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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 10∘from 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(I⋅𝝎)+𝝎×I⋅𝝎=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=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

3G∗Ω2𝜔i(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+1−𝜌p)(rkrk𝛿ij −rirj)dV

≃∫S

(𝜌p+1−𝜌p)(rkrk𝛿ij −rirj)urdS,p=0,1,2,…,N−1

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 m−3) 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

45∘colatitude 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 45∘in 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)

C−A,ΔI23(t)

C−Band

CΔ

.

I13(t)

Ω(C−A)(C−B),CΔ

.

I23(t)

Ω(C−A)(C−B). 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)

C−Aand CΔ

.

I13(t)

Ω(C−A)(C−B)show very

good agreement. For a grid size of 300 km, the ΔI13(t)

C−Aand CΔ

.

I13(t)

Ω(C−A)(C−B)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=

A+ΔI11 ΔI12 ΔI13

ΔI21 B+ΔI22 ΔI23

ΔI31 ΔI32 C+ΔI33

(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=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

I⋅𝝎≈

AΩm1+ΩΔI13

BΩm2+ΩΔI23

CΩ+Cm3Ω+ΔI33 Ω

(7)

𝝎×(I⋅𝝎)≈

−Bm2Ω2+Cm2Ω2−ΔI23 Ω2

Am1Ω2−Cm1Ω2+ΔI13 Ω2

0

(8)

Substituting equations (7) and (8) into equation (1) we have

.

m1=−

C−B

AΩm2+Ω

AΔI23 −Δ

.

I13

A(9a)

.

m2=C−A

BΩm1−Ω

BΔI13 −Δ

.

I23

B(9b)

.

m3=−

Δ

.

I33

C(9c)

Note that now we cannot deﬁne the Eulerian free precession frequency as 𝜎r=C−A

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)

C−A+CΔ

.

I23 (t)

Ω(C−A)(C−B)(11a)

m2(t)= ΔI23(t)

C−B−CΔ

.

I13 (t)

Ω(C−A)(C−B)(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, (C−A)∕C=(C−B)∕C

represents the ﬂattening of the model and the magnitude of (C−A)∕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 0≤t≤T0(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=Ω

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+ti−t

ti−ti+1

𝝎i+1for ti≤t≤ti+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 m−3) 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: ΔI=ΔID+ΔIL,Δ̇

I=Δ

̇

ID+Δ̇

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 45∘colatitude. 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 45∘colatitude 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

Journal of Geophysical Research: Planets 10.1002/2016JE005114

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 45∘colatitude. However, when the rotation axis drifts away from the mass anomaly by 1∘,in

that instantaneous moment, the mass anomaly is actually placed at 46∘colatitude. 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=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

1−𝜔2

2

1+𝜔3

𝜔2

−𝜔1−𝜔2𝜔3

(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=Ω

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

Journal of Geophysical Research: Planets 10.1002/2016JE005114

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′=Ω

i(m1,m2,1+m3)T. We normalize this vector as

𝝎′=Ω

i+1̄

𝝎′where ̄

𝝎′is 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=Ω

i+1Q̄

𝝎′where 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=QTΔIDQ+ΔILinstead of ΔI1=QT(ΔID+ΔIL)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 2∘of 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 60∘colatitude, 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 45∘latitude, 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 180∘colatitude: If we place the same mass anomaly at 10∘colatitude, after a polar wander of 2∘the 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 60∘colatitude. 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

s−si

≈kT

e−

M

i=1

kT

i

si

=kT

f(1−T1s)

(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 45∘colatitude 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,i≠j≪ΔI′

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=2𝜋a3∕GM, 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

′(𝜔′

1,𝜔

′

2,𝜔

′

3)T, where Ω′is 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

3G∗Ω′2𝜔′

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 C≠Aand C≠Bare 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 X′and Z′align 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=

00−1

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′=Ω(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′=Ω

′1+m′

3,m′

2,−m′

1T(26)

Here Ω′is 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 X′and Z′.Let ̄

X′and ̄

Z′be the normalized

vectors of X′and 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 Z′and then we calculate the new Xaxis as ̄

X′′ =̄

Y′×̄

Z′and 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=Ω

i

r(𝜔i

1,𝜔

i

2,𝜔

i

3)Tand the direction of the tidal axis by 𝝎i

t=Ω

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′=Ω

1(m1,m2,1+m3)Tand X′=Ω

2(1+m′

3,m′

2,−m′

1)T.We

normalize these vectors as Z′=Ω

i+1

r̄

Z′and X′=Ω

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 m−3) 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=Ω

i+1

rUV[0,0,1]T(31a)

𝝎i+1

t=Ω

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 1.6×10−5and 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 (60∘colatitude). 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 20∘colatitude, −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 0∘longitude. (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 20∘colatitude, −15, −45, and −75∘longitudes.

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 60∘colatitude, −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

0∘longitude. 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 60∘colatitude, −15, −45, and −75∘longitudes.

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 C−B<B−A. 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 180∘longitude 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(1−e−t∕𝜏G

i)(A1)

where N,gP

i, and 𝜏G

iare material constants. As the equivalence of Maxwell rheology, we have N=1,gP

1=

1−1−10 (Abaqus requires that gP

i<1, so a value very close to 1 is chosen) and 𝜏G

1=𝜇∕Ewhere 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