DYNAMIC TOPOGRAPHY OF CONTINENTS AND ROTATIONAL … · 3-D images of mantle shear wave anomaly...

DYNAMIC TOPOGRAPHY OF CONTINENTS AND ROTATIONAL STABILITY OF PLANETS WITH

LITHOSPHERES

by

A m y Louise Daradich

A thesis submitted in conformity with the requirements for the degree of D octor of Philosophy

Graduate Department of Physics University of Toronto

© Copyright by Amy Daradich (2007)

Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

DYNAM IC T O P O G R A P H Y O F C O N TIN EN TS AND

ROTATIONAL STA BILITY O F PLA N ETS W IT H LITH O SPH ER ES

D octor of Philosophy, 2007, Amy D aradich

D epartm ent of Physics, University of Toronto

Abstract

This thesis exam ines two distinct topics rela ted to th e long-term evolution of te r

restria l planets. T he first, dynam ic topography, is th e vertical m otion of th e E a r th ’s

tectonic plates in response to viscous stresses in th e m antle driven by convective

processes. Using m antle flow modelling, I show how dynam ic topography linked to

p la te subduction can explain a long-wavelength com ponent of sedim ent deposition in

th e Silurian Baltic Basin. Sim ulations constrain th e paleo-dip angle of subduction to

40°-60° and show th a t th e slab-induced m antle flow m echanism provides 40-85% of

th e near-field sedim ent deposition. In ano ther regional study, I use convection simu

lations constrained by seismic tom ography to reconcile th e observed broad tilting of

th e present-day A rabian platform th a t extends from th e Red Sea to th e Persian Gulf.

This area has been cited as a classic exam ple of rift-flank uplift; however th e influence

of rift-flank processes is largely lim ited to uplift w ithin a few hundred kilom eters of

th e m argin. D ensity heterogeneities linked to a m egaplum e, which are responsible

for high topography in Southern A frica and rifting in E ast Africa, can reconcile the

anom alous topography seen in A rabia.

T he second topic in th is thesis deals w ith th e ro ta tional s tab ility of p lanets w ith

lithospheres. Using an equilibrium ro ta tional theory su itable for a p lanet w ith a

lithosphere characterized by long-term elastic strength , along w ith observational con

stra in ts on the figure of M ars, I show th a t th e current ro ta tio n axis of M ars is stable.

I also find th a t developm ent of the massive T harsis volcanic province caused a re-

ii


orientation of the planet tha t was likely less than 15° and tha t the thickness of the

elastic lithosphere at the time of Tharsis formation was at least ~ 50 km. Finally,

I extend the equilibrium theory for a planet with an elastic lithosphere to consider

the effect of a viscoelastic lithosphere on rotational stability. I find th a t for suffi

ciently high lithospheric viscosities (5 x 1024 Pa-s or greater) a viscoelastic lithosphere

can have a significant impact on reducing rates of true polar wander induced by an

uncompensated load. These rates depend on the viscosity of the lithosphere and the

size of the load. t


Table of Contents

A bstract ii

Table o f C ontents iv

List o f Tables vii

List o f Figures viii

A cknow ledgem ents x

Forward 1

Part I: Dynamic Topography o f Continents 3

1. Introduction 4

1.1. Early Numerical Simulations and Gravity Anomalies .......................... 51.2. Dynamic Topography and P late S u b d u c tio n ........................................... 61.3. Dynamic Topography and Mantle Superplumes........................................ 131.4. C o n c lu sio n s....................................................................................................... 15

2. Numerical M odelling 16

2.1. The Axisymmetric Mantle Convection C o d e ........................................... 162.1.1. M athematical Formulation of the M o d e l ..................................... 172.1.2. Non-dimensionalisation of the Hydrodynamic Equations . . . 202.1.3. Streamfunction and Vorticity ......................................................... 222.1.4. Numerical Solutions of Convection in the Mantle ...................... 24

2.2. Calculations of Dynamic T o p o g rap h y ........................................................ 25

iv


3. Silurian Baltic Basin 293.1. Geology and Tectonic S e t t i n g ..................................................................... 293.2. Numerical M o d e llin g ...................................................................................... 31

3.3. R esu lts ................................................................................................................. 333.4. C o n c lu sio n s....................................................................................................... 38

4. Rift-flank Uplift o f Arabia 39

4.1. In tro d u c tio n ....................................................................................................... 394.2. A Mantle-flow Scenario for the Tilting of A ra b ia ..................................... 42

4.3. Numerical Model Formulation and R e s u l t s .............................................. 454.4. C o n c lu sio n s ....................................................................................................... 50

Part II: Rotational Stability o f P lanets with Lithospheres 52

5. Introduction: A Review of Gold (1955) 53

5.1. Mathematical Treatment of Fig. 5 . 1 ........................................................... 555.1.1. The Inertia Tensor: Gold (1955)...................................................... 57

5.2. Rotational Stability of a Planet in the Presence of a Lithosphere . . . 59

6. Rotational Stability and Figure o f Mars 61

6.1. In tro d u c tio n ....................................................................................................... 61

6.2. The Physics of Rotating P la n e ts .................................................................. 64

6.3. R esu lts ................................................................................................................. 686.3.1. The Present-Day Rotational Stability of M a r s ........................... 696.3.2. TPW Driven by Tharsis L o a d in g .................................................. 74

6.4. C o n c lu sio n s ....................................................................................................... 82

7. Rotational Stability o f Planets: T he Influence o f a V iscoelastic Litho

sphere 85

7.1. In tro d u c tio n ....................................................................................................... 857.2. M athematical F o rm u la tio n ............................................................................ 897.3. Numerical Im plem en tation ............................................................................ 947.4. Results and D iscussion ................................................................................... 967.5. Future W o rk ....................................................................................................... 102

Summary 105

v


A. M athem atical Formulation o f Axisymm etric Convection Code 120

A.I. Non-dimensionalisation of the Hydrodynamic E quations....................... 121

A.2. Streamfunction and Vorticity ...................................................................... 126

B. M athem atical Treatm ent o f Fig. 5.1 131

B .l. The Inertia Tensor: Two Case Studies ...................................................... 131B.1.1. Case 1: The Equilibrium Form ...................................................... 132B .l.2. Case 2: A Remnant Rotational B u lg e ............................................ 133

B.2. Stokes Coefficients for Case 2: Axisymmetric L o a d in g ........................... 136

vi


List of Tables

2.1. Typical values for physical param eters used in models of mantle convection.................................................................................................................... 21

6.1. Mars model param eters.................................................................................. 73

7.1. E arth model p a ra m e te r s ............................................................................... 96

vii


List of Figures

1.1. Schematic illustration of subduction induced platform subsidence and

u p li f t ..................................................................................................................... 71.2. Relationship between dip angle of the subducting slab and horizontal

deflections of the overlying l i th o s p h e re ...................................................... 101.3. Numerical simulation of a “slabalanche event” ....................................... 12

3.1. Reconstructed sediment deposition on the Silurian Baltic Basin . . . 303.2. Schematic illustration of the axisymmetric convection simulation . . . 33

3.3. Results of mantle flow-induced dynamic to p o g rap h y ............................. 353.4. Both the slab dip angle and percent contribution to CDF deposition

due to dynamic topography, as a function of trench lo c a t io n ............. 37

4.1. Tectonic setting of East African-Arabian r e g i o n .................................... 404.2. Topography and residual topography of East African-Arabian region . 414.3. 3-D images of mantle shear wave anomaly contours in the mantle un

derlying the Afro-Arabian region ................................................................ 444.4. Shear wave velocity-to-density scaling p ro f ile .......................................... 464.5. Axially symmetric solution space and density heterogeneity from the

Grand (2002) S-wave model. ...................................................................... 484.6. Residual and dynamic topography of the African-Arabian region. . . 49

5.1. Schematic illustration highlighting the physics of T P W ......................... 54

6.1. Global topography of Mars from MOL A ................................................. 626.2. Non-hydrostatic moments of inertia for M a r s .......................................... 716.3. Non-equilibrium moments of inertia for M a rs .......................................... 74

viii


6.4. Range of TPW angles, as a function of the uncompensated size ofTharsis, tha t yield ‘acceptable’ fits to the present-day gravitational

figure of M a r s .................................................................................................... 796.5. Range of lithospheric thickness, as a function of the uncompensated

size of Tharsis, tha t yield ‘acceptable’ fits to the present-day gravitational figure of M a r s ....................................................................................... 81

7.1. Schematic illustration highlighting the physics of T PW for a viscoelastic lith o sp h ere .................................................................................................... 88

7.2. TPW for different initial load colatitudes, with Q' = 1 and lithosphericviscosity of 5 xlO 25 Pa-s . 97

7.3. Time scales of polar wander, as a function of Q ' .................................... 997.4. TPW for different lithospheric viscosities, with Q' — 1 and an initial

load colatitude of 2 0 ° ....................................................................................... 1017.5. Time scales associated with polar wander, as a function of lithospheric

v isc o s ity .............................................................................................................. 102

ix


Acknowledgements

Many thanks to my supervisor, Professor Jerry Mitrovica, for his encouragement and guidance throughout the course of my graduate studies. Your boundless enthusiasm inspired me to pursue a graduate degree in geophysics. I consider myself very lucky to have had you as a supervisor.

I would like to acknowledge my committee members, Professors Dick Bailey and Russ Pysklywec. Thanks to Professors Sabine Stanley and Roberto Sabadini for agreeing to serve as examiners for my final defense.

I also received a lot of support over the years from the following post-docs and graduate students: Mark Tamisiea, Jonathan Mound, Rob Moucha, Roblyn Kendall, Jill Pearse and Zeina Khan. I can’t thank you enough for your friendship and for the many helpful discussions related to my research.

My time spent here has been filled with many happy times and has broadened me as a person in large part due to the graduate students I befriended. Thank you for the brunches, giant lunches in Chinatown, political discussions, jam sessions, knitting sessions, nights at the pub, coffee runs, softball games, concerts, etc. Pm so very glad to have met such kind, diverse and talented people.

Thanks to my family for their love and support from “day one” . My grandparents, Helen and Victor Best, parents, Lynn and Gary Daradich and sister Natalie. You’ve

always believed in me, no m atter what it was I decided to pursue. Also thanks to my little boy who has kept me company for the last seven months while I was finishing this thesis. I can hardly wait to meet you!

Finally, I can never hope to express the depth of my gratitude to my husband, Denis Dufour, for his constant love and encouragement, whether near or far. I dedicate this thesis to him.


Forward

The terrestrial planets are dynamic, evolving systems tha t are subject to a large

range of internal and external forcings with a broad suite of characteristic time scales.

Since these planets have viscoelastic rheologies, their response will depend on the time

scale of the forcing. That is, for a progressively shorter period forcing, the behaviour

becomes increasingly more elastic, while at much longer time scales the same planet

will respond as a viscous fluid.

The Earth provides many examples of this behaviour. Shorter period forcings

include earthquakes, which give rise to the propagation of seismic waves, and the

gravitational attraction of the sun and moon (as well as other bodies in the solar

system) which cause tides in the oceans and solid surface. Seismic normal modes

have periods ranging from seconds to many hours, while tides are clustered into semi

diurnal, diurnal and long-period bands. At intermediate time scales, the E arth has

been subject to glacial cycles of period ~ 100 kyr, and the response to the time

varying (ice plus ocean) surface load gives rise to deformation formally termed glacial

isostatic adjustment. This deformation is viscoelastic; the adjustm ent continues to

the present-day, 5000 years after the end of the final deglaciation event.

This thesis is comprised of two distinct subjects th a t are nevertheless linked by the

very long time scales tha t characterize each of them. P art I of the thesis is concerned

with the response of the Earth to therm al convection driven by internal tempera-

1


ture contrasts th a t have existed for billions of years. This convection is ultimately

responsible for horizontal motions of tectonic plates, which give rise to the processes

of sea-floor spreading and continental drift. However, the same convective flow also

drives vertical motion of the plates, or vertical plate tectonics, and this dynamic to

pography is evident in the geological record of continents. Such motions are known

to have caused the Cenozoic tilting of the western interior of North America, as well

as the present uplift of southern Africa. In this part of the thesis, I will examine

whether dynamic topography is a viable explanation for two enigmatic episodes of

continent-scale deformation; namely, the Silurian development of the Baltic Basin,

and the present-day tilting of the Arabian plate.

In P art II of the thesis I tu rn my attention to the long-term rotational stability

of terrestrial planets. In this case, I adopt a so-called equilibrium rotation theory

in which all regions of the planet, with the possible exception of the high viscosity

lithospheric lid, are treated as inviscid. As an application of the theory I first assume

an elastic plate, and revisit the question of the stability of the rotation axis of Mars. A

common view, in this regard, is tha t the massive Tharsis volcanic province, emplaced

early in M ars’ history (~ 4 billion years ago), led to a dram atic shift in the rotation

pole relative to the surface geography, or true polar wander (TPW ). Finally, I extend

this elastic plate theory to consider the impact, on long-term rotational stability, of

viscoelastic behaviour within the lithosphere.

Given the rather distinct nature of the processes treated in these two sections of

the thesis, a detailed introduction will precede each part. I begin with a discussion of

the growing literature within the mantle convection community associated with the

concept of dynamic topography.


Part I:

D ynam ic T opography o f C ontinents

3


1. Introduction

Plate tectonics (e.g. Wilson, 1963, 1965) is a widely accepted and elegant theory

which holds th a t the E arth ’s surface is composed of rigid plates in relative motion.

The theory subsumes the earlier ideas of continental drift and sea-floor spreading and

provides a framework for explaining a wide array of surface geophysical and geolog

ical features. However, plate tectonics is a purely kinematic theory, and questions

regarding the driving force for plate motions remain a source of active debate in geo

physics. Specifically, while therm al convection and plate buoyancies (i.e., ridge push

and trench pull) are clearly the forces behind plate tectonics, the detailed connection

between interior flow and plate motions remains unclear.

Since the E arth ’s mantle has a tem perature dependent rheology, the plates can

be considered to be the cool upper therm al boundary layers of large-scale mantle

convection cells. The earliest studies exploring the link between plate motions and

mantle flow were generally limited to horizontal motions, as reflected in the terms

‘continental drift’ and ‘sea-floor spreading’. However, if one accepts tha t flow in the

mantle can cause horizontal motions of plates, then it is entirely reasonable to assume

th a t vertical motions could also result from this flow. Concerted efforts to understand

the vertical deflection of plates, or dynamic topography, can be traced back to the

1980’s (Hager, 1984; Mitrovica et ah, 1989; Gurnis, 1992, etc.). These studies were

focused on a broad class of surface observables, including gravity anomalies, conti-

4


1. Introduction

nental subsidence and uplift. The assumption in such studies is th a t the observed

vertical deflection of the E arth ’s surface (after correction for isostatic effects), termed

dynamic topography, results from viscous stresses induced by flow in the mantle.

1.1. Early Numerical Simulations and Gravity

Anomalies

Even the earliest, highly idealised, numerical simulations of mantle convection (e.g.

Richter, 1973; McKenzie et ah, 1974) showed th a t the process resulted in non-trivial

normal stresses exerted at the top (and bottom) of a convection cell. These stresses

would in turn deflect the upper boundary of the Earth.

The first clues to this deflection arose from the study of the E arth ’s gravity field.

Runcorn (1965), among others, argued th a t gravity should be anomalously low over

hot rising mantle ‘currents’. However, a series of observations leading to the seminal

work of Chase (1979), suggested the opposite; tha t is, the gravitational pull actually

increased over such regions. These observations were paradoxical, but they were

ultimately resolved by invoking dynamic topography. T hat is, while the lower density

of the column of hot material does result in reduced gravity, this reduction is more

than offset by the gravitational effect from the upward deflection of the surface.

Interestingly, there appears to be continuing disagreement over who should be given

credit for resolving the paradox. Gurnis (2001) has recently argued th a t Hager (1984)

was responsible for predicting the relative size of the opposing effects on the E arth ’s

gravitational field. However, the same argument appears in a classic paper by McKen

zie et al. (1974), who modelled the E arth ’s mantle as a constant density, constant

viscosity, infinite Prandtl number and incompressible fluid (see his Fig. 18). Regard


1. Introduction 6

less of the history of the argument, the observations and its ultim ate reconciliation

provided the first compelling geophysical evidence for the importance of flow-induced

vertical deflections.

1.2. Dynamic Topography and Plate Subduction

It has long been known th a t the geological record shows evidence tha t continents

periodically experience up and down motions (e.g. Bond, 1978; Veevers, 1984; Mitro-

vica et ah, 1989). These movements, termed epeirogenic motions, have been dis

tinguished from global eustatic sea-level changes by considering contemporaneous

sea-level records from other continents or even other regions on the same continent

(Bond, 1978). Geophysical efforts to explain these epeirogenic motions provided a

second, even more direct, argument for the existence of mantle flow-induced dynamic

topography.

Over the last two decades there has been growing appreciation tha t topography

driven by mantle flow viscously coupled to descending tectonic plates will contribute

significantly to the evolution of basins in the vicinity of convergent margins (e.g. Beau

mont, 1982; Mitrovica et ah, 1989; Gurnis, 1992, 1993; Gurnis et ah, 1998; Pysklywec

and Mitrovica, 1999). A schematic illustration of this mechanism is shown in Fig. 1.1.

In the first frame of the figure, a stable continent experiences the onset of subduction

on its margin. Viscous stresses associated with subduction drive mantle flow (frames

1 and 2) which acts to exert normal stresses on the base of the overlying lithosphere

(and continent) and which causes the lithosphere to subside (frame 1 and 2). This

subsidence leads to a sea-level transgression (or onlap) and the development of a sed

imentary basin on the continental platform. The cessation of subduction (in this case

by continental accretion) weakens the dynamic support for the subsidence and leads


1. Introduction 7

Ait Both "Rifted" Croton i Arc Margin

'Rifted" p. .. Mcegw Ptot{orm

V=Vt

PlatformForedeep

V « V,

-10 Mo X s \ to d ea r n£"\

upper mantle \

4)

v=o

Platform

V= V,

Figure 1.1.: Schematic illustration demonstrating the mechanism of subduction induced platform subsidence and uplift. (Taken from Mitrovica et al., 1989).

to an uplift of the sedimentary basin (frames 3 and 4). The subduction may begin

again, leading to a second cycle of platform sedimentation (frame 5).

Mitrovica et al. (1989) invoked this general mechanism to explain the long-wavelength


1. Introduction

late Cretaceous subsidence of the western interior of North America and its subse

quent uplift during the Tertiary. Specifically, during the Cretaceous, the now extinct

Farallon plate was subducting under the western coast of North America. As the plate

descended, viscous stresses tilted the western portion of the continent downward to

the west producing a large accommodation space for sediments. The end of Farallon

subduction (due to triple junction migration rather than continental accretion) led to

the Tertiary uplift of the basin. This uplift is the cause of the present-day topography

of the western U.S., which increases toward the west and which has led, for example,

to the ‘mile-high’ topography of cities like Denver.

This process of subduction-controlled tilting is distinct from the effects of loading

due to overthrusts, which leads to the development of foreland basins (Beaumont,

1981). Overthrusts, and foreland basins, only result in warping of regions within

~300-500 km of the edge of the craton (Price, 1973; Beaumont, 1981). This is impor

tan t for two reasons. One, it provides a means for distinguishing basin development

due to overthrust and mantle (subduction) loads. Second, it indicates tha t the ‘tra

ditional’ view of a foreland basin as being the response to overthrusts alone must

be broadened to include other mechanisms of ‘tectonic loading’, including mantle

flow-induced dynamic topography (e.g. DeCelles and Giles, 1996; Catuneanu et ah,

1997; Pysklywec and Mitrovica, 1999). The contribution to near-trench basin de

velopment from tectonic loading may in fact be quite significant. Indeed, Pyskly

wec and Mitrovica (1999) have considered the signature of slab-subduction on the

Carboniferous-Triassic development of the Karoo Basin and they showed th a t the

mechanism contributed up to 30% of the subsidence associated with the foreland

environment.

In the work presented by Mitrovica et al. (1989), subducting plates were modelled


1. Introduction 9

by introducing thermal fields from either a continuous or block slab model into a

mantle convection code. These models were able to produce a maximum tilt ampli

tude of ~ 3 km and deflections with horizontal length scales of ~1400 km or more.

The most im portant param eter governing the horizontal scale of the deflection was

the dip angle of subduction. Fig. 1.2 shows the relationship between dip angle and

horizontal deflections of the overlying lithosphere, which indicates th a t the la tter in

creases strongly which a reduction in the former. The vertical scale of subsidence,

however, is influenced by a number of different parameters, including the tem pera

ture contrast between the slab and the surrounding mantle, the flexural rigidity of the

lithosphere, the dip angle of the subduction zone and the age of the subducting slab.

Uplift subsequent to the loss of dynamic support can be initiated by the cessation of

subduction, a roll-back of subduction to larger dip angles (Mitrovica et ah, 1989) or

an introduction of younger lithosphere into the subduction zone (Gurnis, 1992).

The tilting of the interior of North America during the Cretaceous is certainly

not the only example of subduction induced tilting of otherwise stable continental

interiors. The mechanism has been invoked to explain, for example, the Miocene

subsidence of the Taranaki basin (Stern and Holt, 1994), the Ordovician tilt of the

Michigan basin (Coakley and Gurnis, 1995) and the Devonian to Permian tilting of

the Russian platform (Mitrovica et ah, 1996).

Subducted slabs can have an impact on the dynamic topography of continents

even after the cessation of plate consumption. An example of this is given by the

Cretaceous vertical motion of Australia, as described by Gurnis et al. (1998). 130

million years ago, Australia’s eastern edge was bordered by a subduction zone. This

subduction was acting, as in the mechanism described above, to pull the eastern

edge of Australia downward. Subduction ceased about 95 million years ago and


1. Introduction 1 0

2000kmr

1500km -

1000km -xT3

?Oz

500 km -

o * ................................... — — i .........

20 30 45 60 70NEAR-SURFACE DIP (DEGREES)

Figure 1.2.: Plot of the horizontal scale cLh (the distance between the point of maximum subsidence and the point of zero surface deflection which is furthest from the subduction zone) versus near-surface dip for a block model of subduction. Results were computed using a lithospheric flexural rigidity of 5 x 1023 Nm. dfjax is the distance from the point of maximum subsidence to the point of maximum uplift which is furthest from the subduction zone. (Taken from Mitrovica et al., 1989).

A ustralia began to drift eastward. As Australia passed over the remnants of the

inactive subduction zone, it sank ~ 300 m below sea level. Approximately 20 million

years later (70 million years ago), the descending and detached slab had reached

sufficient depth tha t support for dynamic topography began to reduce considerably

and the continental interior began to uplift. Thus local sea-level fell during this period

despite the fact th a t eustatic sea level was a maximum at this time. These epeirogenic

motions are clearly evident in the geological record (Bond, 1978; Gurnis et al., 1998).

i n i t ia l BLOCK T£M R

■ -200 *K

- - 4 0 0 *K

- -800 *K


1. Introduction 11

The connection between subducted slabs and sedimentary basin development may

extend beyond continental margins. The geological record also shows evidence of

transient large-scale intracratonic sedimentary basins; th a t is, the development of

transient basins deep into the interior of continents. It has recently been suggested

th a t these basins are linked to dynamic topography associated with so-called mantle

‘slabalanches’ (Pysklywec and Mitrovica, 1998).

Mantle convection simulations have shown th a t both cold downwelling plumes and

subducting slabs are impeded by the spinel to post-spinel phase boundary at 660

km depth in the mantle (e.g. Christensen and Yuen, 1984). The impedance of flow

by this endothermic phase boundary results in pooling of cold material above the

boundary until an instability gives rise to penetration through the phase boundary.

This mass flux event has been investigated in a number of mantle convection studies

(e.g. Machetel and Weber, 1991; Tackley et al., 1993; Weinstein, 1993; Honda et al.,

1993; Solheim and Peltier, 1994) and has come to be known as a mantle “avalanche

event” . The earliest simulations of the process suggested very dram atic mass flux

events occurring at the 670 km phase boundary, while la tter simulations involving

temperature-dependent viscosity yielded a more moderate mass flux (e.g. Zhong and

Gurnis, 1994). More recently, the influence of the endothermic phase boundary on

slab descent and dynamics has also been investigated (Pysklywec and Mitrovica, 1998;

Tackley, 1998) and, in this case, the pooling and subsequent mass flux has been called

a “slabalanche event” (Tackley, 1998). An example of a numerical simulation of such

an event is given in Fig. 1.3.

Pysklywec and Mitrovica (1998) have argued tha t the mass flux associated with

delayed slab penetration through the 660 km phase boundary (see Fig. 1.3) is capa

ble of driving a pulse of dynamic subsidence in the overlying lithosphere and basin


1. Introduction 12

t = 2Ma t = 5 Ma t = 20 Ma t = 30 Ma

t = 50 Ma t = 70 Ma t = 90Ma t=120Ma

Figure 1.3.: Thermal fields showing the evolution of a “slabalanche event”. The labels in the figure indicate the time elapsed since the injection of a slab into the upper mantle with a dip angle of 20° and a temperature contrast of —500°K. This simulation uses a modified version of the convection code which is described in Chapter 2. The modified code includes the effects of both the 660 km phase boundary (indicated by the thin black line in the figure) and variable (pressure and temperature dependent) viscosity. (Taken from Pysklywec and Mitrovica, 2000).

development well away from continental margins. As an example, they argued that

such an event may have been responsible for the formation of a large intra-continental

basin in the western interior of North America from the the Early Devonian to the

Late Carboniferous (Pysklywec and Mitrovica, 2000). Their analysis suggested a

rather general connection between plate subduction and continental basin systems.

This connection involves foreland basin development due to overthrust loads, tilting

of continental margins due to mantle flow coupled to active subduction, and intracon

tinental basin development produced by slab penetration through the 660 km phase

boundary. The latter can occur either during or after a phase of active subduction at

the continental margin.


1. Introduction 1 3

1.3. Dynamic Topography and Mantle Superplumes

So far, I have discussed the effects of downwellings in the mantle on observed surface

features of the Earth. There are, however, a class of large scale mantle upwellings, or

superplumes, which can have significant effects on surface topography. The distinc

tion between hotspots and superplumes is based on considerations of the area of the

region th a t is affected by the mantle plume. While hotspots can only affect regions of

seafloor tha t are less than 1000 km in cross-sectional diameter (the uplifted region is

then termed a “swell”), superplumes influence sea-floor elevations on scales tha t are

as large as ~7000 km or more (the uplifted region is then termed a “superswell”) (Mc

N utt, 1998). Upwellings causing superswells are large enough to be seen using mantle

seismic tomography, and appear as regions of lower than average seismic velocities.

Since seismic waves travel more slowly in regions th a t are less dense, if the upwellings

are not chemically distinct, then these regions represent thermally buoyant masses.

Two examples of large mantle upwellings tha t have been detected through use of seis

mic tomography are the African superplume and the South Pacific superplume. The

view tha t the buoyancy of both of these superplumes is dominated by thermal effects

is supported by the existence of large numbers of hotspots and rifting in the vicinity

of both of these upwellings (Burke, 1996; Calmant and Cazenave, 1987) and recent

geodynamic studies tha t will be described in this section (e.g. Lithgow-Bertelloni and

Silver, 1998; Gurnis et al., 2000; Forte and Mitrovica, 2001).

In addition to therm al activity, anomalous uplift of the eastern and southern African

plateaus and surrounding oceans can be attributed to the African superplume (Hager

et al., 1985). Presently, the southern African plateau stands more than 1 km above

sea-level, while the average elevation of most cratons is on the order of ~ 500 m. Sur

rounding oceans exhibit a residual bathym etry exceeding 500 m (Lithgow-Bertelloni


1. Introduction 1 4

and Silver, 1998). The possibility tha t anomalous topography of Africa is caused by

dynamic topography has recently been examined, since isostatic considerations alone

cannot reconcile these areas of extremely high topography in Africa (Hager et al.,

1985; Lithgow-Bertelloni and Silver, 1998; Gurnis et al., 2000).

By invoking calculations tha t consider the effects of viscous stresses exerted on

the base of the lithosphere by the African superswell, Lithgow-Bertelloni and Sil

ver (1998), following the original work of Hager et al. (1985), were able to reconcile

the observed residual topography - th a t is, the topography which remains after hav

ing removed effects such as isostatic loading and the square-root age law from the

observed topography. Calculations of dynamic topography associated with the super

swell were made by first converting tomographic data into a density field, which was

then used as input for a numerical simulation of therm al convection in an incompress

ible, Newtonian, viscous fluid. Topography resulting from the low density field was

then computed from (instantaneous) flow calculations of stresses which were induced

on the lithosphere by the density anomaly. Although this work was successful in

reconciling the anomalous topography of Africa, it was later shown tha t calculations

by Lithgow-Bertelloni and Silver (1998) would give erroneous results for other geody

namic parameters, such as the rate of change of dynamic topography (Gurnis et al.,

2000). Lithgow-Bertelloni and Silver (1998) also assumed tha t the entire low-shear

velocity anomaly below Africa was entirely therm al in nature, an assumption which is

now thought to be incorrect. Below 2000 km depth in the mantle, seismic inversions

have shown th a t shear wave speed and bulk sound speed are either poorly correlated

or anticorrelated (Kennett et al., 1998; Ishii and Tromp, 1999). This result sug

gests th a t wide-spread chemical heterogeneity exists within the deep mantle. Recent

work by Ishii and Tromp (1999) indicates the presence of denser than average regions


1. Introduction 15

(by ~1%) of chemical heterogeneity extending up to 500 km above the core-mantle

boundary. Calculations have shown th a t while these dense regions of compositional

heterogeneity have the effect of reducing positive (thermal) buoyancies of both the

African and South Pacific superplumes (Forte and Mitrovica, 2001), the net buoyancy

of both structures is positive, Indeed, calculations which include these denser regions

can still reconcile both the dynamic topography and the rate of change of dynamic

topography produced by the African superplume (Gurnis et al., 2000).

The second large-scale deep mantle upwelling, the South Pacific superplume, has

also had significant impact on the E arth ’s topography. The area of the surface of the

E arth th a t is affected by the superplume is approximately 15 million km2. In areas of

French Polynesia, seafloor depths are 250 to 750 m shallower than lithosphere in the

North Pacific and the North Atlantic th a t is of the same age (McNutt and Fischer,

1987). Higher elevations of the sea-floor in the vicinity of the superplume can also

be largely explained by the viscous forces th a t the plume induces on the overlying

lithosphere (McNutt, 1998).

1.4. Conclusions

The purpose of this introduction was to give the reader a broad overview of the

subject of dynamic topography. Although not all aspects of mantle flow described

here are relevant to the problems of modelling subsidence of the Silurian Baltic basin

or rift-flank uplift of Arabia, the exercise was nevertheless instructive. For example,

while phase boundaries were not included in convection simulations used in this work

(most notably the endothermic phase boundary at 670 km depth), they would need

to be included in future work tha t explores the time evolution of dynamic topography

in both regions.


2. Numerical Modelling

2.1. The Axisymmetric Mantle Convection Code

My convection simulations are based on a spherical, axisymmetric, finite differencing

scheme. The set of equations implemented in the code, which was originally writ

ten by Larry Solheim and later revised by Russell Pysklywec, are those governing

conservation of mass, energy and momentum in a compressible, uni-phase, constant

viscosity, infinite P randtl number, Newtonian fluid. In what is to follow, the hydro-

dynamic field equations will be non-dimensionalised and then reformulated in terms

of both streamfunction and vorticity. This will leave us with the set of equations tha t

are actually implemented in the code.

It should be noted tha t while more complex codes exist which include either depth

or tem perature dependent viscosity, it was deemed instructive to begin with a more

simple model, the results of which can then be compared to later simulations which

would employ a variable viscosity scheme. For a more detailed derivation of the

equations used in the convection code, the reader is referred to Appendix A.

16


2. Numerical Modelling 17

2.1 .1 . Mathematical Formulation of the Model

This derivation begins with the hydrodynamic field equations governing conservation

of mass, momentum and internal energy, respectively (e.g. Landau and Lifshitz, 1959):

Equation (2.4) is a linearised Taylor expansion of the density about a reference state

with density pr, adiabatic tem perature Ts and hydrostatic pressure ph■ These

profiles are assumed to be a function of radius, only.

In this system of four equations, the variable quantities p, p, T and u are the

density, pressure, tem perature and velocity, respectively. In solving these equations,

the coefficient of thermal expansion a, isothermal bulk modulus, K?, gravitational

acceleration, g, thermal conductivity, k, and specific heat at constant pressure, cp,

are assumed to vary with depth alone. The internal heating rate per unit volume, Q,

which is due to the decay of radiogenic elements, is assumed to be constant.

Provided tha t the mantle can be modelled as a Newtonian fluid, the deviatoric

stress tensor r^ , can be related to the velocity field through the dynamic viscosity rj

! + V . ( p u ) = 0 (2 .1)

(2 .2)

(2.3)

where, ^ + u • V is the material or Lagrangian derivative.

These equations are completed by an equation of state:

(2.4)


2. Numerical Modelling 1 8

and the bulk viscosity rj2. This relationship is given by:

(2.5)

Since volume changes in the mantle occur over times tha t are greater than 107

years, we can neglect the last term in equation (2.5). This is because 772 is

associated with rapid changes in volume (Jarvis and McKenzie, 1980). Equation

(2.5) thus reduces to:

Our system of hydrodynamic equations can be further simplified by treating the

mantle as a ‘Boussinesq-like’ fluid. Using approximations implemented by Solheim

and Peltier (1994), we neglect deviations from the reference density pr in every term

except for the body force term, pg, of the momentum balance equation (2.2).

Unlike the Boussinesq approximation, the reference density is not constant, but is

instead depth dependent and can be considered to be th a t of a fluid in hydrostatic

equilibrium. Below 670 km depth, the mantle is very nearly in hydrostatic equilib

rium, as can be seen in the seismic model PREM (Dziewonski and Anderson, 1981).

The reference density is obtained by best-fitting data from this region of the mantle

to an exponential function which increases with depth. The portion of the mantle

residing above 670 km is not included in calculations of the reference density state,

since there are a number of discontinuities in the data for the upper mantle. Instead,

values used for the reference density in the upper mantle are extrapolated from the

density function found for the lower mantle.

(2 .6)



Under the assumption of hydrostatic equilibrium in the mantle, we can use the

hydrostatic equation = —pg, to find a new expression for a T j f i in equation (2.3).

a T « a T ^ + aT(u-Vph)

= aT [u ■ { - p rgf)\

= —a T p rgur (2.7)

Here we have set ^ = 0 since the reference pressure state will not vary with time.

Another simplification arises from the fact tha t the mantle has an extremely high

P randtl number, P r = - m 1024. Here, k = is the therm al diffusivity, andK prCp

v = 11 is the kinematic viscosity. In a thermally convecting, high P randtl numberPr

fluid, viscous dissipative forces are much larger than inertial forces. Traditionally, the

mantle is said to have effectively infinite Prandtl number (McKenzie et al., 1974),

and then the inertial force term p ^ - is neglected in the momentum equation (2.2).

The mantle is thus treated as being momentum free (relative to dissipative forces).

Finally, we can eliminate or filter elastic waves from the model by setting = 0.

This is known as the ‘anelastic approxim ation’ (Jarvis and McKenzie, 1980). This

approximation is highly accurate because the propagation of seismic waves requires

rapid density changes over time scales tha t are orders of magnitude shorter than

mantle convection time scales.

By invoking all of the approximations mentioned previously, our hydrodynamic

field equations reduce to:

V • (Pru) = 0 (2.8)

d , NV p = - p g r + — Tij (2.9)


2 . Numerical Modelling 2 0

D T du-PrCp~Dt + a T PrgUr = ^ ‘ + Q + rii~Q T (2-10)

3

where ry is given in equation (2.6).

2.1 .2 . Non-dimensionalisation of the Hydrodynamic Equations

In mantle convection simulations, variables are often non-dimensionalised. The utility

of non-dimensionalisation is tha t it isolates a set of non-dimensional similarity vari

ables th a t control the basic physics of the system and thus link a disparate set of differ

ent dimensional cases. The following is a set of equations used to non-dimensionalise

the hydrodynamic field equations:

j 2 k p>nr — dr1, t = — — f , u = ^ — u', T = TCT>+T0, K T = p0g0dK'T, (2.11)

K0Ha a

p = p0p', p — p0gdp', a = a 0OL, k — k0k', v = vQv ', g — gDg',

k — kQk , Cp CpQCp.

Here, primed terms are dimensionless. Terms with the subscript zero are reference

quantities adopted from a surface value of the appropriate parameter. The

characteristic tem perature Tc is the difference in tem perature between the upper

and lower boundaries if both boundaries are isothermal. In these equations we have

also introduced the Rayleigh number, Ra = a°Jc °d . Values for most of the

dimensional constants introduced can be found in Table 2.1. All of these values were

taken from the convection model presented in Solheim (1992). The depth of the

convection cell, d, is taken to be equal to the depth of the whole mantle.



Param eter Value

depth scale, d 2.89 x 106 mthermal expansion, a 0 2.5 x 10“5 K -1gravitational acceleration, g 10 m s~2reference density, pQ 4.0 x 103 kg m -3thermal conductivity, k 10.0 W m K_1thermal diffusivity, k0 2.0 x 10-6 m2 s-1heat capacity, cp 1.25 x 103 J k g -1 K - 1dynamic viscosity, p 1021 Pa-skinematic viscosity, u0 2.5 x 1017 m2s_1

Table 2.1.: Typical values for physical parameters used in models of mantle convection.

After applying the non-dimensionalising scheme, equations (2.8) to (2.10) and the

equation of state (2.4) reduce to (dropping the primes):

V ■ [pru] = 0

V p = -p g r + a 0Tc[V2u + -V (V ■ u)

d T~dt

— - V ■ (Tpru) - r (T + T0)ur +p^ i \ cl

V 2T + - — — +Qd2 ! To<P

k0kTc pfCp

p = pr[ 1 - a 0Tca (T - Ts) + — (p - ph)]

(2 .12)

(2.13)

(2.14)

(2.15)

where

= 2 e i j e i j g ( V ' u ) (2.16)

is the non-dimensional viscous dissipation rate per unit volume (Solheim, 1992) and

21 ( duj_ , d u A2 dxi dxj J is the rate of strain tensor.

In the derivation of equation (2.14) we have also introduced the dissipation func

tion, r ( r ) = t 0 — , with its corresponding reference value of t0 = The dissipation



function measures the extent to which frictional heating and work done by compres

sion influence energy balance in the flow.

2.1 .3 . Streamfunction and Vorticity

In this section we shall find new expressions for our non-dimensionalised hydrody

namic equations in terms of vorticity and streamfunction. In what will follow, all

quantities are dimensionless. We can write the axially symmetric velocity field in

terms of the streamfunction, ip, in such a way tha t the continuity equation (2.12) is

automatically satisfied:

/ r>\ 1 dip 1 d ipu — [ur, Uq, 0), ur — - ; , Uq = ; x 'x- (2.17)prr2 sin 9 ov prr sin 9 or

From this, we can determine an expression for the azimuthal component of the

vorticity, w = V x u = (0 ,0,u;). We find that:

u j — — -

d2ip 1 dpr dip cot 9 dip 1 d2ip ■prr sin 9 I d r2 pr dr dr r2 d9 r 2 d92 .

(2.18)

Other expressions involving u> can be found by first taking the curl of both sides of

equation (2.13). When this operation is performed, all gradient terms in the equation

are eliminated. After some algebraic manipulation we obtain:

V 2c u -UJ_______ = 9 dp

r2 sin2 9 ot,0Tcr d9(2.19)



Using equation (2.15), we can find an expression to replace in equation (2.19).

Substituting this expression into equation (2.19),

V 2 u prga d T pr g dpr 2 sin2 6 r 86 K T u 0Tcr 86 }

Next, an expression for | | can be found by equating 6 components on both sides of

equation (2.13). This new expression for can then be expressed in terms of both

streamfunction and vorticity, using equations (2.17) and (2.18). After significant

algebra we obtain:

1 dp du> uj 4 dpr 1ce0Tcr 86 dr r 3 dr p2rz sin 6

Substituting this expression into equation (2.20) gives:

prgdui K t dr

d2V> _ ft8 t L 862 86

(2 .21)

__o prg duo r Prg i i UJ =prga 8 T 4 g dpr 1 'd2i/j dtp'

- c o t0 m ..tK t r2sin26- r 86 2>Kt Pt dr r 3sin# 1862(2 .22)

Finally, equation (2.14) can be rewritten in terms of streamfunction, through use of

equation (2.17). Our final expression for equation (2.14) in terms of streamfunction

becomes:

8 T _ _ k_8t Ra

72rr , 1 d k d T t Qd2vzr + —k dr dr k0kTc. pr I cp r * sm ( r(T + r0) J + J(T,V.)]}

(2.23)

and the8 T /d r 8 T /8 6dd> / dr dip j 89

dissipation function, (p, in spherical coordinates is equal to (Solheim, 1992):

where J{T,ip) is by definition d(T,'ip)/d(r,6) =


2 . Numerical Modelling

1

2 4

prrz sin ( )H( -1 dpr 2^ dtp_ + d_dpr\2+2

dr2

1 cfy .dr dO

pr dr r ) dO dO dr -

(

1 dtp dtp~ co tdK~ r dO dr

(2.24)

pr dr r / 3 r r 2 dO2 r2 dO-

d'tp 1 dtpdr r dO

d dtp dtp l d t p i 2 1 / 1 dpr \ 2 / d t p \ 2^3 \ pr dr ) \ dO ) i

To summarise, our complete anelastic model consists of the following system of

equations:

u> =1 rd2tp 1 dpr dtp cot 0 dtp 1 d2tp

prr sin 0 . d r2 pr dr dr r 2 dO r 2 dO2 .(2.25)

prQdu)K t dr

r Prg i i prga d T 4 g dpr 1 ■d2tp- c o t e ae.-rKx r2 sin2 0. U r dO 3K t Pt dr r 3sin0 [ d o 2

? L ~ J Ldt Ra

V 2T +1 dk d T Qd

+k dr dr k0kTc + -1 (Tn 1

pr t cp r z sin (

(2.26)

-(T + r j ^ + j f r ^ ) ] }

(2.27)

These are our equations (2.18), (2.22) and (2.23), respectively. This set of equations

is completed by the equation for the dissipation function in spherical coordinates,

equation (2.24).

2.1 .4 . Numerical Solutions of Convection in the Mantle

Our derivations have led us to the system of equations solved by the axisymmetric

convection code written by Solheim (1992). It is instructive to summarise the order

in which these equations are solved within the code.

To solve the field equations, an initial streamfunction and tem perature field is

specified. Using these fields, the vorticity may be found from equation (2.26). The

resulting vorticity is then used as a source term for equation (2.25), the solution of

which yields the streamfunction. Ultimately, this streamfunction is then used to up-



date the tem perature field through use of equation (2.27). This process is repeated

as the system evolves in time. Since equation (2.26) contains the vorticity explicitly,

equations (2.26) and (2.25) must be iterated until a self-consistent streamfunction-

vorticity pair is found at each time step. This is im portant, since I begin my simu

lations by specifying a tem perature field which defines a cold slab in the mantle and

I have no prior knowledge of its accompanying streamfunction. Therefore, the initial

streamfunction is set equal to zero everywhere, and the code will then iterate between

equations (2.26) and (2.25) in order to find a self-consistent streamfunction-vorticity

pair corresponding to the initially specified tem perature field.

2.2. Calculations of Dynamic Topography

Dynamic topography is computed from mantle convection simulations by calculating

the normal stresses exerted on the top surface by the fluid flow. ‘Free-slip’ conditions

are applied to the upper boundary layer, and it follows tha t tangential stresses are

equal to zero on this surface. Topography is then computed by determining what

vertical deflection of the top surface is required for these normal stresses to vanish.

We begin our analytic derivation with the dimensional stress tensor for an isotropic,

compressible fluid (e.g. Chandrasekhar, 1961):

2P&ij T HrjCij ~ rj8jje^ (2.28)

The term is the stress acting in the j direction on a surface th a t is perpendicular

to the i direction. As such, we find th a t the stress normal to the surface, arr, is

equal to:

„ dur 2 .Orr = - p + - 3 ^ ( V ' u ) ( 2 -2 9 )



In equation (2.29), the fluid pressure, p, is actually the sum of a hydrostatic com

ponent, p0, and a non-hydrostatic component due to motion of the fluid, p\. Using

our condition th a t normal stresses at the surface vanish, we find that:

Using the hydrostatic equation, pQ can be rewritten in terms of the distance, to, from

the position of the unperturbed upper boundary layer. The following expression is

valid, provided th a t the values for density and gravity evaluated at the unperturbed

surface do not vary greatly near this same surface. Surface values for density and

gravity will be indicated with a subscript ‘s’. Our new expression for to is:

Equation (2.31) can be non-dimensionalised using the non-dimensionalising scheme

outlined in section 2.1.2. The non-dimensional form of equation (2.31) is given by

(dropping primes):

It is im portant here to notice th a t the dynamic topography is independent of the

(2.30)

(2.31)

(2.32)

absolute value of the dynamic viscosity in an infinite P randtl number fluid.

An expression for p x can be derived by first equating 0 terms of the non-dimensional

momentum balance equation (2.13):

(2.33)



In deriving this expression, we have made use of the vector identity:

V 2A = V (V • A) - V x V x A (2.34)

If equation (2.33) is evaluated at the surface, then the hydrostatic pressure, p0,

equals zero and hence p = p\. Our final expression for p\ becomes:

Pi = a 0Tcr / (1 8{ruS) 4

o \ r dr 3V M L) dd (2.35)

Substituting equation (2.35) into equation (2.31), we find th a t m is equal to:

m =Ps9s

a 0Tcr( 2 ' 3 6 )

Using the vector identity V • ( /A ) = / ( V ■ A) + A • V / , and the non-dimensional

continuity equation (2.12), we can express (V • u) as:

„ 1 dprV ■ u = -------— urpr or

(2.37)

Through use of equation (2.37), we can derive the expression for m th a t is actually

computed by the topography code with output from a mantle convection simulation.

We find that:

1TO =

p sgso 0Tcr

f d /1 d{ruj) 4 r 1 dpr /o \ r dr 3 Lpr dr

Ur d0 — 2a,0TC(dur 1 dpr dr ^ 3 p r dr

Ur (2.38)

The term is equal to zero, since ur has a value of zero at the surface.

In order to calculate the dimensional value of the topography, to needs only to be

multiplied by d. As mentioned previously, d is taken to be equal to the depth of the

entire mantle.



Finally, we must consider how topography will change if surface depressions are

to be filled with either sediment or water. If either is the case, then topography

calculated using equation (2.37) must be multiplied by an isostatic amplification term,

. Here, pt is the density of either sediments or water, and pm is the density of

the shallow mantle (« 3.33 x 103 kg/m 3).


3. Silurian Baltic Basin

3.1. Geology and Tectonic Setting

The Baltic Basin is a Late Vendian-Phanerozoic (Caledonian) sedimentary basin that

developed in the western part of the East European Craton. The location and present-

day extent of the Baltic Basin are shown in Fig. 3.1. The basin contains lithified

sediment ranging in age from the Vendian (650-544 Ma) to the Tertiary (65-1.5 Ma)

but the most prominent depositional phase in the basin was during the Silurian (443-

418 Ma)(Ulmishek, 1990). At this time, the region experienced an episode of rapid

subsidence, resulting in the deposition of up to ~5000 m of initially marine sediment

(Ulmishek, 1990; Poprawa et ah, 1999). Significant preservation of the sequence

was supported by subsequent tectonic and/or sea-level events. Isopach contours of

the restored thickness of Silurian accumulation (based on Lazauskiene et al. (2002))

are plotted on Fig. 3.1. There is significant thickening of the sedimentary stra ta

towards the southwest. This wedge of sediment may have extended as far inland as

~900 km from the Caledonian Deformation Front (CDF). Regression began in the

middle Silurian as terrigenous material gradually becomes predominant towards the

top of the stratigraphic succession (Ulmishek, 1990).

The tectonic regime during the Silurian was characterized by convergence along

the western margin of the East European Craton. P late reconstructions suggest tha t

29


3. Silurian Baltic Basin 30

(-j

Baltic Basin /

EastEur opean Craton

A

300km

Figure 3.1.: Geographic extent and reconstructed sediment deposition (in meters) on the Silurian Baltic Basin (after Lazauskiene et al. (2002)). The location of the paleo-trench is uncertain and my numerical models consider positions anywhere along a line from point A-B. The red line superimposed on the figure bounds the location of nine SW-NE trending profiles used to construct the sediment deposition profile (the shaded region) in Fig. 3.3.

East Avalonia began to move towards Baltica in the middle Ordovician (472-461

Ma)(Torsvik et al., 1996; McCann, 1998). The intervening Tornquist Sea began to

close during this time (McCann, 1998), resulting in the subduction of the ocean plate

beneath Baltica (Abramovitz and Thybo, 1998; Poprawa et al., 1999; Lazauskiene

et al., 2002), although the polarity of subduction remains a source of contention (e.g.

Giese and Koppen, 2001). The precise position of the collisional margin between the

two plates is uncertain. Various geological and geophysical interpretations of Cale

donian deformation suggest that the contact is either at the Caledonian Deformation


3. Silurian Baltic Basin 3 1

Front, the Elbe Lineament, or somewhere in between (Tanner and Meissner, 1996;

Lazauskiene et a l , 2002). The points A and B on Figure 3.1 reflect this possible

range for the position of the plate suture.

The correlation between subsidence in the region and convergent tectonics along

the plate margin has led to speculation tha t the Baltic Basin may have formed as

a flexural foreland basin (Poprawa et al., 1999; Lazauskiene et al., 2002). These

studies argue th a t while such supra-crustal loading mechanisms play an im portant

role, there is an anomalous component of observed subsidence in the basin th a t must

be explained by alternative, contemporaneous epeirogenic mechanisms.

The question arises as to whether the ‘anomalous’ subsidence identified in the Sil

urian evolution of the Baltic Basin (Poprawa et al., 1999) may be a consequence

of mantle flow coupled to Caledonian subduction. Lazauskiene et al. (2002) have re

cently provided maps of the reconstructed paleo-thickness of the Silurian Baltic Basin.

Their comparison of these maps with three-dimensional finite difference predictions of

foreland flexure by supracrustal loads indicates a long-wavelength residual subsidence

th a t they ‘tentatively interpreted’ as being due to mantle dynamic loading. In the

next section I will discuss the use of mantle flow models of the subduction process

to directly test whether slab-induced dynamic topography is a plausible mechanism

for reconciling the residual (reconstructed) topography inferred by Lazauskiene et al.

(2002).

3.2. Numerical Modelling

Mantle flow calculations are based on a spherical, axisymmetric finite difference for

mulation of equations tha t govern the conservation of mass, energy and momentum

of a compressible, infinite P randtl number, Newtonian fluid. Details of the numerical



scheme and governing equations can be found in Chapter 2.

The axisymmetric solution space, as shown in Fig. 3.2, involves the entire depth

of the mantle. Free slip boundary conditions are applied across all boundaries of

the solution space. A cold slab, defined by a mean tem perature contrast A T with

respect to the upper mantle, thickness Sh and dip angle a , is introduced into the

mantle model at roughly the location of the model equator (Fig. 3.2). It is necessary

for the case of an axisymmetric code to place slabs as close as possible to the model

equator, since in the extreme case, a slab placed near the pole would result in a

subducting cone, rather than a subducting slab. Simulations were run for dip angles,

a, varying from 25 to 75 degrees, in increments of 5 degrees. The dynamic topography

supported by flow coupled to the descending slab is computed in two steps. First,

normal stresses at the surface of the convecting (whole mantle) layer are computed

from the output of the numerical simulation, using methods described in Chapter

2. Next, these stresses are applied to an over-riding lithosphere of constant flexural

rigidity. The results presented below all adopt a rigidity of 2 x 1024 Nm, or an effective

elastic thickness of 55 km (Lazauskiene et al., 2002). Since the predicted dynamic

topography profiles are of relatively long wavelength, none of the main conclusions

of the work are altered significantly by an order of magnitude change in this rigidity.

Calculations of dynamic topography assumed tha t subsiding regions were being filled

with sediment th a t has a density of ~ 2.5 x 103 kg/m 3. All slabs adopted in this

study have a dip-angle length of ~ 870 km. This length is the inferred length of slab

which would have subducted beneath Baltica by the end of the Silurian, assuming a

conservative average subduction rate of ~ 2 cm per year.



FR

\ d h

AT

Figure 3.2.: Schematic illustration of the axisymmetric convection simulation. The arrow and symbol at the top-right indicate the pole of axisymmetry. The model slab (temperature contrast of AT, thickness of 5h and dip angle of a) is introduced at the model equator. The calculations of dynamic topography assume a lithosphere of flexural rigidity FR overlying the slab.

3.3. Results

The shaded region in Fig. 3.3 shows the reconstructed thickness of Silurian sediments

within the Baltic Basin along a SW-NE trending profile originating a t the CDF. This

range of values was constructed by choosing a sequence of nine SW-NE profiles within

the area bounded by the red line in Figure 3.1. All nine profiles were term inated at



the final, 500 m, deposition contour in Fig. 3.1, and thus the edge of the basin, tha t

is a value of zero on the ordinate scale, would be achieved several hundred kilometres

to the north-east (i.e., to the right) of the shading.

According to Fig. 3.1, the trench was located from 0 km (point B, on the CDF)

to ~ 340 km (point A, on the Elbe Lineament) to the south-west of this deposition

profile (that is, to the left of the origin). Thus, Silurian deposition extended on

the order of 600-1000 km from the trench (see Figures 3.1, 3.3). Models of foreland

basin development due to supracrustal loading in the region (e.g. Lazauskiene et al.,

2002) predict deposition tha t extends no further than ~ 250 km from the CDF,

and thus Fig. 3.3 illustrates the necessity of invoking additional mechanisms for the

reconstructed sediment deposition.

In Chapter 1 I discussed in detail how previous models of slab-induced mantle flow

have noted a strong sensitivity between the dip angle of the slab and the horizontal

extent of the associated dynamic topography (e.g. Mitrovica et al., 1989). My mod

elling of the far-held Silurian deposition evident in Fig. 3.3 takes advantage of this

sensitivity in the following manner. First, for a specific location of the trench between

points A and B on Fig. 3.1, I vary the dip angle of the model slab until the horizon

tal extent of the sediment deposition is matched. W ith this dip angle specified, the

amplitude of the dynamic topography is a function of the negative buoyancy of the

slab, which is proportional to the product AT8h. I vary this product in an attem pt

to reconcile the observed far-held deposition.

As an example of this exercise, the dashed line in Fig. 3.3 represents the best-ht

sediment deposition arising from a slab of dip angle 40° with a trench located at point

A in Fig. 3.1. It is clear from the hgure th a t slab-induced dynamic topography is

capable of reconciling all of the ‘anomalous’ far-held (beyond ~ 200 km on the prohle



2000

3000

4000

5000

6000 700600500300100 400200

Distance from CDF (km)

Figure 3.3.: The shaded region indicates reconstructed Silurian sediment deposition in the Baltic Basin along a SW-NE trending profile. The zero on the abcissa axis refers to the location of the Caledonian Deformation Front (CDF) and the shading bounds the deposition determined by considering 9 SW-NE profiles within the red line superimposed in Figure 3.1. The dashed line on the figure is a prediction of dynamic topography (with amplification due to sediment load) associated with a slab subducting with dip angle 40° at a trench located at point A in Figure 3.1. The discrepancy between the shaded profile and the dashed line within the near-field of the CDF, given by the dotted line, is thus the implied deposition due to supracrustal loading. The dashed-dotted line is analogous to the dashed, with the exception that the dip angle has been changed to 60° and the trench relocated to point B in Figure 3.1.

and greater than ~ 540 km from the trench) Silurian deposition. The mantle flow

also supports a significant fraction of the near-field deposition. For the prediction

treated in Fig. 3.3, dynamic topography accounts for about 40% of the reconstructed

deposition at the CDF (the to tal is ~ 5 km). The nature of this contribution is further

illustrated by the dotted line on the figure, which provides the residual deposition

(observed minus mantle flow-induced dynamic topography) in the near-field. Note

th a t the horizontal extent of this residual deposition is consistent with the length-scale



associated with deflection by supracrustal loads.

I have repeated the exercise in Fig. 3.3 by varying the location of the trench across

a line joining points A and B in Fig. 3.1; as I have discussed, this variation is a

conservative estimate of the current uncertainty in the trench location. In Fig. 3.4a

I plot the inferred dip angle as a function of the adopted location of the trench. For

a given trench location, the shaded region provides the range of dip-angles inferred

by varying A T while simultaneously reconciling the far-field sediment deposition.

Points along the upper boundary of the shaded region in Fig. 3.4a were calculated

using tem perature contrasts th a t were on average 30% higher than those used for

points lying along the lower boundary of the shaded region.

For each specific pair of trench location and preferred dip angle (Figure 3.4a),

Figure 3.4B provides the predicted fraction of reconstructed sediment deposition at

the CDF due to slab-induced dynamic topography. Values calculated for points along

the upper (lower) boundary of the shaded region in Figure 3.4b map onto points on

the upper (lower) boundary of the shaded region in Figure 3.4a. The shaded region

reflects the range of values for CDF sediment deposition obtained by varying AT.

As the trench location is varied from the Elbe Lineament to the CDF, the effective

horizontal length scale of the Silurian sediment deposition is reduced and the preferred

dip angle is increased. As this dip angle increases, the dynamic topograpy associated

with slab-induced mantle flow becomes more localized in the near-field of the trench

and, accordingly, the contribution of this process to the Silurian deposition at the

CDF increases. The dashed-dotted line in Fig. 3.3 shows the dynamic topography

computed for a model simulation in which the trench is located a t the CDF and the

slab dip angle is 60°. In this case, slab-induced flow yields ~ 68% of the near-field

sediment deposition. This percentage can increase to up to 85% for the range of


3. Silurian Baltic Basin 37

70

60

50

40

30

BA90

70

o 5 50V </)5 °■a n.~ 0)

30

10

BALocation of Trench (km)

Figure 3.4.: (a) The inferred dip angle of subduction as a function of the location of the trench (from points A to B; see Figure 3.1). For a given trench location, the range of inferred dip angles arises by varying the temperature contrast, AT, while simultaneously fitting the far-held sediment deposition (Figure 3.3). (b) The percentage of near-held deposition at the CDF provided by the dynamic topography as a function of the trench location. Values computed along the upper (lower) boundary of the shaded region correspond to results along the upper (lower) boundary of the shaded region in frame 3a.



model runs show in Figure 3.4. Thus, supracrustal loading may only be required to

reconcile a small faction of the near-field sediment deposition.

3.4. Conclusions

Results from Figure 3.4a show tha t the Caledonian subduction was characterized by

a dip angle of 40°-60°, with this range reflecting the uncertainty in trench location.

Furthermore, dynamic topography due to mantle flow coupled to the descending slab

contributed a t least 40%, and as much as 85%, of the Silurian deposition along the

Caledonian Deformation Front. Lazauskiene et al. (2002) noted th a t they had to

use unrealistically large orogenic (supracrustal) loads (greater than 10 km) to explain

Silurian sediment deposition near the CDF on the Baltic Basin, and they argued

th a t slab-induced mantle flow would allow them to reduce these loads. These results

suggest tha t this reduction is by a factor of ~ two or more.

The numerical simulations of slab-induced dynamic topography used here have

assumed two-dimensional axisymmetric mantle flow. This assumption appears to be

reasonable for the case of the Baltic Basin, where lateral variations relative to the

main SW-NE trending gradients in deposition appear to be minor (Fig. 3.1). In future

work one could extend this analysis to consider the region to the north-west of the

Baltic Basin, where deposition reflects the changing strike of the CDF (Fig. 3.1). In

this case, three-dimensional mantle flow models may be required.


4. Rift-flank Uplift of Arabia

4.1. Introduction

The present day tectonic setting of the African-Arabian region is shown in Figure 4.1.

The Red Sea marks the beginning of the separation of the Arabian plate from the

African plate, and this separation is accompanied by convergence between the Arabian

plate and Eurasia. Rifting of the Red Sea Margin is thought to have begun in the

early Oligocene (34-28.5 Ma). Uplift of the adjacent Arabian plate began in the early

Miocene, although the most active uplift has occurred since the middle Miocene (i.e.,

after ca. 15 Ma) (Almond, 1986; Bohannon, 1989). Present-day topography across the

Red Sea is asymmetric; with the exception of the Afar-Arabian dome, the topographic

high on the African side of the Red Sea is more localized and subdued relative to

its Arabian conjugate (Fig. 4.2A). In contrast, much of southwest Arabia reaches

elevations of >1000 m, and high topography extends well into the plate interior. In

fact, there is a distinct regional tilt of the entire Arabian plate from the Red Sea to

the Persian Gulf, a distance >1000 km. This pattern of topography with highlands

flanking a lowland across a rifted margin is commonly cited as a classic example of

rift-flank uplift (e.g. Wernicke, 1985).

A variety of therm al and mechanical models have been developed to explain rift-

flank uplift, and these have been applied with varying success to the Arabian margin

39


4. Rift-flank Uplift o f Arabia

30*

INDIANPLATE

Figure 4.1.: Tectonic setting of the African-Arabian region, (from Vita-Finzi, 2001).

(van der Beek et al., 1994). The models are distinguished by the dominance of

thermal, mechanical, geometric, and melt processes. In therm al models, uplift can

result from depth-dependent stretching (Royden and Keen, 1980) or from the heating

of flanks by small-scale convection (Keen, 1985; Buck, 1986). Mechanical modelling

indicates tha t upward flexure may occur if the lithosphere maintains finite strength

during rifting (Braun and Beaumont, 1989; Weissel and Karner, 1989). Geometric

models first presented by Wernicke (1985) explain the asymmetry of uplift in terms of

a single low-angle detachment penetrating the entire lithosphere. Finally, extensive

flank uplift may also result from magmatic underplating due to asthenospheric partial

melting (Cox, 1980; W hite and McKenzie, 1989).

These disparate models generally assume a passive origin for rift formation (i.e.,

rifting in response to a remote stress field, in contrast to rifting in response to a

therm al upwelling from the mantle), and they are largely concerned with flank uplift


4. Rift-flank Uplift of Arabia 41

30*30*

25*■jsj

20*

15'15'

10 '

5'N5'N

55' 60' 35'E 40' 45' 50' 55' 60'35'E 40' 45' 50' 65'

-5000 -4000 -3000 -2000 -1000

ELEVATION(m)

Figure 4.2.: A: Topography of East African-Arabian region from ETOP05 data set (http://www.ngdc.noaa.gov/mgg/global/seltopo.html). RS, PG, ZM - locations of Red Sea, Persian Gulf, and Zagros Mountains, respectively. B: Residual topography derived by correcting raw topography in frame A for crustal isostatic effects using crustal thickness and density model CRUST 2.0 (Laske et al., 2002).

within a few hundred kilometers of the rift basin. Therefore, none of these mechanisms

provide a good explanation for the broad tilting of the Arabian plate that extends

from the flank of the Red Sea to the Persian Gulf at the foothills of the Zagros

Mountains (Fig. 4.2A).

The tilt appears to be a young feature. Although the western Arabian Shield may

have been above sea level for much of the Phanerozoic, it has experienced only a few

kilometers of erosion (Bohannon, 1989) and is not likely to have been elevated until

recently. Furthermore, late Mesozoic and Paleogene marine sedimentary deposits

are preserved near the Red Sea coast in northern and southern Arabia, indicating

post-Eocene surface uplift (Beydoun, 1991).


http://www.ngdc.noaa.gov/mgg/global/seltopo.html

4 . Rift-flank Uplift o f Arabia 4 2

The geologic record provides additional clues to the origin of the tilting. Although

passive rifting is thought to have produced the northwest-southeast trending tholeiitic

to transitional lavas seen in western Arabia, active mantle upwelling may have placed

younger transitional to strongly alkalic lavas along north-south trends in Arabia (Al

mond, 1986; Camp and Roobol, 1992). These younger lavas are largely contempo

raneous with the period of major crustal uplift (12 Ma to the present), and their

alkalinity suggests a deep-mantle origin (Almond, 1986; Camp and Roobol, 1992).

The geologically inferred connection between deep-mantle processes and the devel

opment of topography provides a possible mechanism for the long-wavelength tilting

of the Arabian plate. In this section I argue th a t the tilting represents the dynamic

response of the Arabian plate to viscous stresses associated with active, large-scale

mantle flow. To support this argument, predictions of dynamic topography based on

viscous-flow simulations constrained by seismically inferred mantle heterogeneity will

be presented.

4.2. A Mantle-flow Scenario for the Tilting of Arabia

Global tomographic analyses of seismic data sets have progressively improved the

resolution of models of mantle structure below the Afro-Arabian region (e.g. Grand

et al., 1997; Ritsema et ah, 1999). Figure 4.3 shows 3-D images of two recent seismic

models (Ritsema et al., 1999; Grand, 2002), where the red contour represents a specific

shear-wave velocity perturbation (see caption) relative to a radial reference model.

The plots highlight a large, seismically slow region at the base of the mantle under

southern Africa th a t has previously been interpreted as an upwelling megaplume on

the basis of high topography (Hager, 1984; Lithgow-Bertelloni and Silver, 1998) and

geologically inferred uplift rate (Gurnis et al., 2000) (for a more detailed description


4. Rift-flank Uplift o f Arabia 4 3

of the findings of these studies, the reader is referred to section 1.3). The megaplume

appears to connect at shallower depths with structure th a t is thought to act as the

driving mechanism for rifting in East Africa (Ritsema et al., 1999; Ritsema and van

Heijst, 2000; Lithgow-Bertelloni and Silver, 1998). This shallow heterogeneity reaches

the lithosphere southeast of the Red Sea, near the Afar triple junction, and ultimately

spreads beneath the Red Sea-Arabian region.

Because mantle flow associated with the megaplume has been identified as the

driving force for epeirogenic and tectonic deformation in Africa, it is logical to consider

the implications of the structure, and in particular its shallowest parts, for the already-

described tilting of the Arabian plate. Intuitively, viscous stresses associated with the

material upwelling beneath the Red Sea flank of the Arabian plate would be expected

to dynamically support uplift within this region. In the next section this scenario will

be quantified by using predictions of dynamic topography generated from mantle-flow

simulations.

To isolate dynamic effects, predictions will be compared to residual topography,

i.e., the observed topography (Fig. 4.2A) corrected for crustal-thickness variations

assuming isostatic compensation of the crust. The recent model CRUST 2.0 (Laske

et al., 2002) will be adopted for this purpose. CRUST 2.0 provides both thickness and

density of structures through the crust within 2° X 2° cells. While this model is rea

sonably well constrained within the Arabian region (Laske et al., 2002), uncertainties

in the density structure of the lower crust may introduce errors in the topographic

correction. The residual topography (Fig. 4.2B) features a more pronounced tilt of

the Arabian platform - including a dynamic depression of 1000 m in the vicinity of

the Persian Gulf - and higher peak topographies on both shoulders of the Red Sea.



Figure 4.3.: (A) 3-D image of the —0.7% mantle shear wave anomaly contour, based on the S-wave velocity model S20RTS of Ritsema et al. (1999) within a portion of the mantle underlying the Afro-Arabian region. (B), as in (A), except for the —0.6% shear wave anomaly contour from Grand’s (2002) seismic velocity model. The Cartesian projection extends from the core-mantle-boundary at bottom to the Earth’s surface at top. The contour captures the geometry of a buoyant megaplume originating beneath South Africa, bending as it rises through the mantle toward the East Africa Rift and terminating around the Afar/Red Sea region. The Ritsema et al. (1999) model is based on the analysis of body wave travel times, surface wave phase velocities and normal mode splitting data. The Grand (2002) analysis uses body wave data, and it is the most recent in a sequence of high resolution tomographic models (e.g. Grand et al., 1997).



4.3. Numerical Model Formulation and Results

Mantle flow calculations in this section are based on a spherical, two-dimensional (ax

ially symmetric) finite-difference formulation of the equations th a t govern the conser

vation of mass, energy and momentum of an incompressible Newtonian fluid having

an infinite Prandtl number described in Chapter 2. The model involves the entire

depth of the mantle, from the core-mantle boundary to the surface, and spans an an

gular distance of 140° (Fig. 4.5). The use of a two-dimensional model is motivated by

computational limitations, but it is justified, to first order, by the observed geometry

of the Arabian tilt (Fig. 4.2). T hat is, with the exception of the Afar-Arabian dome,

the residual topography of the region is dominated by variations perpendicular to the

trend of the Red Sea.

First, a density, or buoyancy, field is prescribed within the solution domain using

results from seismic tomography (see subsequent discussion). Second, the governing

viscous flow equations are solved for instantaneous flow fields throughout the domain.

Third, dynamic topography is computed, a-posteriori, by applying surface normal

stresses output from the convection code to a finite-element model of elastic beam

deformation. The finite-element beam-bending code used was w ritten by Phillipe

Fullsack. It is assumed th a t this beam has a uniform flexural rigidity, with the

exception of a break associated with the plate boundary on the Red Sea. Estimated

values for the flexural rigidity of the Arabian lithosphere in the Red Sea region range

from 3 x 1022 Nm to 2 x 1024 Nm (van der Beek et al., 1994). The predicted long-

wavelength topographic variation is relatively insensitive to the choice of rigidity

within this range.

The mantle buoyancy field is prescribed by converting seismic velocity anoma

lies from the tomography models of both Grand (2002) and Ritsema et al. (1999)


4. Rift-Rank Uplift o f Arabia 4 6

(Fig. 4.3). This conversion is performed by using the depth-dependent velocity-to-

density-scaling profile of Forte and Woodward (1997), shown in Figure 4.4. This

scaling was modified from an earlier profile of Karato (1993) using constraints from

a variety of geodynamic observations. Seismic wave speeds are greatly affected by

the density of the material which they travel through, with hotter (or less dense)

material resulting in slower wave speeds, for example. The conversion used here as

sumes th a t the shear-wave velocity anomalies observed in both tomographic models

originate from tem perature variations, although it is im portant note th a t recent work

suggests tha t the African megaplume structure may also partially reflect variations

in iron content (Forte and Mitrovica, 2001; Ni et al., 2002).

0

50Q

1.000fI f 1,500"a.a>

2,000

2.500

3,0000.00 0.15 0.30

0lrtp/.dlnVi

Figure 4.4.: Depth-dependent scaling profile used to convert shear-wave anomalies to density anomalies used in convection simulations. This profile from Forte and Woodward (1997), is a modified version of an earlier profile by Karato (1993), which was based on high pressure mineral physics data for olivine.

Figure 4.5 shows a density heterogeneity field prescribed in this manner superim

posed on the numerical model domain. In the deep mantle, southwest of Arabia,

the cross-section is dominated by the buoyant (hot) megaplume structure described

above. As in Figure 4.3, this feature appears to connect with shallower, buoyant

— Karato (modified)


4. Rift-Rank Uplift o f Arabia 47

material tha t impinges under the Red Sea side of the Arabian plate. The northeast

ern side of the Arabian plate is underlain by dense (colder than average) material

extending through the entire upper mantle.

One final issue in the modelling involves heterogeneity in the shallowest (top 200

km) of the mantle. Because a therm al origin for seismic velocity anomalies is assumed,

negative buoyancy would be ascribed to regions where chemical heterogeneities con

tribute to seismically fast velocity anomalies in the shallow mantle below continents,

i.e., in the neutrally buoyant “tectosphere” (Jordan, 1978). To overcome this diffi

culty, seismically fast heterogeneity is zeroed out in the top 200 km of the mantle.

However, it should be noted tha t simulations with and without this procedure ap

plied showed only minor differences (on the order 100 m) in the computed dynamic

topography.

Two-dimensional simulations were initiated with heterogeneities taken from the

four vertical cross-sections specified in the inset of Fig. 4.5. The average dynamic

topography predicted from these four numerical runs, using heterogeneity based on

the Grand (2002) seismic model, is shown in Fig. 4.6 (dotted line), which is then com

pared to the range of residual topography computed from Figure 4.2B across the same

four profiles (shaded region, Fig. 4.6). The predicted dynamic topography provides a

close fit to the flank uplift and broad tilt of the Arabian platform. Specifically, viscous

flow associated with large-scale mantle density structure supports a to tal differential

deflection of ~2.2 km from the topographic high at the flank of the Red Sea margin

to a low at the Persian Gulf.

The remaining dynamic topography curves in Figure 4.6 explore the sensitivity

of these results to various aspects of the modelling. The dashed curve in Figure

4.6 is analagous to the dotted curve with the exception tha t the former is derived



Figure 4.5.: Main Figure: Geometry of the axisymmetric solution space used in my viscous flow simulations. The model domain extends from the core-mantle boundary to the surface, and across a 140 degree arc relative to the pole of axisymmetry (denoted by the symbol “upper-case phi”). Free-slip conditions are applied across all boundaries of the solution space, and the numerical grid is defined by 257 nodes in the radial direction and 769 nodes in the direction of increasing azimuth. Within this model domain, I superimpose density heterogeneity in a vertical cross section, with orientation given by profile 4 (highlighted in red in the inset). The density field is obtained by scaling the S-wave velocity model of Grand (2002) using the velocity-to-density conversion profile shown in Figure 4.4. Inset: Orientation of four vertical cross-sections through the seismic S-wave models used to initiate my 2-D numerical simulations.

by adopting the seismic model of Ritsema et al. (1999) for mantle heterogeneity

(Fig. 4.3A). The fit to the Arabian residual topography (shaded region) is comparable



3000Residual topography

Observed topography2500Dynamic Topography:Ritsema et al. (1 9 9 9 ) -----Grand (2 0 0 2 ) ................Grand (2 0 0 2 )________0-670 km

2000

1500

E1000 PGJZQ.500SU)oCl

£-500

-1000

RS-1500

-2000200 400 600

Distance (km)1000 1200 1400-400 -200 800

F igu re 4.6.: Minimum and maximum values of residual topography (shaded region) from Figure 4.2B computed across four profiles shown in inset of Figure 4.5. Observed (i.e., raw) topography (solid curve) is averaged across the same four profiles. Dynamic topography (dashed curve and dotted curve) is computed by averaging results from two-dimensional numerical predictions across the same four profiles; these viscous-flow predictions were initiated by using vertical cross sections of S-wave heterogeneity adopted from Grand (2002) or Ritsema et al. (1999) and scaled to density by using the conversion profile shown in Figure 4.4. Dashed-dotted curve is analogous to dotted; however, in this case I have deleted all mantle heterogeneity below 670 km depth (i.e., in lower mantle). RS and PG - locations of Red Sea and Persian Gulf, respectively.

for this model choice. For the dashed-dotted result, the Grand (2002) model is again

used, but this time all mantle heterogeneity below 670 km depth is deleted. The close

agreement between the dotted and dashed-dotted curves indicates th a t upper-mantle

structure is largely responsible for the flow-induced tilting of the Arabian plate.

These calculations do not predict the observed large peak in African residual to-



pography within a few 100 km of the Red Sea. This may be due to various shortcom

ings of the model, including the resolution of the seismic model, the simplicity of the

lithospheric structure adopted in the elastic-beam calculation, or the two-dimensional

modelling apporoach. It should also be noted tha t the raw topography shows signif

icantly less uplift than the residual topography on the African shoulder of the Red

Sea (compare Figs. 4.2A and 4.2B), and thus the misfit in Figure 4.6 depends on the

accuracy of the crustal model at this location.

Finally, the modelling results presented here do not preclude tha t the other rift-

flank processes described in the introduction may contribute to short-wavelength

topography on either side of the Red Sea rift. Similarly, the topographic low observed

in the Persian Gulf is, in part, due to topographical loading by the thickened crust

in the Zagros-Taurus collision zone. Although the portion of the thickened crust tha t

is locally compensated has been accounted for, there is a flexural depression of the

Arabian plate tha t has not been accounted for. This is likely to contribute to the

topographic low in the Persian Gulf, but not in the Arabian plate interior.

4.4. Conclusions

Topography dynamically supported by large-scale viscous flow in the mantle is re

sponsible for the dramatic tilting of the Arabian platform. The high topography of

the Arabian rift-flank is largely driven by upper mantle structure, which appears to

be connected to the seismically slow and thermally buoyant megaplume structure

(Fig. 4.3) tha t has previously been associated with the anomalous uplift of south

ern Africa and rifting in East Africa. The tilting is also enhanced by seismically

fast (cold, dense) mantle beneath northeast Arabia (>200 km depth), which acts to

dynamically depress the overlying plate in this area. These results do not preclude



shorter wavelength topographic effects on the margin of the Red Sea associated with

one or more of the proposed mechanisms for flanking uplift a t passive rifts (discussed

in the introduction). There is also a contribution from loading of the Zagros moun

tains on flexure of the Arabian plate in the vicinity of the Persian Gulf tha t has not

been accounted for.


P art II:

R otation a l S tab ility o f P lan ets w ith L ithospheres

52


5. Introduction: A Review of Gold

(1955)

A planet may experience large excursions of its rotation axis relative to the surface ge

ography. This reorientation of the rotation pole, known as true polar wander (TPW ),

occurs when the inertia tensor of the planet varies as a function of time. Changes to

the inertia tensor may result from surface or internal (i.e., convective) loading and can

occur over different timescales. For example, on Earth, true polar wander is driven by

both glacial isostatic adjustment, which occurs over timescales of tens of thousands

of years, and mantle convection, which redistributes mass and deforms the surface

over millions of years. The physics underlying a planet’s rotational stability has been

the subject of a series of seminal studies. It is instructive to begin here with a review

of the physics described by Gold (1955), who discussed the rotational stability of a

simple, ‘hydrostatic’ planet subject to a surface mass loading.

Figure 5.1 begins with a rotating planet in hydrostatic equilibrium (Fig. 5.1A0).

Subsequently, a surface load is placed off-axis which will act to push the pole away.

In the short term, the rotational bulge will resist this tendency and stabilize the

rotation axis (Fig. 5.1A2). However, in the long term the bulge will relax completely

(i.e., hydrostatically) to any reorientation of the pole position (Fig. 5.1 A3), thus

53


5. Introduction: A Review of Gold (1955) 54

INITIALSTATE

PRIOR TO LOADING

FINALROTATIONAL

STATE

AO A2 A3

B2 B3

C2 C3

• •

A4

Figure 5.1.: Schematic illustration highlighting the physics of TPW. Only panel “A” of this figure is relevant for the discussion found in this chapter. Chapter 6 will address subsequent panels which pertain to planets that are characterized by the presence of a lithosphere. (A) An initially hydrostatic planet (A0,A1) is subject to a surface mass load (green beetle, A2) assumed to remain partially uncompensated. The load will push the pole away (green arrow), while the hydrostatic rotational bulge will, initially, resist this motion (blue arrow). This resistance will vanish as the hydrostatic rotational bulge relaxes to the new rotational state (A3), leading to further load-induced TPW. The process will continue until the load reaches the equator (A4). (B) Rotational stability of an initially spherical planet (BO) with an elastic lithosphere (blue shell) subject to a surface mass loading. Once the model planet is set rotating, it will ultimately achieve the equilibrium form shown in Bl. As in (A), the application of a surface mass load will move the pole away, and TPW will cease when the load reaches the equator. (C) The scenario described by both Willemann (1984) and Matsuyama et al. (2006), in which a young, initially hydrostatic planet (CO) develops an elastic shell through cooling of the interior (Cl). The shell develops without any internal elastic stresses and thus the form of the planet will be identical to the hydrostatic case in Al rather than to the (less oblate) equilibrium form in Bl. The appearance of a surface mass load will act to push the pole away, and this will be resisted by the rotational bulge. However, in contrast to the first two cases, in this scenario the bulge cannot perfectly adjust to the new rotation axis (note that the oblate form in C3 is not symmetric relative to the rotation axis). Ultimately, the load will not reach the equator (C4), but rather a position governed by the balance between the load-induced push and the pull associated with the remnant rotational bulge. For each scenario, the figures are drawn so that the z-axis is fixed to the initial rotation pole in order to be consistent with the mathematical analysis appearing in appendix B.


5. Introduction: A Review o f Gold (1955) 55

erasing all memory of past positions, and perm itting further load-induced TPW . Gold

realized tha t any load, even one as small as a beetle, will eventually reach the equator

(Fig. 5.1A4), the minimum energy state of the system. T hat is, T PW will continue

until the pole aligns with the maximum axis of inertia associated with any surface load

(i.e., non-hydrostatic) forcing. It remains a common view, following Gold’s influential

analysis, tha t the stability of a planet is thus governed by the observed figure of the

planet after correction for a hydrostatic form.

In the next section, standard (fluid) love number theory is used to derive expressions

for the inertia tensor appropriate to the scenario depicted in Fig. 5.1 A.

5.1. Mathematical Treatment of Fig. 5.1

The response of a spherically symmetric, Maxwell viscoelastic planet to the applica

tion of a surface mass or tidal (potential) load are commonly formulated in terms of

viscoelastic Love numbers (Peltier, 1974). In the time domain (t), the viscoelastic

load and tidal (or tidal-effective) k Love numbers a t spherical harmonic degree two

have the form (Peltier, 1974):

jkL(t, L T ) = kL’E5(t) + ^rV j-exp(—Sj£) (5.1)

j=i

andj

kT(t, LT) — kT,E5(t) + 'y ^ r 'jexp (-S jt) . (5.2)i=i

These Love numbers yield the gravitational potential perturbation, at degree two,

arising from the deformation of a spherically symmetric, Maxwell viscoelastic plan

etary model subject to an impulsive surface mass load and gravitational potential


5. Introduction: A Review o f Gold (1955) 56

(tidal) forcing, respectively. The first term on the right hand side (henceforth RHS)

of each expression represents the immediate elastic response to the loading (hence

the superscript E ), while the second term is a non-elastic response comprised of a

series of J normal modes of exponential decay. The modes for the load and tidal

Love numbers have a common set of decay times (sj), but distinct modal amplitudes

(rj and r'j). These Love numbers are dependent on the viscoelastic structure of the

planetary model. For our purposes, the dependence on the elastic thickness of the

lithosphere (LT) is most im portant, and thus this dependence is made explicit. In

the Laplace transform domain, these Love numbers have the form:

k L(s ,L T ) = k L'E + J 2 - ^ j - (5.3)j =i J

and

kT (s ,L T ) = kT’E + Y J J - ^ J . (5‘4)j =i

The so-called fluid Love numbers represent the response of the planetary model after

all viscous stresses have relaxed. These may be derived from the above expressions

either by taking the s = 0 limit of the Laplace-domain equations (5.3)-(5.4), or by

convolving the time-domain equations (5.1)-(5.2) with a Heaviside step loading and

considering the infinite time response. In either case, one would then obtain:

jk f (L T ) = kL'E + (5-5)

;=i Sj

and

k j (LT) = kT'E + Y 1 ~ - (5‘6)j = i s i

The fluid Love numbers are dependent on the density profile of the planetary model,


5. Introduction: A Review o f Gold (1955) 5 7

as well as the thickness of the elastic lithosphere. The latter is, of course, subject to

no viscous relaxation. For the sake of brevity, the dependence on L T in equations

th a t require the fluid Love numbers (e.g., k j , k j ), will be suppressed. However, in the

case of a purely hydrostatic planet, i.e., one which has no long-term elastic strength

(L T — 0), the notation k j * and kj'* will be used.

5 .1 .1 . The Inertia Tensor: Gold (1955)

Let us assume a co-ordinate system oriented so th a t the z-axis is fixed to the rotation

pole of the planet just prior to loading (e.g., Fig. 5.1A1). The initial angular velocity

vector will be denoted by (0 ,0 ,0 ). At any subsequent time, the rotation vector will

be given by cuj(t), i = 1,2,3, with magnitude ui2(t). The symbols a and M will

represent the radius and mass of the planet, respectively, while G is the gravitational

constant.

For the scenario in Fig. 5.1A, i.e., the physics treated by Gold (1955), the total

inertia tensor is (Munk and MacDonald, 1960; Ricard et al., 1993):

oPfTt* I

L j( t) = I0Si:i + [ui(t)ujj(t) - - u 2(t)5ij] + I%(t) (5.7)

where / 0 is the spherical term and I^ (t) is the contribution to the inertia tensor

from the combined effect of the surface mass load and the planetary deformation it

induces. The second term on the right hand side is related to the rotationally-induced

flattening of the hydrostatic model adopted by Gold (1955).

The first two terms on the RHS constitute the inertia tensor for a hydrostatic planet

w ith angular velocity u>i(t). Thus, we can write

5 7 *I * v \ t ) = IoSij + °L^ - [c^(t)wj(i) - . (5.8)



This represents the component of the inertia tensor in Eq. (5.7) th a t perfectly adjusts

(in the fluid limit) to any change in the rotation axis (i.e., as in the fully relaxed cases

shown in Figs. 5.1A1, A3 and A4). As discussed by Gold (1955), this component of the

planet since it provides no memory of any previous orientation of the rotation vector.

Thus, for this planetary model, the reorientation of the pole is governed by the non-

hydrostatic inertia tensor:

(i.e. the maximum axis of inertia in a coordinate system where the inertia tensor is

diagonal). Thus, the adjustment in Fig. 5.1A will continue until the load has moved

to the equator (Fig. 5.1A4).

It will be instructive to consider the inertia tensor in the initial ( t = t o ) , unloaded

state of the system. Applying the initial rotation vector, (0,0, fi), into the expression

[ u j i ( t ) u j j ( t ) — | u 2 ( t ) 5 i j ] within Eq. (5.8) yields:

The orientation in Fig. 5.1A1 is a principal axis orientation, and thus we may use

this expression to derive a formula for the hydrostatic component of the J2 harmonic

(Bills and James, 1999)

inertia tensor is not relevant to the long-term rotational stability of the (hydrostatic)

(5.9)

In particular, the rotation vector is aligned with the maximum principal axis of I™hyd

(5.10)

■hyd __

M a 2 3 G M '(5.11)



As a final point, it should be noted th a t there is an inherent inconsistency in

the Gold (1955) scenario. In a purely hydrostatic planet, the contribution to the

inertia tensor from the load and direct deformation, /£ , will be zero, since the load

will be perfectly isostatically compensated. In the terminology of fluid Love number

theory, the uncompensated fraction of an applied load is given by 1 + k f , but, as L T

approaches zero, kj —► kj'* = —1. Thus, the scenario in Fig. 5.1A only holds if one

makes the ad-hoc assumption tha t the rotational bulge will be perfectly relaxed in

the fluid limit, but tha t the load will not be perfectly compensated.

5.2. Rotational Stability of a Planet in the Presence

of a Lithosphere

The question arises as to whether the non-hydrostatic stability theory described in

this chapter is appropriate in describing rotational dynamics of planets which can be

characterized by non-zero long-term strength within the lithosphere. This problem

was addressed most notably by Willemann (1984), who recognized th a t the presence

of an elastic shell has a potentially stabilizing effect on true polar wander. This is

because a lithosphere which cools and forms on a rotating planet will in effect “freeze

in” part of the planetary oblateness. Movement of the rotation pole on such a planet

will induce stresses in a previously stress-free lithosphere and the viscous bulge due

to rotation cannot perfectly adjust to the new pole position.

In Chapter 6, the rotational stability of Mars will be assessed using constraints on

the present-day inertia tensor of the planet from geodetic data. Since Mars has a

lithosphere which has maintained elastic strength over billions of years, it is appro

priate in this instance to use the theory describing the rotational stability of a planet



with an elastic lithsophere.

While it is known th a t the presence of an elastic lithosphere has a great effect

on true polar wander, it has not yet been shown what effect, if any, the presence

of a viscoelastic lithosphere will have on stabilizing the rotation axis of a planet.

In Chapter 7, a new theory is laid out to address this problem. This problem has

particular significance for Earth, where models which do not include the effect of

a viscoelastic lithosphere must appeal to extremely high lower mantle viscosities or

inefficient excitation geometries to explain the low rates of true polar wander observed

in the last 100 Myr found on the basis of paleomagnetic data.


6. Rotational Stability and Figure of

Mars

6.1. Introduction

A diverse set of studies have argued tha t the rotation pole of Mars has been subject

to large excursions relative to the surface geography, or true polar wander (TPW ),

including 90° reorientations known as inertial interchange events. The observations

supporting these studies include equatorial deposits th a t resemble sediments at the

present poles of Mars (Schultz and Lutz, 1988), hydrogen rich equatorial deposits

which are inferred to be remnants of ancient polar caps (Wieczorek et ah, 2005) and

analyses of crustal magnetic anomalies tha t suggest an early magnetic field existed

th a t is not aligned with the current rotation axis (Arkani-Hamed and Boutin, 2004;

Hood et ah, 2005). Other geological evidence for significant TPW on Mars is found in

the spatial distribution of valley networks th a t appear to be formed by liquid surface

runoff (Mutch et al., 1976) and craters formed by oblique impacts, which may record

the demise of ancient equatorial satellites (Schultz and Lutz-Garihan, 1982).

The most prominent feature in the geology of the M artian surface is the Tharsis

rise (see Fig. 6.1). The figure of Mars, as defined by the gravitational potential field

61


6. Rotational Stability and Figure of Mars 62

-60°

-30°

60°

30°

0°

11-8

-4

0

4

8

Figure 6.1.: Global topography of Mars measured by the Mars Orbiter Laser Altimeter (MOLA) taken from Smith et al. (1999). The Tharsis volcano-tectonic province is located near the equator from 220°E to 300°E. The colour scale in this figure saturates at elevations higher than 8 km.

of the planet, is dominated by the signature of this massive volcanic structure (Smith

et al., 1999) and a rotationally-induced equatorial bulge. Previous analyses of this

figure have commonly focussed on two questions related to the stability and evolution

of the Martian rotation vector. First, to what extent did the development of Tharsis,

which is now located near the equator of the planet (Zuber and Smith, 1997), cause

TPW (Melosh, 1980; Sprenke et al., 2005)? Second, how stable is the current rotation

axis to changes in the surface and internal mass distribution (Bills and James, 1999)?

Chapter 5 highlighted an important conceptual study by Gold (1955). Following

this study, a number of other papers concerned with Martian rotational stability have

assumed that the stability of the Martian rotation pole - both at present-day and in

response to Tharsis loading - is governed by the observed figure of the planet after

correction for a hydrostatic form (Melosh, 1980; Bills and James, 1999; Sprenke et al.,


6. Rotational Stability and Figure o f Mars

2005). In this context, a series of studies since 1980 have reached the conclusion tha t

the present non-hydrostatic form of Mars (i.e., the form which results from removing

the contribution due to planetary rotation in the case where there are no viscous

or elastic stresses) is characterized by a maximum axis of inertia th a t lies along the

current equator, 90° from Tharsis, while the intermediate axis of inertia is aligned

with the present-day pole (Melosh, 1980; Bills and James, 1999). The more recent

of these analyses demonstrate th a t the maximum and intermediate non-hydrostatic

moments of inertia are nearly equal (e.g. Bills and James, 1999), and thus conclude

th a t the current rotation vector of Mars in inherently unstable. T hat is, relatively

small surface mass loads can cause large (order 90°) excursions of the pole along the

great circle joining the present-day pole and the maximum non-hydrostatic inertia

axis. Furthermore, analysis of the non-hydrostatic form, after correction for the signal

from Tharsis loading, has led to a conclusion tha t the development of this volcanic

province must have induced a large (15 — 90°) excursion of the rotation pole (Sprenke

et al., 2005).

As was briefly mentioned in Chapter 5, Willemann (1984) recognized th a t the

presence of an elastic shell has a potentially significant stabilizing effect on TPW .

The incomplete isostatic compensation of the ancient Tharsis load implies tha t Mars

is characterized by non-zero long-term strength within the lithosphere (Zuber and

Smith, 1997), and estimates of the elastic thickness of this region range up to several

hundred kilometres (McGovern et al., 2004; Zhong and Roberts, 2003; Turcotte et al.,

2002; Sohl and Spohn, 1997). The question then arises: Is the non-hydrostatic stabil

ity theory cited above valid for a planet with an elastic lithosphere? (This question

was also posed, without resolution, by Bills and James (1999, p. 9094).)

The Willemann (1984) analysis, recently refined and corrected by M atsuyama et al.


6. Rotational Stability and Figure o f Mars 6 4

(2006), has been neglected in subsequent studies of M ars’ rotational stability (Bills

and James, 1999; Sprenke et al., 2005). There may be two reasons for this neglect.

First, Willemann (1984), and also M atsuyama et al. (2006), considered the general

problem of load-induced TPW , with some emphasis on Mars and Tharsis, but they

did not quantitatively address the implications of their results for the present-day

figure of the planet. Second, Willemann (1984) and M atsuyama et al. (2006) ana

lyzed a specific scenario in which a lithosphere develops through cooling of an initially

hydrostatic planet which is then subject to loading (see below). In any event, the

Willemann (1984) study indicates th a t a non-hydrostatic theory for rotational sta

bility is not appropriate for a planet th a t has a sufficiently thick elastic lithosphere

(e.g., Mars; see also a discussion on p. 28,682 of Zuber and Smith (1997)).

The main goal of this chapter is to derive, using physical arguments supported by

standard mathematical analysis, a new, generalized statem ent of rotational stability

th a t is valid for any planet - whether it has an elastic lithosphere or not. W ith this

generalization in hand, the level of present-day rotational stability implied by M ars’

gravitational figure is reassessed. Next, the analysis of M atsuyama et al. (2006) is

extended to quantify the to tal contribution of load-induced TPW to the planetary

figure. Finally, these new expressions are compared to observational constraints on

the figure of Mars in order to set bounds on the range of Tharsis-induced TPW .

6.2. The Physics of Rotating Planets

In this section, the physics of load-induced TPW on planetary models tha t may or

may not be characterized as having a uniform elastic lithosphere will be discussed

(Fig. 5.1). In the former case, the thickness of the lithosphere is denoted by LT. This

discussion will be supported by the mathematical analysis appearing in Appendix B,



which is based on standard fluid Love number theory (Section 5.1). T hat is, time

scales considered are once again assumed to be long enough such tha t all viscous

stresses associated with the response to surface mass loading and changes in the

rotational (i.e., centrifugal) potential have relaxed completely.

Again, as discussed in Chapter 5, Gold (1955) was concerned with the rotational

stability of a planet in purely hydrostatic equilibrium. Such a planet has no elastic

strength (LT = 0), and in this case the rotational flattening (or oblateness) of the

background hydrostatic form (Fig. 5.1 AO or 5.1A1) is a function of the rotation rate,

Cl, and the internal density structure of the planet (Eq. 5.11; the sensitivity to internal

structure is embedded within the fluid tidal k Love number computed for the model

with no lithosphere, fcj’*). In this scenario, any non-hydrostatic contributions to the

inertia tensor will be associated with the applied surface mass load (Eq. 5.9). Thus,

diagonalization of the non-hydrostatic inertia tensor will yield a maximum principal

axis, tha t is a rotation pole, tha t is oriented 90° from the load, as in Fig. 5.1A4.

The minimum principal axis will pass through the center of the load, and both the

intermediate and minimum axes will pass through the equator.

Next, scenarios in which the planet has an elastic lithosphere are examined, be

ginning with Fig. 5.IB. In this somwhat unrealistic, but nevertheless physically in

structive case, an initially non-rotating planet with a pre-existing elastic lithosphere

(Fig. 5.1B0) is spun-up to its current rotation rate, and the system is allowed to

reach a state in which all viscous stresses below the lithosphere relax completely

(Fig. 5.1B1). This relaxed form will not be in hydrostatic equilibrium, since the

elastic lithosphere has permanent strength, and thus the oblateness of the form will

be less than the hydrostatic flattening in Fig. 5.1 A (Mound et al., 2003). To distin

guish the former from the la tter we will henceforth use the term ‘equilibrium form’,



and note tha t this form approaches the hydrostatic figure as the elastic lithospheric

thickness approaches zero. Mathematically, the oblateness of the equilibrium form is

a function of the rotation rate, fl, and the fluid tidal k Love number, k j (Eq. B.5),

and the la tter is a function of both the internal density structure of the planet and

the thickness of the elastic lithosphere ( k j —► kj'* from below as L T —► 0) (Mitrovica

et ah, 2005; Matsuyama et ah, 2006).

As in Fig. 5.1 A, a load applied to this model planet will ultimately reach the equator

(Fig. 5.1B4) since the equilibrium rotational bulge which defines the initial rotating

state (Fig. 5.1B1) will eventually reorient perfectly to a change in the position of the

rotation pole (i.e., it provides no memory of a previous rotational state; see figure

caption and the discussion between Eqs. B .l to B.2). However, it would be incorrect to

analyze the rotational stability of this system using non-hydrostatic stability theory.

Specifically, if one were to correct the figure in Fig. 5.1B1 for a hydrostatic form, one

would be left with a residual, non-hydrostatic form tha t was characterized by a deficit

in oblateness, or mass excess at the poles (a prolate spheroid). (Mathematically, this

difference arises because k j < kj'* when L T ^ 0.) One would thus erroneously

introduce a spurious tendency for the entire figure to drive a TPW event tha t would

move the pole towards the equator.

It is reasonable to conclude th a t the long-term stability of a rotating planet is

governed by the terms in the inertia tensor which do not perfectly reorient to the

contemporaneous rotation axis (Eq. B.2). T hat is, the hydrostatic form in the scenario

of Fig. 5.1A (Fig. 5.1A1; Eq. 5.8) and the equilibrium form in the scenario of Fig. 5.IB

(Fig. 5.1B1; Eq. B.2) are irrelevant to the rotational stability. Thus, the stability

of the system in Fig. 5.IB is governed by the non-equilibrium (rather than non

hydrostatic) inertia tensor. This statem ent provides a fundamental extension of the


6. Rotations,! Stability and Figure o f Mars 6 7

Gold (1955) stability theory to the case of planets, like Mars, with non-zero elastic

strength in the lithosphere.

Fig. 5.1C shows a more complicated and realistic scenario for such planets which

has been considered by both Willemann (1984) and M atsuyama et al. (2006). An

initially hydrostatic, rotating planet (Fig. 5.ICO) cools and develops an elastic litho

sphere (Fig. 5.1C1). Lithospheric formation will not disturb the hydrostatic form

since the elastic lithosphere will grow in a fully relaxed state. In contrast to Fig. 5.IB

(or Fig. 5.1A), the rotational bulge cannot reorient perfectly to a change in the pole

position since there would be no way for the elastic lithosphere to re-establish a hy

drostatic form around the new pole position: TPW will introduce stresses in the

previously stress-free lithosphere. The system thus has a memory of the initial rota

tional state (Fig. 5.1C1) and any departures from this state would be resisted. The

final load position (Fig. 5.1C4), which is not at the equator, represents a balance

between this resistance and the tendency of the load to drive TPW . Unless the load

is of the same order as the mass associated with the rotational bulge, little TPW can

occur.

Is the generalized statem ent tha t rotational stability is governed by non-equilibrium

components of the inertia tensor appropriate to the scenario depicted in Fig. 5.1C? To

answer this requires tha t the change in shape between the initial (Fig. 5.1C1) and final

(Fig. 5.1C4) rotational states are separated into a contribution th a t perfectly reorients

as the pole moves around and a residual term. Physically, the la tter can be inferred by

simply switching off rotation in the case of Fig. 5.1C1 and determining the departure

from sphericity th a t would result. This departure would be the difference between

the hydrostatic form (Fig. 5.1C1) and the equilibrium form associated with a planet

having the same rotation rate and elastic lithospheric thickness (as in Fig. 5.1B1).



This difference, which is termed the remnant rotational bulge (Willemann, 1984;

M atsuyama et ah, 2006), is frozen into place relative to the initial pole position.

Thus the oblateness in Fig. 5.1C1 can be decomposed into a component from the

remnant rotational bulge, which stays fixed relative to the initial rotation axis, and

an equilibrium rotational form th a t will adjust perfectly (in the long-time limit) as

the pole moves from the initial to final (Fig. 5.1C4) state. Therefore, as in Fig. 5.IB,

the rotational stability of the planet is governed by non-equilibrium components of

the inertia tensor. As in previous scenarios, these components include the surface

mass load, but, in the case of Fig. 5.1C they also include a remnant bulge.

The same separation of the figure of the planet into: (1) an equilibrium form

th a t adjusts perfectly to the change in the orientation of rotation and thus has no

bearing on the rotational stability; and (2) a non-equilibrium, remnant rotational

form oriented with the initial pole position, is derived mathematically in Appendix

B.1.2 (see Eqs. B.8 to B.10).

As a final point, M atsuyama et al. (2007) have analyzed the rotational stability of

the scenario in Fig. 5.1C by writing expressions for the total energy in the system

and finding the TPW th a t minimizes this energy. Their expressions provide an in

dependent confirmation th a t it is the diagonalization of the non-equilibrium inertia

tensor th a t governs the rotational stability.

6.3. Results

The non-equilibrium theory described above provides a generalized framework for

assessing the rotational stability of a planet on the basis of its gravitational figure

or, equivalently, its inertia tensor. In this section two issues th a t were previously

investigated by applying a non-hydrostatic rotation theory to the figure of Mars are



" a " ' fthyda - - gnhyda -

b = 5hydb + ^nhydjj

c _ 5hydc _ <nhydc

reassessed. First, how stable is the present-day rotation vector of the planet? Second,

w hat level of T PW was driven by the development of the Tharsis volcanic province?

6 .3 .1 . The Present-D ay Rotational Stability o f Mars

The to tal inertia tensor of Mars can be written as:

(6 .1)

where a, b, and c are the non-dimensional moments (non-dimensionalized by the mass

and mean radius of Mars) in the principal axis system (a < b < c) and the superscripts

hyd and nhyd denote hydrostatic and non-hydrostatic contributions. On Mars, these

three moments refer to axes on the equator at the same longitude as Tharsis, on

the equator 90° from Tharsis, and the current rotation axis, respectively. Embedded

within the hydrostatic contribution to the total inertia tensor is a spherical term

which results in a trace, for this contribution, tha t is non-zero.

The non-hydrostatic moment increments are commonly expressed in terms of the

observed harmonic (Stokes) coefficients of the gravitational potential at degree two,

J2 and J 22, in the same principal axis system (Bills and James, 1999):

- 5 nhyda - ' - 1 /3 • " - 2 'finhydfo = - 1 /3 (j 2 _ jM ) + 2f inhydg 2/3 0

22 (6 .2 )

where the param eter is a correction to the observed J 2 harmonic associated with

the hydrostatic form of the planet. Satellite-based measurements (Smith et al., 1999)

have yielded the estimates: J 2 = (1.960T0.02) x 10~3 and J22 = (6.317±0.003) x 10-5 .



This decomposition makes no assumption regarding a connection between the rota

tional stability and the inertia tensor. However, let us proceed by assuming th a t the

non-hydrostatic inertia tensor governs the rotational stability. In this case, once the

observed harmonics J2 and J22 are specified, assessing the rotational stability reduces

to estimating J%yd in Eq. (6.2). Bills and James (1999) combined a satellite-based

estimate of J2 with a constraint on the spin-axis precession rate (or precession con

stant) based on radio tracking from the Pathfinder mission (Folkner et al., 1997), to

estimate the non-dimensional polar moment of inertia as c = 0.3662 ± 0.0017. They

then used the Darwin-Radau relationship to convert this value to an estimate for

jhyd _ pgqg _j_ Q Q21 x 10~3 (see their Eq. 88). Using this value in Eq. (6.2) yields:

- Snhyda - ' -166.06 ± 7 .2 0 '£nhyd,Jj = 86.63 ± 7.20 x 10~6finhydc 79.43 ±14.27

Bills and James (1999) concluded, since 5nhydb > 5nhydc. th a t the current rotation pole

of Mars is 90° from where it should be on the basis of the non-hydrostatic stability

theory of Gold (1955). Moreover, since 5nhydb ~ 5nhydc, they also concluded tha t the

pole is unstable; small mass loads would be capable of moving the pole along a great

circle joining the maximum and intermediate axes of inertia (i.e., along the great

circle 90° from Tharsis).

Yoder et al. (2003) have derived a more recent estimate of c = 0.3650±0.0012. The

Bills and James (1999) analysis was repeated for this range of values and the result

is shown in Fig. 6.2. (One difference in this analysis is th a t the polar moment c is

corrected for a small non-hydrostatic contribution before the Darwin-Radau relation

is applied. Under the assumption tha t these non-hydrostatic contributions to the

inertia tensor are axisymmetric about a point on the equator at the same longitude



110100

105

-100 .nhyd.

-2000.3660.364 0.365

70------------1------------1------------1--------- 1---------1------------0.3635 0.364 0.3645 0.365 0.3655 0.366 0.3665

Polar Moment of Inertia

F igu re 6.2.: Non-hydrostatic moments of inertia, computed using Eq. (6.2), as a function of the total polar moment of inertia, where the latter is varied within the uncertainty (0.3650 ±0.0012) cited by Yoder et al. (2003). The moments 5nhydc and 5nhyda refer to axes in the direction of the current rotation pole and the equatorial location of Tharsis. The axis associated with the moment 5nhydb is aligned with a point on the equator 90° from Tharsis. All moments are normalized by M a2, where M and a are the mass and mean radius of Mars. I adopt the observed values of J 2 = 1.960 x 10-3 and J22 = 6.317 x 10-5 cited in the main text.

as Tharsis, the correction is simply f J 22; Bills and James (1999)). Over this entire

range of c values 5nhydb ~ Snhydc, and therefore one would again conclude on the

basis of a non-hydrostatic theory th a t the M artian rotation pole is unstable. Note

th a t the c value adopted by Bills and James (1999) falls at the high end of the range

considered in Fig. 6.2, and tha t Snhydc > 5nhydb when c < 0.3657.

As noted in the last section, this non-hydrostatic theory is not appropriate for



the analysis of M ars’ rotational stability if the M artian lithosphere is characterized

by non-zero elastic strength. Accordingly, the non-equilibrium, rather than non

hydrostatic form needs to be analyzed to properly assess the stability. Simply put,

the hydrostatic correction removes too much flattening from the observed form and

will thus imply a significantly less stable planet than the correct, non-equilibrium

approach.

Let us begin by re-writing equation (1) in the form:

(6.4)

where the superscripts eq and ne denote the equilibrium and non-equilibrium contri

butions to the total inertia tensor. The second of these contributions can be written

in term s of J2 and J 22 using a modified form of Eq. (6.2):

~ 0 ’ 5eqa ' " 5nea 'b = 5eqb + Snebc . 5eqc . . Snec .

' 5nea ' ' - 1 /3 ■(j 2 _ j W ) +

■ - 1 /3 ■(J ^ d - J ? ) +

' - 2 '5neb — - 1 /3 - 1 /3 2

. $nec . 2/3 2/3 0J22 (6.5)

The second term on the right hand side of this equation is the difference between the

hydrostatic and equilibrium contributions to the inertia tensor and it can be written

in the form (Eq. B.6):. . o 2 „ 3 _

(6 .6)7hyd jeq Cl . t + ? j . .Joy ~ J y = — kf

' 2 “ 3G M L / /J

where a and M are the radius and mass of the planet, respectively, and G is the

gravitational constant. The quantity within square brackets represents the difference

between fluid Love numbers computed for no lithosphere (i.e., the hydrostatic form)

and a lithosphere of thickness L T (the equilibrium form).

In Fig. 6.3 Eqs. (6.5) and (6.6) are used to compute the non-equilibrium moments



L T (km) hL k l30 -0.951 1.14670 -0.910 1.091100 -0.875 1.053200 -0.764 0.899

Table 6.1.: Effects of lithospheric thickness (LT) on the fluid k Love numbers of Mars. We adopt the 5-layer model of Martian structure described by Sohl and Spohn (1997). The tidal fluid Love number for LT = 0 is kj* = 1.18955.

5nea, 6neb and 5nec, as a function of the elastic lithospheric thickness. The Love

numbers were computed using the Mars model of Sohl and Spohn (1997) (see Table

6.1). Furthermore, in evaluating the first term on the right hand side of Eq. (6.5),

the hydrostatic correction of Bills and James (1999) is used. For L T = 0, the non

equilibrium theory collapses to the old non-hydrostatic case ( k j ’* = k j in Eq. (6.6)

and therefore the second term on the RHS of Eq. (6.5) vanishes) and the results

suggest an unstable rotation pole (i.e., Sneb ~ 5nec, as noted in Fig. 6.2). However,

the estimate of 5nec increases rapidly relative to 5neb as L T is increased above zero.

Indeed, a value of L T = 100 km yields a moment difference Snec — 5neb which is

comparable to the difference Sneb — 5nea, and a highly stable rotation pole. I therefore

conclude tha t the current orientation of the M artian rotation pole is stable for values

of L T th a t are consistent with widely cited estimates (e.g. McGovern et al., 2004;

Zhong and Roberts, 2003; Turcotte et al., 2002; Sohl and Spohn, 1997).



400

300

' o 200T“XW•McoEo5

100

E3£1 -100 o-LLiIC

Z -200

-300

-40050 100

E lastic L ithospheric T h ick n ess150

T h ick n ess200

Figure 6.3.: Non-equilibrium moments of inertia, computed using Eqs. (6.5) and Eq. (6.6), as a function of the adopted thickness of the elastic lithosphere, LT (km). The moments 5nea, Snec and Sneb refer to axes in the direction of Tharsis, the current rotation pole and a point on the equator 90° from Tharsis, respectively. All figures for the moments are normalized by M a2, where M and a are the mass and mean radius of Mars. I adopt the observed values of J 2 = 1.960 x 10-3 and J22 = 6.317 x 10~5 cited in the main text. The fluid Love numbers required in Eq. (6.6) are given in Table 1 for various values of LT.

6 .3 .2 . T PW Driven by Tharsis Loading

An im portant, outstanding issue in the long-term evolution of Mars concerns the

extent to which the development of Tharsis changed the orientation of the rotation

vector. For example, there have been numerous inferences of Tharsis-driven TPW

based on tectonic patterns, geomorphologic features, magnetic anomalies and grazing

impacts (see introduction and also Sprenke et al., 2005). In addition, there have been



theoretical predictions of polar wander driven by a surface mass loading consistent

with the size and current location of Tharsis (Melosh, 1980; Willemann, 1984; M at

suyama et al., 2006, 2007). These theoretical analyses admit both small and large

polar wander solutions (TPW angle, 5, of ~ 10° or ~ 80°, respectively; see discussion

below Eq. B.19).

Sprenke et al. (2005) analyzed the observed figure of Mars using a non-hydrostatic

stability theory and concluded th a t Tharsis induced a polar wander of 15°-90°. Specif

ically, they began by adopting the non-hydrostatic form given by Eq. (6.3) and then

corrected this form for Tharsis using the load model of Zuber and Smith (1997).

They next performed a search through all possible (pre-Tharsis) pole positions tha t

satisfied the following stability equation (Bills and James, 1999):

j n h y S > 2 J nhyd t

where the superscript f denotes the residual non-hydrostatic field after correction for

Tharsis. The collection of acceptable pole positions defined a pre-Tharsis stability

field, and the range of TPW angles th a t moved the pole from within this stability

field to the present position yielded the inference of 5 = 15 — 90° (Sprenke et al.,

2005).

To understand the origin of Eq. (6.7), consider a special case where the correction

for Tharsis does not alter the principal axis orientation determined from the figure of

Mars. In this case, Eq. (6.2) could be revised to remove the Tharsis load using:

6 n h y S a r - 1 / 3 1 ' - 2 '

f inhydtfo = - 1 / 3 1 ft. 1

+ 2

f inhyd i c 2 / 3 0

( j22 - 4) (6.8)



where j \ and j \ 2 are the Tharsis contributions to these coefficients. If we define

jnhyS _ j 2 _ jhyd _ j t ancj __ j ^ ^ en ^ equation (6.7) is

seen simply as a condition th a t 8nhyd1 c > 8nhyd'b.

There are various assumptions inherent to the procedure adopted by Sprenke et al.

(2005). First, th a t the non-hydrostatic form governs the rotational stability (see their

discussion on p. 488). Second, related to the first, tha t the remnant bulge dynamics

discussed by Willemann (1984) may be ignored. This assumption is implied by the

procedure of searching through possible pre-Tharsis pole positions. In the physics of

Fig. 5.1C, each reorientation of the pole in this manner would introduce a remnant

bulge contribution to both the J2 and J22 tha t should be accounted for. A further

assumption of Sprenke et al. (2005) is th a t the figure of Mars at spherical harmonic

degree two has not changed, with the exception of a simple rotation, subsequent to

the end of the development of Tharsis (i.e., the values of J 2 and J 2 2 used in Eqs. (6.7)

are present-day values).

In this section I revisit inferences of Tharsis-driven TPW based upon the observed

figure of Mars by using the non-equilibrium rotation theory appropriate to the physics

of Fig. 5.1C. In Appendix B.2 expressions are derived for the to tal Stokes coefficients

J 2 and J 2 2 (in the principal axis system) arising from the loading of a planet within

this scenario (Eqs. B.20 and B.21). For the benefit of the reader, I repeat these

equations here:

fl2a3 r 1 — 3cos2#'f >3 G M

.t ,* _ i .T\“ a r1 ~'f f ) 3 GM' - 2

T Ft2a3 r 1 — 3cos2<5



where the TPW angle 5 is given by (Eq. B.18):

(5 = ^arcsin[Q/Q;sin(20j[y)]

and a is a param eter which depends on the planetary model and the adopted litho

spheric thickness (Eq B.19):1 +

1 — fcj/fcj’*

These expressions yield the Stokes coefficients for the final state given by Fig. 5.1C4.

In these equations, 9fL denotes the final colatitude of the Tharsis load, which is taken

to be 83° (Zuber and Smith, 1997). For a given model of M ars’ density structure,

which yields kJ ’* (Table 6.1), there are two free parameters on the RHS of these

equations. The first is the lithospheric thickness. Specifying L T sets the values of

the fluid tide and load Love numbers, k j and fcj, respectively (Table 6.1), as well

as the param eter a. The second is the uncompensated size of the Tharsis load, Q'

(Eq. B.15), defined as the ratio of the gravitational potential perturbation at degree

2 due to the direct effect of the load and the hydrostatic bulge (The latter, together

with LT, sets the angle 5 in Eq. B.18).

The procedure for determining the range of acceptable T PW angles S driven by the

Tharsis load is as follows. First, some tolerance within which the predictions should

fit the observed values of the Stokes coefficients J2 and J 22 is specified. Next, a value

of L T is chosen. For this lithospheric thickness, a search is performed through a wide

range of Q' values and all predictions th a t fit the Stokes coefficients are noted. This

procedure is then repeated for different choices of LT. For each Q ' , this provides a

range of 5 values (this range can be zero).

Eqs. (B.20) and (B.21) assume th a t the only contributors to the non-equilibrium



planetary form are the Tharsis load (represented by Q') and the remnant bulge (whose

contribution depends on the level of TPW ). Although Tharsis dominates the observed

gravitational form of Mars, not including other loading contributions and their associ

ated TPW will introduce some error in predictions of both J2 and J 22, and the misfit

tolerance discussed above is an attem pt to explore the sensitivity of these assumptions

to this error.

Fig. 6.4 shows all acceptable solutions for 5 as a function of the uncompensated size

of the load when a misfit of up to 10% of the observed value of J 2 and up to 25% of

the value of J 22 is allowed. (These different values reflect the fact th a t the bacgkround

rotational bulge dominates the J2 observation; as an example, the non-hydrostatic

figure of Mars inferred by Bills and James (1999) is ~ 6% of the observed value.)

Fig. 6.4A maps out the variation in the misfit to the J 2 coefficient (as indicated

by the colour bar) within the range of acceptable solutions, while Fig. 6.4B is the

analogous map for the J22 harmonic. Embedded within these calculations are elastic

lithospheric thicknesses ranging from 30 to 200 km.

No solutions exist below a Q' value of 0.45. Moreover, these calculations adopt an

upper bound Q' value of 3.0, which is significantly larger than the bound cited by

Willemann (1984). As discussed in section B.2, there are, in theory, two possible true

polar wander solutions for a given load size and final colatitude when Q'a > 1; in this

case, if 5 is a solution of Eq. (B.18), then 90° — 5 is also a solution. However, both

solutions are not necessarily able to reconcile the additional constraint tha t has been

imposed in regard to the fit to the observed Stokes coefficients. For example, high

T PW solutions (i.e., greater than 45°) do exist, but only for Q' > 2 and a mismatch

to the observed value of J 2 > 4% (for Q' < 3). As a progressively greater mismatch

to the J2 coefficient is allowed, high TPW solutions are found for progressively lower



0.5 1 1.5 2 2.5 3Q'

Q'

Figure 6.4.: The full range of TPW angles, 6, as a function of the uncompensated size of Tharsis, Q', that yield ‘acceptable’ fits to the observed Stokes coefficients J2 and J22 for the present-day gravitational figure of Mars. The calculations are based on the non-equilibrium stability theory summarized in Fig. 5.1C and by Eqs. (B.18)-(B.21). These predictions adopt a final Tharsis colatitude of 83° (Zuber and Smith, 1997), and a lithospheric thickness, LT, which varies from 30-200 km. The range of solutions includes all predictions which fit the observed J2 and J2 2 coefficients to within 10% and 25%, respectively. The colour contours in (A) show the variation in the J2 misfit across the range of acceptable solutions. (B) is the analogous result for the J22 misfit.



values of Q'.

A comparison of Fig. 6.4A and Fig. 6.4B indicates tha t the J 2 coefficient provides a

more stringent constraint on the acceptable range of TPW than J 22. As an example,

while high TPW solutions only exist for a misfit tolerance greater than 4% of J 2, these

solutions span a wide range of J 22 misfits (i.e., from less than a percent upwards).

Therefore, under the assumptions inherent to the present analysis, the development

of Tharsis could only have driven a large excursion of the pole if a significant fraction

of the present-day J2 observation is due to signals from sources other than Tharsis

and its associated remnant bulge reorientation. The size of this required contribution,

which reaches ~ 10% for Q' = 1.7 (Willemann, 1984), is larger than any surface load

found on Mars (Smith et al., 1999). This suggests tha t the only plausible source would

be related to internal, convectively-driven dynamics. To rephrase this conclusion, the

J 2 signal associated with a large TPW event (and remnant bulge reorientation) driven

by Tharsis differs significantly from the present-day observation of this harmonic.

The relationship between the lithospheric thickness and misfit within the suite of

acceptable TPW solutions shown in Fig. 6.4 is plotted in Fig. 6.5. That is, for a

given Q', Fig. 6.5 provides the range in L T embedded within the solutions for 5

on the associated frame of Fig. 6.4. Note th a t for both the low and high TPW

solutions, the elastic thickness of the lithosphere which produced a solution for the

Stokes coefficients within the specified misfit tolerances tends to decrease as Q' is

increased.

This trend reveals some interesting physics. Consider, first, the small TPW branch

of solutions. The results in Fig. 6.4 indicate tha t a small Tharsis load (Q' ~ 0.5)

emplaced at a colatitude very close to its final colatitude (i.e., 5 of a few degrees) will

yield a good fit to both the J 2 and J 22 observations. Increasing the uncompensated



200

•2 120

2 100

200

1 8 0

?* 1 6 0(0§ 1 4 0 | 120

- 100

2 . 5 31 .5 20 . 5 1

- 1 5

Figure 6.5.: The range of lithospheric thickness, as a function of the uncompensated size of Tharsis, that yield ‘acceptable’ fits to the observed Stokes coefficients J2 and J22 for the gravitational figure of Mars. The associated range in TPW angles, 6, is given in Fig. 6.4. The details of the calculation are discussed in the caption to Fig. 6.4 and in the text. (A) and (B) show the variation in the J2 and J22 misfit across this range of acceptable solutions.



size of the load will increase the TPW angle S (Fig. 6.4). To maintain a similar fit to

the gravitational field as Q' is increased, th a t is, to maintain a similar effective load

size and remnant bulge signal, requires, in this case, th a t the lithospheric thickness

be reduced. Next, within the high T PW branch, decreasing Q' leads to a higher, not

smaller, level of TPW (Fig. 6.4). Once again, one can maintain a similar contribution

to the Stokes coefficients from the surface load by increasing L T as Q' is decreased

since this will reduce the level of isostatic compensation. However, in this case,

the signal from remnant bulge reorientation will not remain the same; rather it will

increase because both the TPW angle and the size of the remnant bulge will increase.

The result is an increasing level of J 2 misfit as Q' is decreased in the high TPW

branch.

Under the assumptions adopted in the analysis, Figs. 6.4 and 6.5 may be used to

constrain the thickness of the M artian lithosphere at the time of the formation of

Tharsis and the TPW driven by this formation. As discussed above, if an acceptable

upper bound on the (uncompensated) size of Tharsis is the one cited by Willemann

(1984), Q' — 1.7, then high TPW solutions are ruled out unless the misfit to J 2 is

well over 10%. In this particular case, the TPW angle is limited to less than 15°

(Willemann, 1984; Matsuyama et al., 2006) and the minimum elastic thickness of the

M artian lithosphere at Tharsis formation would be 45 km. For a Q' value of 1, the

T PW angle would be less than 10°, and L T > 90 km.

6.4. Conclusions

Numerous analyses of the rotational stability of Mars, either a t present-day or in

response to loading by Tharsis, have been based on the assumption th a t this stability

is governed by the non-hydrostatic gravitational figure of the planet. This work has



demonstrated tha t such a treatm ent is incorrect. In particular, for any planet with

long-term elastic strength within the lithosphere, the stability of the rotation vector

is governed by the gravitational figure after correction for an ‘equilibrium’, rather

than hydrostatic form. The former is defined as the shape achieved by an initially

non-rotating planet with an elastic outer shell after all viscous stresses below the shell

have relaxed subsequent to the onset of rotation (e.g., Fig. 5.1B1). The equilibrium

form depends on the thickness of the elastic plate as well as on the rotation rate

and the internal density structure of the planet, and it is the component of the

gravitational figure which will perfectly reorient (in the fluid limit) to a change in

the rotation vector; thus, it provides no long-term memory of any previous rotational

state. The ‘non-equilibrium’ theory provides the necessary extension of the oft-cited

non-hydrostatic theory of Gold (1955) to the case of planets with elastic lithospheres.

The observed figure of Mars, after correction for the equilibrium form, indicates tha t

the present-day rotation axis of the planet is stable for adopted elastic thicknesses

of the M artian lithosphere well below current estimates (Fig. 6.3). This counters

previous conclusions, based on a non-hydrostatic theory of planetary rotation, tha t

the present-day orientation of the pole is unstable and will move easily on a great

circle defined by the arc joining the current pole and a point 90° from Tharsis on the

M artian equator.

Finally, a version of the non-equilibrium theory valid for the scenario of planetary

evolution considered by Willemann (1984) and M atsuyama et al. (2006), in which

a lithosphere develops on an initially hydrostatic form, has been used to estimate

the range of possible Tharsis-driven TPW . The analysis presented here, based on

a comparison of predictions of the Stokes coefficients J 2 and J 22 with present-day

observational constraints, suggests tha t Tharsis drove less than 15° of polar motion.



These calculations also indicate th a t L T a t the time of Tharsis formation was at

least ~ 50 km, though a more likely lower bound (if one accepts tha t Q' ~ 1) is

~ 100 km. This bound suggests tha t the formation of Tharsis did not markedly

reduce the strength of the M artian lithosphere.

This inference of Tharsis-driven T PW assumes th a t the figure of Mars has been

altered by a relatively minor amount, defined by an imposed misfit tolerance, since

the end of Tharsis formation. I can clearly not rule out th a t other loads, in particular

internal heterogeneity related to convection, provide a significant contribution to the

present-day form. However, my intent was to specifically reassess the conclusions of

previous work th a t had assumed th a t these contributions were small. In this regard,

my results dem onstrate th a t arguments th a t Tharsis-induced TPW was at least 15°,

based on a non-hydrostatic theory of M artian rotational stability, are not robust.


7. Rotational Stability of Planets:

The Influence of a Viscoelastic

Lithosphere

7.1. Introduction

The last chapter dealt with the stabilizing influence on T PW of an elastic lithosphere

and the application to Mars was motivated by the observation th a t th a t planet was

characterized by very long term elastic plate strength. The physics described by

Willemann (1984) has not been applied to the Earth, by him or later workers, for two

reasons. First, the E arth ’s lithosphere is relatively thin and viscoelastic, and thus the

planet will relax completely to the presence of loading or a change in the centrifugal

potential over sufficiently long time scales. Second, the E arth ’s lithosphere is broken

into plates, which presumably further increases its tendency for load compensation

and bulge relaxation. However, it is im portant to note th a t the results in Willemann

(1984) and M atsuyama et al. (2006) indicate tha t even a relatively thin lithosphere

can exert a significant stabilizing effect on planetary rotation. W ith this in mind,

in this chapter 1 present an extension to the M atsuyama et al. (2006) theory to

85


7. Rotational Stability o f Planets: The Influence o f a Viscoelastic Lithosphere 86

focus on the potential impact on polar motion of a viscoelastic lithosphere. The

associated time-dependent theory replaces the equilibrium theory summarized in the

last chapter.

Continental drift, sea-floor spreading, and the large-scale dynamic topography of

continents all reflect an active mantle convective regime within the post-Jurassic

Earth, yet excursions of the rotation pole relative to the hot spot reference frame

are remarkably muted. Over the last 100 Myr, true polar wander (TPW ) has been

less than 10° (Besse and Courtillot, 1991), and this stability has been explained in

terms of a high viscosity, and thus sluggish, lower mantle (Steinberger and O’Connell,

1997), and/or as a consequence of an inefficient excitation geometry (Richards et al.,

1997).

Steinberger and O ’Connell (1997) used mantle convection simulations constrained

by seismic tomography, in which the equatorial bulge is assumed to be perfectly

relaxed to the contemporaneous pole position at each time step (i.e., the Gold (1955)

scenario), to find a time-dependent inertia tensor for Earth. W ithin this framework,

perturbations to the load inertia tensor (driven by internal loading due to mantle

convection), no m atter how small, dictate the location of the principal axis of inertia

and hence the rotation axis. Since a small change in the inertia tensor can cause a

relatively large reorientation of the planet, reducing the rate of true polar wander

necessitates slowing down the rate at which the load inertia tensor changes. One way

to do this is to make the thermal convection more sluggish by increasing the viscosity

of the mantle.

Calculations of T PW driven by tectonic plate subduction th a t incorporate some

time dependence of bulge relaxation (Spada et al., 1992; Ricard et al., 1993), also

require relatively high values of lower mantle viscosity to reduce rates of TPW . In


7. Rotational Stability o f Planets: The InEuence o f a Viscoelastic Lithosphere 87

this case, a higher lower mantle viscosity slows down the movement of density hetero

geneities within the mantle, but it also suppresses relaxation of the rotational bulge.

Richards et al. (1997) assumed th a t the tectonic history of plate subduction has been

the main driving force of TPW since the Cenozoic. They argued th a t the slow change

in tectonic patterns during this time produced a geometry of slab forcing th a t was

inefficient at driving TPW .

Is there some other way to explain the muted excursions of the rotation axis on

E arth over the last 100 Myr? The question arises as to what impact, if any, would

a broken, viscoelastic lithosphere have on the long-term rotational stability of the

Earth? To explore this issue I begin by considering the schematic diagrams in Fig. 7.1,

which illustrate various scenarios for the rotational response of a planet subject to

an uncompensated surface mass load. The first two rows repeat cases treated in the

last two chapters, namely T PW in the case where the rotational bulge will relax com

pletely to a change in rotation (Gold, 1955) and the case where an elastic lithosphere

yields a remnant rotational bulge (Willemann, 1984), respectively. The third row

treats the situation of a viscoelastic lithosphere, which I consider in detail here.

Let us assume th a t viscous stresses within the viscoelastic lithosphere have a char

acteristic relaxation time of ruth, and th a t the relaxation of the equatorial bulge minus

lithosphere in response to TPW is Uody Let us also assume th a t the viscoelastic struc

ture within the lithosphere and body of the planet is such th a t Tuth » Vwdy for the

principal normal modes tha t govern the relaxation. In response to a load (Fig. 7.1C2),

the pole will initially follow the sequence of events shown in Figs. 7.1B3-7.1B6. T hat

is, for periods shorter than the relaxation time of the lithosphere, the perturbed cen

trifugal potential will “see” an effectively elastic surface plate, and the pole will reach

a point where the impact of the load will be balanced by a resistance associated with


7. Rotational Stability of Planets: The Influence of a Viscoelastic Lithosphere 88

A2 A6A3 M

C4C3

c »

Figure 7.1.: Schematic illustration of the physics of TPW as treated in previous (A,B) and the present (C) analysis. The blue disk represents an internal (density) load, the solid- green outer shell is an elastic lithosphere, the hatched green outer shell is a viscoelastic lithosphere, the solid arrow (with spin and TPW directions specified at the head and tip, respectively) is the rotation vector, and the long-dashed arrows are previous rotation vectors within the same series. On each frame, a dashed line denotes the plane of the rotational bulge, while a dotted line is the rotational equator. When these two lines are aligned (A2, A4, A6, B l, B2, Cl, C4), the bulge is perfectly relaxed. The labels ruth and Uody represent characteristic time scales over which viscous stresses within the viscoelastic lithosphere and body (minus lithosphere) of the planet, respectively, will relax in response to a change in centrifugal potential associated with TPW. (A) TPW on a planet in which the rotational bulge will ultimately adjust completely to load-induced TPW, and the load will ultimately migrate to the equator (A6) (Gold, 1955). (B) TPW when an initial, hydrostatic form (B2) includes an unstressed elastic lithosphere. In this case, the elastic shell will permanently resist excursions of the rotation pole and the final state (B6), in which the load has not reached the equator, will represent a balance between this resistance and the load-induced impact on the pole (Willemann, 1984). (C) As in (B), except the lithosphere is treated as viscoelastic. For times t < ruth, the TPW will follow the scenario in row B (C3=B6). However, viscous relaxation within the lithosphere for times t > ruth will ultimately weaken the resistance to TPW and the pole will, as in scenario A, ultimately migrate to the equator. The amount of time this will take depends both on rmh and the size of the load.

the imperfect reorientation of the lithosphere/bulge (Fig. 7.1C3). Next, after suffi

cient time has elapsed for the lithosphere to experience viscous relaxation (t > Tuth),


7. Rotational Stability o f Planets: The Influence o f a Viscoelastic Lithosphere 8 9

the pole will continue through the sequence shown in the later stages of Fig. 7.1A,

and the load will eventually migrate to the equator (Fig. 7.1C4).

This scenario is simplified for several reasons. First, the time required for the

load to reach the equator will depend both on the viscosity of the lithosphere and

the size of the load. Indeed, for the case of a small uncompensated load (i.e., a

real beetle placed on the Earth), this time scale may be greater than the age of the

universe (D. Stevenson, pers. communication). This is also, of course, true for the

two other scenarios treated in the figure, and it represents an underlying limitation of

the equilibrium theory treated in the last chapter. The dependence of the TPW time

scale on the size of the load will be apparent in the time-dependent theory outlined

below.

While Fig. 7.1 shows T PW in response to internal density loads, this chapter will

deal with surface loads. The physics of both scenarios is the same if, as in Fig. 7.1,

the internal uncompensated load is not moving through the mantle as a function of

time. In any event, the placement of a stationary load on the surface will allow us to

focus the physical discussion on how the T PW time scale may be influenced by the

presence of a viscoelastic lithosphere.

7.2. Mathematical Formulation

This section begins with the inertia tensor for an initially rotating planet which has

developed a lithosphere th a t does not alter the initial hydrostatic shape of the planet.

This initial case was shown schematically in Figure 5.1 and described in Section B.1.2



(see also Fig. 7.1C1). The inertia tensor after applying a surface load is given by:

0 2 5 ] T , * ,/«(*) = +

* 1 h (* )" i W - ^ 2(‘)«.j] - !12(& - | ) M + ■ # ) (7-1)

where a is the E arth ’s radius, G is the gravitational constant, and S is the Dirac

delta-function. The first two terms on the right-hand side (RHS) represent the form

of a hydrostatic planet rotating with angular velocity, G. The third term on the RHS

is the time dependent response of the planet to a change in the centrifugal potential.

In contrast to the potential forcing term found in Eq. B.7, in Eq. 7.1 there is a

time convolution between the viscoelastic tidal Love number at spherical harmonic

degree 2, kT(t), and a term related to the centrifugal potential (curly brackets). This

difference arises because in previous chapters, we were only concerned with the fluid,

or infinite time limit of the inertia tensor. In the infinite time limit, kT(t) for the

case of a perfectly elastic lithosphere is k j (the fluid tidal Love number) and Eq. 7.1

reduces to:

Q2 5 Lb* 1I , M = + - 3^ - ( f e - 3 ) ^

($ 1 ^ ' 1 1

+ l o d M * )w' (i) - - 3 >««} + r « ( 0

n 5k T 1 O 2n 5 1= V « + - 3 ^ h ( t V i ( t ) - 3 ^ 2 ( t ) < 5 « ] + - 3 ) ^ [ k f - kf ]

+ r « w (7-2)

which is identical to Eq. B.8.

In order to simplify Eq. 7.1, it is useful to find an analytical solution for the con

volution term present in this equation. Let us assume th a t the term related to the



centrifugal potential can be written as a sum of Heaviside step functions:

1 1 ^ p , ( t ) w y ( f ) - - S i 2 ( f a - 3 ) S , j = £ - ( , ) = * „ ( « ) (7.3)

here, the Heaviside function, H (t — ti), is zero for t < ti and equal to one for t > L.

Recall th a t the expression for the viscoelastic tidal Love number is given by:

K

kT(t, LTV) = k T,ES(t) + ^ 2 r jexp (-S k t) (7.4)k=1

Although the form of this expression is identical to Eq. 5.2, the above equation makes

explicit tha t we are now dealing with a lithosphere of a given thickness and viscosity

(denoted by “LTy”). In previous chapters we were only concerned with lithospheric

thickness since the fluid limit was adopted, i.e. after all modes of viscous relaxation

had relaxed. Additionally, in previous chapters, the lithosphere was assumed to have a

perfectly elastic component which would never relax, therefore the tidal Love number

kT(t) —» k j in the fluid limit. This is not the case for a viscoelastic lithosphere.

In this case, in the fluid limit kT ( t,L T v ) —> /cj’*; i.e., all stresses would relax and

a hydrostatic form would be achieved. The response of a viscoelastic lithosphere

becomes increasingly more elastic as the viscosity approaches infinite values or the

timescales of forcing become shorter.

Using Eqs. 7.3 and 7.4, the convolution in Eq. 7.1 can be performed analytically to



obtain:

kT(t,LTv) * { - j | V(f)(%] - fil2(^3 -

= kT’E Y &ijH{t ~ti ) + Y — i1 - exp(-sfc(t - ti))]1=0 1=0 k = i S k

= kT'E$ i j (t) + Y & i j H i t - t i ) [ l -exp( -3fc( t - i j ) ) ] (7-5)i=o k=i Sk

We will proceed by making the assumption tha t the viscosity of the lithosphere is

high enough tha t the relaxation time scale for the lithosphere is much longer than

other modes of adjustment. Moreover, we will assume th a t all modes in the system

have relaxed except for those associated with the lithosphere. (That is, we assume

th a t an equilibrium theory holds for all modes except the mode of adjustment asso

ciated with the viscoelastic lithosphere.) The RHS of Eq. 7.5 then reduces to:

kT’E^ i j ( t ) + Y ^ i j H (t - t i) ^ 2 — [ ! - exP i-sm h it-L ))] (7.6)i= o k= l Sk i= o S l i ih

ky^lith

where the subscript “lith” indicates the normal mode associated with the lithosphere.

Using the definition of the fluid tidal Love number from Eq. 5.6, Eq. 7.6 can be

rearranged into the following form:

L

k f ^ t ) - Y - *0— [exp (-s lith(t - t,))] (7.7)i=o Slith

where, as mentioned previously, the tidal fluid Love number for the case of a viscoelas

tic lithosphere is k j ’* (i.e., identical to the case where no lithosphere is present).

Since the mode strength associated with the lithosphere can be thought of as the

difference between the response of a planet with and without a lithosphere to tidal



forcing, the term can be expressed as the difference between fluid tidal Lovelith

numbers for these two cases, i.e., k j ’* — k j . Using this result, which we have confirmed

numerically, along with Eq. 7.7, the total inertia tensor for the system (Eq. 7.1)

becomes:

a5kT’* 1Iij (t) = IoSij + [ui{t)uj (<) - - J 2 (t)Sij]

5 ^

- T n ( kf* ~ kJ) E ~ tl) b M s u t h ( t - U))] + I^ i t) (7.8)1=0

One may consider the limits of this inertia tensor as they relate to the viscosity

of the lithosphere. As the viscosity decreases, suth —> oo (i.e., the time scale of

adjustment for the lithosphere approaches zero) and Eq. 7.8 becomes:

Iij(t) — IoSij H— [a)i(t)uj(t) - + lfj(t) (7.9)

which is the inertia tensor for a purely hydrostatic planet plus a loading term. This

is the case treated by Gold (1955), in which diagonalization of the non-hydrostatic

term yields a load positioned a t the equator.

As the viscosity of the lithosphere increases, suth —> 0 and Eq. 7.8 becomes:

^5 fT,* -. Lh ( t w o - 3<s(t)s„) - ( k p - k j) ^ w

1=0-i q 2^5 I

= W - 3 ^ W « « ] + - g g f f e - j )■ % V ,'' ~ }

+Ik(t) (7.10)

which is identical to Eq. 7.2. This is the inertia tensor for the case treated by Wille-

mann (1984); i.e., an elastic lithosphere provides a rem nant rotational bulge and


7. Rotational Stability of Planets: The Influence o f a Viscoelastic Lithosphere 9 4

the pole position (in this equilibrium theory) is governed by a balance between this

remnant bulge term and the perturbation in inertia associated with the load.

In the next section, I will discuss how the inertia tensor for the general viscoelas

tic case (Eq. 7.8) is numerically evaluated to investigate the influence a viscoelastic

lithosphere can have on time-dependent solutions of TPW .

7.3. Numerical Implementation

To examine the time-dependence of polar wander, we begin by considering a partially

uncompensated load (i.e., the load is only compensated elastically) emplaced on a

rotating planet with a viscoelastic lithosphere at some initial colatitude. The effect

of allowing the load to isostatically compensate as viscous stresses in the lithosphere

relax will be discussed in a later section of this chapter.

The size of the uncompensated load and its initial colatitude specifies the value of

the load inertia tensor, ih . The components of the load inertia tensor for a given

uncompensated load size are found using Eq. B.16. The full inertia tensor in this case

(Eq. 7.8), at a time t = to immediately after applying the load is equal to:

aSkT £7 (2 1= I 0Sij + [u i ( to ) U j ( t o ) - -UJ2( t0)Sij] + - g £ r (^ 3 - gMij [&/’* - k j ] + i f j

(7.11)

which is identical to the case for a perfectly elastic lithosphere (Eq. 7.2). In order

to find to find the new rotation axis th a t corresponds to this time we need only

diagonalize the m atrix equal to the sum of the last two terms of Eq. 7.11 (i.e., the

non-equilibrium inertia tensor). Thus, the first step in the solution is the TPW

computed using Eq. B.18 (Matsuyama et al., 2006).

In order to move forward in time, a series of evenly spaced increments are taken



such th a t at least ten timesteps occur per decay time of the viscoelastic lithosphere.

In order to compute the inertia tensor for the next timestep, t\ , we must first find

the value of <f>h. From Eq. 7.3:

= { [ u j i i t o ^ j i t o ) - ^co2( to)<5ij] - f l 2 (6i3 -

- { - ^U!2(tl)5ij] - d,2(Si3 -

= [cJi(t0)wj(to) - ^uj2{t0)5ij ] -[u ji {ti)ujj(t1) - iw 2(ti)5y] (7.12)

Substituting this expression into Eq. 7.8 yields:

of kT 1I i j ( t l) = l o^ i j 3 j ) — —a; (ti)5p]

q5 ^- g q [^'* - kJ] { M * o V j(t0) - ^ ( m j ) - Q,2(8i 3 - -)<5ij } e x p ( - s iitft(t1 - 10))

+ ^G ~ ^ M *oV i(*o) - ^ 2(*o)<5ii] + Iij (7-13)

As with Eq. 7.11, the first two terms of the inertia tensor have no bearing on the

reorientation of the rotation axis. The location of the rotation axis at t = t\ can

be found by diagonalizing the sum of the last three terms of Eq. 7.13. The form of

the inertia tensor at t \ is similar to th a t a t to in the sense th a t the “remnant bulge”

term (i.e., the stabilizing term tha t results from forming a lithosphere on a rotating

body) present in Eq. 7.11 is present in Eq. 7.13; however this term is now oriented

about the to rotation axis. There is an additional term in Eq. 7.13 tha t represents

a stabilization term provided by the lithosphere tha t viscously decays over time. In

subsequent timesteps, lithospheric resistance terms introduced from previous pole

positions will viscously decay, while a remnant bulge stabilizing term is continuously


7. Rotational Stability o f Planets: The In fluence o f a Viscoelastic Lithosphere 9 6

updated.

7.4. Results and Discussion

In this section we use the numerical procedure outlined above to consider the time-

dependent response of the pole to the emplacement of a surface mass load. In par

ticular, we consider the sensitivity of the TPW to lithospheric viscosity, the size of

the uncompensated load and its starting location (i.e., initial colatitude). The results

presented below assume a thickness of the viscoelastic lithosphere of 100 km and the

E arth model parameters listed in Table 7.1.

Figure 7.2 shows the predicted TPW angle as a function of time for different ini

tial load colatitudes (as labelled). A positive TPW angle represents a movement of

the rotation axis in a direction away from the load. For this family of curves, the

uncompensated load size, Q' (see Eq. B.15), is equal to 1.0 and the viscosity of the

lithosphere is 5 x l0 25 Pa-s. At t = 0, immediately after the load is applied, the re

sponse of the lithosphere is elastic and the position of the new rotation axis is given by

the solution to Eq. 7.11. T hat is, at t — 0 the TPW versus the initial load colatitude

Param eter Value

radius, a 6.37 x 106 mangular velocity, Ll 7.29 x 10-5 rad s-1gravitational acceleration, g 9.81 m s-2gravitational constant, G 6.67 x 10-11 m3 kg-1 s~2mass, M 5.97 x 1024 kgtidal fluid love number (LT=0), k j ’* 0.9354load fluid love number (LT=0), Ay* —1.000tidal fluid love number (LT=100 km), k j 0.9237load fluid love number (LT—100 km), k j —0.9915

Table 7.1.: Values for physical parameters used in modelling true polar wander on Earth



70

60

50 40

40

6030

20

10

050 100 150 200 250 300 350 400 450 5000

Time (Ma)

Figure 7.2.: TPW for different initial load colatitudes. The uncompensated load size in all cases is Q' = 1 and the viscosity of the lithosphere is equal to 5 x l025 Pa-s. Positive values of TPW indicate a movement of the rotation axis in a direction away from the load. Each curve corresponds to a load placed at different initial colatitudes, as indicated on the figure. After the load is emplaced at t=0, the lithosphere behaves elastically and the pole reaches some stable position (Matsuyama et al. (2006)). This is followed by a relaxation of the lithosphere that allows the load to move toward, and ultimately reach, the equator. The final TPW angle corresponding to each initial colatitude is equal to 90° minus the starting colatitude of the load.

reflects the result described by M atsuyama et al. (2006), whereby the pole position is

governed by a balance between the forcing associated with the uncompensated load

and the resistance provided by the remnant bulge.

However, as the system evolves with time, the viscoelastic lithosphere relaxes (i.e.,

the remnant rotational bulge, and its associated resistance to TPW , diminishes) and

the rotation axis moves progressively away from the load. Eventually, the load moves



to the equator; the final TPW angle for each initial colatitude is equal to 90° minus

this colatitude. At this stage, the bulge provides no memory of the previous rotational

state, and the pole position is governed solely by the inertia tensor perturbation asso

ciated with the uncompensated load. Thus, the family of curves in Figure 7.2 reflect

a time-dependent transition between the physics of rotational stability described by

W illemann (1984) and Matsuyama et al. (2006) (t = 0) and by Gold (1955) (t » 0).

For the case of Q' = 1, the to tal time it takes for the load to reach its final position

does not vary greatly as a function of the initial colatitude. However, the initial

rate of TPW (i.e., speed) does depend strongly on this colatitude, increasing as the

initial colatitude decreases. This trend is related to the residual TPW in each case.

Consider, for example, an initial colatitude of 20°. The results in Figure 7.2 indicate

th a t the pole will move about 23° right after the load is emplaced, and this will

move the load to a colatitude of 43°. As the viscoelastic lithosphere relaxes (i.e., the

rem nant bulge diminishes), the uncompensated load will be sufficiently far from its

final state at the equator th a t it will continue to force a relatively large movement of

the pole in subsequent time steps.

In Fig. 7.3 we explore the dependence of the time scale of T PW on the size of

the uncompensated load. Specifically, the figure shows the time to travel half of the

to tal TPW (i.e., 90° minus the initial load colatitude) as a function of Q' for different

initial load colatitudes. Again, a lithospheric viscosity of 5 x l0 25 Pa-s is adopted in

all cases.

For Q' = 1.0 the time scale of ‘to ta l’ TPW , as it is defined on the figure, is not

strongly dependent on the initial load colatitude, as was apparent from the results in

Figure 7.2. However, the dependence of this time scale on the initial load colatitude,

and indeed on Q' itself, becomes significant as Q' is progressively reduced below 1.0.



550

500

450

400

350

300

j= 250

200

150

100

0.7 0.8 0.90.3 0.4 0.5 0.60.1 0.2Q‘

Figure 7.3.: The time to travel half of the total TPW as a function of Q' for different initial load colatitudes. The lithospheric viscosity in all cases is 5 x l0 25 Pa-s. The ‘total TPW’ is equal to the difference between the final pole position (i.e., 90°) and the initial colatitude.

In particular, as Q' is reduced below ~ 0.3, the time scale increases dramatically.

Thus, as we discussed above, while Gold (1955)’s beetle will eventually reach the

equator, the time scale required to get there can exceed the age of the Earth for

sufficiently small loads.

What is the origin of this dependence within these calculations? A smaller load

will result in a smaller driving force for TPW, and hence within each time step in the

numerical evolution the pole displacement will be small. Each small displacement

of the rotation axis will lead to a small perturbation in the centrifugal potential,



and the associated resistance of the rotational bulge will relax according to the time

scale of the lithospheric relaxation. Thus, the to tal pole path (90° minus the initial

colatitude) will be covered by a large number of small pole displacements, each of

which will be subjected to a resistance th a t relaxes with a time scale governed by the

lithospheric viscosity. Larger loads will drive greater excursions of the rotation axis.

These excursions will also be limited by the viscous relaxation of the lithosphere;

however, not as many “steps” need to be taken to reach the final pole position. Note

th a t as Q' increases, the time for the pole to move half the to tal path approaches the

decay time of the lithosphere, which in this calculation is Tiith ~25 Ma.

Figure 7.4 illustrates the dependence of T PW on the viscosity of the lithosphere.

For all cases the uncompensated load size is Q' = 0.1 and the initial load colatitude

is 20°. The viscosity of the lithosphere was varied by two orders of magnitude, from

5x 1023 to 5x 1025 Pa-s. As one might expect, as the lithospheric viscosity is increased,

the relaxation of the lithosphere (and any remnant rotational bulge) is inhibited; thus,

it takes longer for the load to reach its final destination. The decay times, ruth, in

order of increasing lithospheric viscosities, are approximately 0.25, 2.5 and 25 Ma.

Fig. 7.5 shows the time scale of TPW , once again defined as the time for TPW

to reach half the ultim ate pole displacement, as a function of lithospheric viscosity.

Results are shown for a series of initial load colatitudes and in each case a linear

relationship between the time scale and the lithospheric viscosity is clearly evident.

Mathematically, this relationship is expected from the form of the inertia tensor in

Eq. 7.13; an order of magnitude increase in the viscosity will result in a ten-fold

decrease in the value of suth- We conclude th a t the ability of the lithosphere to

stabilize the E arth ’s rotation pole will be a strong function of lithospheric viscosity.

In this section we considered the case of a load th a t remained uncompensated



5e23 Pa-s 5e24 Pas 5e25 Pas

Q- 30

300 350 400150 200 Time (Ma)

250100

Figure 7.4.: TPW for different lithospheric viscosities, with an uncompensated load size of Q' = 1 and a load emplaced at an initial colatitude of 20°. Main: The curve corresponding to a lithospheric viscosity of 5x 1025 Pa-s is identical to the 20° curve shown in Fig. 7.2. The time taken for the load to reach its final position at the equator increases as the viscosity of the lithosphere increases. Inset: Same curves as in the main frame for the time window 0-5 Ma. In the case of the lowest value of lithospheric viscosity, the polar wander is essentially complete by t = 5 Ma.

throughout the evolution of the rotating system. It is straightforward to incorporate

a time-dependent viscoelastic compensation of the surface mass load by introducing

the load (rather than tidal) k-Love number (Eq. 5.1) into the expression for the

load inertia tensor perturbation. In this case, assuming all modes other than the

lithospheric mode are fully relaxed (to be consistent with our treatment of the remnant

bulge), the load will reach isostatic equilibrium in a time scale governed by Tuth• Test

calculations that followed this approach showed that TPW will cease after the first



1000

100.0

CO2o>eF

10.00

1.0001.00e24 1.00e26

Lithospheric Viscosity (Pa-s)

Figure 7.5.: Time to travel half of the total TPW as a function of the lithospheric viscosity for an uncompensated load size Q' = 0.1 and different initial load colatitudes.

time step (since the bulge resistance and the load driving force will decay at the same

rate). These calculations are not relevant to the real Earth since the surface mass load

treatment introduced here is intended as a proxy for the case in which internal loads

associated with advection of density heterogeneities are continuously being generated.

7.5. Future Work

The results presented in this chapter have dealt with an idealized situation in which

a load, which is never perfectly isostatically compensated, is placed on a planet with

an unbroken, viscoelastic lithosphere. Examining the behaviour of this particular


7. Rotational Stability of Planets: The Influence o f a Viscoelastic Lithosphere 1 0 3

system has been useful in bridging the end-member solutions first described by Gold

(1955) and Willemann (1984). Moreover, this simplified system may be appropriate

for some terrestrial planets, e.g., Mars, which are characterized by very long-term

isostatic disequilibrium in response to surface mass loads. However, an application of

the present results to the case of Earth will require further work.

First, the E arth is not composed of a single viscoelastic plate but is instead broken

into twelve major, in addition to many smaller, tectonic plates. The results described

here show tha t a single viscoelastic lithosphere (of sufficiently high viscosity) can have

a potentially significant stabilizing effect on TPW , can the same be said for a broken

one? To address this issue, some preliminary calculations were undertaken using a

viscoelastic finite-element code developed by K onstantin Latychev. P late boundaries

of the twelve major plates were treated as zones of extremely low viscosity and tidal

forces were applied in order to determine the long term response (i.e., the fluid tidal

love number appropriate to this situation). These finite-element results indicate tha t

even a broken lithosphere can maintain strength in response to a tidal forcing. In

this case, the resistance of the remnant rotational bulge depends on the direction

of the forcing, since plate boundaries have a complex geometry, and this direction-

dependent response would have to be taken into account in models of TPW driven

by time-dependent loads.

Secondly, a more realistic time-dependent load inertia tensor for the Earth needs

to be used. This can be done either by using convection simulations initiated using

mantle density heterogeneities derived from seismic tomography or, following Spada

et al. (1992) and Ricard et al. (1993), by dropping masses into the mantle a t speeds

comparable to those found at plate convergence zones.

The time scale of TPW depends on the viscosity of the lithosphere and on the size


7. Rotational Stability o f Planets: The Influence o f a Viscoelastic Lithosphere 1 0 4

of the load. In the internal loading case, the time scale will also depend on the rate of

advection of density heterogneities, which in turn will be a function of the viscosity of

the mantle. As noted in previous work (Ricard et al., 1993), sufficiently high viscosity

within the E arth ’s mantle may increase the decay time of internal modes of viscous

relaxation (e.g., associated with the density jump at 670 km depth) to values tha t

are relevant to the long-term TPW considered herein (i.e., greater than million year

time scales). In this case, the theory described here would have to be extended to

incorporate these modes using the approach we have developed or the methodology

described, for example, by Ricard et al. (1993).


Summary

This thesis has dealt with two distinct areas of research related to the long-term

evolution of terrestrial planets and was therefore divided into two sections. While each

section contains associated conclusions, this summary is included for completeness

and will serve to highlight some of the particularly im portant results from each section.

The focus of P art I was two case studies of regional dynamic topography. Dy

namic topography is the vertical motion of the E arth ’s surface in response to viscous

stresses in the mantle. In Chapter 3, a study of the formation of the Silurian Baltic

Basin demonstrated tha t continental tilting driven by mantle flow coupled to plate

subduction can produce sediment deposition with a spatial wavelength well beyond

w hat could be caused by supra-crustal loading alone. Mantle convection simulations

of this process were able to constrain the dip angle of the slab during the Silurian to

40° to 60° and show th a t dynamic topography can provide 40-85% of the near-field

sediment deposition. The second regional study, described in Chapter 4, revisited

the hypothesis th a t the high topography seen across the Red Sea margin is a clas

sic example of rift-flank uplift. Since the flanking topography is highly asymmetric,

with much broader regions of uplift on the Arabian side of the margin extending all

the way to the Persian Gulf, I examined the possibility tha t dynamic topography

105


might play a role in producing the observed asymmetry. The driving mechanism for

this topography is a thermally buoyant upwelling linked to a seismically observed

superswell originating from the core-mantle boundary; the same upwelling has been

linked to anomalously high topography in Southern Africa and rifting in East Africa.

Using mantle convection simulations initiated using density anomalies derived from

seismic tomography, I was able to reconcile the observed, long-wavelength tilting of

the Arabian platform (after correction for crustal effects).

P art II of this thesis examined the rotational stability of planets with lithospheres.

I demonstrate that, in the case of a planet characterized by long-term lithospheric

strength, the non-hydrostatic theory of Gold (1955) fails and it must be extended

to a so-called non-equilibrium theory. The latter theory was applied to examine

the rotational stability of Mars. In Chapter 6, using observational constraints on

the figure of Mars, I show th a t the current rotation axis is highly stable for the

range of lithospheric thicknesses cited in recent studies. The chapter also highlights

a second question im portant to the evolution of Mars; namely, to what extent did

the development of the massive Tharsis rise drive true polar wander? Again, using

equations based on a non-equilibrium rotation theory, I find tha t reorientation of Mars

due to Tharsis was likely less than 15° and tha t the thickness of the elastic lithosphere

at the time Tharsis formed was at least ~ 50 km. This result counters previous studies

th a t concluded tha t Tharsis drove a reorientation of the planet in excess of 15° and up

to 90° (the la tter is a so-called inertial interchange true polar wander event). Finally,

Chapter 7 extends the non-equilibrium rotation theory to include the effect th a t a

viscoelastic lithosphere would have on the planet’s rotational stability. I found tha t

for sufficiently high viscosities of the lithosphere, (i.e., 5 x l0 24 Pa-s or greater), the

plate can significantly reduce rates of true polar wander caused by an uncompensated

106


surface mass load. The rate of true polar wander induced by the load was found to

depend on both the viscosity of the lithosphere and the size of the load. I note tha t

a beetle will cause a reorientation of the rotation axis tha t ultimately moves the

equator, however it might take the age of the universe to do so. In any event, the new

theoretical description provides a bridge between the stability theories developed by

Gold (1955) and Willemann (1984).

107


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A. Mathematical Formulation of

Axisymmetric Convection Code

This appendix provides the reader with a more extensive derivation of the equations

derived in Chapter 2. Some equations appearing in this appendix are repeated in the

main text. These equations are included here for the sake of completeness.

By invoking all of the approximations discussed in section 2.1.1, our hydrodynamic

field equations were reduced to:

V • (pru) = 0 (A.l)

(A.2)

L J JL ( J U 'PrCp~Ot + a T prgur = V ■ (k V T ) + Q + (A.3)

where, 7*

120


A. Mathematical Formulation o f Axisym m etric Convection Code 121

A.I. Non-dimensionalisation of the Hydrodynamic

Equations

The following is a set of equations used to non-dimensionalise the hydrodynamic field

equations:

rP k Tinr = dr', t = — ~ t ' , u = - V u ' , T = TCT '+ T 0, K T = p0godK 'T, (A.4)

K0tia d

P = PoP1, P = Pogdp', a = o l0 o l , k = k0k', v = v0v', g = g0g ',

h — k0k , Cp — CpaCp.

Here, primed terms are dimensionless. Terms with the subscript zero are reference

quantities adopted from a surface value of the appropriate parameter. The charac

teristic tem perature Tc, is the difference in tem perature between the upper and lower

boundaries if both boundaries are isothermal. In these equations we have also in

troduced the Rayleigh number Ra = Q° . Values for most of the dimensional

constants introduced can be found in Table 1. All of these values were taken from

the convection model presented in Solheim (1992). The depth of the convection cell,

d, is taken to be equal to the depth of the whole mantle.

The non-dimensional form of equation (A .l) is derived from the following steps:

V • (pru) = 0

^ r / , K 0R u j- J ' [ P°Pr{—^ - u ) \ = 0

V -[ p 'ru!} = 0 (A.5)

where V ' is the non-dimensional form of the V operator.


A. Mathematical Formulation o f Axisym m etric Convection Code 122

Before non-dimensionalising equation (A.2), we first derive a more general expres

sion for gf-Ty. This generalisation will serve to simplify calculations outlined in the

next section of this paper.

d d , ( dui du j 2 duu ,d x jTlj dxj ^ dx j ^ dxi 3 l jdxC

= 77V • [Vu + uV - | 7 V • u]O

= 7?[V • (Vu) + V • (uV) - |V ■ (7 V • u)]o

= t?[V2u + V (V ■ u) — ^ V (V • u)]o

= 7?[V2u + ^ V ( V - u )] (A.6)o

In this derivation, we have made use the vector identities V ■ (V u) = V 2u,

V • (uV ) = V (V • u) and V • ( / V • u) = V (V • u). Substituting equation (A.6) into

equation (A.2), we find that:

V p = - p 5f + 77[V2u + ^ V ( V - u ) ] (A.7)o

Non-dimensionalising equation (A.7) gives us:

^ - ( p0g0dp') = - P o p W ^ + ? ? ( ^ ^ ) ( ^ ) [ V ,2u / + ^ V ,(V/ -u /)]

r-7/ / 1 1 * 1 ( a 0Tcg0d ^ r /2„/ , ^V7' 7r 7' ,,'MPogoV p = J[V u + g V C V - u ) ]

PoQoVp' = - p 0p'g0g 'f + [rjK0a 0Tcg ^ (J - 'j [V 'V + ^ v '(v ' ’ u ')l

p0g0V p ' = -Pop'g0g 'f + ( ga0Tcg ( J ^ j [V 'V + ^ V '(V ' ■ u')]

W = - p 'g 'f + a 0Tc[V 2u ' + ^ V '( V '■ u')] (A.8)

As a preliminary to non-dimensionalising equation (A.3), we first seek a more


A. Mathematical Formulation o f Axisym m etric Convection Code 1 2 3

general expression for:

d u i / d u i d u j 2 duk \ du iT l j d x j ^ \ d x j ^ d x i 3~Wy d x k J d x j

In order to simplify the notation used, let dj —

dmTi j g — h ( d j V i ”t“ d i U j i j d k U k ) d j l l i

= V d j U i d j U i + d i U j d j U i - - 5 i j d k U k { d j U i ) O

Since ‘i’ and ‘j ’ are simply dummy indices, this is equivalent to:

dujTijdXj = V

= rj

= V

1 1 2- ( d j U i d j U i + d i U j d i U j ) + - ( d i U j d j U i + d j U i d i U j ) - - 5 i j { d j U i ) d k U k A A o

1 2~ ^ Uj + d jU i ) (yOiUj + d j U i ) —[djUj'jdk'Uk

2[^{diUj + d ju j^ d iU j + djUi)] - | ( V ■ u)2

If we let = \{diUj + d ju j ,

c i i i ■ r 1

Tiidx- =r]^ = r] 2 lev ev ” 3^V ' u )2]

Non-dimensionalising this equation, we find that:

(A.9)

duj i jdx] ( 3 ) [ 2 I 4 4 - 3 ( V ' . u ' ) =

rjKgRa2 ,

d4(A. 10)

Using equation (A.10) and the vector identity V • ( /A ) = / ( V ■ A) + A • V /,

where A is some vector and f is some scalar, the non-dimensionalised form of equation



(A.3) is then derived from the following steps:

f d TPrCp\~di

du •V - ( f c V T ) + Q +u • VT^ + aT prgur

( ^ ) + r »> = - i ^ r ) ( 3 ) [u' • ^ + ™

+ r.) ( ^ ) „ ; + * [v“p y r + r.) + + r.) +

[ pn2Ra2 ,prCpdd

k J

ar; ap

dT 'dt'

■Tc(u' ■ V T ') - ( ^ ) (Tcr + T0)u'r\ Cp /

r]K0FLa ,

, kTc r + K0i?a

. 1 dk dT ' Qd2V T H---------------- h ——

k dr' dr' kTc

d2prcp

V'

r]K0Ra ,

- ( u ' . V T ' ) - r ( r + A < + Ki?a

2 i a f c a r QcP f c a r 'a r ' UT.

Ted PrCp

(u' ■ V T ') - t ( T ' + T'0)u'r +

r]K0 / u 0Tcg0d3\ f t Tcd2p0cPo \ k0v0 ) p'rdp

K0Rav/2r +, 1 dk' dT ' Qd2 ]

+k! dr' dr' kak'Tc

= - ( u '- V 'T /) - r ( T / + T > ; +

+ - ( — J l - ,cPo ) \ v0) p'rdp71 f u 0g0d \ ( 1 \ (j)'

KRa

K

2 1 dk’ dT ' Qd2k' dr' dr' k0k'Tc.

= - ( u , - V ' T ' ) - t ( T ' + T ' K + — V 2T +, 1 d k’ & r Qd2 1 t04>'

k' dr' dr' k0k'Tc PrCp

Using equation (A.5) and the vector identity V ■ (6A) = (V6) ■ A + 6V • A , where

A is some vector and b is some scalar, our expression transforms into:



a r~oF

d rdt'

P;K,'

+ R a

7 [V • (T p W ) - T V ■ ( p y ) ] - r { T + t > ; .

V T +„ 1 dk' dT ' Qd

+'2 n T„d>'

k! dr' dr' k0k'T, P'rdp

- k v - i T M - T i r + n n + A v 2r +„ 1 dk' dT ' Q d'2

k' dr' dr' k0k'Tc

P'rdp(A .ll)

In the derivation of equation (A .ll) we have introduced the dissipation function,

r _ gsi' The dissipation function measures the extent to which frictional heatingCp

and work done by compression, influence energy balance in the flow.

The final equation tha t we shall non-dimensionalise in this section is the equation

of state:

p 0p ' = P o p 'A 1 - a 0a'(TcT ' + T0 - TCT'S - T0) + P°f , (p' - p'h)\P o90d K 'T

p’ = p'r[ l - a 0Tca ' ( T ' - r s) + — ( p ' - p ’h)}A t

(A. 12)

We now have our final system of non-dimensionalised equations (dropping primes):

V ■ [Pf-u] = 0 (A. 13)

Vp = -p g r + a 0Tc[V2u + ■ u)

f)fT 1 LJ*_ = v ■ (Tprii) - r ( T + To)ur + —dt pr Ra

1 dk d T Qd2v2r h h ——k dr dr k0kTc P Cp

(A. 14)

(A. 15)

P = P r [ l - a 0Tca (T - Ts) + ~ Ph)] (A. 16)



A.2. Streamfunction and Vorticity

In this section we find new expressions for our non-dimensionalised hydrodynamic

equations in terms of vorticity and streamfunction. In what will follow, all quantities

are dimensionless. We can write the axially symmetric velocity field in terms of the

streamfunction, ip, in such a way tha t the continuity equation (A. 13) is automatically

satisfied:

t AN 1 dip I dipu = [ur ,u„ 0), Ur = - ^ — g - , (A.17)

From this, we can determine an expression for the azimuthal component of the

vorticity, uj = V x u = (0,0,w).

u> = -

r1r1

l {rue)

i ! ( - -i

dur '~ ~de.1 dip'

pr sin 0 d r . dpr dip

sin#1

(ap.2 dr dr d2ip

prr sin 6 . d r2

d / 1 dip yd9 V prr2 sin 9 d6 ) .1 d2ip\ 1 ^ cos9 dip

pr d r2 Jdpr dip

~ dr dr

prrz sin 9 dOcot 9 dip 1 d2ip

r2 dO r2 dO2 .

1 d2ip y sin# dO2 ) .

(A. 18)

Other expressions involving u can be found by first taking the curl of both sides of

equation (A. 14). When this operation is performed, all gradient terms in the equation

can be eliminated.

V x Vp = V x [—pgr] + a 0TcV x [V2u + ttV (V • u)O

Using the vector identity V 2A = V (V • A) — V x V x A and since

V x V {scalar} = 0,



a 0Tc

0 = V x [-pgr] + a 0TcV x [- 'V x V x u]

V x \pgf\ = a 0Tc[—V x V x w]

V x H = a 0Tc[V2a; - V (V • w)]

V x [pgr] = [V2u;] (A.19)

since V • to = V • (V x u), and the divergence of a curl is zero.

Equating the azimuthal components of both sides of equation (A.19), we find that:

V2w -UJ = 9 dp

~2 sin2 0 0 ioTcr 80(A.20)

The left-hand side of this equation is simply the <j) component of the Laplacian of a

vector.

Using equation (A. 16), we can find an expression to replace in equation (A.20).

In doing so, we should recall th a t pr , Ts, K t and Ph are purely radial functions:

dp80

t 8 T 1 8p —a 0l ca —— + —"xx

80 K t 801

Substituting this expression into equation (A.20),

(A.21)

V 2cu —u>

'2 sin2 0 u

,2 sin2 0

a 0Tcr prga 8 T p,

n r* T r -U Pr 8ppra 0Tca dd + K ^ Qe

g 8pr 80 K t a 0Tcr 80

(A.22)

Now, by equating 0 components on both sides of equation (A. 14), we can find an

expression for However, in order to do this, we must first find another expression

for V (V • u).



Using equation (A. 13) and the vector identity V ■ ( /A ) = / ( V • A) + A • V /,

where A is some vector and f is some scalar,

V ■ (pr u)

Pr ( V • U )

Pr(V • U )

Pr(V ■ U )

(V -u )

= P r ( V • u) + U • (Vpr) — 0

= -U • (V pr)

'dpr( dpr"u ‘ V d F 7')

= —u,dpr

r dr1 dpr

ur — pr dr

(A.23)

Now we can evaluate the 6 components of the terms present in equation (A. 14):

V p ’ ° = ~ % r 86

[v2u] ■ e = v 2ue ue+

2 dur 8u<p

V (V • u)1 / 1 dp7

VI ur 3 V pr dr

r2 sin2 0 r2 86

)

d(f>

p — 1 dPr N3r 86 V Ur pr dr j

Equating the 6 components on both sides of equation (A. 14), we find that:

(A.24)

(A.25)

(A.26)

1 dp r 86

1 dp a 0Tcr 86

a 0T(

V 2ue

V 2ueUg

+2 dUr

r2 sin2 6 r2 861 5 / 1 dpr \

?>r 86 \ r o„ dr )1 dpr

pr drUg 2 8ur 1 1 dpr 8 , '

+ - - - - 7 X 3 Wr 2 sin2 6 r 2 86 3r pr dr 86(A.27)

Using equations (A. 17) and (A. 18), the right hand side of equation (A.27) can be

expressed as a function of vorticity and stream function. After significant algebra we

obtain:

1 » = ^ + C . J S - c o t 0^ 1 (A.28)1

a 0Tcr 86 dr r 3 dr p2r3 sin 6ft^ t

862 861



Substituting this expression into equation (A.22) gives:

V u —LU prga d T prg j dcu u 4 dpr

r2 sin2 9 r 89 K t I dr r 3 dr p2r 3 sin 9

jd u K t dr

'd 2tp ndip ~89\

_ 9 prg dui V 2w + ^ A — + [ Pr9 1 prga 8 T 4 g dpr 1 ■d2ip

-co t 9 ^ }L rK r r 2 sin2 9- ^ r 89 3 K t Pt dr r 3 sin 9 1892 COtU 891

(A.29)

Finally, equation (A. 15) can be rewritten in terms of streamfunction, through use

of equation (A. 17):

d T _ J ^ _ dt Ra

V T + -1 dk d T Qdl

+k dr dr kakTc.

prr (T + T0)ur - V • (Tpru ) \Pr Cp

Now,

V ■ (pruT) = V ■ (T dip

r2 sin 9 89 ’1 d / T dtp\r2 dr Vsin 9 89 ) ' r sin 9 89

dr V 89 J 89 V dr J .

o')r sin 9 dr ’ /1 d ( T dhp\

dr )■(— 1V r

r 2 sin 9 1

r2 sin 9 1

r 2 sin 9 1 _r2 sin (

8 T dtp 82ip d T dtp d2tpdr 89 ^ dr89 89 dr 89drdT d ip _ 8T_dijrdr 89 89 d r .

J(T , ip) (A.30)

8 T /8 r 8 T /8 9 dip/dr dtp/89

expression for equation (A. 15) in terms of streamfunction now becomes:

Where J(T ,ip ) is by definition d(T ,ip )/d (r ,9 ) = Our final

8 T _ _ k_ d t Ra

V 2T + 1 dk 8 T Qd2+

k dr dr kakTc Pr Cp ! sin#-(r + T0) J + J (T,V>)]}

(A.31)

The dissipation function, (p, in spherical coordinates is equal to (Solheim, 1992):



= (1

prr2 sin 6 !H( 1 dpr 2 Oip d dip

+ 2

pr dr rJ 06 06 O r.

/ 1 dpr 2 \ dipV pr dr rJ Or

cot 6

+21 <9V> ,0ip

ir 06 °° Or J(A.32)

2r0 2ip ^ ( 1 dpr 2 \0 ip 1 cPip ( c o t9 OipOr2 V pr dr rJ Or r2 062 r2 06

0 Oip Oip 1 Oipi2 1 / 1 dpr \ 2 /0 i p \2-Or 06 Or r 06 J 3 \ pr dr J \0 & ) 1

To summarise, our complete anelastic model consists of the following system of

equations:1 r02ip 1 dpr 0ip cot 6 Oip i 1 02ip~\ (A 33)U! =

2 prgOu V u + K-T f r +

prr sin 6

Pr9 1

Or2 pr dr Or

PrQ O i OTrK x r2 sin2 1

U!

r2 06 r2 092 -

4 g dpr 1

& L - 1 LOt Ra

lO kO T Qd2k Or Or k0kTc

r 06 ZKt Pt dr r 3sin(?

1 ( r 0 . 1+ - p ^ -

Pr Cp r2 sin 6' / x dipt (T + T0) ^ ~ +

02ip Oip~062 COt W

{A M )

(A.35)

These are our equations (A.18), (A.29) and (A.31), respectively. This set of

equations is completed by the equation for the dissipation function in spherical

coordinates, equation (A.32).


B. Mathematical Treatment of

Fig. 5.1

This appendix follows the analysis of section 5.1 and uses the same fluid Love number

theory described there to derive expressions for the inertia tensor for each of the cases

depicted in Fig. 5.IB and 5.1C. In each case, the connection between these expressions

and the J% and J22 harmonics within the principal axis system is considered. The

reader is asked to note tha t prior to Eq. (B.18), the symbol 5 refers to the Kronecker

delta and not the T PW angle.

B .l . The Inertia Tensor: Two Case Studies

The co-ordinate system used is oriented so th a t the z-axis is fixed to the rotation pole

of the planet just prior to loading (e.g., Figs. 5.1A1, B1 or C l). The initial angular

velocity vector will be denoted by (0,0, fi). At any subsequent time, the rotation

vector will be given by i — 1,2,3, with magnitude a>2(t). Finally, a and M are

the radius and mass of the planet, respectively, while G is the gravitational constant.

131


B. Mathematical Treatment o f Fig. 5.1 1 3 2

B.1.1. Case 1: The Equilibrium Form

Next, we turn to the scenario in Fig. 5.IB. The relevant expression for the to tal inertia

tensor is trivially derived from Eq. (5.7) by replacing the hydrostatic (L T — 0) fluid

tidal Love number with the more general case k j . This yields

Iij(t) = I05ij + [uji(t)ujj(t) - -u j2(t)Sij] + /y (t). (B.l)

In this case, the component of the inertia tensor tha t perfectly reorients to a change

in the rotation vector (as in Figs. 5.1B1, B3, and B4) is given by the first two terms on

the right-hand-side of Eq. (B .l). These terms represent the equilibrium (i.e., relaxed)

form achieved by a rotating planet with an elastic shell (Mound et al., 2003). We will

denote this equilibrium form as:

= I„S„ + p . ( t )^ ( t ) _ |u /2(t)i«] ■ (B.2)

Once again, this component of the to tal inertia tensor does not play a role in the long

term rotational stability of the planet since it provides no memory of any previous

rotational state. Therefore, the reorientation of such a planet is governed by the

non-equilibrium component of the inertia tensor:

J T ( t ) = I v ( t ) - J g ( t ) = l i ( t ) . (B.3)

As in the first scenario (Fig. 5.1A), the pole will be aligned with the maximum

principal axis of /-(£ ), and thus the load will ultimately move to a position on the

equator (Fig. 5.1B4). However, in contrast to the scenario in Fig. 5.1A, no ad-hoc

assumption tha t the load will never be perfectly compensated needs to be made. This



incomplete compensation is assured by the presence of the elastic lithosphere, and in

this sense the scenario in Fig. 5.IB is a more self-consistent illustration of the physics

th a t Gold (1955) was highlighting.

In analogy to the case treated in Section 5.1.1, an expression for the equilibrium

inertia tensor in the initial configuration of Fig. 5.1B1 is:

i y { t 0) = - !)■% (b.4)

and thus the equilibrium component of the J 2 harmonic is given by:

m _ M Q2azkT

~ “ 3G M ' ('B '5'1

In the main text an expression for the difference between the hydrostatic and

equilibrium components of the J 2 harmonic is required. This expression is obtained

by subtracting Eq. (B.5) from (5.11):

n 2 _ 3

F " - J ? = - kh ( a 6 )

This equation is identical to Eq. (6.6) within the main text.

B . l . 2. Case 2: A Remnant Rotational Bulge

The rotating form of a planet will be established early in its history, prior to the

development of an elastic lithosphere. Accordingly, Willemann (1984) suggested tha t

the rotational stability of a planet with a lithosphere will be governed by the physics

summarized in Fig. 5.1C. The scenario assumes th a t the initial rotating form will

be hydrostatic (Fig. 5.ICO); the subsequent development of the lithosphere through



cooling of the planet will not alter this hydrostatic form (Fig. 5.1C1) since no elastic

stresses will be introduced within the plate. However, any subsequent surface mass

loading and perturbation to the centrifugal potential (Fig. 5.1C2 onwards) will intro

duce such stresses within the lithosphere. Thus, the planetary model tha t governs

the response to such loads (L T ^ 0) will be different from the model th a t governs

the initial form (L T = 0).

In this case, the to tal inertia tensor of the planet subsequent to the application .of

the surface load is:

0 2 5 LT,* ,

I,j( t) = W ii +

+ i § - { h ( fH (‘) - ^ 2(i)^] - f ! 2( f e - i ) % } + ^ (t). (B.7)

The first two terms on the RHS represent the original hydrostatic form of the planet

(Fig. 5.1C1; compare this equation with Eq. (5.10) and note th a t this form is identical

to th a t in Fig. 5.1A1). The third term is the contribution from the response of the

planet (with L T / 0) to the perturbed centrifugal potential.

Eq. (B.7) can be re-arranged into the following form

a5kT 1 fl2(75 1m = (B-8)

As discussed above, any term in the inertia tensor th a t perfectly adjusts to a change

in the rotation vector will provide no memory of a previous rotational state and will

thus have no bearing on the long-term stability of the rotation pole. In this regard,

the form of Eq. (B.8) allows a natural separation between terms which adjust perfectly

(first and second terms on the RHS) and those th a t do not (third and fourth term s).



The former is simply

It-(t) = I0Sij + [ui(t)wj(t) - ^ 2( f ) ^ ] (B.9)

which is identical to the equilibrium form defined in Eq. (B.2). T hat is, in the scenario

of Fig. 5.1C, the component of the inertia tensor th a t adjusts perfectly (in the fluid

limit) to the change in pole position is the same as it was in Fig. 5. IB - namely,

the equilibrium form for a rotating planet with an elastic shell (Fig. 5.1B1). Thus

the rotational stability is once again governed by the non-equilibrium inertia tensor.

From Eqs. (B.8) and (B.9), this term is given by

J T W = [ * / ‘ - kf ] + w - (R 1 °)

The first term on the right hand side of this equation is aligned with the initial form

of the planet (i.e, the orientation at the time of the development of the lithosphere)

and it is known as the remnant rotational bulge (Willemann, 1984; Matsuyama et ah,

2006). The long-term reorientation of the pole is thus governed by a balance between

this term, which acts to resist (and thus stabilize) the motion of the pole, and the

loading term, which acts to push the pole away. The result is a final load position

th a t lies less than 90° from the pole (Fig. 5.1C4).

In the scenario of Fig. 5.1C, the determination of the long-term T PW reduces to a

diagonalization of the non-equilibrium inertia tensor given by Eq. (B.10). This is the

procedure followed by both Willemann (1984) and M atsuyama et al. (2006), though

they did not explicitly identify their expressions as representing non-equilibrium

forms. Indeed, M atsuyama et al. (2006) referred to a diagonalization of the non

hydrostatic form, where ‘hydrostatic’ was intended to mean the form in which all



viscous relaxation in the region below the elastic lithospheric region was complete.

We prefer here to use the term non-equilibrium for the L T ^ 0 case in order to avoid

confusion with the hydrostatic (L T = 0) terminology inherent to Case 1 (Fig. 5.1A).

We can use the equilibrium inertia tensor to derive an expression for the equilibrium

J 2 harmonic. As in Case 2, this expression is:

eo 41 -* = = W - (B 11)

B.2. Stokes Coefficients for Case 2: Axisymmetric

Loading

W ithin the main text the remnant bulge scenario is used to explore the range of

pre-Tharsis orientations th a t yield a to tal inertia tensor consistent with observational

constraints on the J2 and J 22 harmonics. In this section, expressions are derived

for the total (principal axis system) J2 and J22 harmonics arising from the Case 2

scenario. For this purpose, following Zuber and Smith (1997), Tharsis is modelled as

an axisymmetric load.

To begin, an arbitrary spherical harmonic decomposition (degree £, order m) of a

scalar field A is defined as:

OO i

A (M ) = E E ( M ) (B.12)£=0 m=~£

where 9 and (j) are the colatitude and east-longitude, respectively, and the Ypm are

surface spherical harmonics normalized such tha t

[ Yem'(^> 4>)ytm(6,4>)dS = 47rfe'5mm'- (B.13)Js



The symbol f denotes the complex conjugate and S is the complete solid angle.

Let us assume th a t the axisymmetric surface mass load, if placed a t the north

pole, is characterized by a degree-two spherical harmonic coefficient of L '^0. ff this

load is placed at an arbitrary position (&i, (fi), then one can show th a t the harmonic

coefficients at degree two for this load orientation are given by:

L 2m = ^ Y l m(9L,<i>L). (B.14)

ft will be convenient, following Willemann (1984) and M atsuyama et al. (2006), to

represent the size of the load by the ratio of the degree two gravitational potential

perturbation due to the direct effect of the load and the hydrostatic rotational bulge,

ff we use the symbol Q' to denote this ratio, then (Matsuyama et a l , 2006)

j ^ a 2£l2kf ’(B.15)

The degree two components of the load, as represented in Eqs. (B.14)-(B.15), may

be converted into inertia tensor perturbations I^( t ) using the following mapping (M at

suyama et al., 2007):

r>2„5 i4 = W + - efef] (B.16)

Here, e f is a unit vector in the direction of the load and is equal to:

e f = (sin cos < l, sin sin <?!>£, cos 0 l) (B.17)

This expression for the load inertia tensor can be w ritten in terms of the final

colatitude of the load (i.e., the colatitude in a reference frame in which the z -axis is



aligned with the rotation pole). This form of (B.16) can then be substituted into an

analogously modified version of our equation for the non-equilibrium inertia tensor

(B.10). Diagonalizing the result yields the following expression for the TPW angle 8

(Matsuyama et al., 2006)

5 = ^arcsin[Q'a:sin(2#{)] (B.18)

Here, 0[ is the final colatitude of the load and a is a param eter dependent on the

planetary model (and, in particular, LT):

1 + kf1 — k j / k j ’a ~ „ , r / , T , * ■ (B.19)

As a consequence of the symmetry of the load, the reorientation of the pole occurs

along the great circle tha t includes the initial load longitude (i.e., = </>£,). The sign

convention is such th a t 8 is the angle directed away from the load when Q' > 0 (as

implied by Fig. 5.1C).

The solution in Eq. (B.18) is an extension of the expression derived by Willemann

(1984), who applied approximations tha t led him to conclude th a t the TPW angle

was independent of the lithospheric thickness (i.e, a = 1 in this earlier study). If we

define Q ef f = Q'ol, then when Q ef f < 1 there is one admissible solution for a given

value of the final load colatitude; in contrast, there are two solutions, <5 and 90° —

when Qef f > 1 (Matsuyama et al., 2006, Fig. 2). As an example, Willemann (1984)

estimated an upper bound value of Q' — 1.74. Using this value in Eq. (B.18), and

the fluid Love numbers in Table 1, yields solutions of 8 ~ 10° or ~ 80° for a range of

L T values (Matsuyama et al., 2006).

M atsuyama et al. (2007) also derived Eq. (B.18) by minimizing the total energy of

the system in the case where elastic energy stored within the deformed lithosphere



is ignored. Their derivation is an independent confirmation th a t the non-equilibrium

inertia tensor governs the long-term stability of the rotation pole. (The extension to

the case where elastic energy within the lithosphere is included in the minimization

is also treated by Matsuyama et al. (2007).)

Finally, diagonalizing the to tal inertia tensor and combining the principal moments

appropriately, yields the following expressions for the Stokes coefficients:

Note th a t when Q' — 0, 5 = 0 from Eq. (B.18); thus J 2 = fl2a3k j ’*/ (3GM) and

J 22 = 0, as required for the initial hydrostatic form of Fig. 5.1C1.

The appropriate value of L T to be used in these expressions is the elastic thickness

a t the time Tharsis developed. Following the scenario shown in Fig. 5.1C, the equa-

and th a t any change in L T during this form ation/TPW was negligible. If the latter

was not the case, then the total remnant bulge would have to be computed by taking

into account the changing pole position for each incremental change in LT. However,

these expressions do not preclude tha t the elastic thickness continued to increase once

TPW -driven by Tharsis ceased.

fi2a 3 r1 — 3cos25n 3G M ^ 2 ^

(B.20)

and

tions (B.20)-(B.21) assume tha t TPW was driven by the formation of Tharsis alone,


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