwell
rheology. In Abaqus, Prony series expansion is defined by the dimensionless relaxation modulus gRas
gR(t)=1
N
i=1
gP
i(1et𝜏G
i)(A1)
where N,gP
i, and 𝜏G
iare material constants. As the equivalence of Maxwell rheology, we have N=1,gP
1=
1110 (Abaqus requires that gP
i<1, so a value very close to 1 is chosen) and 𝜏G
1=𝜇Ewhere 𝜇and Eare
material viscosity and elasticity.
Both options give similar results with the same accuracy for the time history of the tidal Love number or the
individual components of the inertia tensor (Figure 3). However, when the terms which determine the TPW
(like in Figure 4) are calculated, which are the combinationsof the components of the iner tia tensor, the results
obtained with the Creep option, as can be seen in Figure A1, show a much larger error compared to those
obtained with the option Viscoelastic as shown in Figure 4. This demonstrates that the Viscoelastic option in
Abaqus is a better choice to represent a Maxwell material. This suggests that also comparisons between results
from Abaqus and spectral models [Wu and van der Wal, 2003; vanderWaletal., 2015] might be improved.
Figure A1. Result of the same test as that in Figure 4 with the viscous deformation being defined with the option Creep
in Abaqus.
HU ET AL. A NUMERICAL METHOD FOR REORIENTATION 20
Journal of Geophysical Research: Planets 10.1002/2016JE005114
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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 pro-
gram of the Netherlands Organization
for Scientific Research (NWO). We are
grateful to Hermes Jara Orué for useful
discussions during the development
of this paper. All data used to produce
the figures, the input files for the finite
element model, and the codes for the
three algorithms can be obtained from
the author (email: h.hu-1@tudelft.nl).
HU ET AL. A NUMERICAL METHOD FOR REORIENTATION 21
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Satellite and recent Earth-based observations of Io's surface reveal a specific spatial pattern of persisting hotspots and sudden high-intensity events. Io's major heat producing mechanism is tidal dissipation, which is thought to be non-uniformly distributed within Io's mantle and asthenosphere. The question arises to what extent Io's non-homogeneous heat production can cause long-wavelength variations in the interior and volcanic activity at the surface. We investigate dissipation patterns resulting from two different initially spherical symmetric visco-elastic rheological structures, which are consistent with geodetic observations. The spatial distributions of the time-averaged tidal heat production are computed by a finite element model. Whereas for the first rheological structure heat is produced only in the upper viscous layer (asthenosphere-heating model), the second rheological structure results in a more evenly distributed dissipation pattern (mixed-heating model) with tidal heating occurring in the deep mantle and the asthenosphere. To relate the heat production to the interior temperature and melt distribution, we use steady-state scaling laws of mantle convection and a simple melt migration model. The resulting long-wavelength thermal heterogeneities strongly depend on the initial tidal dissipation pattern, the thickness of the convective layer, the mantle viscosity, and the ratio between magmatic and convective heat transport. While for the asthenosphere-heating model a strong lateral temperature signal with up to 190 K peak-to-peak difference can remain, convection within a thick convective layer, as for the mixed-heating model, can reduce the lateral temperature variation to <1 K, if the mantle viscosity is sufficiently low. Models with a dominating magma heat transport preserve the long-wavelength pattern of tidal dissipation much better and are favoured, because they are better to explain Io's thick crust. The approach presented here can also be applied to investigate the effect of an arbitrary interior heating pattern on Io's volcanic activity pattern.
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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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Seismic data indicate that there are large viscosity variations in the mantle beneath Antarctica. Consideration of such variations would affect predictions of models of Glacial Isostatic Adjustment (GIA), which are used to correct satellite measurements of ice mass change. However, most GIA models used for that purpose have assumed the mantle to be uniformly stratified in terms of viscosity. The goal of this study is to estimate the effect of lateral variations in viscosity on Antarctic mass balance estimates derived from the Gravity Recovery and Climate Experiment (GRACE) data. To this end, recently-developed global GIA models based on lateral variations in mantle temperature are tuned to fit constraints in the northern hemisphere and then compared to GPS-derived uplift rates in Antarctica.
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A recently proposed model links the formation and early evolution of the Tharsis volcanic province on Mars to the preexisting hemispheric dichotomy (Zhong, 2009). A key aspect of this model is the assumption of a deep lithospheric root below the thicker crust of the southern highlands. We implemented a parameterization of partial melting into the 3-D spherical shell mantle convection code CitcomS in order to investigate whether the required lithospheric thickness variation can be generated self-consistently by partial melting when stiffening of the melt residue due to devolatilization is considered. The rate of melt production strongly depends on the mantle temperature, and additional strong coupling between the flow and partial melting is introduced through the stiffening effect on the melt residue. We find that it is possible to generate a lithospheric keel by partial melting above a single upwelling that excites a relative rotation between the one-plate lithosphere and the mantle below while producing the amount of melt distributed in a broad region constrained to one hemisphere that is necessary to form the crustal dichotomy. This scenario thus offers an internal mechanism for the Martian dichotomy formation and validates the hypothesis of Zhong (2009).
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This book explains how it came to be that Venus and Earth, while very similar in chemical composition, zonation, size and heliocentric distance from the Sun, are very different in surface environmental conditions. It is argued here that these differences can be accounted for by planetoid capture processes and the subsequent evolution of the planet-satellite system. Venus captured a one-half moon-mass planetoid early in its history in the retrograde direction and underwent its “fatal attraction scenario” with its satellite (Adonis). Earth, on the other hand, captured a moon-mass planetoid (Luna) early in its history in prograde orbit and underwent a benign estrangement scenario with its captured satellite.
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The Solar System is a complex and fascinating dynamical system. This is the first textbook to describe comprehensively the dynamical features of the Solar System and to provide students with all the mathematical tools and physical models they need to understand how it works. It is a benchmark publication in the field of planetary dynamics and destined to become a classic. Clearly written and well illustrated, Solar System Dynamics shows how a basic knowledge of the two- and three-body problems and perturbation theory can be combined to understand features as diverse as the tidal heating of Jupiter's moon Io, the origin of the Kirkwood gaps in the asteroid belt, and the radial structure of Saturn's rings. Problems at the end of each chapter and a free Internet Mathematica® software package are provided. Solar System Dynamics provides an authoritative textbook for courses on planetary dynamics and celestial mechanics. It also equips students with the mathematical tools to tackle broader courses on dynamics, dynamical systems, applications of chaos theory and non-linear dynamics.
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The static deformation of an elastic half-space by surface pressure is reviewed. A brief mention is made of methods for solving the problem when the medium is plane stratified, but the major emphasis is on the solution for spherical, radially stratified, gravitating earth models. Love-number calculations are outlined, and from the Love numbers, Green's functions are formed for the surface mass-load boundary-value problem. Tables of mass-load Green's functions, computed for realistic earth models, are given, so that the displacements, tilts, accelerations, and strains at the earth's surface caused by any static load can be found by evaluating a convolution integral over the loaded region.