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Journal of Geophysical Research: Planets

A Full-Maxwell Approach for Large-Angle Polar

Wander of Viscoelastic Bodies

H. Hu1, W. van der Wal1, and L. L. A. Vermeersen1

1Department of Aerospace Engineering, Delft University of Technology, Delft, Netherlands

Abstract For large-angle long-term true polar wander (TPW) there are currently two types of nonlinear

methods which give approximated solutions: those assuming that the rotational axis coincides with the

axis of maximum moment of inertia (MoI), which simpliﬁes the Liouville equation, and those based on the

quasi-ﬂuid approximation, which approximates the Love number. Recent studies show that both can have

a signiﬁcant bias for certain models. Therefore, we still lack an (semi)analytical method which can give

exact solutions for large-angle TPW for a model based on Maxwell rheology. This paper provides a method

which analytically solves the MoI equation and adopts an extended iterative procedure introduced in

Hu et al. (2017) to obtain a time-dependent solution. The new method can be used to simulate the eﬀect

of a remnant bulge or models in diﬀerent hydrostatic states. We show the eﬀect of the viscosity of the

lithosphere on long-term, large-angle TPW. We also simulate models without hydrostatic equilibrium

and show that the choice of the initial stress-free shape for the elastic (or highly viscous) lithosphere of a

given model is as important as its thickness for obtaining a correct TPW behavior. The initial shape of the

lithosphere can be an alternative explanation to mantle convection for the diﬀerence between the observed

and model predicted ﬂattening. Finally, it is concluded that based on the quasi-ﬂuid approximation, TPW

speed on Earth and Mars is underestimated, while the speed of the rotational axis approaching the end

position on Venus is overestimated.

Plain Language Summary The North and South Poles of the Earth are slowly moving. This is

because a large mass can form on the Earth’s surface on geologic time scales, which changes how the

Earth rotates. Examples of large masses are ice sheets and large mountains. This phenomenon of moving

poles takes place on many planets or moons and is referred to as polar wander. Polar wander can be the

explanation for why we ﬁnd surface features at certain locations. For instance, we often observe mountains

near the equator of a planet or moon. This is usually the consequence of polar wander: these mountains

could have formed anywhere on the planetary body, but due to the polar wander, they eventually end up

on the equator. This is the case for the Tharsis plateau on Mars. The mathematical description of polar

wander on planetary bodies is diﬃcult, and only approximated solutions have been obtained in the past.

Our study establishes a new method which can give an accurate prediction of how the polar wander

proceeds through time. We show that previous studies have incorrectly estimated the speed of polar

wander on Mars and Venus. With the new method, polar wander on a wide range of planetary bodies can

be simulated. Predictions of our method can be compared to observations of surface features to get a

better understanding of the interior structure of planets and moons.

1. Introduction

Concerning the study of large-angle true polar wander (TPW) on a viscoelastic body such as terrestrial planets

like the Earth, Mars, and Venus, there are currently two types of nonlinear approaches to obtain a time-

dependent solution. One of them is from Nakada (2007) which applies an iterative scheme but simpliﬁes the

Liouville equation by ignoring the time derivative of the MoI term. This approximation is equivalent to assum-

ing that the rotation axis coincides with the axis of the maximum moment of inertia during the process of

TPW. The validity of this assumption was discussed in detail in Cambiotti et al. (2011) who showed that even

for the Earth this assumption is not always appropriate. Another approach which is more commonly applied

in recent studies was formulated originally by Sabadini and Peltier (1981) and further developed by Sabadini

et al. (1982), Spada et al. (1992), and Ricard et al. (1993) which is based on the quasi-ﬂuid approximation.

RESEARCH ARTICLE

10.1002/2017JE005365

Key Points:

• A semianalytical method for

large-angle true polar wander that

can deal with a complete scheme

of multilayer Maxwell rheology

is presented

• We ﬁnd that using the quasi-ﬂuid

approximation leads to a large error

for TPW on the Earth, Mars, and Venus

• We show the eﬀect of the permanent

shape of an elastic lithosphere and

a model that is not in hydrostatic

equilibrium on TPW

Correspondence to:

H. Hu,

h.hu-1@tudelft.nl

Citation:

Hu, H., van der Wal, W., &

Vermeersen, L. L. (2017). A full-Maxwell

approach for large-angle polar wander

of viscoelastic bodies. Journal of

Geophysical Research: Planets,122.

https://doi.org/10.1002/2017JE005365

Received 9 JUN 2017

Accepted 10 NOV 2017

Accepted article online 22 NOV 2017

©2017. The Authors.

This is an open access article under the

terms of the Creative Commons

Attribution-NonCommercial-NoDerivs

License, which permits use and

distribution in any medium, provided

the original work is properly cited, the

use is non-commercial and no

modiﬁcations or adaptations are made.

HU ET AL. A FULL-MAXWELL APPROACH FOR TPW 1

Journal of Geophysical Research: Planets 10.1002/2017JE005365

Mathematically, this approximation is the ﬁrst-order Taylor expansion of the Love number in the Laplace

domain. The consequence of adopting the quasi-ﬂuid approximation is that the elastic response of a Maxwell

model is missing, and it also simpliﬁes the individual viscous relaxation of diﬀerent modes. Hu et al. (2017)

tested the validity of the quasi-ﬂuid approximation and showed that it can lead to a large error in the tran-

sient behavior of TPW for a model whose strong modes have very diﬀerent relaxation times. This could lead

to erroneous conclusions when the model-predicted TPW speed is compared with the speed that is observed

or inferred from surface features (e.g., on Mars, Bouley et al., 2016). The method developed by Hu et al. (2017)

is a numerical approach which requires that the change in the inertia tensor is calculated either by convo-

lution or by a ﬁnite-element package. Both the numerical convolution and the ﬁnite-element package are

not suitable for studies of models containing layers with very diﬀerent viscosities since the large contrast in

viscosity results in a large increase in computational time for numerical methods. The increase in the com-

putational time is caused by the fact that the total integration time has to be long to account for the long

relaxation time while the integration step size must be small to accurately simulate the layers with short

relaxation time.

Another issue which is intensively studied in recent years is the eﬀect of an elastic or highly viscous layer on

TPW (Cambiotti et al., 2010; Chan et al., 2014; Harada, 2012; Harada & Xiao, 2015; Mitrovica et al., 2005; Moore

et al., 2017; Willemann, 1984). The existence of such a layer can create a delayed readjustment of the equa-

torial bulge (often called remnant bulge) which signiﬁcantly changes the behavior of TPW as discussed by

Willemann (1984) and Mitrovica et al. (2005). Mitrovica et al. (2005) show the importance of a correct choice

for the initial hydrostatic state when the TPW is estimated. They used the ﬂuid tidal Love number which cor-

responds to the observed ﬂattening instead of the model predicted ﬂattening. Recent studies often assume

that this extra ﬂattening comes from mantle convection (Cambiotti et al., 2010; Mitrovica et al., 2005). Alter-

natively, if we do not assume that the model is in hydrostatic equilibrium, this diﬀerence can also come from

the elastic lithosphere which has its background shape corresponding to a faster rotational speed. As far

as we know, this issue has not been discussed yet. Recent studies concerning the time-dependent solution

of long-term large-angle TPW with an elastic or highly viscous lithosphere (Chan et al., 2014; Harada, 2012;

Moore et al., 2017) are all based on the method developed by Ricard et al. (1993) and adopt the quasi-ﬂuid

approximation.

Compared with the linear approach (e.g., Wu & Peltier, 1984), which can only simulate TPW for small-angle

changes, the method from Ricard et al. (1993) enables the study of issues such as the coupling of the rotational

perturbation in the Xand Ydirections. This means that in the body-ﬁxed frame, a mass distribution imbal-

ance in the X-Zplane would cause a rotational perturbation not only in the X-Zplane but also in the Y-Z

plane. This coupling eﬀect increases as the rotational speed of the object decreases and can turn TPW into

a mega-wobble for some objects like Venus which rotates very slowly (Spada et al., 1996). The phenomenon

of the mega-wobble is caused by the increase of the contribution from the mass anomaly itself compared to

that of the equatorial bulge readjustment. When the contribution from the equatorial bulge readjustment is

dominant, the periodic behavior, often called Chandler wobble, damps out quickly and its secular eﬀect on

the long-term TPW can be ignored. However, when the rotational speed decreases which causes the equa-

torial bulge to decrease, the change in the inertia tensor becomes dominated by the mass anomaly itself.

As a result, the rotational behavior resembles the free nutation of a rigid body (Lambeck, 2005). This cou-

pling eﬀect, or periodic behavior, is almost always neglected in the linear scheme for the study of the Earth

(e.g., Cambiotti et al., 2010; Wu & Peltier, 1984).

To conclude, a semianalytical approach which can accurately calculate TPW of a Maxwell model in diﬀerent

hydrostatic states is missing. It is the main purpose of this paper to develop such a method and show how

more accurate solutions are obtained. We also show if the diﬀerence in results between the methods has a

signiﬁcant impact on planetary studies (e.g., observation and modeling) in the following cases: TPW for slowly

rotating objects: mega-wobble of Venus; eﬀect of a remnant bulge caused by an elastic or highly viscous

lithosphere on large-angle TPW; and TPW on a body that is not in hydrostatic equilibrium.

In section 2, the inﬂuence of the quasi-ﬂuid approximation is discussed in more detail and our new method

will be presented. Sections 3–5 will cover the above listed issues. Sections 3 and 4 contain a case study of

Venus and Mars, respectively, which compares the results obtained in previous studies and that from our

new method.

HU ET AL. A FULL-MAXWELL APPROACH FOR TPW 2

Journal of Geophysical Research: Planets 10.1002/2017JE005365

2. Method

The governing equation for the rotation of a rigid body in the body-ﬁxed rotating reference frame is the

well-known Euler’s equation. As the body becomes deformable, it is often referred to as the Liouville equation.

For a torque-free case, it reads (Sabadini et al., 2016)

d

dt(I⋅𝝎)+𝝎×I⋅𝝎=0(1)

where Iis the inertia tensor and 𝝎is the rotational vector whose magnitude is the rotating speed. Both val-

ues are deﬁned in a body-ﬁxed coordinate system. When the moment of inertia of the body is perturbed by a

geophysical process which causes mass redistribution, the rotational axis shifts, and consequently, the equa-

torial bulge readjusts. Analytically, given a rotational vector as 𝝎=𝜔1,𝜔

2,𝜔

3T=Ω̄𝜔1,̄𝜔2,̄𝜔3T, where Ω

is the angular speed of the rotation and ̄𝜔1,̄𝜔2,̄𝜔3Tis a unit vector which represents the direction of the

rotation, the total moment of inertia attributable to such a process is given by (Ricard et al., 1993)

Ii,j(t)=I𝛿ij +kT(t)a5

3G∗𝜔i(t)𝜔j(t)− 1

3Ω(t)2𝛿ij

+𝛿(t)+kL(t)∗Ci,j(t)

(2)

where Iis the principle moment of inertia of the spherical body in hydrostatic equilibrium, Gis the gravita-

tional constant, and ais the radius of the planet. kT(t)and kL(t)are the degree 2 potential tidal Love number

and load Love number, respectively. Love numbers are obtained by the normal mode method and based on

the Maxwell rheology (Farrell, 1972). The asterisk denotes convolution in the time domain. Ci,jrepresents the

change in the moments and products of inertia without considering the deformation and this is the triggering

load for the TPW. The most diﬃcult part of solving equations (1) and (2) is the convolution of the tidal Love

number and the centrifugal potential, in particular, the part kT(t)∗𝜔i(t)𝜔j(t). In the following subsection, we

ﬁrst show how this problem is tackled by adopting the quasi-ﬂuid approximation and the inﬂuence of this

approximation on calculation of the inertia tensor. Following this subsection a new approach is presented to

calculate the MoI equation analytically. Section 2.3 demonstrates how to use the developed algorithm and

provides initial results from our method.

2.1. Conventional Approach Based on the Quasi-Fluid Approximation

The tidal Love number in the Laplace domain for a given harmonic degree is expressed as (Peltier, 1974)

kT(s)=kT

e+

m

i=1

kT

i

s−si

(3)

where kT

eis the elastic Love number, kT

iare the residues of each mode, and siare the inverse relaxation times.

This form of the Love number contains all the information about how a multilayered Maxwell body deforms:

an instantaneous elastic response characterized by kT

efollowed by viscous relaxation of separate modes char-

acterized by their diﬀerent inverse relaxation time siand mode strength −ki∕si. We call this form of the Love

number the full-Maxwell rheology scheme, as opposed to the quasi-ﬂuid approximation introduced below.

In order to solve equation (2), Ricard et al. (1993) took the quasi-ﬂuid approximation which approximates the

tidal Love number with its ﬁrst-order Taylor expansion:

kT(s)≈kT

e−

m

i=1kT

i

si

+kT

is

s2

i

=kT

f(1−T1s)

(4)

where kT

fis the ﬂuid Love number which is the sum of the mode strength:

kT

f=ke−

m

i=1

kT

i

si

(5)

The time constant T1is

T1=1

kT

f

m

i=1

kT

i

s2

i

(6)

Thus, by taking the quasi-ﬂuid approximation, all information for a viscoelastic layered model is combined into

one constant T1. Because of the ki∕s2

iterm, this constant is dominated by the slowest modes. Thus, applying

HU ET AL. A FULL-MAXWELL APPROACH FOR TPW 3

Journal of Geophysical Research: Planets 10.1002/2017JE005365

Figure 1. O-X<mathdollar>YZis the body-ﬁxed frame where the Zaxis

is the original rotational axis. O-X’Y’Z’ is the bulge-ﬁxed frame where

the Z’ axis is the instantaneous rotational axis and the X’ axis lies within

the Z-Z’ plane.

the quasi-ﬂuid approximation on a physical model which contains modes

with both very long and short relaxation times will result in a large bias. It

can also be seen that with this approximation, the elastic response as well as

the viscous relaxation characterized by the function 1∕(s−si)in the Laplace

domain (esitin the time domain) are missing (as will be shown in equation

(7)); therefore, the ongoing deformation does not agree with the complete

Maxwell rheology scheme. The convolution of equation (4) with a linear load

function F(t)=a+bt, where a,bare constants, results in a time domain

response of the form

R(t)=kT

fF(t)−F′(t)kT

fT1(7)

where the derivative F′(t)=band R(t)is the response function. This response

demonstrates the eﬀect of the quasi-ﬂuid approximation: for a near-constant

load (F′(t)≈0), the response reaches its ﬂuid limit kT

fF(t)immediately with-

out the time-dependent viscous behavior. This means that when the speed of

TPW is very slow compared to the characteristic relaxation speed of the body,

the results based on the quasi-ﬂuid approximation approach those which are

obtained from the ﬂuid limit method which diagonalizes equation (2), such as

in Matsuyama and Nimmo (2007). For other loads which change linearly in

time, the instantaneous ﬂuid limit response kT

fF(t)is shifted by a value which

is proportional to the speed of the change of the load, as given by the sec-

ond term of the equation (7). For the complete Maxwell rheology scheme, the

original form of the Love number (equation (3) needs to be convoluted with the loading function, resulting

in damping of this part with a function of e−At , where Ais a positive constant. As a result, compared to the

original Maxwell rheology, the response of a Heaviside or fast changing load based on the quasi-ﬂuid approx-

imation will likely result in a very diﬀerent TPW path. For example, the change in the inertia tensor due to an

impact crater which appears instantly and is preserved afterward is a Heaviside load to the planet.

Next, we quantitatively show that adopting quasi-ﬂuid approximation can either underestimate or overesti-

mate the equatorial readjustment for certain components of the inertia tensor in the normal polar wander

case and in the mega-wobble case. Substituting equation (4) into (2) gives the change in the moment of inertia

as (Ricard et al., 1993)

ΔIi,j(t)= kT

fa5

3G𝜔i(t)𝜔j(t)− 1

3Ω(t)2𝛿ij

−kT

fa5

3GT1̇𝜔i(t)𝜔j(t)+𝜔i(t)̇𝜔j(t)− 2

3𝜔l(t)𝜔l(t)𝛿ij+Eij

(8)

where Ei,j(t)and ̇

Ei,j(t)are obtained by convolving Ci,j(t)and ̇

Ci,j(t)with 𝛿i,j+kL(t). We compare the change in

the moment of inertia calculated by equations (8) and (2) to show the inﬂuence of adopting the quasi-ﬂuid

approximation. The latter is obtained by numerical calculation of the convolution. As a representative of

terrestrial planets, we use a SG6 Earth model (which represents a multilayered model with interior density,

rigidity, and viscosity change) and let the rotational axis move in two ways:

1. The rotational axis drifts with a constant speed along the X-Zplane in the body-ﬁxed coordinates for 90∘.

This is to simulate the normal polar wander case for fast-rotating planets such as the Earth and Mars.

2. The rotational axis initially stays at 30∘colatitude and 0∘longitude in the X-Zplane and moves longitudinally

with a constant speed along the 30∘colatitude circle for 720∘. This is to simulate the mega wobble for very

slowly rotating objects such as Venus.

In the ﬁrst case, the drift speed of the rotational axis is chosen to be fast enough to view the eﬀect of the

quasi-ﬂuid approximation. We deﬁne a bulge-ﬁxed frame whose Z’ axis coincideswith the instantaneous rota-

tional axis and whose X’ axis lies within the Z-Z’ plane as shown in Figure 1. So a pure rotation (around Y’ axis)

can transform the body-ﬁxed frame into the bulge-ﬁxed frame.

Figure 2 gives the change in the six components of the MoI tensor in the bulge-ﬁxed frame. In this ﬁgure, the

diﬀerences in the I11,I22,andI33 are small, but the most important component I13 calculated with the quasi-

ﬂuid approximation is signiﬁcantly larger than the accurate value. The magnitude of the I13 in the bulge-ﬁxed

HU ET AL. A FULL-MAXWELL APPROACH FOR TPW 4

Journal of Geophysical Research: Planets 10.1002/2017JE005365

0 200 400

-7.15

-7.14

-7.13

-7.12

-7.11

-7.1 x 1034 I11

ka

kg m2

0 200 400

-7.1423

-7.1422

-7.1421

-7.142

-7.1419 x 1034 I22

ka

0 200 400

1.424

1.425

1.426

1.427

1.428

1.429

1.43 x 1035 I33

ka

Quasi

Numerical

-fluid

0 200 400

-2

0

2

4

6x 1033 I13

ka

kg m2

0 200 400

-1

-0.5

0

0.5

1

I23

ka

0 200 400

-1

-0.5

0

0.5

1

I12

ka

Figure 2. Normal polar wander case: change in the six components of the MoI tensor when the rotational axis initially

stays at 30∘colatitude and 0∘longitude in the X-Zplane and precesses with a constant speed along the 30∘colatitude

circle for 720∘in 500 ka. Red lines are the accurate results and blue lines are from the quasi-ﬂuid approximation.

0 200 400 600

-7.15

-7.14

-7.13

-7.12

-7.11

-7.1 x 1034 I11

ka

kg m2

0 200 400 600

-7.15

-7.1

-7.05

-7

-6.95

-6.9 x 1034 I22

ka

0 200 400 600

1.4

1.41

1.42

1.43

1.44 x 1035 I33

ka

Quasi

Numerical

-fluid

0 200 400 600

0

1

2

3

4

5

6x 1032 I13

ka

kg m2

0 200 400 600

-4

-3

-2

-1

0

1x 1034 I23

ka

0 200 400

-6

-4

-2

0

2x 1032 I12

ka

Figure 3. Mega-wobble case: change in the six components of the MoI tensor. The rotational axis initially stays at 30∘

colatitude and 0∘longitude in the X-Zplane and precesses longitudinally with a constant speed along the 30∘colatitude

circle for 720∘in 600 ka. Red lines are the accurate results, and blue lines are from the quasi-ﬂuid approximation.

HU ET AL. A FULL-MAXWELL APPROACH FOR TPW 5

Journal of Geophysical Research: Planets 10.1002/2017JE005365

frame determines how fast the equatorial bulge readjusts. A larger value of I13 suggests a slower readjustment

(if the readjustment is complete, then the rotational axis coincides with the new principle axis and I13 would

be zero). So adopting the quasi-ﬂuid approximation in the normal polar case causes a large underestimation

for the speed of the equatorial bulge readjustment.

For the second case where the rotational axis tends to wobble around a ﬁxed point, in the bulge-ﬁxed frame,

the I23 implies the readjustment in the direction along the track of the rotational axis (called along-track direc-

tion in the following) and I13 gives the readjustment in the direction which is perpendicular to the plane that

contains the along-track direction and the rotational axis (called normal direction in the following). The results

are shown in Figure 3. For components I11,I22 , and I33, the quasi-ﬂuid approximation misses the small oscilla-

tions but the diﬀerences are still very small between the two methods; thus, for these three components the

error introduced by the approximation can be ignored. However, the magnitude of the along-track compo-

nent I23, is largely underestimated just like the I13 component in Figure 2 (I13 is the along-track component

for the normal polar wander case). On the other hand, the normal directional component I13 in Figure 3 is

underestimated by the quasi-ﬂuid approximation which suggests an overestimation of the equatorial bulge

readjustment in this direction.

The key information obtained in this subsection is that by adopting the quasi-ﬂuid approximation, the speed

for equatorial readjustment can be, depending on the model and load, largely underestimated in the along-

track direction but overestimated for the mega wobble case in the normal direction. These are the main rea-

sons for the diﬀerence of the TPW path calculated by diﬀerent methods which will be discussed in sections 3

and 4.

2.2. A New Approach

In order to eliminate the convolution in the part kT(t)∗𝜔i(t)𝜔j(t)while staying consistent with the fundamen-

tal rheology of the system, we adopt the strategy of approximating the load term 𝜔i(t)𝜔j(t)instead. Within

the considered time period Tn, at time t=Tp,p=0,1,…,n, values of 𝜔i(t),i=1,2,3are known, then we have

𝜔i(Tp)=Wi,p. Assuming that 𝜔i(t),i=1,2,3changes linearly between each time step, 𝜔i(t)can be written as a

piecewise linear function:

𝜔i(t)=

n

p=1

𝜔i,p(9)

where

𝜔i,p=Wi,p−1+Wi,p−Wi,p−1

Tp−Tp−1t−Tp−1Ht−Tp−1HTp−t(10)

and H(t)is the Heaviside step function. With this form, k(t)∗𝜔i(t)𝜔j(t)and its derivative can be expressed

analytically by applying the Laplace transformation

kT(t)∗𝜔i(t)𝜔j(t)=−1[[kT(t)∗𝜔i(t)𝜔j(t)]] (11)

where and −1stand for the Laplace and inverse Laplace transformation, respectively.The explicit expression

of equation (11) can be found in Appendix A as equations (A1a)– (A1c). Substituting equations (A1a) – (A1c)

into (2), the inertia tensor and its derivative at time t=Tncan be expressed analytically as

Ii,j(Tn)=I𝛿i,j+a5

3G(Ai,j(Tn)+Bi,j)− 1

3

3

k=1

(Ak,k(Tn)+Bk,k)+Ei,j(Tn)(12a)

̇

Ii,j(Tn)= a5

3G(Ci,j(Tn)+Di,j)− 1

3

3

k=1

(Ck,k(Tn)+Dk,k)+̇

Ei,j(Tn)(12b)

where expressions for Ai,j(t),Bi,j(t),Ci,j(t)and Di,j(t)can be found in Appendix A. When t=Tpwith p=0,1, ...n

and 𝜔i(Tp)=Wi,pwith p=0,1,2...n−1are given, equations (12a) and (12b) express the moments and products

of inertia and its derivative as a function of Wi,n. Then Wi,ncan be solved by equation (1). To this end, it helps to

see the problem as a global optimization problem: we seek the value of Wi,nin the neighborhood of Wi,n−1so

that the value of d

dt(I⋅𝝎)+𝝎×I⋅𝝎is minimized. As a result, the method which is introduced in Hu et al. (2017)

as algorithm 2 (p. 10) can be applied. This method applies the linearized form of the Liouville equation and

an iteration procedure to obtain Wi,n. It will be brieﬂy explained in the following and outlined in Appendix B.

HU ET AL. A FULL-MAXWELL APPROACH FOR TPW 6

Journal of Geophysical Research: Planets 10.1002/2017JE005365

This algorithm is validated in (Hu et al., 2017) by both the linear (Wu & Peltier, 1984) and the nonlinear method

(Ricard et al., 1993) as shown in Figure D1 in Appendix D. We deﬁne the perturbed rotational vector as

𝝎′=Ω(m1,m2,1+m3)T(13)

where m1,m2, and m3are small real numbers. The Liouville equation can be linearized to obtain the form

(Hu et al., 2017)

m1(t)= ΔI13(t)

C−A+CΔ

.

I23 (t)

Ω(C−A)(C−B)(14a)

m2(t)= ΔI23(t)

C−B−CΔ

.

I13 (t)

Ω(C−A)(C−B)(14b)

m3(t)=−

ΔI33

C(14c)

In equations (14a)–(14c), the terms of the inertia tensor, A,B,Cand I13,I23 ,̇

I13,̇

I23 are not in the body-ﬁxed

frame but in the bulge-ﬁxed frame. The transformation matrix from the body-ﬁxed frame to the bulge-ﬁxed

frame by a pure rotation can be obtained as

Q=

𝜔3+𝜔2

2

1+𝜔3

−𝜔1𝜔2

1+𝜔3

𝜔1

−𝜔1𝜔2

1+𝜔3

1−𝜔2

2

1+𝜔3

𝜔2

−𝜔1−𝜔2𝜔3

(15)

A coordinate transformation is required before we can substitute the value of the inertia tensor calculated

from equations (12a) and (12b). The detailed procedure of algorithm 2 in Hu et al. (2017) is given in Appendix B.

In general, the only assumptions we make in the entire calculation in this study are two linear approximations:

the changes in the rotational vector and the inertia tensor (see equation (10) in Hu et al. (2017)) are small in

each step and can be treated as linear. These assumptions are valid when the step sizes (Δtp=Tp−Tp−1,p=

2,3,…,n) are small enough. Since we do not approximate Love numbers, our method gives the TPW path

for a viscoelastic body which is consistent with the complete scheme of Maxwell rheology. We will label our

method in the following as full-Maxwell method.

2.3. Initial Setting and Validation

One of the major factors that controls the TPW behavior is the shape of the equatorial bulge (and the tidal

bulge which is discussed in Hu et al., 2017). When the interior model and rotational speed is given, this shape

is controlled by the hydrostatic state of the model. Due to the limitation of the method, previous studies

based on either linear (Sabadini et al., 1982; Wu & Peltier, 1984) or nonlinear (Chan et al., 2014; Moore et al.,

2017; Ricard et al., 1993) approaches can only simulate TPW on a model which is assumed to be in hydrostatic

equilibrium. However, as will be shown in section 5, the choice of the hydrostatic state can have a signiﬁcant

impact on the TPW behavior. With our method, we can choose the hydrostatic state of the model at which

the TPW starts.

To simulate the TPW of a body at a certain hydrostatic state, we need to apply a centrifugal force to the model

for a certain length of time. If the rotational vector at the start of the simulation is given by 𝝎0=Ω(0,0,1)T,

applying centrifugal force to the model for a duration of This expressed in our scheme as

T0=0(16a)

T1=Th(16b)

W1,0,W2,0,W3,0T=W1,1,W2,1,W3,1T=𝝎0(16c)

The triggering load Ei,j(t)needs to be applied at t=Thto start the TPW. To simulate a model in hydrostatic

equilibrium, Thneeds to be large enough so that all modes of the model are suﬃciently relaxed. In order to

achieve this, we can choose a Thso that the slowest mode is relaxed more than 99.999%. Assuming s1is the

slowest mode,

1−es1Th>0.99999 (17)

HU ET AL. A FULL-MAXWELL APPROACH FOR TPW 7

Journal of Geophysical Research: Planets 10.1002/2017JE005365

0 2000 4000 6000 8000 10000 12000 14000

-60

-50

-40

-30

-20

-10

0

Time (ka)

Colatitude (degree)

T =2e3 ka

N, T =2e3 ka

T =1e6 ka

Quasi-fluid

Figure 4. Polar wander in the X-Zplane for the SG6 Earth model

triggered by a point mass of 2×1019 kg attached at the surface at

30∘colatitude.

translates into

Th>−11.513

s1

(18)

This choice is similar to that in Hu et al. (2017), which sets Thso that

kT(Th)>99.95%kT

f, but much more strict. If the model contains a very slow

and strong mode, we can obtain a very large value of Thand this will cause an

extremely long calculation time for a pure numerical method (Hu et al., 2017).

By expressing the inertia tensor analytically, a very large Thcan be dealt with.

Eventually, there is no limit for the choice of Thas well as the initial loading

𝝎0as long as numerical errors (e.g., truncation error) are avoided. As a result,

TPW for a body in a diﬀerent hydrostatic state can be obtained.This makes our

method suitable to study the eﬀect of a remnant bulge or TPW on a model

without hydrostatic equilibrium as will be shown and discussed in detail in

sections 4 and 5.

The algorithm, as shown in Appendix B, was developed in Hu et al. (2017). The

main idea is to decouple the two governing equations. The MoI equation was

solved by either direct convolution or from a ﬁnite element method and the

result is fed back into the linearized Liouville equation and solved by an iterative procedure. Such algorithm

has been validated by both comparing to the linear (Wu & Peltier, 1984) and nonlinear method (Ricard et al.,

1993) in Figures 6 and 9 of Hu et al. (2017). The diﬀerence between the method in this paper and that of Hu

et al. (2017) is that here the MoI equation is solved analytically with the assumption that the rotational vector

changes linearly during each time step of TPW. If the step size is set to be much smaller than the relaxation

time of the dominant modes, which is the same requirement for calculating TPW, the analytical solution of the

MoI equation with the linear assumption will be suﬃciently equivalent to the result from direct convolution.

Therefore, the TPW solution generated in this paper can approach that from Hu et al. (2017) for a small enough

step size. We ﬁrst demonstrate the result of TPW calculated with Th=2,000 ka, which is the choice in Hu et al.

(2017) for a six-layer incompressible Earth model SG6 (Table 2 in the same paper), and a much higher value

(Th=106ka) for which the model can be considered in hydrostatic equilibrium (equation (18) holds). Weplace

a stationary (kL(t)=0) mass anomaly, which means that the mass anomaly does not “sink” into the body, at

the surface at 30∘colatitude. Here we do not consider the eﬀect of the remnant bulge (which will be discussed

in detail in section 4), so we ignore the slowest mode generated by the lithosphere (its viscosity is set to 1031

Pa s to calculate the Love numbers). We also include the result obtained by the quasi-ﬂuid approximation

according to Ricard et al. (1993).

As can be seen in Figure 4, the result obtained forTh=2,000 k a by the full-Maxwell method is very close (within

0.05% diﬀerence) to the result of the numerical method from Hu et al. (2017), for a step size of 5 ka. It is also

clear that choosing Th=2,000 ka still shows a TPW path that is diﬀerent from that of a body which can be

considered to be in hydrostatic equilibrium (Th=106ka). This suggests the sensitivity of the TPW to a small

deviation from its hydrostatic equilibrium. So choosing Thlarge enough is the ﬁrst guarantee that the correct

TPW path for models with hydrostatic equilibrium is obtained.

The computational cost of our method depends on three factors: (1) the complexity of the layered model, or

more precisely, the number of the modes in the Love numbers; (2) the number of iteration necessary to obtain

a convergent result in each step; and (3) the number of time steps. The computational time increases roughly

linearly with these factors. For the results shown in Figure 3, which is for a SG6 Earth model that contains

12 modes, the number of time steps is 2,600 (each step is 5 ka) and every step requires 50 (ﬁrst step) to 18

(last step) iterations. The program is written in MATLAB and the total computational time on our desktop com-

puter is about 3 min. Compared with the numerical computation (Hu et al., 2017) which is about several hours,

the speed from the semianalytical approach is much faster.

Based on the analysis in section 2.1, the largely underestimated TPW speed by adopting the quasi-ﬂuid

approximation is caused by the underestimation of the speed of the equatorial readjustment for the case of

Earth and Mars. For the mega-wobble case in Venus, the situation can be quite diﬀerent as will be discussed

in the following section.

HU ET AL. A FULL-MAXWELL APPROACH FOR TPW 8

Journal of Geophysical Research: Planets 10.1002/2017JE005365

3. Mega-Wobble: TPW on Venus

The linearized Liouville equation (equations (14a)– (14c)) is obtained by two fundamental assumptions. First,

during the TPW, the components of the inertia tensor satisfy I33(C)>I11(A)and I33 (C)>I22(B). Second, the peri-

odic terms which represent the Chandler wobble can be ignored. While these assumptions are true for cases

like Mars and the Earth, it can be invalid for some slowly rotating objects such as Venus. Since the rotational

speed of Venus is so slow that its equatorial bulge is also extremely small, the diﬀerence of the two principle

moment of inertia: C−A, which can be calculated by

C−A=kT

fa5Ω2

3G(19)

for Venus is less than 1.5×10−5of that for Earth. For magnitudes of a mass anomaly of 10−5or 10−6of the

total mass of the planet considered in Spada et al. (1996), depending on the depth and position, the moment

around the rotational axis may not be the largest of the diagonal components anymore (C>A,Bare not

satisﬁed). Furthermore, the period of the wobble, which can be estimated as 2𝜋

ΩAB

(C−A)(C−B)(see equation (26))

when C>A,B, is about 4 months (depending on the interior model) on Earth or Mars but can be, depending

on the interior model, over 10 Ma on Venus. Because of such low frequency, the periodic terms will have a

secular eﬀect for TPW on Venus and cannot be ignored as for the Earth and Mars. Therefore, in order to study

TPW on Venus, it is necessary to ﬁrst derive a new set of linearized Liouville equations suitable for a body with

a very long wobble period. The linearized Liouville equation for a triaxial body reads (Sabadini et al., 2016)

̇

m1=−

C−B

AΩm2+Ω

AΔI23 −Δ̇

I13

A(20a)

̇

m2=C−A

BΩm1−Ω

BΔI13 −Δ̇

I23

B(20b)

̇

m3=−

Δ̇

I33

C(20c)

We ﬁrst deal with the cases of C>A,C>B, and C<A,C<B. By assuming that the change in the moment of inertia

is linear, equations (20a)– (20c) can be solved analytically. The result contains the non-periodic terms in

equations (14a)–(14c) and periodic terms

̄

m1(t)=B

A(C−A)3(C−B)Ω2sin (C−A)(C−B)

AB ΩΔt(A−C)ΔI23(t)+CΔ̇

I13(t)

−cos (C−A)(C−B)

AB ΩΔtΔI13(t)

C−A+CΔ̇

I23(t)

Ω(C−A)(C−B)(21)

̄

m2(t)=A

B(C−A)(C−B)3Ω2sin (C−A)(C−B)

AB ΩΔt(C−B)ΔI13(t)+CΔ̇

I23(t)

−cos (C−A)(C−B)

AB ΩΔtΔI23(t)

C−B+CΔ̇

I13(t)

Ω(C−A)(C−B)(22)

When the period of the wobble becomes very long and the step size Δtis small enough, the magnitude of

(C−A)(C−B)

AB ΩΔtin the trigonometric functions is very small, and we can apply sin(𝜃)≈𝜃and cos(𝜃)≈1−𝜃2∕2.

Applying these approximations, combining equations (14a)– (14c), (21), and (22) and ignoring the derivative

terms of the inertia tensor gives

m1(t)= (C−B)Ω2ΔI13(t)Δt2+2BΩΔI23 (t)Δt

2AB (23a)

m2(t)= (C−A)Ω2ΔI23(t)Δt2−2AΩΔI13 (t)Δt

2AB (23b)

m3(t)=−

ΔI33

C(23c)

HU ET AL. A FULL-MAXWELL APPROACH FOR TPW 9

Journal of Geophysical Research: Planets 10.1002/2017JE005365

Tab le 1

Venus Model

Outer radius (km) Density (kg m−3) Shear modulus (Pa) Viscosity (Pa s)

6,052 2,900 0.36 ×1011 ∞

6,002 3,350 0.68 ×1011 0.6×1021

5,500 3,725 0.93 ×1011 1.6×1021

5,200 4,900 2.07 ×1011 6.4×1021

3,250 10,560 0 0

The derivation for other situations such as B<C<Aor B>C>Aare shown in Appendix C. Note that in

equations (23a)–(23c), as well as in equations (C5a), (C5b), (C6), and (C6b), we do not have the C−Aor C−B

terms in the denominator; thus, these expressions have no singularity problem for C=Bor C=A. When TPW on

a very slowly rotating object like Venus is calculated, equations (23a)– (23c) instead of equations (14a)–(14c)

should be applied. Basically, equations (14a)– (14c) and equations (23a)–(23c) give two extreme situations

for calculating the rotational perturbation. When the step size of the calculation Δtcan be set to much larger

than the Chandler period, equations (14a)– (14c) should be used to give the secular behavior. When the step

size Δtis chosen to be much smaller than the period of the Chandler wobble, equations (23a)– (23c) can give

the periodic “short”-term behavior which leads to the mega-wobble on Venus or, when the step size is set to

days, the Chandler wobble on Earth or Mars.

Next, we apply our method to a model of Venus and test some results obtained in Spada et al. (1996) who

apply the quasi-ﬂuid approximation. We create a ﬁve-layer Venus model which approximates the density and

rigidity proﬁle used in Armann and Tackley (2012), and the viscosities are chosen similar to those of Earth.

The interior properties are shown in Table 1. The eﬀect of a remnant bulge is not included here and will be

discussed in the next section.

Figure 5. A point mass of −5×1018 kg is attached at the surface at 45∘colatitude. (top row)The displacement of

the rotational axis in the along-track and normal direction. (bottom row) The movement of the mass anomaly in the

bulge-ﬁxed frame where the rotational axis is always pointing upward at the center. Blue lines are obtained by applying

the quasi-ﬂuid approximation, and red lines are from the full-Maxwell method. The black dots are the original locations

of the mass anomalies in the bulge-ﬁxed frame.

HU ET AL. A FULL-MAXWELL APPROACH FOR TPW 10

Journal of Geophysical Research: Planets 10.1002/2017JE005365

For this model, the period 2𝜋

ΩAB

(C−A)(C−B)is about 11.45 Ma. The step size needs to be much smaller than this

value; therefore, a value of less than 5,000 years is suﬃcient to obtain an accurate solution. One way to test if

the chosen step size is indeed small enough is to recalculate the TPW with half of the step size. If the chosen

step size is indeed suﬃciently small, the result will not change signiﬁcantly.We ﬁrst simulate the TPW on Venus

driven by a negative mass anomaly of magnitude 5×1018 kg (which is about the maximum value simulated

by Spada et al. (1996)) which is attached at the surface at 45∘colatitude. We decompose the displacement of

the rotational axis in the direction which lies in the plane that contains the mass anomaly (normal direction)

and the along-track direction. The result is shown in Figure 5. As we can see, for the along-track displace-

ment which describes the long-term wobble of the rotational axis around the mass anomaly, the results from

the two methods diﬀer less than 1%. However, for the displacement in the normal direction, representing

the movement of the rotational axis toward the negative mass anomaly, the result obtained based on the

quasi-ﬂuid approximation overestimates the speed. The agreement in the along-track direction displacement

and a disagreement in the normal direction displacement between the two methods can be explained by the

small contribution of the equatorial bulge readjustment to the rotational perturbation for Venus. As pointed

out by Spada et al. (1996), for Venus, the rotational behavior is largely dominated by the long-term wobble

which resembles the free nutation of a rigid body. The long-term wobble is mainly caused by the mass

anomaly itself, while the equatorial readjustment contributes very little. Based on the discussion of Figure 3,

the component I23 also contributes to the along-track speed of the wobble. This component is largely over-

estimated by the quasi-ﬂuid approximation, but since its magnitude is much smaller than the contribution

from the mass anomaly itself (less than 1% in this case), the diﬀerence between the two methods can be

ignored. However, the damping of the oscillatory motion, or the displacement of the rotational axis in the

normal direction, is solely controlled by the viscous relaxation of the body, speciﬁcally the component I13 in

Figure 3. Since this component is underestimated by the quasi-ﬂuid approximation, the rotational behavior

on Venus obtained by adopting the approximation results in too much damping.

In the study of terrestrial planets whose tidal bulge can be ignored, like Earth, Mars, Venus, and Mercury, we

may see certain geographic features (e.g., the supercontinent Pangaea on the Earth or the Tharsis plateau

on Mars) which have the potential to cause (or have caused) polar wander. Based on an interior model and

TPW history, we can estimate the age of the feature or the history of its relocation by its (estimated) former and

current latitude since we know that positive mass anomaly tends to relocate toward equatorand negative one

toward the pole. The latitudinal information in many situations is much more important than the longitudinal

one if the body is not tidally locked. As a result, correctly estimating the TPW speed in the latitudinal (normal)

direction is crucial for a better understanding of the planet reorientation. In the following the same Venus

model is used and several diﬀerent magnitudes of mass anomaly are tested. We compare the diﬀerence in

speed as a function of colatitude of the mass anomaly in the normal direction between the two methods.

Magnitudes of mass anomaly 1×1016,1×1017 ,1×1018, and 5×1018 kg are chosen, which are about 10−5to

10−6times of the total mass of Venus, similar to the values chosen in Spada et al. (1996). The results are shown

in Figure 6. Generally, methods based on the quasi-ﬂuid approximation overestimate the normal-directional

speed by a factor of 3 to 5.

Spada et al. (1996) state that for the same mass anomaly, the instantaneous velocity of rotational pole on

Venus is about 30 times larger than that of Earth and Mars. But they compared the complete rotational

behavior of Venus whose largest part is the wobble with the secular rotational behavior of Earth and Mars in

which the Chandler wobble has been ﬁltered out. A more proper comparison would be either between the

speed of Chandler wobble on Earth or Mars with the mega-wobble on Venus, or between the secular speed

of rotational variation on Earth or Mars and the normal-directional speed of the rotational axis on Venus.

In the latter case, for an Earth, Mars, and Venus model with the same average viscosity (ranging from 1020 to

1022 Pa s), it can be shown that TPW on Earth and Mars is 10 to 15 times larger than the normal-directional

speed of Venus’ rotational axis for the mass anomalies considered in this study. This means that for the same

scale of mass anomaly, it will take much longer on Venus than on Earth or Mars before it can reach the pole

or equator. The knowledge of Venusian viscosity is very limited, if the observed normal-directional change of

Venus’ rotational axis is out of this range (1/10 to 1/15 times of the Earth’s TPW), then the average viscosity of

Venus must be lower or higher than that of the Earth.

HU ET AL. A FULL-MAXWELL APPROACH FOR TPW 11

Journal of Geophysical Research: Planets 10.1002/2017JE005365

Figure 6. Normal directional speed of the rotational axis toward the mass anomaly as a function of the colatitude.

The results we obtained in this section (e.g., the speed in Figures 5 and 6) are, of course, dependent on the

interior model of Venus. With diﬀerent viscosities (1019 Pa s to 1022 Pa s), the TPW speed can be very diﬀerent

but adopting the quasi-ﬂuid approximation always largely overestimates the normal-directional speed.

4. Eﬀect of a Remnant Bulge on TPW and a Study of Mars

For a viscoelastic model which can be suﬃciently relaxed, a positive mass anomaly with any magnitude will

end up at the equator, while a negative mass anomaly will eventually reach the poles. However, most of the

observed geophysical features which are thought to have triggereda reorientation, such as the Tharsis plateau

on Mars (Bouley et al., 2016) or Sputnik Planitia on Pluto (Keane et al., 2016), are not located exactly at the

equator. A common explanation is that certain parts of the planet, usually the lithosphere which is considered

to be elastic or to have a very high viscosity, have not yet relaxed, preventing the mass anomaly from being

relocated further. The eﬀect of such elastic or highly viscous lithosphere on TPW has been studied for the

linear scheme (e.g., Cambiotti et al., 2010; Mitrovica et al., 2005), and the nonlinear scheme with the quasi-ﬂuid

approximation (e.g., Chan et al., 2014; Harada, 2012; Moore et al., 2017). Here we demonstrate the eﬀect of a

remnant bulge on the large-angle TPW with the full-Maxwell method.

First, the origin of the remnant bulge is shown using the normal mode method. This has also been discussed

by Moore et al. (2017). The remnant bulge, either formed by an elastic layer or a highly viscous layer, appears

because of a certain mode(s) which has a much longer (or inﬁnite) relaxation time compared to other domi-

nant relaxation modes of the model. We demonstrate this with a simple two-/three-layer Earth models with

and without a lithosphere of varying viscosity. In Table 2, the physical properties of the models are shown.

Tab le 2

Properties of the Two-/Three-Layer Earth Models (M1– M5)

Layer Outer radius (km) Density (kg m−3) Shear modulus (Pa) Viscosity (Pa s)

Lithosphere 6,371 4,448 1.7×1011 1×1021,24,26,29,∞

Mantle 6,361 4,448 1.7×1011 1×1021

Core 3,480 10,977 0 0

HU ET AL. A FULL-MAXWELL APPROACH FOR TPW 12

Journal of Geophysical Research: Planets 10.1002/2017JE005365

Tab le 3

Potential Tidal Love Number of Models M1– M5

M1 (1021 Pa s) M2 (1024 Pa s) M3 (1026 Pa s) M4 (1029 Pa s) M5 (∞Pa s)

Modes si−ki∕sisi−ki∕sisi−ki∕sisi−ki∕sisi−ki∕si

ke0.3449 0.3449 0.3449 0.3449 0.3449

T1−0.5341e−1 0.4731e−2−0.5343e−3 0.4204e−2−0.5343e−6 0.4200e−2

T2−0.7886e−1 0.3045e−2−0.2764e−1 0.2035e−2−0.2713e−1 0.2027e−2−0.2712e−1 0.2027e−2

C0−0.4086 0.2376 −0.4208 0.2337 −0.4193 0.2351 −0.4193 0.2352 −0.4193 0.2352

M02.225 0.4591 −2.2344 0.4552 −2.2343 0.4553 −2.2343 0.4553 −2.2343 0.4553

kf1.0416 1.0416 1.0416 1.0416 1.0374

Model M1 has a lithosphere viscosity of 1021 Pa s which is the same as the mantle; thus, this is actually a

two-layer model without lithosphere. The lithosphere of models M2– M5 have viscosities of 1024,1026 ,1029,

and ∞Pa s, respectively.

The degree 2 potential tidal Love numbers of models M1– M5 are shown in Table 3, where the inverse relax-

ation time sihas unit 1/ka. Following Sabadini et al. (2016), we can see that the two-layer model M1 only

contains two relaxation modes, C0corresponding to the core-mantle boundary and M0corresponding to the

surface. When a viscoelastic layer with the same density is added to the model (M2– M5), two additional tran-

sition modes, ̄

T1and ̄

T2, are triggered if the Maxwell time on either side of the boundary is diﬀerent. In the case

of model M2, these two additional modes have relaxation times not too diﬀerent from the dominant modes

(C0and M0in this case). Consequently, the delayed relaxation of both ̄

T1and ̄

T2is not large enough to cause a

remnant bulge. However, in the case of M3 and M4, as the viscosity of the lithosphere increases, the relaxation

time for one of the ̄

Tmodes increases with the same order as the viscosity. It is this ̄

Tmode that determines if

the remnant bulge is present. As the viscosity of the lithosphere increases further and eventually approaches

inﬁnity, as is the case in model M5, the ̄

T1mode disappears and its mode strength −k1∕s1becomes absent in

the ﬂuid Love number kf. It can be seen that the diﬀerence in the ﬂuid Love number between M5 and M4 is

almost the same as the mode strength of ̄

T1in M4: kM4

f−kM5

f≈(k1∕s1)M4. The remnant bulge is dealt with in

previous studies by either those dealing with an elastic lithosphere (Cambiotti et al., 2010; Chan et al., 2014;

Harada, 2012; Mitrovica et al., 2005) or viscoelastic lithosphere (Cambiotti et al., 2010; Moore et al., 2017) by

isolating this part of the Love number and formulating the inﬂuence of it separately.

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000

-45

-40

-35

-30

-25

-20

-15

-10

-5

0

Time (ka)

Colatitude (degree)

LT=30km

LT=20km

LT=10km

Figure 7. Polar wander in the X-Zplane for the Earth models with

diﬀerent lithosphere thickness triggered by a point mass of 2×1019 kg

attached at the surface at 20∘colatitude. Solid lines show results

obtained with the full-Maxwell method from this paper, the line dots

are calculated with the method by Chan et al. (2014) which is based

on the quasi-ﬂuid approximation.

One of the advantages of the full-Maxwell method is that we can choose any

value for the initial loading time Th. This enables us to simulate the inﬂuence

of the remnant bulge without any extra formulation. To include such a bulge

caused by a very slow relaxation mode, we only need to set the initial loading

time Thto a value large enough so that this slow mode is fully relaxed to the

centrifugal force, according to the condition in equation(18). For instance, for

the model M4 in Table 3, we can set Th=1×109ka which guarantees that the

̄

T1mode is relaxed. In practice, we do not need to simulate the case with a fully

elastic layer. Instead, we can always set the viscosity of the layer high enough

to guarantee that its relaxation within the considered time can be ignored. In

that case, the lithosphere is eﬀectively elastic.

Now we demonstrate the eﬀect of a remnant bulge with the full-Maxwell

method and compare the results with those obtained by applying the

quasi-ﬂuid approximation. A more realistic SG6 model is used, in contrast with

those shown in Figure 4 where the slowest mode is ignored. Three cases with

diﬀerent thickness of the lithosphere are considered and the viscosity of this

layer is set to 1031 Pa s so that within the considered time span (10 Ma), the

relaxation of the slowest mode can be ignored. We compare our results with

those obtained by using the method of Chan et al. (2014) which is based on

the quasi-ﬂuid approximation. The results are shown in Figure 7. While pre-

dicting the same end position of TPW, the quasi-ﬂuid approximation gives a

much slower transient response.

HU ET AL. A FULL-MAXWELL APPROACH FOR TPW 13

Journal of Geophysical Research: Planets 10.1002/2017JE005365

00.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

x 104

-60

-50

-40

-30

-20

-10

0

Time (ka)

Colatitude (degree)

LT=20km

LT=10km

LT=0km

Figure 8. Apointmassof2×1019 kg attached at the surface at 30∘

colatitude and removed after 10 Ma. Lines are for models with

lithosphere viscosity of 1031 Pa s and line crosses are for models with

lithosphere viscosity of 1027 Pa s.

Next we show the eﬀect of a viscoelastic lithosphere whose viscosity has a

relaxation time which is comparable with the considered time span. This issue

is important because whether or not the lithosphere can relax during the con-

sidered period can aﬀect the TPW behavior both in the short term (e.g., the

speed) and the long term. The comparison between the eﬀect of an eﬀectively

elastic and a viscoelastic lithosphere on small-range TPW (less than 2∘) has

been done by Cambiotti et al. (2010) using a linear scheme. Here we show the

eﬀect when large-angle TPW is considered. For this issue Moore et al. (2017)

extend the theory in Chan et al. (2014) and consider the slowest mode(s) sep-

arately while applying the quasi-ﬂuid approximation to the rest of the modes.

Apart from the bias introduced by the quasi-ﬂuid approximation as shown in

Figure 7, another problem of this approach is that for a complex multilayered

model like the SG6 Earth model with its lithosphere viscosity smaller than a

certain value (e.g., 1028 Pa s for SG6 model), the slowest mode might not be

the ̄

T1mode from the lithosphere but one of the Mmodes (M2,M3,…) which

is generated by a density diﬀerence of the inner layers. Moreover, there might

not be a large diﬀerence in the relaxation time between two modes (such as

cases M3 and M4 in Table 3); as a result, it is not clear which modes need to

be modeled separately. Our approach does not have this limitation. Similar to

the case of an eﬀectively elastic lithosphere, we only need to choose a large

enough value for Th. Here the SG6 model with lithospheric thicknesses of 0, 10, and 20 km and with viscosi-

ties of 1027 and 1031 Pa s is used. A mass anomaly of 2×1019 kg is placed at 30∘colatitude for 10 Ma, then

removed. Within the considered time span of 20 Ma, the lithosphere with viscosity of 1031 Pa s can be con-

sidered as eﬀectively elastic while that with a viscosity of 1027 Pa s is partially relaxed. The result is shown in

Figure 8. We see that the behavior of the TPW is very sensitive to the thickness of the lithosphere. A thicker

lithosphere gives stronger resistance against TPW as well as a faster rebounding of the rotational axis when

the triggering load is removed. Due to the partial relaxation of the ̄

T1mode, which allows the equatorial bulge

Figure 9. TPW on Mars triggered by a mass anomaly of 3.5×1019 kg which is attached at the surface at 45∘latitude

for four diﬀerent values of mantle viscosity. This magnitude is about the same as for Q=1of the normalized load

parameter Qdeﬁned in Willemann (1984).

HU ET AL. A FULL-MAXWELL APPROACH FOR TPW 14

Journal of Geophysical Research: Planets 10.1002/2017JE005365

to adjust faster for the high viscosity lithosphere, the TPW for models with a low viscosity lithosphere is larger

and the rotational axis cannot go back to its original position after the mass anomaly is removed.

We also apply our method to a model of Mars to compare the speed of TPW on Mars calculated by the two

methods. We establish a ﬁve-layer model whose density and rigidity approximates the model in Zharkov

and Gudkova (2005) which contains a 50 km lithosphere. The viscosity of the model is divided into an eﬀec-

tively elastic lithosphere and a uniform mantle of viscosities 1019 Pa s, 1020 Pa s, 1021 Pa s, and 1022 Pa s

which covers the value used in most recent studies (Breuer & Spohn, 2006; Hauck & Phillips, 2002). We load the

model with a surface mass anomaly of magnitude 3.5×1019 kg (which is about the magnitude for Q=1for

the normalized load parameter Qdeﬁned in Willemann (1984) and used by Chan et al. (2014) and Matsuyama

and Nimmo (2007)) placed at 45∘latitude. The results are shown in Figure 9. Similar to the Earth model, the

quasi-ﬂuid approximation has a large underestimation of the speed for most of the duration of the TPW.

The instantaneous speed from the full-Maxwell method is, for all four viscosities, about 4.6 times as large as

those obtained based on the quasi-ﬂuid approximation. The reason for this is, as mentioned before, the under-

estimation of the equatorial bulge readjustment when the approximation is adopted. When the rotational

axis approaches its end position, the speed of TPW obtained from the full-Maxwell method drops faster than

for the quasi-ﬂuid approximation which results in the end position being reached later. Generally, the method

based on the quasi-ﬂuid approximation underestimates the time it takes for a mass anomaly to reach its end

position by about half, compared to the full-Maxwell method.

5. TPW on a Model Without Hydrostatic Equilibrium

In practice, a physical model (consisting of layers with given density, rigidity, and viscosity) can be derived from

a geochemical model and the density proﬁle matches the total mass and/or gravitational data. However, it can

happen that the predicted tidal ﬂuid Love number based on the physical model does not match the observed

value for the present-day rotational speed Ω. By assuming that the model is in hydrostatic equilibrium, the

ﬂuid Love number can be estimated from the observed diﬀerence in the polar and equatorial moments of

inertia C−A(Mound et al., 2003):

kobs

f=3G

a5Ω2(C−A)obs (24)

For the Earth, this issue appears to have been studied ﬁrst by Mitrovica et al. (2005) who introduced the 𝛽

correction term to the tidal ﬂuid Love number when the present-day TPW speed is estimated. Usually, it is con-

sidered that this extra nonhydrostatic contribution stems from mantle convection. Here we simulate another

possible cause for this contribution and its eﬀect on TPW. Before the TPW starts, the lithosphere is not in

hydrostatic equilibrium, or more speciﬁcally, the permanent shape of the elastic lithosphere does not match

the present-day rotational speed. The stress-free ﬂattening of the elastic layer can be either larger or smaller

because the rotational speed during the formation of the planet (or moon) was either faster or slower than the

present-day value. We demonstrate here the inﬂuence on the TPW of a lithosphere with the same thickness

but in diﬀerent hydrostatic state.

Since the inﬂuence of each relaxation mode siis formulated separately in terms ai,j,p,q(t)and ci,j,p,q(t)in

equations (A1a)–(A1c), we can set both the initial loading potential characterized by 𝜔i(t)and the load-

ing period diﬀerently for each relaxation mode to simulate the model being in a diﬀerent hydrostatic state.

For each relaxation mode si, we set the initial loading period as Tihand the rotational speed as 𝝎i0. In contrast

to equations (16a)–(16c), we now have

T0,i=0(25a)

T1,i=Tih(25b)

W1,0,W2,0,W3,0i=W1,1,W2,1,W3,1i=𝝎i0(25c)

In the following, we show that the choice of the hydrostatic state is as important as the choice of the model.

We demonstrate this concept using SG6 models with a thin (10 km) and thick (20 km) lithosphere. As shown

in Figure 8, when the models are in hydrostatic equilibrium, diﬀerent lithospheric thicknesses result in sig-

niﬁcantly diﬀerent TPW behavior. This SG6 model is put into two categories which results in four scenarios

in total.

HU ET AL. A FULL-MAXWELL APPROACH FOR TPW 15

Journal of Geophysical Research: Planets 10.1002/2017JE005365

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

x 104

-45

-40

-35

-30

-25

-20

-15

-10

-5

0

Time (ka)

Colatitude (degree)

I. a

I. b

II. a

II. b

Figure 10. TPW for a point mass of 2×1019 kg attached at the surface

at 30∘colatitude and removed after 10 Ma. Red and blue lines are

models with an eﬀectively elastic (viscosity 1031 Pa s) lithosphere with

thickness of 10 km and 20 km, respectively, which are in hydrostatic

equilibrium. Green and black lines represent models with a lithosphere

thickness of 10 km and 20 km, respectively, and their lithospheres

initially contribute the same to the moment of inertia as the models

in hydrostatic equilibrium.

Category I: Scenario Ia. A thin-lithosphere model in hydrostatic equilibrium

with the present-day rotational speed.Ib. A thick-lithosphere model whose

slowest mode has relaxed so that its mode strength is equal to that of the thin

lithosphere model:

k1

s1thick

×1−es1Th1=k1

s1thin

(26)

Since other modes are much faster than the slowest mode, this Th1is set for

all the modes. In this way, all other modes except the slowest one are fully

relaxed. The partial relaxation of the slowest mode of the thick-lithosphere

model has the same strength as the slowest mode of the thin-lithosphere

model.

Scenario Ib is created to simulate that the rotational speed during the forma-

tion of the lithosphere was slower than present-day.

Category II: Scenario IIa. A thick-lithosphere model in hydrostatic equilibrium

with the present-day rotational speed.IIb. A thin-lithosphere model whose

slowest mode has fully relaxed for a faster rotational speed of

𝝎Thin

10=

k1∕s1Thick

k1∕s1Thin 𝝎0(27)

where 𝝎0is the present-day rotational speed. k1∕s1Thick and k1∕s1Thin are the mode strengths of the

slowest modes of the thick and thin lithosphere model, respectively.

Scenario IIb corresponds to the situation that the rotational speed during the formation of the lithosphere is

faster than the present-day value. With equations (26) and (27), we conﬁgure the scenarios of models without

hydrostatic equilibrium, Ib and IIb, such that the inﬂuence of the slowest mode has the same contribution to

the inertia tensor of the entire model at the start of the simulation in each category. We test the model with

an eﬀectively elastic lithosphere (viscosity 1031 Pa s) with the same loading as those in Figure 8. The results

are shown in Figure 10.

As we can see in Figure 10, although the models in each category have diﬀerent thicknesses of the lithosphere

and hydrostatic states, the performance of TPW is almost identical. This can be understood as follows: when

the lithosphere is thin enough (e.g., less than 100 km for the Earth’s case), the inﬂuence of changing the prop-

erties of this layer (thickness, rigidity, and viscosity) is very small on modes other than ̄

T1and ̄

T2generated

from the nonlithosphere part of the model. Its largest eﬀect is, as shown in Figure 4, a remnant bulge which

resists the readjustment of the equatorial bulge. Once we choose a proper hydrostatic state so that the same

contribution to the inertia tensor is guaranteed from the lithosphere of a diﬀerent model, the path of TPW

will also be almost the same. As for the issue concerning the diﬀerence between the observed and model

predicted ﬂattening, it can be seen from equation (27) that if the elastic layer generates a mode with strength

−k1∕s1and the rotational speed during its formation is Ω0, and Ωcat present-day, then this elastic layer would

cause an extra ﬂattening represented in the observed ﬂuid Love number as

Δkobs

f=−

k1

s1Ω0

Ωc2

−1(28)

For the Earth model, for instance, considering that 6 Ma ago the rotational speed of Earth was about 15%

faster (Zahnle & Walker, 1987), the rotational speed during the formation of the lithosphere would be even

higher considering that tidal dissipation started when the Earth-Moon system formed about 4.5 Ga ago.

An elastic lithosphere of 80 km with a rotational speed during formation that is 30% faster than the present-

day speed would cause the observed ﬂuid Love number to increase by about 0.01.

We conclude that for the study of TPW for a model with a lithosphere, it is necessary to know both the initial

stress-free shape of the lithosphere as well as its thickness before a correct TPW behavior can be predicted.

HU ET AL. A FULL-MAXWELL APPROACH FOR TPW 16

Journal of Geophysical Research: Planets 10.1002/2017JE005365

This can be signiﬁcant for the study of TPW on Mars since it has less tectonic activity compared to Earth

to release the stress in its lithosphere, which results in a higher chance that the Mars is not in hydrostatic

equilibrium.

6. Conclusion

We have established a new semianalytical method for calculating large-angle true polar wander (TPW) which

is consistent with the complete scheme of Maxwell rheology, meaning that both fast and slow modes are cor-

rectly taken into account, in contrast with previous studies which adopt the quasi-ﬂuid approximation that

approximate the Love number. We extend the scheme of the linearized Liouville equation in Hu et al. (2017)

which can also be used to simulate the mega-wobble on Venus. Theinﬂuence of the delayed relaxation of elas-

tic or highly viscous layers as well as models in diﬀerent hydrostatic states (e.g., not in hydrostatic equilibrium)

has been studied. With the help of our model, the following conclusions can be drawn:

1. The quasi-ﬂuid approximation can introduce a large error in the transient response for a time-dependent

solution. The TPW speed on Earth and Mars based on the quasi-ﬂuid approximation can be underestimated,

while the speed of the rotational axis approaching the end position on Venus is overestimated.

2. Depending on the considered period, the TPW for a model with a viscoelastic lithosphere can have a

larger displacement and a weaker restoring response compared to a model with an (eﬀectively) elastic

lithosphere. Due to the sensitivity of TPW to the shape of the lithosphere, the viscosity of the lithosphere

needs to be taken into account when the studied period is not much shorter than the relaxation time of

the lithosphere.

3. The permanent shape of the elastic lithosphere can have a large eﬀect on the TPW and this can be

an additional explanation for the diﬀerence between the observed and model-predicted ﬂattening of a

rotating body.

Thus, studies involving TPW can beneﬁt from our method to achieve better accuracy in the pole path,

especially while taking into account a viscous lithosphere or nonhydrostatic background.Our method can also

be extended to tidally deformed models. In this paper we applied algorithm 2 of Hu et al. (2017) to calculate

the rotational perturbation. In the same paper, a new iterative procedure was also presented to combine

both the rotational and tidal perturbation (algorithm 3). Combining this algorithm with the analytical solution

for the change in the moment of inertia presented in the present paper can give a semianalytical solution for

large-angle reorientations of rotating tidally deformed bodies.

Appendix A: Expressions for Ai,j(t),Bi,j(t),Ci,j(t),andDi,j(t)

kT(t)∗𝜔i(t)𝜔j(t)t=Tn=Ai,jTn+Bi,j(A1a)

dkT(t)∗𝜔i(t)𝜔j(t)

dtt=Tn=Ci,jTn+Di,j(A1b)

where

Ai,j(t)=

n

p=1

m

q=1

ai,j,p,q(t)(A2)

Ci,j(t)=

n

p=1

m

q=1

ci,j,p,q(t)(A3)

and

Bi,j=kT

eWi,nWj,n(A4a)

Di,j=kT

eWi,n−1Wj,n+Wi,nWj,n−1−2Wj,n

Tn−1−Tn

(A4b)

HU ET AL. A FULL-MAXWELL APPROACH FOR TPW 17

Journal of Geophysical Research: Planets 10.1002/2017JE005365

where for t>Tp

ai,j,p,q(t)= kq

Tp−1−Tp2s3

qesq(t−Tp−1)Wi,nWj,n−1Tp−Tp−1sq−2+2Wj,n

+Wi,n−1Wj,n−1Tp−1−TpsqTp−1−Tpsq+2+2+Wj,nTp−Tp−1sq−2

−esq(t−Tp)Wi,n−1Wj,nTp−1−Tpsq−2+2Wj,n−1

+Wi,nTp−1−Tpsq−2Tp−1−TpsqWj,n+Wj,n−1+2Wj,n

(A5)

ci,j,p,q(t)= kqe−sqTp−1

Tp−1−Tp2s2

qetsqWi,nWj,n−1Tp−Tp−1sq−2+2Wj,n

+Wi,n−1Wj,n−1Tp−1−TpsqTp−1−Tpsq+2+2+Wj,nTp−Tp−1sq−2

−esq(Tp−1−Tp+t)(Wi,n−1Wj,nTp−1−Tpsq−2+2Wj,n−1

+Wi,nTp−1−Tpsq−2Tp−1−TpsqWj,n+Wj,n−1+2Wj,n

(A6)

and for t≤Tp

ai,j,p,q(t)= kq

Tp−1−Tp2s3

qWi,nWj,n−1sqsqt−Tp−1t−Tp+Tp−Tp−1esq(t−Tp−1)

−Tp−1−Tp+2t−2esq(t−Tp−1)+2

+Wj,n2esq(t−Tp−1)−1−sqt−Tp−1sqt−Tp−1+2

+Wi,n−1Wj,n−12esq(t−Tp−1)−1+sqTp−1−TpTp−1−Tpsq+2esq(t−Tp−1)

+t−TpsqTp−t−2

+Wj,nsqsqt−Tp−1t−Tp−Tp−1−Tp+2t+Tp−Tp−1sq−2esq(t−Tp−1)+2Wj,n

(A7)

ci,j,p,q(t)= kqe−sqTp−1

Tp−1−Tp2s2

qetsqWi,nWj,n−1Tp−Tp−1sq−2+2Wj,n

+Wi,n−1Wj,n−1Tp−1−TpsqTp−1−Tpsq+2+2)+Wj,nTp−Tp−1sq−2

−eTp−1sq(Wi,nWj,n−1sqTp−1+Tp−2t−2+2Wj,nsqt−Tp−1+1

+Wi,n−12Wj,n−1sqt−Tp+1+Wj,nsqTp−1+Tp−2t−2

(A8)

Appendix B: Iterative Algorithm for Calculating TPW

1. Assume that 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 being 𝝎i+1. For the ﬁrst iteration we assume that the vector of the

rotation does not change: 𝝎i+1=𝝎i.

2. Obtain ΔIand its derivative Δ̇

Ifrom equations (12a) and (12b). With Qas deﬁned in equation (15) being the

coordinate transformation matrix from the body-ﬁxed coordinates to the local coordinates where the Zaxis

aligns with the direction of the rotation, the inertia tensors in the transformed coordinates are obtained by

ΔI1=QTΔIQ and Δ̇

I1=QTΔ̇

IQ.

3. Substitute ΔI1and Δ̇

I1into equations (14a) –(14c) and obtain 𝝎′=Ωi(m1,m2,1+m3)T. Normalize this vector

as 𝝎′=Ωi+1̄

𝝎′, where ̄

𝝎′is the direction of the perturbed rotational axis in the local coordinate system and

transform back into the body-ﬁxed frame to obtain 𝝎i+1=Ωi+1Q̄

𝝎′.

4. Substitute 𝝎i+1into step 2 until the result converges.

Appendix C: Governing Equation for the Case B<C<Aand B>C>A

For the case B>C>A, we assume that the changes in the inertia tensor are linear:

ΔI13(t)=a1+b1t(C1a)

ΔI23(t)=a2+b2t(C1b)

HU ET AL. A FULL-MAXWELL APPROACH FOR TPW 18

Journal of Geophysical Research: Planets 10.1002/2017JE005365

Then equations (20a)– (20c) can be solved analytically which results in

m′1(t)=B

A(C−A)3(B−C)Ω2sinh (C−A)(B−C)

AB ΩΔt(C−A)ΔI23(t)−CΔ̇

I13(t)

−cosh (C−A)(B−C)

AB ΩΔtΔI13(t)

C−A+CΔ̇

I23(t)

Ω(C−A)(B−C)

+ΔI13(t)

C−A+CΔ̇

I23(t)

Ω(C−A)(B−C)

(C2)

m′2(t)=A

B(C−A)(B−C)3Ω2sinh (C−A)(B−C)

AB ΩΔt(C−B)ΔI13(t)+CΔ̇

I23(t)

+cosh (C−A)(B−C)

AB ΩΔtΔI23(t)

B−C−CΔ̇

I13(t)

Ω(C−A)(B−C)

−ΔI23(t)

B−C−CΔ̇

I13(t)

Ω(C−A)(B−C)

(C3)

where sinh(u)and cosh(u)are hyperbolic functions

sinh(u)= eu−e−u

2(C4a)

cosh(u)= eu+e−u

2(C4b)

When Δtis small enough, we can apply approximation sinh(𝜃)≈𝜃and cosh(𝜃)≈1−𝜃2∕2. Ignoring the

derivative terms of the inertia tensor simpliﬁes equations (C2) and (C3) into

m′

1(t)= (B−C)Ω2ΔI13(t)Δt2+2BΩΔI23 (t)Δt

2AB (C5a)

m′

2(t)= −(C−A)Ω2ΔI23(t)Δt2−2AΩΔI13 (t)Δt

2AB (C5b)

The same procedure can be applied to the case A>C>B, which results in

m′′

1(t)= (C−B)Ω2ΔI13(t)Δt2+2BΩΔI23 (t)Δt

2AB (C6a)

m′′

2(t)= −(A−C)Ω2ΔI23(t)Δt2−2AΩΔI13 (t)Δt

2AB (C6b)

Figure D1. (left) The polar wander path in the x-zplane of the two-layer (blue) and SG6 (red) Earth models triggered by

a mass anomaly of 2×1019 kg attached at 45∘colatitude in the x-zplane. Lines show the results with the semianalytical

method of Wu and Peltier (1984) (Figure D1, left). Circles represent our method. (right) The polar wander path in the x-z

plane of the two-layer Earth model triggered by a mass anomaly of 2×1019 kg attached at 30∘colatitude in the x-z

plane. Lines show the results with the semianalytical method of Ricard et al. (1993) (Figure D1, left). Circles represent

our methods.

HU ET AL. A FULL-MAXWELL APPROACH FOR TPW 19

Journal of Geophysical Research: Planets 10.1002/2017JE005365

Appendix D: Validation of the Iterative Algorithm

Figure D1 contains the validation done in Hu et al. (2017). The iterative procedure shown in Appendix B has

been compared for short-term small-angle TPW, where linear method (Wu & Peltier, 1984) is accurate. For

large-angle TPW we compare results against the nonlinear method from Ricard et al. (1993). That method can

only simulate TPW for models without large relaxation time contrast in the modes (Hu et al., 2017). It can be

seen that our iterative procedure matches the results obtained with the methods from Wu and Peltier (1984)

and Ricard et al. (1993) very well.

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Acknowled gments

We thank two anonymous reviewers

for their review and constructive

suggestions. This research has

been ﬁnancially supported by the

GO program of the Netherlands

Organization for Scientiﬁc Research

(NWO), project ALW-GO-12/11. All data

used to produce the ﬁgures and the

codes for the algorithm have been

uploaded to 4TU Center for Research

Data (https://data.4tu.nl/) under the

name of this paper.

HU ET AL. A FULL-MAXWELL APPROACH FOR TPW 20