Long-Term Rotational Effects on the Shape of the Earth and Its Oceans

LONG-TERM ROTATIONAL EFFECTS ON THE SHAPE OF THE EARTH AND ITS OCEANS

Jonathan Edwin Mound

A thesis submitted in cooforrnity with the reqnhents for the degree of Doctor of Philosophy Gradiiate Depart ment of P hysics University of Toront O

@ Copyright by Jonathan E. Mound 2001 . .. ilbitionsand et 9-Bib iogrephic SeMces -Iiographiques 395 WeIlington Street 395, ni6 Wellington ûttawa ON K1AOW OtEaweON KtAW Canada Canada

The author has granted a non- L'auteur a accorde une licence ncm exdusive licence allouing the exclusive permettant à la National Library of Cana& to Bibliotheque natiode du Canada de reproduce, Ioan, distriibute or seli reproduire, prêter, distniuer ou copies of this thesis in microform, vendre des copies de cette thèse sous paper or electronic formats. la forme de microfiche/film, de reprociucbon sur papier ou sur format électronique.

The author retains ownership of the L'auteur conserve la propriété du copyright in this thesis. Neither the droit d'auteur qui protège cette thèse. ttiesis nor substantial extracts &om it Ni la thèse ni des extraits substantiels may be printed or otherwise de celle-ci ne doivent être imprimés reproduced without the author's ou autrement reproduits sans son permission. autorisation. LONG-TERM RQTATLONAL E-FFECTS ON THE SHAPE OF THE EARTH AND ITS OCEANS

Doctor of Philosophy, 2001. Jonathan E. Mound Department of Physics, University of Toronto

Abstract

The centrifuga1 potent ial associated with the Eart h's rotation influences the shape of both the solid Earth and the oceaos. Changes in rotation t hus deform both the ocean and solid surfaces. The equilibrium form of the rotating Earth is generally computed using hydrostatic theory that treats the planet as an inviscid body. 1 have found that a thin elastic shell acts to reduce the flattening of the equilibrium form relative to the value obtained from the tradi t ional hydrostat ic calcuiation. Futhermore, t his perturbation is large enough that the excess non-hydrostatic fl attening of the Eart h, defined as the difference between the observed and hydrostatic forms, may be a significant underestimate of the departure of the observed form from its equilibrium state. The ocean surface and the Earthk solid surface deform by different arnounts in response to an applied potential load and thus a relative sea-level change is produced. Due to the presence of the elastic lithosphere and slowly decaying modes of viscous relaxation within the mantle, even geologically long potential forcings can produce a non-negligible sea-level signal. Using the geologically constrained history of rotat ional variations over the past 130 million years (the Cretaceous-Tertiary) I have found that the sea-level signal induced by truc polar wander may be as large as the sea-level change that has been observationally inferred for that the. True polar wander produces a long wavelength global pattern of sea-leuel trends that may be mistaken for a uoiforrn signal. The welC documented, and presumed global, Cretaceous-Tertiary sea-level cycle should therefore be reinterpreted as a combination of a globally uniform and a spatially varying true p* lar wander signal. Furt hermore, the distinct pattern of the induced sea-level variations enables the use of regional sea-level records as a test of, often contentious, true polar wander events proposed on the basis of paleomagnetic data. .4 prelirninary sea-Ievel test of a proposed Early Cambrian inert ial interchange t rue polar wander event found good general agreement bet ween predicted and observed sea-level t rends. O bservational sea-level constraints ivere not found to provide consistent support for a rapid true polar wander event proposed for the Late Cretaceous. Acknowledgements

I'd like to thank a whole bunch of people, but I'd also like to keep the acknowledgements under one page so 1 won't be able to thank everyone by narne. 1'11 start by t hanking Jerry, for getting me into my studies and back out again, and Professors Bailey and Edwards for serving on rny committee over the years. 1 am also grateful to Professor Dunlop for serving on rny final examination committee, and Professor Sabadini for serving as my external examiner. Thanks to Russ. Glenn. Hamid and Xin for passing along the ever-evolving B'files used to produce this thesis, and Rad and Khader for their help with graphics both here and for various presentations. Thanks to my parents for so much for so long, my brother for (among other thiugs) 19 words, numerous faculty, post-docs, staff and fellow students from bot h physics and geology for being both helpful and/or diversionary as the case might be, roornmates and officernates for putting up with me, ultimate teammates for hammers thrown and received. patrons of the Foaming Boot for knowing where that is, and basically everybody who made my graduate life a little better. 1 received generous financial support from bot h the Ontario Graduate Scholarship and NS ERC Postgraduate Scholarshi p programmes. Contents

Abstract

Acknowledgements

List of Tables vi

List of Figures vii

1 Introduction 1 1.1 Sea-Level Change ...... 3- 1.2 True Polar Wander ...... 5 1.3 Rotational Variations and Sea-Level Change ...... 8 1.4 Contributions and Outline of Thesis ...... , . . . . . 12

2 Theoreticai Formulation of Rotation Induced Sea-Level Variations 14 2. Introduction ...... 14 2.2 The Sea-Level Equation ...... 15 2.2.1 Surface Slass Loads ...... 16 2.2.2 PotentiaI Loads ...... 18 2.2.3 Solving the Sea-Level Equation ...... 19 2.2.4 A Simplified Sea-Level Equation ...... 39-- 2.23 The Semi-Inviscid Approximation ...... 25 '2.3 Constant True Polar Wander: .4 Case Study ...... ZY

3 The Rotational Perturbation to the Shape of the Earth 37 3.1 Introduction ...... 37 3.2 The EKick of an Elde lithosphere on the Eqniiibriurrr Shape of the Earth 38 3.3 The Fossil Bulge ...... 4%

4 True Polar Wander Induced Sea-Level Change: The Cretaceous-Tertiary Second-Order Cycle 46 4.1 Introduction ...... 46 4.2 0bsermtional Constraints ...... 47' 4'2.1 Sea Level ...... 47 3.2.2 Earth Rotation ...... 49 4.3 Mode1 Results and Discussion ...... 53

5 nue Polar Wander Induced Sea-Level Change: A Test of Early Cam- brian Inertial Interchange True Polar Wander 58 5.1 Introduction ...... 58 5.2 Modelling IITP W-induced Sea-level Trends ...... 59 5.3 A Preliminary Cornparison Wit h Cambrian Sea-level Records ...... 69

6 True Polar Wander Induced Sea-Level Change: A Test of Rapid Ttue Polar Wander During the Late Cretaceous 72 6.1 Introduction ...... t2 6.2 Results of Sea-Level Modelling ...... 73 6.3 Comparison With Observed Late Cretaceous Sea-Level Trends ...... 79

7 Discussion and Conclusions 83 7.1 Variations in Rotation .Variations by Rotation ...... S3 7.2 Speculations. Future CVork aod Final Remarks ...... 85 List of Tables

1.1 S t rat igraphic cycles and t heir causes ...... 3

3.1 Geologic constraints on rotation rate ...... 14 3.2 Magnitude of the fossil bulge ...... 14

6.1 Predicted relative sea-level change at selected sites for the proposed Late Cretaceous TPtV event ...... 76 6.2 Observational constraints on Late Cretaceous sea-level change . Y0 List of Figures

Mid-ocean ridge volume and spreading rate ...... Geometry of potential perturbations associated with changes in rotation . .

Schernatic illustration of a restoring buoyant force ...... Dependence of the inviscid limit Earth response on effective thickness of the elastic lithosphere ...... Sensitivity of relative sea-Ievel predictions to TPW rate and Earth model structure for constant TPW events ...... Sensitivity of reiat ive sea-level predictions to TP W duration and Eart h model structure for constant TPW events ...... , ...... Semi tivity of relative sea-ievei predictions to TPW rate and duration for constant TPW events ......

Dependence of the flattening of t he geoid on effective t hickness of the elastic lithosphere ...... - ......

The Vail curve...... True polar wander since the Early Cretaceous ...... Rotationalfy indnced sea-tevet change and site distribution ...... Relative sea-level predictions at four sites for Cretaceous-Tertiary TPW . . Eart h model sensi t ivi t ies of relative sea-level predict ions for Cretaceous- Tertiary TPW ......

Early Cambrian paleogeography ...... Sea-level predictions at three sites for a 25 MwwIITPW event ...... Sea-level predictions for different initial positions of the site in Laurentia . Sea-level predictions for different initial positions of the site in Baltica . . . Sea-level predictions for different durations of the IITP W event ...... Sea-level predictions for different evolutions of the IITPW event . . . . .

vii 5.7 Sensibivity of IWPW-i-indueedsea-levd pdictious to changes in the Earth mode1 parameters ...... 68

6.1 Predicted sea-level response at t hree sites for the proposed Late Cretaceous

RP TPWevent ...... 13 6.2 Sensitivity of the predicted Late Cretaceous TPW-induced sea-level re-

m- sponse tochanges iinlithospheric thickness ...... (4 6.3 Sensitivity of the predicted Late Cretaceous TPW-induced sea-level response to changes iii mantle viscosity ...... 79 Chapter 1 Introduction

As usual, and this should be obvious to anyone whose eyes have not been blinded by the false light of Western science, the pie that we see is not the real pole, for the real pole is the one that cannot be seen, ezcept by some adepts, whose lips are sealed.

Urnberto Eco, Foucault's Pendulurn

The planet Earth is a dynamic body which is constantly deformed by processes that act on a wide range of timescales. Tidal potentials deforrn not only the oceans, but also the solid Earth, wit h periodicities ranging from hours to years. Continental drift. driven by convection in the mantle, has led to the formation and break-up of supercontinents over timescales measured in the hundreds of millions of years. These are two examples of processes that not only deform the solid Earth but also cause variations in the planetary rotation vector. Because t here is no direct measure of t he location of the paleorotat ion pole, its position must be inferred from the position of the paleomagnetic pole. The difficulty in accurately determining changes in pateorotation has caused the importance of these changes to be somewhat overlooked. The extent to which the paleorotation pole has wied though time can be used to constrain models of long-term geodynamic processes and t hus improve our understanding of the Earth's evolution. Additionally. changes in planetary rotation have implications for bot h sea-level and climate change. It is well established that present day short-term rotational variations drive global-scale sea-level fluctuations. yet relatively little work has been done on the influence of long-term changes in the Earth's rotation on sea level. This thesis will be prirnarily concerned wit h establishing the importance of rotat ionally-induced changes in long-term sea-level trends. Furt herrnore. because t here is no direct measure of the past location of the Earth's rotation pole inferences of its mot ion from paleomagnet ic st udies are often content ious, records of sea-level change may 1.1 Sea-Level Change

Like changes in rotation. changes in sea level occur on a wide variety of timescales and have a number of different causes. Tides are an obvious example of sea-level fluctuations which occur with relatively short periods, from a geologic viewpoint. However. tides have been recorded in the geologic record as variations in the physical properties of laminae in sedimentary deposits such as tidal rhythmites and banded iron formations (eg. Williams, 1997) as well as in the growth patterns of stromatolites (eg. Pannella, 1976) and rnaiine invertebrates (eg. Rosenberg and Runcorn, 1975). These records have enabled the determination of length-of-day values as far back as .- 2.5 Ga (billion years ago) and thus are of value to st udies of paleorotation (Williams, 1997). In general t hough. stratigraphic sequences record variations in sea level that occur on timescales which are much longer t han tidal periods. Stratigraphic analysis of sedimentary sequences bas revealed a great deal about the spatial and temporal extent of sea-level variations. Depending on the mechanism, sea- level change can occur ei t her regionally or globally, wi t h periodicit ies ranging lrom tens of thousands to hundreds of millions of years (eg. see Miall, 1990). Cycles of sea-level rise and fall, as recorded in stratigraphic cycles, are classified according to t heir duration (see Table 1.1). Since t his thesis is mainly concerned with changes in rotation t hat occur over a few million to order one hundred million years, 1 will locus on second and third order cycles of sea-level change. Before continuing it is important to clarify certain terms used in the discussion of the stratigraphic record. Stratigraphic cycles, and the sea-level variations inferred from them, should not be interpreted as purely periodic changes. Cycle, in this sense, refers only to a depart ure from and return to a starting condition, generally a given sea-level position. Such changes, and the underlying mechanism, may be periodic but not necessarily so. Transgression refers to landward motion of the shoreline, and regression to a seaward motion; motions which are generally, although not exclusively. due to the rise and fa11 of sea level respectively. The term 'eustatic' was introduced as an adjective referring to sea-level changes which occur with equd magnitude and at the same time throughout the globe (Suess. 1906). In an attempt to find a fked point of reference eustasy is often measured wit h respect to the centre of the Earth. CVhat is recorded at a given site however is not eustasy per se but the relative motion between the ocean surface and the planet ary surface ( which is i tself deformable), a quanti ty referred to as relative sea-level Chapter 1: fntroduction

I Type Duration Probable Cause (My4

First order Major eustat ic cycles caused by format ion and breakup of supercont inents Second order Eustat ic cycles induced by volume changes in global mid-ocean spreading ridge system Third order Possibly produced by ridge changes and continental ice growth and decay Fourth order Milankovi t ch glacioeust atic cycles, ast ronomical forcing Fifth order Milankovitch glacioeustatic cycles. ast ronomical forcing

Table 1.1: St ratigraphic cycles and t heir causes. (Adapted from Miall (1990))

change. T hus strat igraphic sequences record some combinat ion of eustatic (ie. global ) and regional signals. Clnha l strat igra phir cycles OF difkrent order- have different causal rncchanisms (surnmarized in Table 1.1); higher order cycles are generally attributed to ice mass fluctuations and ast ronomical forcing, while lower order cycles are li kely ult irnately linked to mant le convection (eg. Niall. 1990). Change in mid-ocean ridge volume is widely accepted as the mechanism responsible for second order global sea-level cycles, and is also believed to be responsible for some third order cycles (Hallam, 1963). Oceanic lithosphere is cre- ated at spreading ridges; as the lit hosphere moves away from the ridge it cools. becoming thicker and more dense, and isostatic subsidence leads to the well known square mot age dependency for the depth of oceanic sea Boor (eg. Turcotte and Schubert. 1982). The faster the spreading rate. the further from the ridge lithosphere of a given age will be and the greater the ridge volume (see Figure 1.1). Pitrnan (1978) showed that rneasured changes in spreading rates would produce ridge volume changes large enough to account for observed second order global sea-level trends. This explanation of long-term sea-level trends has been called into question however as it focuses on the creation of plates at rnid- ocean ridges while ignoring the consumpt ion of plates at subduction zones ( Hager, 1980). Consumpt ion rates are correlated to creat ion rates and numerical models of subduction have found t hat altering the rate of convergence can produce topographie changes t hat counteract the effect of variations in spreading ridge volume (eg. Gurnis. 1990: Mitrovica Distance from ridge axis (km)

Figure 1.1: Plot of ridge depth versus distance from the ridge axis. Dotted line and open symbols for a spreading rate of 1 cm/yr, solid line and filled symboIs for a spreading rate of 3 cm/yr. SymboIs indicate the position of lithospher~ t.hat. is 0 (circles), 5 (triangles), 10 (inverted triangles), 25 (diamonds), 50 (squares), 75 (stars) or 100 (hexagons) million pars old.

Eustasy. whatever its origin, is but one component of the total sea-Ievel signal and regional sea-level change can great ly influence, or even dominate, the st rat igraphic record at a given location (eg. Mitrovica et al., 1989; Gurnis, 1990. 1993a,b; Mitrovica et al.. 1996). T hus, in order to construct a global sea-levd curve it is necessary to find either sites free of regional sea-level signals or a method of removing regional influences from the data set. àlany mechanisms of long-tem regional sea-level change have been proposed, most of which are related to tectonism and are t herefore an expression of mantle convection. In order to avoid most regional effects, estimates of long-term sea-level trends are generally constructed using sites in tectonically stable locations (eg. Vail et al.. 1977; Hailam, 1978: Haq et al.. 1987). Rotationally-induced sea-level signais are not absent at tectouically stable sites however. As will be discussed more thoroughiy below, sea-level changes arising from perturbations in Earth rotation have a distinctive pattern which is not uniform but is of sufficiently long wavelength to be easily mistaken for a globally uniforrn signal. This is one reason why ses-levef changes itssotiated witlr Wmaf variations are of import once. In this section I have discussed some general issues related to iong-term sea-level change; in subsequent chapters specific sea-level records will be discussed in more detail as needed.

1.2 True Polar Wander

Redistribution of mass either within the Earth or on its surface alters the planetary inertia tensor and t hus, t hrough the conservation of angular momentum, the planetary rotation vector. Changes in the magnitude of the rotation vector, that is the rate of rotation, are generally described as changes in the 'length-of-day'. Changes in the orientation of the rotation vector with respect to the solid Earth are referred to as true polar wander (TPW). Since there are no external forces being considered, the orientation of the rotation vector does not change wit h respect to an inert id frame of reference and t herefore i t may be more accurate to describe TP W as motion of the solid Earth wi t h respect to the planetary rotation axis. The two descriptions are equivalent and rnerely depend upon one's frame of reference. Generally, studies of TPW treat the E~rthas fixed and therefore speak of motion of the pole. This section will surnmarize previous theoretical and observat ional st udies of TP W which occurs over geologically long timescales. TPW has been a subject of speculation wit hin geophysics for over one hundred years, being one of the rnechanisms considered in order to explain observations of large climatic changes in the geologicai past. Darwin (1577)argued that appreciable amounts of TPW are possible if the Eart h is 'plastic' but not if it is a rigid body, with the latter possibility being favoured. Gold ( 1955) argued that the stability of the rotation axis depends upon the ellipticity of the Earth and its response to the changes in centrifugal. or 'rotational', potential associated wit h TPW. For a perfectly spherical planet "the smallest beetle ivatking over it would be able to change the axis of rotation relative to the rnarkings on the sphere by an arbitrarily large anglen (Gold, 1955). If the equatorial bulge is free to move in response to a small change in the rotational potential then large amounts of TPW would still be possible since a small change in the orientation of the rotation vector would deform the Earth leading to a greater change in the position of rotation followed y continued deformation of the Earth and so on, &as with the ass and the carrot hanging from a stick held by the rider" (Gold. 1955). üItimatelyt Gold (19.55) argued t hat the rate and magnitude of polar motion would depend upon the rheology of the Earth and t hat improvements in paleomagnetic data would be required to determine the relative contributions of both TPW and continental drift to chaages in regional paleolatitudes. The mainly qualitative arguments of Gold (1955) were confirmed by more analytical Chapter I : Introduction sbudies of the Ea& h's delornation luici rotath appearing at approximatekythe same time (eg. Burgers, 1955; Inglis, 1957) and summarized by Munk and MacDonald ( 1960) who concluded that while TPW was possible for realistic Earth models it was not necessary based upon the available observationa1 evidence. The possibility that TPCV could be an important long-term geodynamic process gained new support when Goldreich and Toomre (1969) considered the change in orientation of the rotation axis of a secalled 'quasi-rigid body' using an updated version of the 'beetle on a sphere' model of the Eart h. In this model the Euth is considered to be a rigid sphere upon the surface of which a nurnber of beetles, representing mass anomalies, travel along random paths. Given the distribution of beetles at each tirne step the changes in the moments of inertia, and hence rotation. of the system were t hen calculated. For this admittedly simple rnodel it was found that the rotation axis could undergo large and rapid (relative to the motion of the beetles) periods of TPW, and that the rate of TPW was negatively correlated with the difference between the maximum and intermediate moments of inertia (Goldreich and Toomre, 1969). The results of this simple model also showed that changes in the rate of TPW do not necessarily require an associated change in the rate of rnass redistribution, which was statistically uniform throughout the evolution of the model (Goldreich and Toomre, 1969). The work of Goldreich and Toomre (1969) was extended by Fisher (1974) who considered the TPW of a viscoelastic Earth model. This study confirmed the results of the earlier work and found that the style of TPW observed for the 'quasi-rigid' Earth also occurred in the viscoelast ic case (Fisher, 1974). In addition, Fisher ( 1974) detailed a new style of TPW. called inertial interchange true polar wander (IITPW), which is caused not by the reorientation of the moments of inertia but instead results from the crossing of the magnitudes of the maximum and intermediate moments of inertia. Inertial interchange produces a different s tyie of TPW because the magnitudes of the moments of inert ia can change wit hout changing t heir orientation. Thus, the rotation axis could remain relatively still until the interchange between the magnitudes of the maximum and intermediate me ments at which point the rotation pole moves -- 90° at rates that are approximately ten t imes faster t han t hose reached during 'traditional' TP W (Fisher. 1974). On million year timescales and longer it is density anomalies in the Euth's interior. for example subducted slabs. rather than surface mass anomalies, such as wandering beetles. which drive TPW. Spada et al. (1992) and Ricard et al. (1993) have shown that subducting slabs in a radially stratified, viscoelastic Earth model could produce episodes of TPW at rates similar to observed long-term trends, the rate of TPW was found to depend on both the location of the slab and the viscosity contrast between the upper and lower mantle. Furthermore, ebanging pst terns of sahdion mre fonnd to tead to produce varying rates and directions of TPW, with rapid IITPW style events occuring only for models wit h an isoviscous mantle. These t heoretical st udies dealt only wit h the general question of whet her or not TPW was possible and did not make any attempt to predict the history of the changes in rotation of the Earth. Such a prediction has recently been made in a study by Steinberger and O'Connel1 (1997). Present day mantle density heterogeneities, determined from global seismic tomographie st udies, were advected backwards in time allowing the computation of changes in the planetary inert ia tensor over the pst 65 Myr (million years). Steinberger and 07Coniie11(1997) Found t hat the amount of TP W was a strong function of the assurned mantle viscosity structure and TPW in excess of 10' over 65 Myr was easily obtained in their simulations. In addition, the predicted TPW path, for certain combinations of mantle density heterogeneity and viscosity structure, was found to be in good general agreement with observational constraints on TPW over the last 65 Wyr (Steinberger and O'Connell, 1997). Richards et al. (1997) performed a similar analysis considering only the density anomalies associated wit h subducted slabs. The pattern of mantle density heterogeneity attributable to subduction. derived from plate reconstructionst was found to be relatively stable over the past 100 Myr resulting in TPW rates of < 1°/Myr over that time (Richards et al., 1997). A more recent study has shown that made models which inciude an endotherrnic phase change between the upper and lower mantles are capable of producing both extended periods of relatively slow (- O..jO/Myr) TPW as well as rapid (up to - 5'/Myr) IITP W style events associated with secalled mantle avalanches (Richards et al., 1999). The observat ional const raints on long-term TPW are derived from paleomagnet ic studies. Due to the importance of the Coriolis effect in the geodynamo process of field generat ion, the di po te axis of the Eart h's t ime averaged magnet ic field wil1 be aligned wit h the planetary rotation axis over the relatively long timescales of interest in this t hesis (eg. MerriIl and McElhinny, 1983). Therefore, changes in the position of the rotation pole (ie. TPW) will be accompanied by a change in the position of the magnetic pole. provided that the TPW occurs over sufficiently long tirnescales. Apparent polar wander paths ( AP WP's) show the relative motion between a given region and the magnetic pole and can be used, at least in principle. to determine the history of TPW. However, since .4PWPTs record the motion of the magnetic pole with respect to tectonic plates, which are not fixed with respect to the solid Earth. it is necessary to account for the effects of continental drift in order to isolate the TPW signai in paleornagnetic reconstructions. Through the use of kinematic plate reconstructions and dated hotspot tracks it is possible to remove the influence of plate motiiona Fmm APWP's end th&mct a history of TPW (eg. Harrison and Lindh, 1982; Livermore et al., 1984; Andrews, 1989). More recent studies have refined this technique and found 10-20' of TPW over the past 100-200 Myr; either through relatively continuous motion of the pole (Besse and Courtillot, 1991) or via a series of rapid events separated by periods of standstill (Sager and Koppers, 2000a; Prévot et al.. 2000). However. due to the difficulty in determining TPW from paleomagnetic data there is a great deal of disagreement on the magnitude of TPW in the geologic past (eg. Tarduno and Gee, 1995; Tarduno and Cottrell, 1997; Tarduno and Sniirnov. Y001 ). For more ancient times it is still possible, in some cases, to place observational constraints on TPW despite the lack of reliable plate reconstructions and dated hotspot tracks, provided that other coastraints on plate motion can be found. If sufficiently large motion of the rotation pole occurs over a period that is too short to allow a comparable amount of continental drift then the effect of TPW will dorninate the APWP's. Such a signal has been inferred from APWP's from the Early Cambrian (544-518 Ma)

and interpreted as an IITPW event in which the rotation axis shifted 5 90' in as little as 10 Myr (Kinchvink et al., 1997). However, the study of Meert (1999) suggests that the paleomagnetic evidence is more consistent with rapid rates of continental drift along with an increased TPW rate that, while large, is not as dramatic as IITPW. Geologic constraints on relative plate motions and APWP's from the PermeTriassic have been analysed by Marcano et al. (1999) who concluded that TPW occurred at an average rate of .- OAO/EuIyr for the interval of 295-205 Ma. Thus, bot h t heory and observation indicate t hat the Eart h experiences TP W and t hat the rate at which the pole moves can vary greatly with time. Large changes in the Earth's rotation could have an important influence on climate and sea-level trends. alt hough the difficulty in determining the history of polar motion has made this link difficult to estabtish. [ndeed, since large TPW events wouid induce significant sea-IeveI fluctuations. sea-level records may be used as an independent test of TPW events proposed on the basis of paleomagnetic evidence. Because it is often difficult to conclusively establish the existence of TP W events using paleomagnetic st udies alone. the associated sea-level signal t hus provides a potent ially important observational constraint on the process.

1.3 Rotat ional Variations and Sea-Level Change

Changes in the Eart h's rotation pert urb the associated centrifugd (dso called rotational) potential leading to variations in the radial position of both the geoid and solid Eart h. The geoid is generally defined as the equipotentiai surface which corresponds to the Chapter 1: htroduction 9 rimeaveraged ocean surface (or ma& develsdacek the position of the geoid is oken described by the geoid height, which is the distance between the geoid and the reference ellipsoid (eg. Fowler, 1990). However the shape of the reference ellipsoid is dependent on the rotation of the Earth and my thesis is concerned wi th changes in Earth rotation. It is t herefore more useful in t his context to descri be changes in the geoid relative to its initial, unperturbed, position rather than relative to the reference ellipsoid which is no longer a fixed quant ity. Changes in rotation have a direct effect on the geoid through changes in the centrifugal potential, and an indirect effect associated with the redistribution of mass as the solid Earth and oceans deform to a new equilibrium shape. The theory used to calculate the total change in the geoid, and solid surface? associated with a given change in the rotation vector ivill be detailed in Chapter 2, however a general sense of the geometry of these changes can be obtained by examining Figure 1.2. The centrifugal potential associated with the Earth's rotation (dso called the rotational potential) acts as a load which deforms both the geoid and the solid Earth. The amplitude of the rotational potential at the Earth's surface varies with latitude in the same manner as does the radius of an oblate spheroid. The shape of the solid Earth and geoid are thus perturbed from perfect spherici ty towards the geometry of the rotational forcing. For t his reason the equatorial radius of the Earth, a, is slightly larger than the polar radius, c. The degree of oblateness of the Eart h's shape. as described by the geoid, is measured by the flattening, c = a - c/a, which has a present day value of - 1/%98 (Turcotte and Schubert, 1982). Changes in the planetary rotation will change the geometry of the associated centrifuga1 potential and thus deforrn the Earth's solid surface and geoid. For example, a decrease in the rate of rotation produces a potential perturbation which is negative at low latitudes and positive near the poles (see Figure 1.2A). In contrast. changing the orientation of the rotation axis acts to produce s potential perturbation that is positive in tmo hemispheric quadrants and negative in the remaining two (se Figure l.2B). As this thesis is mainly concerned wit h the sea-level signal t hat arises from TP W, t hat is changes in the orientation of the rotation vector, let us consider this case somewhat more fully. The equation of an ellipse in polar ceordinates (r.B) that has a semi-major axis of length a oriented paralle1 to the x-axis is r = a(1 - ~sin~(0))~where as above c = a - c/a is the flattening. If the ellipse is rotated about the origin by a small angle C, then the governing equation becomes r' = a(1 - csin2(0 - 5)). The difference between these two ellipses is given by r' - r = acCsin(2B) to fist order accuracy in E. Expanding to t hree dimensions. the potential perturbation associated with a shift in the orientation of the rotation axis has the form of the surface spherical harmonic of degree two and Chapter 1: Introduction

Figure 1.2: Schernatic illustrations of the geometry of the potential perturbations associated with changes in rotation. (A) A decrease in the rate of rotation, from time t (dashed ellipse) to t+ (solid ellipse), leads to a decrease in the Rattening. (B) A change in the orientation of the rotation axis, from time t (dashed ellipse) to t+ (solid ellipse), results in a 'quadrential' pattern of offset.

order one, Y2,,O( sin(20) exp(i4). An integral of this expression over the sphere vanishes. and thus it contributes no mean perturbation when averaged over the eotire Earth. As illustrated in Figure 1.2B. when the local rotation pole (eg. the nort h rotation pole in the northern hemisphere) is moving towards a site the perturbation to the rotatiooal potential is negative; conversely the perturbation is positive if the local rotation pole moves away from the site. The maxima of the potential perturbation are located on the meridian of polar motion at an angular distance of 45' from the pole position. The potential perturbation is zero on two great circles? one located 90" away frorn the position of the pole (the 'rotational equator') , and the ot her oriented ort hogonally to bot h the direct ion of TPW and the rotational equator. On a spherically symmetric planet wit h a globe-encircling ocean the geometry of the TPW-induced sea-level change would be the sarne as the geometry of the applied potential load (see Figure 1.2). The TPW-induced sea-level signal will not be eustatic since the magnitude, and even sign, of the forcing varies. However, the TPW-induced relative sea- level signal would be of a sufficient ly long wavelength that it codd be mistaken as globally Chapter 1: Introduction 11

unihm if one was considering a pdydistributcd set of observations. The indnced relative sea-lrvel signal depends on the response of botb the geoid and the solid surface to the applied potential load. If the Eart h behaved as a perfectly elastic body, t hen the geoid and solid surface would bot h deform instantaneously in response to TPW: however, the magnitude of their deformations would be different resulting in a relatively large sea-level signal. If the Earth behaved as a completely inviscid Buid, an approximation which may not seem unreasonable in view of the long timescales considered in this thesis, the geoid and solid surface would again deforrn instantaneously in response to the applied load. but in this case the magnitude of deformation would be the same for bot h surfaces and no relative sea-level change would result. The response of the real Earth is neither perfectly elastic nor inviscid. The Earth's mantle behaves as a high viscosity Auid, resulting in a lag between the timing of an applied load and the deformational response of the geoid and the solid surface. Thus, the presence of slow modes of viscous relaxation in the mantle and of a thin elastic lithosphere allow for a non-zero sea-level signal, even for loads applied over geologically long periods. One example of the influence of short timescale rotational variations on sea level is the so-called pole t ide which is associated wit h the Chandler wobble. The Chandler wobble is a continuously excited free oscillation of the Earth's rotation axis which has an amplitude of order Ot!l and a period of - 434 days ( Lambeck, 1980). This oscillation of the rotation axis induces a corresponding sea-level Buct uation, wit h an amplitude of up to a few centimetres, which has been observed in tide records (eg. Haubrich and Munk? 1959: Miller and Wunsch, 1973). Although small, the pole tide is geophysically significant as it provides a mechanism of dissipation for the Chandler wobble and because it increases the period of the wobble by about 27.5 days with respect to an ocean free Earth (Lambeck. 1980). Turning to longer tirnescales. tidd decereration of the Earth's rotation has been proposed as a mechanism for the production of regionally varying sea-level trends since the beginning of the Late Cretaceous (5 100 Ma). Tidal dissipation has acted to transfer angular rnomentum from the Eart h to the lunar orbit causing a gradua1 decrease in the rate of rotation of the Earth and an increase in the separation between the Earth and the Moon. Eardley ( 1964) calculated t hat the deceleration of the Earth's rotation could produce a pole-teequator differential sea-level change of - 400 rn over the past 100 Myr, and that this influence codd account for an apparent latitudinal dependence in sea-level trends over that tirne. The estimated magnitude of the maximum sea-level effect which could be induced by this mechanism was reduced to -- 100 m by Flatté (196.5). Even the earliest t heoretical studies of TF W recognized that it would lead to changes in sea level and dimabe, however the magnitude of thchanges was not mderstood (Darwin, 1877; Gold, 1955). M6rner (1981) argued quditatively for 'rotational tilt' as a possible mechanism for producing the latitudinal dependence seen in a set of Cretaceous (142-65 Ma) sea-level observations dthough the influence of this tilt on sea level is merely described as 'large.' Sabadini et al. (1990) were the first to calculate the sea-level effect of relatively long-term TP W for a realistic Earth model. They considered a general TPW event, chosen to be consistent with estimates of historical TPW rates, with a constant rate of polar motion amounting to a one degree displacement of the pole over a period

of one million years and found a peak sea-level signal of 5 20-50 m depending upon the Earth model used. Sabadini et al. (1990) recognized that TPW could play an important role in the generation of third order sea-level cycles and that since TPW produced coeval rises and falls in sea level this effect had important implications for the study of eustatic sea-level trends. The work presented in this thesis will extend the analysis of Sabadini et al. (1990) both by considering TPW that occurs over a far broader range of time periods and amplitudes and by analyzing the sea-level signal associated with specific episodes of TP W recent ly inferred from paleomagnet ic st udies.

1.4 Contributions and Outline of Thesis

The importance of long-term changes in the Eart h's rotation vector to sea-level variations is the primary focus of this thesis. Chapter 2 outlines the t heory 1 used to calculate the changes in the shape of the Earth, and hence sea level, that result from variations in the planetary rotation vector. The t heory t reats the Eart h as a spherically symmetric Maxwell viscoelastic body whose elastic and density structure are taken from the preliminary reference Earth model (PREM; Dziewonski and Anderson, 1981). I win derive an anaiytic solution to the scxalled sea- level equation for the special case of constant TPW and will investigate the sensitivities of the calculated sea-ievel response to parameters related t O site location, Eart h structure and the magnitude and duration of TPW. The sea-level signal induced by TPW arises due to the perturbation of the geoid and solid surface from their equilibrium shapes, however those shapes depend on the viscoelastic structure and rotational state of the planet. In previous predictions of the present day equilibrium form of the geoid the Eart h is generally treated as a fluid body in hydrostatic equilibrium with the present day rotationd potential. In Chapter 3. I will investigate the influence t hat the inclusion of a non-zero effective elastic lit hospheric t hickness has on this hydrostatic form. I wil1 also consider the excess flattening of the Chapter 1: Introduction f 3

geoid due ko ê hg bekweea the rnponse of the vismus Eartk and the changing potentiat associated with the gradua1 slowing of the planetary rotation. Once this framework has been established I will demonstrate the importance of TPW to second order sea-level trends. Paleomagnetic studies have inferred - 20' of net polar motion over the last 130 Myr and in Chapter 4 1 model the sea-level variations that would be associated wit h t his amount of TPW. Both the magnitude and geographic variation of the TPW-induced sea-level fluctuations are studied and cornpared with published records of sea-level change since 130 Ma. No type of polar motion is more dramatic than IITPW. During an IITPW event the rotation avis moves at high rates over a large distance. narnely 90' in as little as 10 My. An IITPCV event has been proposed for the Early Cambrian (544-518 Ma) and in Chapter 5 I investigate the sea-level signal that would be associated with that event and compare it with available records of Early Cambrian sea-level change. Rapid polar motion is possible over shorter durations and distances than those which characterize IITPW events. Recent studies of Late Cretaceous (99-65 Ma) TPW have proposed a period of rapid polar motion during this time. The proposed duration and rate of the event varies depending upon the data set considered, and some studies appear to indicate that there was virtuallv no TPW during the tirne in question. In Chapter 6 I study the sea-level signal that would be associated with the proposed Late Cretaceous event in an attempt to place constraints upon the magnitude of TPW during this time. The main results and implications of the thesis are summarized in the final chapter which also includes an outline of some outstanding issues and possible directions for future researc h. Chapter 2 Theoretical Formulation of Rotation Induced Sea-Level Variations

2.1 Introduction

The earth, the ve y emblem of solidity, has moved beneath our feet Iike a thin crust over a puid.

Charles Darwin. The Voyage of the Beagle

The above quote is a description of ground motion during the great Concepcion earthquake on the twentiet h of February, 1835; it could however apply equally well to the long-term behaviour of the Earth. When considered over millions of years the made can be accurately rnodelled as a viscous Buid overlain by a comparatively thin elastic lithosphere. In this chapter 1 will describe the so-called sea-level equation and how it is used to determine the perturbation to the geoid and solid surface, and hence sea Ievel, that is induced by a given load (eg. Mitrovica and Peltier. 1991; Milne and àlitrovica, 1996, 1998). The load may arise from a redistribution of surface mas anomalies. a change in potential. or a combination of mass and potential cornponents; the primary load of interest wit hin this thesis is the potential perturbation associated with a change in rotation. although surface rnass loads also arise from the redistribution of ocean mass associated with rotation-induced effects. Following an explanation of simplifications to the sea-level equat ion adopted when dealing wit h the sea-level signal arising from long timescale TPW, I will derive the solution to the sea-level equation for the special case of constant polar motion. This constant TPW solution will subsequently be used to explore the general form of the induced sea-level signal and its sensitivity to Earth structure and the magnitude and duration of TPW. Cbapt er 2: Sea- Level Theory 15

Materid in this chepter is reprint& with penmssim fmm Witrovica et ai. f-000), copyright Edi trice Compositori.

2.2 The Sea-Level Equation

The sea-level change associated with variations in Earth rotation is computed using the secalled sea-level equation. The sea-level equation was initially developed from theory governi ng the load-induced deformat ion of a spherically symmetric, self-gravi t at ing. Maxwell viscoelastic Earth mode1 and has been used to compute the sea-level signal wsociated with the melting of continental ice sheets and the resulting glacial isostatic adjustment (GIA) (Farrell and Clark, 1976: Peltier and Andrews, 1976; Mitrovica and Peltier, 1991). Subsequent studies have expanded the theory to include the effect of GIA- induced rotation variations on sea level (Han and Wahr, 1989: Bills and James. 1996: Milne and Mitrovica, 1996, 1998; Peltier, 1998). 1 begin wit h

where S, G and R are the perturbations to the relative sea level, geoid and solid surface, respectively, evaluated at a given CO-latitude, 8, east-longitude. b. and time. t. Relative sea-level perturbations are computed over the entire globe, not just over the oceans. It may seem paradoxical to speak of sea level at sites not located within the oceans. hoivever relative sea-level change is defined on the bais of fields (the geoid and solid surface position) t hat are globally defined. Furt hermore, the amplitude of t hese sea-level variations relative to contemporaneous continental hypsometry governs the extent of offlap and continental flooding (Sloss, 1963). In general, relative sea-level change can arise due to any surface mas redistribution (eg. the growt h and ablation of ice sheets) or potent ial variation (eg. the luni-solar t idal potentials). In this t hesis the loads of interest are the changes in centrifugai potential associated with changes in planetary rotation, and the redistribution of ocean mas. Before considering the specific form OF these loads 1 will develop the sea-level response to general mas and potential loads. G and R are computed by the convolut ion of the Green's function response of the planet wit h the applied mass and potentiai loads. The rnass and potential load responses are determined by separate Green's Functions and 1 thus use

and in eqt~atimf 2. t 1, mhere the snpersnipts t anct T, respectively, indicate t hat mass and potent id loads are being considered.

2.2.1 Surface Mass Loads

The response of the solid surface and geoid to surface mass loads may be found using the s-called hL and kL surface load Love nurnbers which can be written as (eg. Peltier.

and

The first term on the right-hand side of these equations represents the elastic response of the planet, denoted by the superscript E, and the second term the non-elastic response described by a sum of J exponentially decaying modes of relaxation. 6 is the Dirac delta function and e denotes the spherical harmonic degree. The modes are characterized by inverse decay t imes, s; t hese decay times are the same for bot h the hL and kL Love numbers but the aiiipli~udeuf lie associateci modes, r and rf respectiveiy, difier. These modes are linked to discontinuities in the structure of the Earth. On the timescales considered in this thesis the most important are mantle buoyancy modes retated to density discontinuities such as those that occur at the 410 km and 670 km depth phase boundaries. Although the high viscosity of the mantle is partially responsible for the long decay times of these modes, more important is the size of the density discontinuity. Consider trvo materials of density pl and pz separated by a horizontal boundary (see Figure 2.1); deformation of this boundary will produce lateral variations in density resulting in a restoring buoyancy force. Fb. The magnitude of this force, per unit volume, is (eg. Turcotte and Schubert.

Figure 2.1: X schematic illustration in which the line represents the boundary between two materials of densities pl and p?, where pl < pz, in (A) the equilibrium state, and (B) the deformed state with buoyant restoring force Fb. Chapter 2: Sea-Level Theory

Thus. for a small density difference the restoring force is weak and the rate of relaxation slow. The Green's functions describing the response of the solid surface and geopotential to an impulse surface mass load are given by Legendre polynomial expansions (eg. Mitrovica and Peltier, 19911,

and

w here

and

The angular separation between the applied load and the site OF observation is denoted by 7.The constants !CI, and a are the mass and mean radius of the Earth respectively, g is the acceleration due to gravity at the Earth's surface which is assumed to be constant. and Pt are the Legendre polynomials. The first delta function on the right-hand side of equation (3.10) accounts for the direct perturbation to the geopotential t hat arises due to the presence of the mass load. The perturbations to the solid surface and geoid are found by convolving in bot h space and time the applied mass load. L. with the Green's function responses. This exercise yields

and

where the spherical harmonic coefficients R:, and Gi, are Chapter 2: Sea- Level Theory and

The Lrm are the spherical harrnonic coefficients in the expansion for the surface mass load. The final terrn in equation (2.14) is a uniforrn shift in the position of the geoid which arises in order to conserve mass when there are changes in the volume of the ocean basins or in the amount of water within the oceans. The spatial convoiut ions have been performed analytically making use of (eg. Mi trw vica and Peltier, 1991)

where denotes the complete solid angle. y is any field defined everywhere over the surface of the Earth (for exarnple R and G in equations (9.11) and (2.12)). which can be represented by a spherical harmonic decomposition

w here

The surface spherical harrnonics basis functions, Y' are given by

i represents the imaginary number fi as required, the imaginary components of the fields considered cancel w hen summed over al1 degrees and orders as descri bed in equat ion (2.16). The spherical harrnonics are normalized so t hat

i;!,,.fi,, sin BdOdw = ls6ct,t6,t., , where denotes the complex conjugate.

2.2.2 Potential Loads

The theory describing the response of the solid surface and geoid to potentiai loads is sirnilar to the formulation used for mass loads. The so-called hT and kT tidal (or tidal-effective) Love numbers can be written (eg. Milne and Mitrovica, 1998) Chapter 2: Sea-Level Theory

and

The inverse decay times for a given mode are the same for the tidal and mass load cases; however, the amplitude of the modes differ. From the tidal Love numbers the Green's function response. at spherical harmonic degree C. of the solid surface and geoid to an applied potent ial load can be const ructed as (eg. Milne and Mitrovica. 1998)

and

In analogy with equations (3.11) and (2.12) I write

and

w here

and A is the applied potential load. Note that, in contrast to the surface mass loading case, a spatial convolution is not required in the derivation of equations (2.26) and (2.27). Equations (2.11). (2.12), (2.24) and (2.25) are, via equations (2.2) and (XI),used in equation (2.1) to compute sea-level perturbations.

2.2.3 Solving the Sea-Level Equation

In order to evaluate the sea-level equation it is necessary to prescri be the mas and potential loads and t hen to evaluate the temporal convolutions in equations (2.13). (2.14). (2.26) and (2.27). In general the appiied Ioads can be any surface mass distribution or potential variation, however within this thesis the ody loads to be considered are the potential load due Chapter 2: Sea-Levei Theory 20 to changes in rokakion and the mws la& th& mises due to redistribation of the means. Changes in sea level represent a change in the distribution of mass on the surface of the Earth and therefore load the planet. This loading acts to amplify an induced sea-level signal. The mass load which causes this amplification can be written

where p, is the density of water. The term SL represents the total change in ocean height and it may be computed from the global sea-level variation. S of equation (2.1). by suitable projection. The sea-level perturbation, S, t hus enters into bot h sides of equation (2.1) and t his equat ion is generally solved using iterative met hods (where a first guess to S is iteratively refined unt il convergence). In order to describe the potential load associated with rotational variations it is necessary to first define the frarne of reference. I will use a right-handed Cartesian coordinate system wit h its origin at the centre of the Eart h and axes denoted by xi. XI will be aligned so as to pass through the equator at the 0" longitude; sa will pass through the equator

90" to the east of 11; 13 thus points towards the north pole. In equilibrium the planetary rotation vector, W. can therefore be written as wo = (0,O, O), where R is the reference rotation rate. Subsequent perturbations to the rotation vector take the form (eg. Munk and MacDonald, 1960) w(t)= wo + Qm(t). (2.29) üsing t his formulation the potential load associated with perturbai ions in the rotation vector are (eg.. Milne and Mitrovica, 1998)

Note t hat the potent ial load induced by variations in rotation is completely descri bed using spherical harmonic degrees O and 2. A key difference between the analyses to be Chapter 2: Sea-Level Theory 21

that in the present case the TPW is the primary rnechanism of sea-level change and it is prescribed a priori. In GIA studies the rotational variations are computed from the applied surface (ice plus sea level) loads. In GIA studies the standard procedure used to evaluate the temporal convoiut ion in the Green's function formulation of the Earth's response is to describe the applied hads at each spherical harmonic degree and order using a series of Heaviside step functions. H(t - to). For a general function, X, this approach takes the form

By applying t his procedure the temporal convolut ion can be performed analyt ically and the sea-level equat ion for the case of TP W-induced sea-level change can be writ ten as

+ Z. [F,~,ssL;,A(~- t,) + -6A;m9 ,#(t - t.) H(t - t,) n= l 1

is a degree dependent factor arising from the spatial convolut ion in the mass load Green's func t ions,

are factors governing the elastic response of the Eart h to the mass and potential loads.

are the non-elastic responses to the mas and potential loads. The SLcm and 6SLt,, represent spherical harmonic coefficients in expansions for the change in ocean height and the time increments of this change. With the potential load prescribed a-priori, the sea-level equation (2.36) yields the resulting sea-level perturbation? S. Chapter 2: Sea-Level Themy

I will now desctibe a common simplification used when solving the sea-level equation. Specifically, I ignore the mass load associated with the redistri but ion of water wi t hin the ocean by renioving the ocean height perturbation, SL,from the right-hand side of equation (2.36). Sabadini et al. (1990) have shown that this assurnption results in a relative sea-level perturbation which is at most 10% less than that which would be obtained if the self-attraction and loading of the redistributed ocean are included. Applying this approximation to equation (2.1) (which includes equations (2.1 1)-(2.14) and (2.24)-(2.27). via equations (2.2) and (2.3)) yields the following integral form of the sea-level equation

in which it is now only necessary to include spherical harmonic degrees zero and two. By making use of a Heaviside formulation, as in the previous section, the temporal convolution can be solved for a general TPW-induced potential load giving

This form of the sea-level equation can be solved much more easily than equation (2.36) since it does not require an iterative approach to solve for S. It will be instructive to consider two limiting cases of the sea-level equation (2.43). The first of these deals wit h an elastic Eart h mode1 and the second wit h the long-t ime limit of the equation. To begin, if 1 neglect non-elastic deformations 1 obtain the simple equation

The sea-level response due to a reorientation of the rotation vector with no change in its magnitude will include degree two terrns only (see equations (2.30)-(%.33)),and as discussed in Chapter 1 it will be dominated by the degree two, order one component. The degree two elastic tidal Love nurnbers, k:*= and hTvE,obtained using the prelirninary reference Eart h mode1 ( PREM; Dziewons ki and Anderson. 198 1), are 0.2993 and 0.6053. respectively. Accordingly. following equation (2.39), Ez = 0.694. Since this number is positive, it indicates that the sea-level perturbation due to TPW will have the same sign as the applied rotational potentid load. That is, sea-level will fdl if the local rotation pole is moving towards the site of observation (eg.? see Figure 1.2). Chapter 2: Sea-Level Theory 23

Next, E take the tongtirrre limit of eqnatiorr (2.43) so that t - t, + 00, for att n. Making use of equation (2.41) 1 obtain

The fluid Love numbers hi and k{ are defined as (eg. Wunk and MacDonald, 1960; Wu and

and

Usirig t hese definitions in equation (2.15) gives

This equation governs the sea-level response an infinite tirne after the final rotational potential perturbation has been applied. The Earth models 1 adopt in this thesis are characterized by an elast ic lit hosphere of some prescri bed t hickness. LT. overlying a viscoelastic mantle (and inviscid core). In this case, equation (2.48) can also be interpreted as the instantaneous sea-level response of a version of the Earth model in which the rnantle is treated as an inviscid fluid (hence the appearance of the fluid Love numbers). I have cornputed the terrn (1+k{ -hi) fm a suite of Earth rnodels characterized by the PREM density structure and an elastic lithospheric thickness varying from O to 120 km. (As 1 discuss in the next section, the calculation of k{ and hi is actually not perforrned by summing modes. as in equations (2.46) and (2.47), but rather is based on a consideration of the so-called inviscid lirnit.) The results, shown in Figure 2.2, raise several points that will prove to be fundamental to this thesis. First, the term 1 + k{ - hi = O in the case where the elastic lithospheric thickness is O km (ie. the model is completely fluid). This demonstrates t hat TPW-induced sea-level changes will be zero on an inviscid planet. The presence of an elastic lit hosphere of non-zero thickness ensures that the planetary response will depart from that of a inviscid model, and as a consequence the sea-level perturbation will also be non-zero. On a purely inviscid pianet, the geoid and solid surface perturbations dlbe equal and the resulting relative sea-level change (according to equation (2.1)) will be zero. Any departures from t his inviscid state, for example associated wit h the presence of a t hin elast ic shell, will lead to unequd geoid and solid surface perturbations. Chapter 2: Sea-Level Theory

Lithospheric Thlckneu, (km)

Figure 2.2: Dependence of the inviscid lirnit Earth response, 1 + k[ - hi, on the effective thickness of the elastic lit hosphere.

Note that for an elastic thickness of 120 km. the largest elastic lithospheric thickness considered in Figure 2.2. the term 1 + k{ - hi z 0.01. As in the case of ET, the positive sign of this term means that the sea-level perturbation will have the same sign as the perturbation to the rotatiooal potential. However, the magnitude of the term is approxirnately 70 times smaller than E:. This indicates t hat viscous compensation reduces the TPW-induced sea-level change considerably relative to t hat which would be obtaitied in the purely elastic case. However, the sea-level change is not zero when the lithosphere has non-zero elastic thickness and. as 1 will show, the amplitude of the sea-level change can be important to the analysis of the geological record. In considering the Buid limit 1 have taken a rather extreme case. In the next section I show that departures of the Earth models fiom a purely inviscid response can also occur when the timescale of the TPW is not long cornpared to the long-decay times t hat characterize certain interna1 modes of deformation. These modes t herefore also contribute to the predicted non-zero TPW-induced sea-level response. Nevertheless. as I will demonstrate, the computed sea-level response on a viscoelastic Earth is significantly smaller than the response which would be deterrnined for an elastic planet. Chapter 2: Sea-Level Theory

1x1 the sea-levei equation the non-elastic response of the Eart h to eit her a mas or potential load is described by a sum of exponentially decaying modes characterized by

their inverse decay time, S. In the Heaviside formulation of the applied load, the non- elastic response is given by

where 1 have ignored the distinction between the mass load and potential load responses. as well as the responses at different harmonic degrees, since the mathematical forrn of the response remains the same. The inverse decay times, as well as the mode strengths. r and r'. are found for a given viscoelastic Earth model using the theory and algorithrns outlined by Peltier (1974) and Wu(1978), a procedure commonly applied in, for example, studies of GIA. At t = t,, the time at which the step load is ~pplied.P = O and t here is no non-elastic response. In contrast, when t » t,, p + x:,, b;foc each mode, the rate of approach J to this final value is governed by sj. In applying the theory outlined by Peltier (1974) and Wu (1978) eoverning the r+ sponse of a Maxwell viscoelastic planet, each numerical boundary in the Earth rnodel generat es modes. Some of t hese numerical boundaries correspond to distinct discontinuities in the Earth (eg. the density discontinuities at the surface, at 670 km depth and at the core-mantle-boundary), while many others arise as a consequence of the discretization of a srnoothly varying density field. Buoyancy modes arising due to major density discontinuities in the Earth have a wide range of decay times and relatively large amplitudes. In contrast, the modes associated wit h the srnoothly varying density field are much weaker and do not contribute significantIy to the planetary response. These weak modes include both very long decay time buoyancy modes and a set of modes with relatively short decay times close to the Maxwell tirne of the individual mode1 layers. It is the strong bouyancy modes associated wit h major physical discontinuities wit hin the Earth that cont rol the response of the mode1 to a given forcing; however, the presence of short decay time weak modes (which nurnber several hundred in a numerical model discretized using 150 radial regions, as is the case for the models 1 have used) makes it numerically challenging to isolate al1 of the important strong buoyancy modes. If a mode is omitted from the sum in equation (2.49) then its contribution to the viscous compensation of the planetary response will not be included. As I discussed in the last section, this viscous compensation is nearly complete in the case of the long tirnescat TPW tht serves as the bsof this thesis. That is, the computed response on a viscoelastic planet wiil be a small fraction of the response that would be cornputed on an elastic planet. Thus, the omission of even a relatively small amplitude 'strong' mode may lead to large errors in the computed TPW-induced sea-level perturbation. To counter t his problem, 1 developed a secalled semi-inviscid approach which assumes that al1 but the long decay t ime 'st rongt modes have completely relaxed. This approach, w hich does not rely ori the isolation of al1 of the short decay time 'strangt modes, is described in detail below. The secalled inviscid limit is the response the Earth would have in the limit of very long timescale forcing; that is, the response under the assumption that the Earth behaved entirely as an inviscid fluid with the exception of the elastic lit hosphere. There are two numericd approaches for computing this limit. One met hod is to assume a priori that the entire planet, with the exception of the lithosphere, is an inviscid Auid. This greatly simplifies the equations governing the response of the Earth and it allows for a highly accurate calculation of the total response that does not involve the calculation of modes. The second method is to use the modes determined frorn the viscoelastic theory and assume that they have reached their inviscid lirnit by setting t = oo in equation (3.19). This is the same approach 1 used in the last section to derive a mode-based expression for the fluid Love oumbers (see equations (2.46) and (3.47)). When calculating sea-level change over millions of years, sj(t -Ln) » 1 for modes of relat ively short t imescale, t hus the inviscid limit will be reached for al1 but the longest decaying modes. Therefore? instead of calculating the response from each and every mode, the semi-inviscid approximation st~ts from an assumption of inviscid deformation (which can be computed accurately), then subtracts out the contribution to that limit which arises from the long timescale modes, leaving the total inviscid response of al1 the short-term modes. This expression is then cornbineci wit h the time-dependent contribution of the long timescale modes to obtain the total non-elastic response function. This method not only alleviates the problem of finding al1 of the short decay time 'strong? modes, it is also faster t han the 'traditional' method of computing the mode spectra since it requires the summation of only a few long-term modes. To illustrate I begin from equation (2.49) and write

where /3, (ie., the LHS of the equation) is the inviscid limit of the Eart h mode1 response. The subscript 'short' indicates a summation including ody normal modes that have a bytime whkh iS shwith respect to the timescde of the forcing and the subscript 'long' indicates a summation over al1 the remaining modes. Now, 1 also rewrite the expression for P(t - t,) (equation (2.49)) using the same 'short' and 'long' decomposition under the assumption that the long decay time modes have not reached their inviscid

rfj- ~(t- tn) = c "j - 'j + C "(1 - exp(-s,(t - tn))I* short long Si Substi t uting for the 'short' mode contribution, which is obtained from equat ion (2.50). and simpli fying, yields

This is the generalized form of t he non-elastic response for the semi-inviscid approximation which is used in place of equations (2.40) and (2.41) in the sea level equation. The term ,û, is computed using a simplified t heory t hat does not require modes, while the second term on the RHS of the equation is accurately determined through the standard mode search algorit hm since t hese modes lie in an unclut tered region of the decay t ime domain. When the semi-inviscid approximation is applied, for example. to the simplified form of the described abow (equation 1.13) the rnponsc bccomcs

r T " 64, r t,j - r& C-C exp(-stVj(t - 1,)) ~(t- t,) n=i g long %J Note the inviscid response Pm, like the elastic response, depends only on the total load at the t ime of observation, whereas the viscous component of the non-elastic response depends upon the entire history of Ioading. In the previous section, 1 dernonstrated t hat TP W-induced relative sea-level changes on a purely inviscid Earth would be zero, and departures from this state would be pr* duced by the presence of an elastic lithosphere. Equation (2.53) demonstrates a second point made in that section; narnely. that contributions to TP W-induced relative sea level can also arise from buoyancy modes with characteristic decay tirnes that are not short relative to the tirnescale of the TP W. If the time scale of the TPW were sufficiently long that there were no such modes. t hen the summation over 'long' in equation (2.53) would go to zero and the expression would converge to equation (2.45) or (2.48). As 1 discuss below. long-term TP W is a continuous process. Thus, both the A and 6A terms in equation (2.43) or (2.53) contribute to the predicted relative sea-level pertur- Chapter 2: Sea-Level Theory 28

bation. Hence, the induced sedevek pert ttrbabim is mksirrrpty a mm of exponentidly decaying signals. Rather, the signal consists of both 'new' variations driven by the ongoing TPW and the decay of previously induced perturbations due to viscous relaxation of the mantle.

2.3 Constant True Polar Wander: A Case Study

In this section 1 will derive the sea-level response to a TPW event in which the pole nioves at a constant rate. In this derivation 1 will make use of the simplifications described in section (2.2.4). Once the sea-level response has been derived. 1 wil1 investigate the nature of the TPW-induced signal and its sensitivity to variations in the prescribed TP W-event and Earth mode1 parameters. The TPW event considered here begins with the rotation pole in its present day equilibriurn position, ie. wo = (0,0,0). TPW wander will begin at time t = O and proceed along the 90' east meridian, the movement thus occurs in the x2-xl plane and is directed towards the xl axis. In this case, the celatitude of t he rotation pole, 0 (rneasured in radians), is given by 6(t) = at, where a is the rate of TPW, and the time-dependent rotation vector becornes, for t 2 O,

By comparing this equation with equation (2.29) it can be seen that for this TPW scenario the perturbation to the rotation vector after the onset of TPW is

Substitution of these mi values into equatiom (2.30)-(2.34) gives the potential toad for t his event, namely iio.o(f) = O, (3.56) In this case: the degree zero component of the potential load vanishes and the global average of the degree two signal is zero. The TPW-induced potential load can not be expressed by a pure degree two, order one geometry since the orientation of the surface spherical harmonic functions are fixed with respect to the ceordinate axes whereas the orientation of the induced potential load is determined by the position of the rotation pole and the direction of TPW. The Heaviside functions indicate t hat t here is no perturbation prior to time t = 0. When this potential load is substituted into equation (2.42) the temporal convolution can be performed analyt ically, wi t hout recourse to the Heaviside function represent at ion introduced in equatioo (2.35), and the induced sea-level perturbation is found to be

where,

At any site the sea-level change induced by TPW is governed by a number of factors. As I discussed with reference to Figure 1.2, the location of the site with respect to the position of the rotation pole and the direct ion of TPW cont rols the sign and rnagni tude of the potentid peft twbaiion, and tmrrr the TPW-inchceci sa-ievet trend. The magnitude of the sea-level variation increases as the response of the Earth departs from that of an inviscid body. This departure depends upon the viscoelastic structure of the Eart h and the magnitude of the sea-level signal grows as eit her the ef€ective elastic t hickness of the lithosphere or the viscosity of the mantle is increased. An increase in the rate of TPW also leads to an increased sea-level response, and in results described below 1 will investigate each of t hese sensit ivit ies. My calculations are performed using spherically symmetric. self-gravitating, Maxwell viscoelastic Earth models characterized by the density and elastic structure of the model PREM (Dziewonski and Anderson, 1981). The Earth model has a fluid core and a radial viscosity profile that is defined by a high viscosity (effectively elastic) Iithospheric lid, with a thickness denoted by LT, and isoviscous upper and lower rnantle regions with viscosity values denoted by vu, and VI,, respectively. I will use an Earth model defined by LT = 100 km, y,, = 102' Pa s and vr, = 3 x 1022 Pa s as my 'reference' model and consider variations from t his mode1 to determine the sensitivity of the sea-level response to the assumed viscoelast ic Eart h structure. Figure 2.3 explores the sensitivity of predicted TPW-induced sea-level change to variations in the rate of polar motion and Earth structure. In al1 cases the sea-level change is predicted after a total of 1" of TPW away from a site initially located 45' from the rotation pole (the site, which is used for al1 predictions in this section, is t hus initially at the location of maximum potential perturbation). Each frame shows the results for TPW rates of 0.2O/Myr. 1°/Myr and JO/Myr, and the different frames represent a suite of calculations in ivhich either the lithospheric thickness. upper mantle viscosity or lower rnantle viscosity is varied from my reference model values. If the Earth rvere purely elastic, then the predicted sea-level change would depend onIy on the total shift of the rotation pole and not the rate (or history) of TPW. 1 have found that for an elastic mode1 the sea-level rise at the chosen site after 1' of TPW is

.Y 133 m (see equation (2.34)). On a rigid Earth this value would be about 50% higher. The right ordinate scale in Figure 2.3 provides the sea-level response as a percentage of the response for an elastic Earth. As discussed in section 2.2.4 the elastic response represents an extreme upper bound on the predicted sea-level variation. As the effective elastic thickness of the lit hosphere or the viscosity of the made is increased. or as more rapid rates of TPW are considered. the response of the Earth model trends toward this elastic response (ie., the magnitude of the predicted sea-level change increases). 1 also showed in section 2.2.4 that the predicted sea-level response obtained for a purely inviscid Earth mode1 is zero. However, this lower bound wodd not be obtained for the models

Chopter 2: Sea-Level Theory 33

considered in Figure 2.3,ewn for very srnit11 &es of madeviscosity, since the ek&c lit hospheric t hickness is dways non-zero. In Figures 2.3B,C the lower limi t , indicated by the horizontal, dashed-dotted line, represents the response of an Earth mode1 with LT = 100 km overlying an inviscid mantle. This response is -. 1.7 rn, or 1.3% of the purely elastic Earth model response. The response for a model with an inviscid mantle. as in the purely elastic case! is independent of the rate (or history) of TPW, as can be seen from an examination of equation (2.48). The TPW rates considered in Figure 2.3 span geophysically important cases. For example, the TPW pat h inferred by Besse and Courtillot (1991) suggests mean TP W rates of 0.2-0.3'/Myr over the last - 130 Myr; 1 will examine in greater detail the relative sea-level signal associated wit h this TPW pat h in Chapter 1. This range includes the case modelled by the dotted line on the figure. The predicted response for the case of the reference Earth model is roughly 5 m after l0 of TPW. or - 1% of the analogous response for a purely elast ic Eart h model. Predictions for ot her Earth models differ by only a few metres from this value. Accordingly, I conclude that viscous compensation acts to reduce the sea-level response predicted for such slow rates of TP W by about 95% from the case on an elastic Earth. Next, consider an IITPW scenario where TPW rates on the order of 5-10°/Myr may be obtained (eg. Kirschvink et al., 1997). In this case of relatively rapid TPW (see dashed lines, Figure S.3), the level of viscous compensation is between - 60 and 90%. with most of the variation due to a sensitivity to the choice of lower mantle viscosity. This means that modest (but rapid) shifts in the orientation of the rotation pole can drive sea-level changes in excess of 50 rn in less than 1 Myr. The sea-level perturbations of rapid TPW events are the focus of C hapters 5 and 6. Figure 3.4 explores the sensitivity of the predictions to variations in the duration of TPW for a fixed rate of polar mot ion (namely I0/Myr). The solid lines show the prediction calculated 1 Myr after the onset of the 1°/Myr TPW (these lines, which ail correspond to a total shift of 1°, are identical to the solid lines in Figure 2.3). In contrast. the dashed-dotted lines represent the predicted sea-level change 3 Myr after the onset of TPW. Changes in the duration of the TPW alter the sensitivity of the predicted sea-level perturbation to variations from the reference Eart h model, alt hough some caution must be applied to t his observation. For example, increasing the lit hospheric t hickness frorn 25 km to 200 km increases the sea-level prediction for the 5 Myr (and hence 5" polar offset) case by a factor of -. 2.7 and the t Myr (and l0 offset) case by a factor of - 2. Thus, the difference in dope between the solid and dashed-dotted lines in Figure 2.U is due both to a real change in Earth model sensitivity and to a difference in the magnitude Lithospheric Thickness (km)

58+20 1-21 2e+21 Upper Mantle Viscosity (Pa s)

1-21 1-22 t e+23 Lower Mantb Viscosity (Pa s)

Figure 2.4: Sensitivity of the predicted relative sea-level variations to changes in (A) lithospheric thicknes, (B) upper mantle viscosity, and (C) Iower mantte viscosity as indicated on the x-axis. Each frame shows the resufts for a constant TPW rate of 1°/hilyr after eit her 1 Myr or 5 Myr as labetled. The y-& gives the net sea-level perturbation in metres. of the pmtieted sea-iwel prtmbation for the hm TPW htions. The dependence of the sea-level predictions on changes in TPW duration is most sensitive to variations in lower mantle viscosity. with the relative sensitivity to vi, decreasing as the TPW event progresses. In al1 cases the magnitude of the induced sea-level signal grows wi t h increasing depart ure from an inviscid Eart h model. Furt hermore, the magnitude of the sea-level prediction increases rnonotonically as the TPW progresses, at least over the rangc of time shown. This observation is considered further in Figure 2.5A. In Figure 2.5A 1 consider only the reference Earth mode1 and plot the predicted sea-level change as a function of time for the three different TPW rates considered in Figure 2.3 (namely, 0.2O/Myr, 1°/Myr and JO/Myr, as labelied). Consider, for exarnple. the case of relatively rapid j0/iLlyr TPW. The results for the reference Earth model in Figure 2.3 indicate that after 0.2 Myr (or a total shift in the pole of 1") the predicted sea-level change is - 30 m. This duration coincides with the period of rapid sea-level rise at the start of the TPCV event evident in the dashed line in Figure 2.5A. After this period the sca-level rise continues at a progressively reduced rate, event ually peaking, after about 6 Myr of TPW, at a value of - 106 m. This sea-level trend arises as a consequence of two contributing effects. At the initiation of TPW. the observation site is located 4.5" away from the local rotation pole; this angle increases as the pole moves away from the site and thus the rotational driving potential progressively decreases. On an elastic Earth the sea-level rise would reach a maximum once the site reached the rotational equator (ie. 90' from the pole), which in this case would take 9 Myr. However, the maximum relative sea-level perturbation is reached more quickly in the viscoelastic case since viscous adjust ments to earlier rotational forcing act to drive down the relative sea-level changes. As a consequence, the maximum is achieved on the reference Earth mode1 when the site is about 75' from the pole. Sea-tevei maxima are not evident for the slower rates of TPW considered in Fig- ure 2.5A because the plotted tirne series only extend for a period of 10 Myr after the onset of TPW and the site has therefore not moved far enough from the location of the maximum potent ial load t O allow viscous compensation to overcome the ongoing loading. Nevertheless, the labels at the right of Figure 2.5A provide the peak relative sea-level perturbation which can be obtained at that site for each TPW rate considered. These values understate the total sea-level variation which can be experienced at a given si te since in al1 cases the calculation begins with a site located 45' From the rotation pole. If I considered a si te ini tially located at extreme high latitudes t hen the sea-level variation experienced by the site once it reached the equator (through TPW) would be approximately twice the value listed on the figure. Furthemore, if one considers differential sea-level trends 1 1 t 1 1 1 O 1 2 3 4 5 TPW Rate (Ol~yr)

Figure 2.5: (4)Relative sea-level change as a function of time over 10 Myr For TPW rates of either O.'ZO/Myr, 1°/bIyr or jO/Myr as labelied. Also shown in each case is the maximum relative sea-level perturbation which can be obtained. (B) Relative sea-level change as a function of TPW rate after either 1 Myr, 5 Myr or 10 Myr of polar motion, as labelled. at pairs of sites. these trends can be as high as four times the values on the labels. Figure Z.5B shows predictions of relative sea-level change as a function of the TPW rate for tirnes 1 Myr, 5 Myr and 10 Myr after the onset of TPW. As in Figure 2.5A. relatively high TPW rates lead to a non-monotonicity in the predicted sea-Ievel trend over the full 10 Myr, whereas for slow TPW rates, the trends remain rnonotonic over 10 Myr. Figure 2.5 has been designed so that one can quickly estirnate the maximum sea-level signal that would be expected in association with a wide range of TPW events. however it rnust be emphasized that complexities in TPW-induced sea-level trends have been suppressed by the adoption of a simple trajectory of the rotation pole relative to the observation site. TLmagnit ades of tkepdictdsite-spmificorMdai sea-levet trends ci ted above are a function of the viscoelastic structure of the Eart h mode1 and the evolving position of the rotation pole relative to the site(s) of interest. In the case of relatively slow rates of TPW (eg. 0.2*/Myr), consistent wit h the infened long-term (order 100 Myr ) reorientat ion of the Earth's rotation vector, 1 bave demonstrated that viscous relaxation reduces the predicted sea-level trends to a very small fraction of the trends that would occur on an elastic or rigid Eart h. For example, the peak value of 49 m cited on Figure UAfor the 0.2°/tv1yr case may be compared to the peak of 3.8 km t hat is obtained in the elastic case. The key is that this departure from complete compensation (as would occur for an inviscid Earth), though small, is still sufficient to yield geologically important sea-level trends. As 1 have discussed. the departure from the inviscid case (in which the sea-level perturbation vanishes) is due to the presence of both an elastic lithosphere and relatively long-t imescale buoyancy modes wit hin the sub-li t hospheric mant le. As the t imescale of the TPW forcing increases, the latter contribution diminishes. Hence, it is important to emphasize that the magnitude of predicted sea-level trends in the case of very slow rates of TPW will be small in the event t hat the lit hosphere possesses no elastic strengt h over timescales of 100's of millions of years. Chapter 3 The Rotational Perturbation to the Shape of the Earth

3.1 Introduction

You say you want a revolution. Well, you knout, we al1 want to change the world.

The Beatles, Revolution

Sydney J Hamis, Tribute to an Egg

This t hesis is prirnarily concerned wit h perturbations to t be Earth's equilibrium shape induced by variations in rotation. In this chapter 1 examine what the "equilibrium shape" of the planet is; or, more specifically, 1 examine the sensit ivity of t his shape to the presence of a lit hosphere wit h long-term elastic strength. A non-rota&ing Ruid Eart h would be sphericat in fom. However the Earth is rotating and this causes its shape is perturbed to that of an ellipsoid; the ellipticity of the planet depends on the interna1 density distribution and on how closely the long-term behaviour of the Earth matches that of a fluid. Determining the equili brium shape of the Eart h is one of the classic problems in geophysics. Clairaut (1743) was the first to derive the relation between the latitudinal variation in gravity over the Eart h's surface and the flattening of the rotating planet. Today. the shape of the geoid is rneasured to great accuracy from the motions of satellites. The flattening of the Earth is generally predicted using a theory in which the plzi.net is treated as a fluid in hydrostatic equilibrium (eg. de Sitter. 1924; Kopal, 1960: Jeffreys, 1963; Nakiboglu 1979,1982); however, as discussed in the previous chapter, the Earth Chapter 3: The Sbape of the Rotating Earth 38

does no& behave as an inviseid Buid men mer pbgicafty hgtirnescates. The effective elastic thickness of the lithosphere is both laterally heterogeneous and dependent upon the timescale of the applied forcing (eg. Watts and Daly, 1981). Regardless. the presence of a lithosphere with non-zero elastic thickness, in sorne globally averaged sense, over geologically long timescaies will clearly have implications on the equilibrium figure of the rotating pianet. The present day ellipticity of the Earth is also influenced by the gadual deceleration of the Earth due to lunar and solar tides. If the behaviour of the Earth is that of an inviscid fluid, or of an inviscid fluid within an elastic shell, the shape of the planet depends on the present rate of rotation. On the other hand, if the viscosity of the mantle is sufficiently large. the history of the Earth's rotation rate will be important. The decreasing magnitude of the Earth's rotation vector has acted to reduce the flattening of the Earth: if the viscosity of the mantle is sufficient to produce a non-negligible lag in the planet's response, then the present day flattening of the Earth would be larger than expected. This excess Bat tening, comrnonly referred to as a "fossil bulge", has been the subject of rnuch debate within the literature and is generally believed to be relatively srnall (eg. Goldreich and Toomre, 1969; Ricard et al., 1993). The difference between the measured flattening of the Earth and the flattening predicted on the basis of hydrostatic theory is usually expressed in tems of the so-called non-hydrostatic geoid of degree two, order zero, G!;. In the spherical harrnonic normalization adopted in C hapter 2 the value of G,": is -27.64 rn (Forte, persona1 communication, 2001). This departure of the rneasured geoid from that expected for a Buid body in hydrostatic equili brium has been attributed to the long-wavelengt h density perturbations associated wit h mantle convection (eg. Forte et al., 1993; Forte et al., 1995; Forte and Mitrovica, 1997; Panasyuk and Hager, 2000). In the following sections I will determine to w hat extent t his conclusion might be affected by the presence of an elast ic lit hoûphere. and/or a fossil bulge.

3.2 The Effect of an Elastic Lithosphere on the Equi- librium Shape of the Earth

The shape of the geoid that arises due to rotation rnay be described by Legendre poly nomial expansion (eg. Nakiboglu lg'i9,1982) Chapber 3: The Shape of the Rotating Earth 39 in whkk the Legendre pdywmids lue wrmdkd so tht &(cos@) = $(3co$% - 1). The ellipticity of the geoid, c, is defined as

where r,, and r,re are the radius of the geoid at the equator and the pole respectively. Nakiboglu (1979,1982) has shown that the ellipticity can be related to the Legendre polynomial coefficients of equation (3.1 ) by

The value of c is - &. Nakiboglu (1979,1982) has shown that f2 is of order E. and j4of order c2. The perturbation to the geoid induced by the rotational potential can be derived from equation (2.25). In Chapter 2 1 adopted a normalization such that &,0(9, $) = $(3 cos29 - 1). Cornparison of equations (2.25) and (3.1 ) gives

and to first order from equatioo (3.3) one obtains

The perturbation to the degree two, order zero geoid in response to an arbitrary potential load is, from equation (2.27),

The load to be applied in this case is the potential associated with the rotation of the Earth, rather than a perturbation to the rotational potential associated with TPW (ie. m(t)in equation (2.29)). The potential load is thus,

Let us assume chat the present rate of rotation was imposed instantaneously at some tirne, t*, in the distant past. The rotationai potential in equation (Xi)can thus be written Cbapter 3: The Shape of the Robatiog Earth

For t » t*, and recalling t hat g = 9,t his expression becomes

where k,/ is the degree two Ruid Love nurnber as defined by equation (2.47). The Battening of the geoid is thus controlled by the rate of rotation and the dueof k{ which depends on the density and elastic structure of the Eart h. Shown in Figure 3.1 is the cornputed value of as a function of elastic lithospheric thickness, LT, for a suite of Earth models with the density structure of PREM (Dziewonski and Anderson, 1951). Al1 material beneath the elastic shell is assumed to behave as an inviscid Buid. Also plotted

O 20 40 60 80 1O0 120

Lithospheric Thicùness (km)

Figure 3.1: The dotted line, and left ordinate axis' show the dependence of k{, on the effective t hickness of the elastic lit hosphere for an Eart h mode1 whose density is constrained by PREM (Dziewonski and Anderson, 1981). The solid Iine. and right ordinate ais, show the corresponding change in G& with respect to the value for a completely Buid Earth. This expression assumes t hat the fluid limit of the Eart h mode1 has been reached and t hus represents the change to the equilibrium figure of the Earth arising from the presence of an elastic lithosphere of thickness, LT. As the elastic lithosphere is thickened. the value of kiLT decreases. Equation (3.5) t herefore implies t hat the magnitude of the ellipticity decreases, that is the planet becomes more spherical, as the effective elastic lithosphere is thickened. For an effective elastic t hickness of - 80 km, the perturbation to the degree two. order zero hydrostatic geoid is of the same magnitude as. but opposite in sign to, the observed non-hydrostatic value

(528 m). The presence of an elastic lithosphere thus increases the difference between the predicted equilibriurn geoid and the observed geoid. The effectiveelastic thickness of the lithosphere varies laterally and with the timescale of the applied forcing. Est imates of short-term elast ic lit hospheric t hickness from seismic studies (eg. Leeds, 1975; Forsyth, 1977; Evans and Sacks, 1979: Clark, 1983; Woods et al.. 1991; Debayle and Kennett, 2000) are generally iarger than estimates of the long-term elastic thickness based on flexure and gravity anomaly measurements in regions of crustal loading (eg. Turcotte et al., 1978; Watts and Daly, 1981; Turcotte and Schubert. 1982: Riad and El-Etr. 1985; Watts and Cox, 1989; Zorin et al., 1989; Pilkington, 1991; Krishna et al., 2000) although in both cases the the lithosphere is found to be thinner in regions wliere it is young and/or hot. The estimates of long-term elastic Iithospheric thickness for oceanic settings generally fa11 within the 10 to 40 km range (with older lithosphere being generally thicker), whereas for continental regions the elastic lithospheric thickness may be greater t han 100 km fespecielly in old, cold coatimutal interiors}. It is diffrcnlt to determine a global "averagen elastic lit hospheric t hickness from this data, particularly due to the presence of plate boundaries: however it is clear that the lithosphere is sufficiently t hick to have a non-negligi ble impact on the shape of the equilibrium geoid relative to the magnitude of the so-called non-hydrostat ic geoid. As an example, 45 km is not an uareasonable estimate for the average thickness of the long-term elastic lithosphere, and in this case AG:, = 14 m. This 14 rn change in the degree two. order zero coefficient of the equilibrium geoid is approximately the same size as the second order corrections of Nakiboglu (1979. 1982). It furt hermore represents an increase of -- 50% in the discrepancy between the measured geoid coefficient and the coefficient predicted using the tradition equilibrium (hydrostatic) theory. T herefore, the Cbapter 3: The Shope of the Rotating Earth 42

change io the Battening of 6 h equilibriunt @ associated wit k the presence of an etastic lithosphere, although small with respect to the total flattening, is important in cornparison to the difference between the observed flattening and the hydrostatic value.

3.3 The Fossil Bulge

In the previous section 1 assumed t hat the Eart h was in equili briurn wi t h the potent ial associated wit h the present day rate of rotation. However, the rate of the Earth's rotation has been gradually slowing t hrough time and it is possible t hat the present shape of the planet is influenced by the faster rotation rates of the past. Because the mantle is not an inviscid fluid, t here will be a lag between its response and any change in the rotational potential, and the present flattening of the geoid will be geater than expected from equilibrium theory. whet her t his t heory incorporates an elastic lithosphere or not. This excess flattening, also referred to as a fossil bulge, will depend on the viscoelastic structure of the pianet and on the rate at which the Earth's rotation is slowing. The potential load associated wit h an arbitrary change in the Eart h's rotation vector is described by equations (2.29)-(2.34). In this section I will assume that the rate of the Eart h's rotation has been decayi ng exponent ially t hrough t irne so t hat

and t hus,

H(t) is the Heaviside step function. The rotation is thus held at a constant rate, R, until time t = O at which point it begins to decay at an exponential rate, A. The flattening of the geoid depends only on the degree two, order zero potential perturbation which? through the substitution of equations (3.13) and (3.14) into equation (2.31). is Found to

Substitution of this expression into equation (3.6) gives the degree two, order zero perturbation to the geoid associated with the deceleration of the Earth's rotation

J 1 - exp(-q;t) exp(4Xt) - exp(-szjt) - j=l S2. j s*,; - 2X Chapber 3: The Shape of the Rotating Esrth 43

This expression gives the totd change in et,wer the meof the deceieration for a given viscoelast ic Earth model; the fossil bulge will be the difference between t his value of G:, G:, and the value obtained for an equivalent Earth model in which the mantle is assumed to be inviscid, G,,,T,inv . The inviscid response can be found either by rnodifying @(t - t') in equation (3.6) and performing the new convolution, or by assuming that 1X < sz, for al1 j in equation (3.16). Either approach gives

And the size of the fossil bulge is

Since the change in rotation is occurring over geologically long timescales it is again useful to employ the serni-inviscid approximation, as outlined in Chapter 2. when solving for Glo(t).Under this approximation the expression for the fossil bulge simplifies to

Where the subscript long indicates that the sum need only be performed for those modes which have geologically long decay t imes. The above formulation of the fossil bulge magnitude is equivalent to assuming that the planet is in hydrostatic equilibrium with the rotation rate until t = O. The result is therefore independent of the assumed rotation rate prior to t = O and gives the magnitude of the fossil bulge acquired since t hat tirne. From the analysis of the variations of laminae properties in tidally influenced sedimentary deposits it is possibIe to determine the rate of the Earth's rotation during the geologic past (eg. Williams, 1997). Given the rate of rotation for the present day and at some point in the past the value of X can be determined. Table 3.1 summarizes the rotationd data [ have used to compute the fossil rotationai bulge. Note that R in equations (3.12) - (3.19) represents the initial rotation rate prior to the exponential decay: it is not the present day rate. The rotation data must be combined with an Earth model in order to determine the magnitude of the fossil bulge. Following the discussion in the previous section 1 have considered a suite of Eart h models characterized by an effective elastic lit hospheric t hickness of 45 km; the upper made viscosity was wied from 2 x to 2 x 1021 Pa s and the iower mantle viscosity from 2 x 102' to 10W Pa S. Table 3.2 gives the largest fossil Cbapter 3: The Shape of the Rotating Earth

Age of Sidered Days Rotation Rate Decay Rate Observation Per Year n(rad/s) Wyr-')

620 Ma 401 f: 7 7.98 x IO-' 1.46 x 10"O 900 Ma 420 8.36 x IO-' 1.52 x 10-l0 2.5 Ga 366 zk 16 9.28 x 10'" 9.63 x IO-"

Table 3.1: Geologic constraints on rotation from Williams (1997). Xote that there was insufficient data to determine error bars for the 900 Ma estimate of the number of sidereal days per year. bulge magnitude obtained from this suite of models for the various geologic constraints on the rotational slowing. The value of AG;,:(T)~=(column 3 in Table 3.2) does not correlate wit h X as might be expected because different arnounts of time have elapsed in each case. In al1 three cases. the size of the fossil bulge is found to be approximately one order of magnitude smaller than the effect of the elastic lithosphere on the equilibricm geoid coefficient ( which is - 14 m for t his elastic lit hospheric t hickness). Furt hermore, the sign of the fossil bulge indicates that the geoid is flatter t han the hydrostatir rqtiivalmt. Of course the Earth's rotation has been slowing throughout its hist,ory and 1 have thus also estimated the fossil bulge acquired over the past 1.5 Gyr. To do t his I have used the geologically determined rates of deceleration and in each case projected backwards in time to find the rotation rate 4.5 Ga. 1 have once again determined an upper bound on the magnitude of the fossil bulge (column 1 in Table 3.2). Since, in this case, al1 calculations

Table 3.2: Maximum magnitude of the fossil bulge from a suite of Earth models using the rotational data summarized in Table 3.1. AG!,:(T), is the upper bound on the hi1bulge acquired since the tirne of observation (in each case the rotation rate was assumed to be constant prior to this tirne). ~~:,:(4.5~a), is the upper bound on the fossii bulge associated wit h rotational deceleration over the entire age of the Eart h. Chapter 3: The Shape of the Rotating Eartb 45 invohe the same chation of rotation raite changes, the amptitude of the fossit bdge correlates with A. ~Gt,:(4.5Ga), is found to be as large as -. 15% of the observed non- hydrostatic geoid. However, I have assumed no change in the viscoelastic properties of the Eart h mode1 over the ent ire history of the Eart h. Due ta the greater concent rat ion of radioactive nuclei, the temperature of the ancient Eart h was li kely significantly hotter tban at present; it is t herefore likely that both the effective elastic thickness of the lithosphere and the viscosity of the made would have be smaller in the past t han at present. The fossil bulge magnitude of - 15% of the observed non- hydrostatic geoid t herefore represents an extreme upper limit. Chapter 4

True Polar Wander Induced Sea-Level Change: The Cretaceous-Tert iary Second-Order Cycle

4.1 Introduction

The esplanatzon oj the cause of transgression cycles in the histoy of the earth will represent one of the most important, but also one of the most dificuit tasks of future geological and geophysical research.

rlljred Wegener, The Origin of Oceans and Continents

The wealth of recent evidence on the dr$ of the earth's continents relative to one anoiher seems to have obscured the role of polar wandering as a distinct though compli- ment~pltenomenon.

Peter Goldreich and Alar Toornre, Some Remarks on Polar Wandering

Alt hough the earliest discussions of TP W argued qualitat ively t hat rot at ional changes would influence sea level. and climate, over geologically long timescales (eg. Darwin. 1Y77; Goid? 1955) the lack of obser~tional evidence for long-term TPW precluded quantitative estimates of its contribution to long-timescale sea-level variations. In this chapter 1 make use of recent estirnates of TPW. made possible t hrough improvernents in paleomagnetic data sets, and consider the cycle of sea-level change t hat occurred during the Cretaceous and Tertiary (- 130 Ma to present). My goal is to establish the importance of TPW as a contributing rnechanism for secondsrder sea-level cycles. Sabadini et al. (1990) Chapter 4: Cretaceous-Tertiitry TPW and Second-Order Sea-Level Change 47

hare shown that episodes of TPW that mc mpid retative to the kmg-term mgerate of polar motion may influence or control third-order sea-level trends, and M~rner(1981) has argued qualit at ively that an apparent latitudinal dependency in long-term sea-level trends may be due to the influence of "rotational tiltn. Tidal deceleration of the Earth's rotation has also been proposed as a mechanism capable of inducing a pole-to-equator differential sea-level change of 100 rn, or more. over the last 100 Mpr (Eardley, 1964: Flatti, 1965). Global-scale second-order sea-level variations are well documented (eg. Vail et al., 1977; Hallam, 1975; Haq et al., 1987); however, the geophysical mechanism for these variations remains uncertain. One clue is that the well-known Cretaceous sea-level transgression apparently occurred when the mean rate of spreading at mid-ocean-ridges increased (Hays and Pit man, 1973). This correlation has led to the widespread view (eg. Turcotte and Schubert, 1982) that changes in plate creation rates alter the net volume of ocean basins and lead to global (that is eustatic) sea-level fluctuations (see Figure 1.1). But changes in mid-ocean-ridge spreading rates are also linked to subduction-induced sea-level variations (Mitrovica et al., 1989; Gurnis. 1990, 1993a,b; Mitrovica et al., 1996). Thus, the extent to which variations in rates of plate creation have influenced eustatic sea-level trends over geological history is unclear. Material in this chapter is reprinted with permission from Mound and Mitrovica (1998), copyright American Association for the Advancement of Science.

4.2 Observational Constraints

4.2.1 Sea Level

In order to gauge the relative importance of TPW to long-term sea-level change it is necessary to have some record of that change. One such constraint is the eustatic curve commonly referred to as the Vail curve (Vail et al., 1977; Hag et al., 1987). which is based primarily on the results of seismic stratigraphy; the last 130 Myr of this curve is plotted in Figure 4.1. For the purpose of this study I have set the relative sea level to zero at 130 Ma, roughly 10 Myr after the start of the Cretaceous. To first order, the curve can be characterized by a sea-level rise over the first - 50 Myr. followed by a drop in sea level which begins gradually, becoming more rapid - 40 Ma and proceeding to the present day. The amplitude of the sea-level oscillation in Figure 4.1 is 5 200 m and has been filtered to remove the signal associated with glaciation. Although there have been questions raised about the accuracy of seismic stratigraphy in general and the Vail curve Chapter 4: Crebaceous-Terbiary TPW aad Serond-Order Sea-Level Change 48

Figure 1.1: A long-term sea-level curve inferred from seismic stratigraphy (Haq et al.. 1987). The oldest datum on the plot is adopted as the zero for the relative sea-level fluctuation. in particular (for a review see Miall, 1990), the curve provides a workable estirnate of' the size and sense of second-order sea-level change, at least in the North Atlantic, over the last 130 Myr. One of the concerns raised in reference to the Vail curve is the distribution of the sites used in its construction; for example. the majority of the sites from the study of Vail et al. ( 1977) were located in central and western Europe. or the United States (see Figure 4.3C). As will be discussed below, t his particular bias in site locations is important when considering seô-level changes due to TPW over the past 130 W~T. The Vail curve is not the the only published ioference of long-term global. or regional? sea-level change. nor is seismic stratigraphy the only method used to determine sea-level curves, whether global or site-specific. Local sea-level curves are generally deterrnined using sequence stratigraphic analysis of sedimentary outcrops, and if one has many local rneasurements of sea-level change that can be correlated in time they can be combined to create a regional or global sea-level curve. This procedure is cornmonly used to produce estimates of global sea-level changes over long periods of time (e.g. Hallam. 1978). How- ever, as in the analysis which produced the Vail curve, the data sets from which these curves have been constructed are generally not global in distribution: but instead highly biased towards a single region with only a lew cornparisons with data from other regions. Chapter 4: CretaceousTertiary TPW and SecoizhOrder Sea-Level Change 49

AO an example, the influentid ~tvitssie(t2&142 hl* sedeve! nwcr of Hahm f 1978) was based rnainly on analysis of site-specific sea-level variations in Europe. -4 second common method of determinhg sea-level changes on regional or larger scales is to study the percentage flooding of continents throughout time, where high sea levels are assumed to correlate with a large amount of continental Eiooding. This method has been used to determine both global (e.g. Fischer, 1981) and regional (e.g. Yanshin, 1973) estimates of sea level. The advantage of this rnethod is that it provides a more broadly sampled view of sea-level change; however there are two main disadvantages. First , i t is very difficult to determine the magnitude of sea-level change fiom this approach. so that only periods of relative highs and lows can be inferred rather than an absolute sea-level curve. Second, it is now recognized that continental Booding events rnay reflect long-wavelength lithospheric response to mantle convection (eg. subduction) rather than absolute sea-level changes (eg. Mitrovica et al., 1989).

4.2.2 Earth Rotation

Through cornparison of hotspot tracks, paleomagnetic measurements and kinematic plate reconstructions, TPW paths can be reconstructed over geological time (eg. Liv- crmorc et al., 1954; Andrews, 1955; ksead Cour~illoI,, i 991j. Figure 4.2 shows the last 130 Myr of the reconstruction of Besse and Courtiliot (1991) which I will adopt to compute TPCV-induced sea-level change. There can be signi ficant difierences bet ween t his TPW pat h and ot her independent inferences (eg. Andrews, 1985)' but t hese differences are mainly restricted to the polar motion prior to 130 Ma. Additionally, these TPW paths rely upon accurate reconstructions of plate mot ion wit h respect to a hotspot reference frame which is presumed to be fixed with respect to the mantle. For example, the reconstruction of Morgan (1983) was used in the determination of the TP W path shown if Figure 4.2. It has been shown that significant relative motion of hotspots. may have occurred (eg. Steinberger. 1996; DiVenere and Kent, 1999), leading to an overestimate of the amount of TPW since the beginning of the Cretaceous (eg. Tarduno and Ge. 1995; Tarduno and Cottrell. 1997; Tarduno and Smirnov, 2001). However, a more recent infer- ence of TPW (Prévot et al., 2000), which attempts to avoid problems related to hotspot fixity through the use of both updated paleomagnetic data (McElhinny and Lock. 1997) and plate reconstructions (Royer et al., 1992; Müller et ai., 1993), is in good agreement ivith both the direction and magnitude of TPW shown in Figure 4.2. The amount of TPW shown in Figure 4.2 thus represents a reasonable upper bound on the total polar motion over the past 130 Myr and 1 will use it to estimate the sea-level Chapter 4: CretaceousTertiary TPW ruid Second-Order Sea-t evel Change

Figure 4.2: Location of the north rotation pole, with age indicated in millions of years, in the hotspot reference frame over the last 130 Myr (adapted from Besse and Courtillot, 199 1) superimposed on present day coastline geometrv. The shaded dashed circles are the 95% confidence ellipses. signal induced by TP W during the Cretaceous and Tertiary. The general motion of the pole can be divided into two main trends. From 120-50 Ma the rotation pole moved away from North America at - 0.4°/Ma. From 30-0 Ma the polar motion reversed direction in the hotspot reference frame. The pole positions shown in Figure 4.2 are found by averaging al1 paleornagnetic data within a 20 Myr window centred on the indicated age (Besse and Courtillotl 1991) and 1 have inte~pdatedthe TP W path msuming constant motion of the pole between each of these locations. Alt hough the long-term TPW path appears to be relatively smoot h in Figure 4.2 it is highly unli kely that the shorter timescale motion of the pole followed a simple path between the time-averaged positions. Indeed, records of historical TPW (eg. Stoyko, 1968; Markowitz, 1970) as well as predictions of the TPW associated with successive glacial cycles (cg. Wu and Peltier, 1984) show large deviations of the TPW path away from the average longer-terrn motion. To specify completely the rotation vector of the Earth over the last 130 million years it is also necessary to have an estimate of changes in the length-of-day, ie. rate of rotation of the Eart h, over this tirne. I adopt a geologically inferred tirne series of tidal-deceleration based upon the study of sedimentary rocks known as tidal rhythmites (Williams, 1997). Chapter 4: Cretaceous-Tertiw TPW and Sefond-Order Sea-Levef Change 51

low tide currents leads to the deposition of dternating laminae of distinguishably coarser and finer sediments, for example sandy versus clayey layers (Williams, 1990). By analysis of the variations in the thickness of these layers in ancient deposits, and comparison with similar analyses of present day deposits, it has been estimated that there were 401 k 7 sidereal days per par 620 Ma (Williams, 1997). By assuming a 1inea.r change in the Eart h's angular momentum between the present day and 620 Ma, and t hat the Lengt h of the year has remained constant, the change in the Earth's rotation rate since 130 Ma can be easily calculated. The spatial geometry of rotation-induced changes in sea-level has been discussed in general above and 1 will briefly review it here making specific reference to the geometry associated with the last 130 Myr of TPW. Variations in rotation rate (Figure 4.3A) induce a lati t udinal sea-level perturbation; t idal-decelerat ion increases relative sea leve: at high latitudes and decreases it at low latitudes. TP W-induced sea-level changes have a signal characterized by four semi-hemispheric quadrants, with zero sea-Ievel perturbation on two great circles (dashed lines in Figure 4.3B), the first at 90' from the instantaneoiis pole of rotation. and the second oriented perpendicular to both the first great circle and to the instantaneous motion of the pole. When the local rotation pole (that is the north pole in the Northern Hemisphere and the south pole in the Southern Hemisphere) is moving toward a quadrant, the sea level falls in that quadrant. Conversely. when the local rotation pole is rnoving away from a quadrant the sea level rises in that quadrant. The great circles of zero sea-level change associated with the mean motion of the rotation pole over the past 10 million years lie roughly on the equator and the geat circle defined by the 51°E and 231°E lines of longitude (Figure 4.3C). A cursory comparison of Figures 4.1-4.3 suggests that TPW may have influenced the second-order sea-Ievei cycle over the tast 130 Myr. The change in the direction of TPW at -50 Ma (Figure 4.2) roughly coincides with the reversal in the long-term sea- level trend (Figure 1.1). Quadrants containing sites in North America, Europe, 'lorth Africa and Australia should have experienced sorne level of TPLV-induced sea-level rise from 130 to 50 Ma (as the local pole moved away from these quadrants) followed by a sea-level fa11 since JO Ma (as the pole changed direction in the hotspot frame), consistent with the geological record of sea-level change over the same period (Figure 4.1). The trend would be reversed for sites in the remaining two quadrants. The sea-level curve of Figure 4.1 is generally interpreted as a measure of global sea-level change; however as seen in Figure 4.3 the distribution of sites used in the construction of such a curve are strongly biased towards a single quadrant of the TPW-induced signal. Chap ter 4: Cretaceous-Tertiary TP W and Second-Order Sea- Level Change

Figure 4.3: Schematic illustration of the approximate spatial geometry of sea-level changes induced by perturbations in the magnitude (A) or orientation (B) of the EarthS rotation vector from time t to t+. The shaded regions represent areas of sea-level rise, while the unshaded regions experience a sea-level fall, and the dashed lines represent the boundaries on which sea-level change is zero. (C) The dotted lines represent great circles of zero TPW-induced sea-level variation associated with the mean TPW path over the last 10 Ma. The collection of continents within each quadrant is not significantly altered either by the re-orientation of the quadrants over the last 130 Ma (as polar wander proceeded) or by plate motions. Sites used in the seismic stratigraphic analysis (Vail et al., 1977) of short-term (third-order and higher) sea-levet trends are shown by solid circles and letters A-D. The letters are used to identify sites considered in Figure 4-4. Chapter 4: Cretaceous-Tertiary TPW and Smd-Order Sea-Level Change

To quant ify the TP W effect 1 solved the gravi tat ionally self-consistent sea-level equation (2.36). The timescale being considered (ie. the last 130 Myr) requires t hat I incor- porate plate motions to prescribe the evolution of both the ocean distribution (CoCGn et al., 1992) and site locations (Irving, 1983). For the initial calculation of second-order TP W-induced sea-level variations (Fig- ure 4.4) 1adopted the reference Earth model prescribed in the last chapter (LT = 100 km. vu, = 102' Pa s and VI, = 3 x 1O2* Pa s) which was also used to consider s horter timescale effects of TPW (Sabadini et al., 1990). In al1 cases the contribution to the sea-level variation from changes in the rotation rate (compare the solid and dotted lines) was rninor and TP W-induced sea-level effects dominate the total sea-level predict ion. The cusps and discontinuities that appear in the sea-level curves of Figure 4.4 are due to discontinuities in the model description of the TPW path or continental evolution. The sense of the predicted sea-level perturbation was positive for the North American, European and Australian sites, and negative for the site in Japan. The maximum sea-level fluctuations predicted for the North American and Japanese sites are - 50-60 m for this Earth rnodel. while the peak sea-level change for the European and Australian sites is about a factor of two smaller. This difference refiects the distribution of the sites with respect to the TPW path (compare Figures 4.2 and 4.3). The polar motion induces a significantly larger change in the rotational celatitude (which 1 define as the angdar distance of a site from the instantaneous north rotation pole) of the North American and Japanese sites than the sites in Europe and Australia. The longer-term sea-level trends predicted for the sites in North America, Europe and ..\ustralia, show a significant relative sea-level rise from 130 to 50 Mat followed by a sea-level hl1 that ofkets a dominant port ion of the eartier transgression f Figme 4.4). As expected, the sense of the sea-level change at these sites is the same as that seen in the observed second-order sea-level curve (Figure 4.1), although there is an imperfect correlation between the timing of the observed and predicted changes. The timing of the predicted relative sea-level peak is, of course, controlled by the adopted TPW path and ot her estimates of TPW show more rapid polar motion earlier in the Cretaceous (eg. Livermore et al., 1984; Prévot et al.. 2000) which would reduce the difference between the times of the predicted and observed transgressive peaks. The predicted amplitude of the dif'ferential second-order signal between the North American and Japanese sites exceeds 100 m, for t his Eaxt h model ( Figure 4.4), alt hough the magnitude of the sea-level predictions wili depend on the adopted viscwlastic Earth model. Chapt er 4: Cretaceous-Tertiary TP W and Second-Oder Sea-Level Change 54

t I 1 1 1 I 60 1 ' C

Figure 4.4: Predicted rotation-induced sea-level change (solid Iines) for four sites (A-D) in Figure 4.3C. The calculations adopt the TPW path and the tirne series of rotation rate changes described in the text. The viscoelastic mode1 is characterized by a lithospheric thickness of 100 km, and upper and lower rnantle viscosities of lo2' and 3 x 10~~Pa s, respectively. The present day geographic coordinates of the sites are: (A) 35ON, 76" W (North America); (B) 50°N, 10°E (Europe); (C)40°S, 148OE (Aus- tralia); and (D) 36ON, 138OE (Japan); however, the predictions inciude the influence of continental drift on site locations. The dotted line in each figure is analogous to the solid, except that changes in the rotation rate have been ignored. .4s in Figure 4.1, we plot sea-level fluctuations relative to the value at 130 Ma.

There is a growing consensus that mantle viscosity increases with depth by a factor close to the value adopted in the construction of Figure 4.4 (see for example Hager, 1984; Richards and Hager, 1984; Nakada and Lambeck. 1989: Ricard and \rVuLningl 1991; Mitrovica, 1996; Forte and Mitrovica, 1996). Reasonable variations in eit her the upper or the lower mantle viscosity of the adopted mode1 do not produce significant changes in the predicted TPW-induced sea-level signal (Figures 4.5B,C). The predictions are however sensitive to variations in the adopted lithospheric t hickness (Figure 4.5A). Doubling t his thickness from 100 km to 200 km increases the predicted peak sea-level rise for the North American site by a factor of two to - 100 m. which is comparable to the sea-level rise estimated from the geologic record (Figure 4.1). In the case of a 200 km elastic lit hosphere. the differentid sea-level signal between North America and Japan is also doubled to -- 'LOO m. The sensitivity to wiations in Earth mode1 parameters has been discussed above (see Section 2.3), and follows from the arguments regarding the sensitivity of the Chapter 4: Cretaceous-Tertiary TP W and Second-Order Sea-Levet Change

Figure 4.5: Predictions of rotation-induced sea-level change for the North American site (as in Figure 4.4 curve A). The calculations involve a suite of viscoelastic Earth models which systernatically vary the free parameters of the model specified in Fig- ure 4.4 (LT = 100 km, vu, = IO*' Pa S. vl, = 3 x 1oZ2 Pa s). (A) Lithwpheric thicknesses of either 50 (dashed line), 100 (solid) or 200 km (dotted). (B) Upper mantle viscosities of either 5 x 1020 (dashed line), 102' (solid) or 2 x 1021 Pa s (dotted). (C) Lower mantle vixobities of either lon (dashed line), 3 x 10~~(solid) OC 102= Pa s (dotted). Chapter 4: Cretaceous-Tertiary TPW and Second-Order Sea-Level Change 56

respouse to departmes fmm arr irmiscict ptanet. Oo - tûû Myr timescaies changes in madeviscosity produce a Iess dramatic departure from the inviscid response than changes in lithospheric thickness as the period of forcing is long compared to al1 modes of relaxation - regardless of the rnant le viscosi ty chosen. The mode1 does not include lateral variations in Earth structure, and hence the appropriate choice for LT in the calculations is unclear but a value of LT > 100 km would not be unreasonable. The North American craton, upon which sea-level study sites are often located, is t hought to have a thick continental root (eg. Jordan, 1975; Grand. 1987; Forte et al.. 1995). Furt hermore, many sites used in sea-level analyses are located on stable continental margins in proximity to old, and therefore thick, oceanic lithosphere. This bias of site locations towards regions of the Ewth wherc the lithosphere is relatively thick is evident in Figure 4.3C. The majority of sites in Figure 1.3C fa11 within two quadrants that would have a TPW-induced sea-level signal of the same sign over the last 10 Myr. Furthermore, 1 have found that motion of the continents and rotation axis over the last 135 Myr does not alter this bias; hence the similar trends of curves A, B and C in Figure 4.4. It would t herefore not be possible to dist inguish the second-order TPW-induced sea-level signal from a global sea-level signal usine the site distribution shown in Figirr 4.3C. This clearly has implications for the interpretation of the Vail curve as a measure of eustatic sea-level change. If the Vail curve is greatly influenced by a TPW-driven signal, then a study of sites in the underrepresented quadrants (as defined by the TPW-induced potential perturbation) should reveal a sea-level history opposite to that seen in Figure 4.1. Indeed, the continental Rooding histories of the Eastern and Western portions of the former Soviet Union appear to be anti-correlated over the last 130 Myr (Yanshin. 1973) in a sense that is consistent with the hypothesis of an important TPW contribution to second-order sea- lever change. Kowever, conclusive evidence for this TPW-induced signal would require an extensive study of a large distribution of sites chosen to sample the full spatial variability of long-term sea-level trends. The resdts of this chapter suggest that TPFV-induced sea-level changes can contribute significantly to second-order sea-level cycles. Furt hermore, TP W effects can not be ignored when comparing or combining sea-level data from different geographic regions. It should also be noted t hat the oft-cited correlation bethveen spreading rates and sea-level Buctuations (Hays and Pitman, 1973) is consistent wit h a sizeable TPW-induced sea- level signal. Recent modelling studies (eg. Steinberger and O'Connell, 1997) have shown that the observed TPW path is consistent with predictions obtained by back-advecting seisrnically inferred densi t y het erogenei t ies. Thus, TP W speeds (and the associated sea- Chapter 4: Cretaceous-Tertiary TP W and Second-Order Sea-Level Change 57 kvek ftnctuatiom) are tike!y tmM ta sprradmg rates, which reffect the rate of advection. The second-order sea-level cycle since 130 Ma is likely a combination of a TPW-induced (quadrential) signal and a eustatic trend. A careful anôlysis of sea-level data. that includes a globally distributed network of sites, will be required to distinguish the relative importance of each contributor. Chapter 5

True Polar Wander Induced Sea-Level Change: A Test of Early Cambrian Inertial Interchange True Polar Wander

5.1 Introduction

Shibiiify is Me a gyroscope. . . . it is dificult to turn jrom its course, but once started il can hardly be stopped.

Phiiip K Dick. Sta b ilit y

The stability of the Eart h's rotation axis depends upon the shape of the planet, or more accurately, the values of the principal moments of inertia of the Eart h. If the planetary rotation axis and the axis of greatest moment of inertia (the major axis) are closely aligned at a given point in time they will remain closety digned regardtes of how the inertia axis moves wit h respect to the planet (eg. Goldreich and Toomre. 1969). Further- more, the rate at which the ais moves is inversely related to the difference between the magnitudes of the maximum and intermediate moments of inertia of the planet (Goldreich and Toomre, 1969). Thus if the shape of the Eart h is strongly oblate, or strongly triaxial? the major axis. and hence rotation axis, will remain relatively fixed wit h respect to the planet; whereas if the Earth is only weakly oblate, or weakly triaxial, the major inertial and rotation axes, although remaining aiigned with each other, may move significantly wit h respect to the planet. However the amount of polar wander wiH not necessarily be large when the major and intermediate moments of inertia are of similar magnitude, and it is possible for the Chapter 5: Early Cambrian 1ITP W and Sea-tevel Change 59

magnibude of bhe primipzd axes of inertia tu chqe&but a change in their orientation. T herefore, the rotation axis could experience lit t le mot ion until the magnitudes of the maximum and intermediate moments of inertia are the same, or even reversed, at which point the planet is no longer in a stable configuration and the rotation axis will move rapidly to align with the new axis of maximum moment of inertia (Fisher. 1974). This possibility is referred to as inertial interchange true polar wander (IITPW) and it leads to -. 90' of polar motion at speeds that may b'e much faster than can be obtained when the rotation axis follows the motion of a single axis of greatest moment of inertia. If IITPW events have occurred they would produce dramatic climatic changes as regions are moved from the poles to the tropics and uice versa. Recent evidence suggesting that an IITPW event occurred during the Early Carnbrian. -. 525 Ma. has renewed interest in this issue (Kirschvink et al., 1997). The timing of the proposed event is especidly intriguing as it coincides with the secalled Cambrian explosion in the diversification of life on the planet (eg. McMenamin and McMenamin, 1990; Schopf et ai., 1992; Narbonne et al., 1994; Grotzinger et al., 1995: Narbonne, 1998) as well as large changes in oceanic water chemistry (eg. Magaritz et al., 1986; Demy et al., 1992; Hoffrnan et al., 1998). However, the interpretation of the paleomagnetic evidence advanced in support of t his IITP W event remains a matter of debate (Torsvik et al., 1998; Evans et al., 1998; Meert. 1999). The large and rapid polar motion of an IITPW event would also drive sea-level variations which may therefore provide a duable independent test for the occurrence of IITPW events. In this chapter 1 will investigate the sea-level signal that would be associated with an IITPW event and make a prelirninary comparison between these predictions and Early-Middle Cambrian sea-level records as a check on the proposed IITPW event. Material in this chapter is reprinted with permission from Mound et al. (1999), copyright Blackwell Science.

5.2 Modelling IITP W-induced Sea-level Trends

.An IITP W event occurs when the magnitudes of the intermediate and maximum moments of inertia of the Earth evolve to near proximity. During such an event, the orientation of the Earth's rotation pole is predicted to undergo a shift of - 90' over some tens of millions of years. Since IITPW events consist of rapid polar motion of several degrees per million -or persisting over tens of millions of years? they provide a link between the fast but short-duration TPW-induced effects studied by Sabadini et al. (1990) and the slower polar motion of longer duration considered in the previous chapter. Due to the large speed and displacement of the pole during an IITPW event Chapter 5: Early CarnbriaD IITP W and Sea-Level Change 60 it k txpeded that these events are capabte af prodrrcing rage sea-tever variations. ne predictions presented below wiil examine the magnitude of sea-level change that con be produced by IITPW as well as the sensitivity of IITPW-induced sea-levei variations to the duration and evolution of the event, the geographic location of the observation site. and the details of the viscoelastic Earth model. -4s a case st udy, 1 will adopt IITP W geometries recently proposed for the Cambrian (Kirschvink et al., 1997). Based on a synthesis of the global Vendian-Carnbrian paie* magnetic database, Kirschvink et al. (1997) have argued that an IITPW event may have occurred during 45Myr of Early Cambrian time (see Figure 5.1). Because their age estimates are based primarily on the numerical calibration of the fossii record, subsequent revisions to that calibration (e.g., Landing et al., 1998; Bowring and Erwin, L99S) rnay alter the precise duration of their IITPW model. In addition, the proposed Cambrian event rnay be one of several rapid TPW episodes during Vendian and early Paleozoic time

Figure 5.1: Equal-angle projection of pre-IITPW Early Cambrian paleogeography, after Kirschvink et al. (lgW), wit h a 30' paleoiatitude-paleolongitude grid. For reference, continental blocks contain 5' tick marks of the present geographic grid. The axis of the propmed IITPW event is shown by the antipodal '+' symbols lying along the equator (darker CU rve) . The large arrow denotes the hypot hesized uniform motion of continents relative to a reference fixed to the rotation axis. Black dots mark the cratonic locations at w hich local sea-level responses have been modelled in t his st ud y. Chapter 5: EKly Cambrian [ITPW and Sea-Levet Change 61

[Evans, 19%). For these reasons, t generdize the rnocîet to encompass severd possible IITPW durations. I will use the pre-IITPW configuration of Kirschvink et al. (1997) and follow the modelled sea-level histories of several sites as t hey rotate 90' uniformly wit h the rest of the paleogeographic elements (Figure 5.1). .4ccording to the IITPW hypothesis. central Balt ica moved t hrough the tropics to mid-southerly latitudes, central Austrdia rot ated counter-clockwise to arrive slightly sout h of the equator, and central Laurent ia translated nearly directly from polar to tropical latitudes on the opposite hemisphere from that depicted. This is a slightly different approoch from that employed by Kirschvink et al. (1997), who obtained a different pdeogeography for their post-IITPW reconstruction because t hey accepted al1 the selected paleomagnetic data at face value. The IITPW event proposed by Kirschvink et al. ( 1997) remains cont roversial (Torsvi k et al., 1998; Evans et al., 1998: Meert, 1999). The controversy is centred primarily on the paleornagnetic data; for example, Meert (1999) has argued that the apparent polar wander paths ( APWP's) considered do not display shifts of the appropriate length or synchronicity to support the IITPW hypothesis. Instead, Meert (1999) prescribes the shifts in the APWP's to enhanced rates of plate motion (up to the order of 20-40 cm/yr) with perhaps a TPW contribution less dramatic than that associated with IITPW. The results presented here therefore represent not only an investigation into the sea-level influence of IITP W events in generd but also testable predictions for the Early Cambrian event in particular. The sea-level predictions were obtained t hrough numerical modelling of the sea-level response to changes in the rot ational potential on a spherically symmet ric, self-gravit at ing, Maxwell viscoelastic Earth model as described in Chapter 2. 1 will be ignoring the loading and self-attract ing effects of the redistributed ocean loads and t herefore the results underestimate the sea-level signal by - 10% (Sabadini et al., 1990). Furthermore, 1 will generally be assuming a constant rate of TP W and thus the sea-level predictions are based upon equation (2.61). The predictions involve a suite of Earth rnodels having the elastic and density structure of the seismic model PREM (Dziewonski and Anderson, 1981). Once again, the models are distinguished on the basis of the thickness of the purely elastic lit hosphere ( LT), and the (assumed uniform) viscosities of the upper and lower mant le regions (yumand ul, respect ively ). Figure 5.2 shows predictions of the sea-level variation at three sites resulting frorn the hypothesised Cambrian IITPW event shown in Figure 5.1. In this case I have adopted a duration of 25 Myr for the event, and thus a TPW rate of 3.6O/Myr. The predictions are based on the reference Earth mode1 defined in Chapter 2 (LT = 100 km, vu, = 102' Pa S. and = 3 x 10~~Pa s). In this case, the site in central Laurentia experiences a sea-leveI Chapter 5: Early Cambriaa IITP W and Sea-tevel Change

Time (Myi)

Figure 5.2: Predicted sea-level responses to the 25 Myr IITPW event at the three sites shown in Figure 5.1. Calculations are based on an Earth mode1 characterized by LT = 100 km. y,,, = 102' Pas and VI, = 3 x 10~~Pa S. Lines correspond to sitw in central Baltica (dashed), central Laurentia (solid) and central Australia (dotted). Time and relative sea level are set to zero at the start of the IlTPW event. rise of- 150 m over the fist 5 13.5 Myr of the IITPW event, followed by a - 100 m sea- level fa11 over the next 11.5 Myr. In contrast, the site in central Australia experiences a sea-level perturbation of less than 15 m from its initial value over the same time period. This difference arises from the distinct positions of the sites relative to the pole during the IITPW event. The change in the rotational celatitude (ie. the angular distance between a site and the north rotation pole) is one determinant of the magnitude of the sea-level variations associated with TPW. The Laurentian site lies relatively close to the great circle dong which the rotation pole is proposed to have moved during the IITPW event; thus the event produces a iarge change in the rotational CO-latitudeand a large amplitude sea-level signal. The Australian site remains close to the rotational equator throughout the IITPW event: hence almost no sea-level change is predicted for this site. Geographic position relative to the rotation pole also controls the sign of the sea-level trend. As a site crosses the rotational equator, the sea-level trend at the site reverses. This explains the shape of the predicted sea-level curves in Figure 5.2. Consider Baltica, where an initially rapid sea-level rise slows and t hen reverses trend to a rapid sea-level fall. Chapter 5: Edy Cambrian IITPW aad Sea-tevel Change 63

At theonset of the EITPW ewnt,the site is in the mrthern hemisphere at a longitude such t hat the local (Le. oort h) rotation pole is moving away from Baltica. As the IITP W event progresses, the site moves across the rotational equator. Once t his occurs (.- 7 Myr after the onset of IITP W), the local (i.e., sout h) rotation pole is rnoving toward the site. This change is reflected by the reversai of the sea-level trend. -4s a consequence of viscous effects (see the discussion in section 2.3), the reversai of the sea-level trend in Baltica actually occurs before the site crosses the equator. In Figures 5.3 and 5.4 I explore the change in the sea-level predictions when the initial positions of the sites in Laurentia and Baltica are moved by 10' in either co-latitude or longitude. The IITPW event and Earth mode1 considered are the same as in Figure 5.2. T hese variations in position may represent eit her uncertainty in the Early Cambrian paleogeographic reconstruction or the efTect of considering different sites across a given continent. For both sites, changes in the initial co-latitude have a greater effect than equivalent changes in longitude. Of course a change in celatitude produces a greater

Figure 5.3: Sensitivity to changes in the initial position of the site in central Laurentia from its 'standard' paleogeographic location of 160' CO-latitudeand 75' east longitude. ('4) Sea-level response of sites with CO-latitudesof 150° (dotted line), 160' (solid line). and 170" (dashed line). (B) Sea-level response of sites with east Iongitudes of 65' (dotted line), 75' (solid line), and 85' (dashed line). In dl cases LT = 100 km. vu, = IO*' Pa s and VI, = 3 x toZ2 Pa S. Chapter 5: Edy Cambrian IITPW and Sea-Level Change

Figure 5.4: Sensitivity to changes in the initial position of the site in central Baltica from its 'standard' paleogeographic location of 55' CO-latit ude and 235' east longitude. (A) Sea-level response of sites with CO-latitudesof' 4.5' (dotted lint'!, 55' (solid linr!, and 6j0 (dashed line). (B) Sea-level response of sites with east longitudes of 225O (dotted line), 235' (solid line), and 245' (dashed line). In al! cases LT = 100 km, 4Arn - 1021 Pa s and ul, = 3 x 10~~Pa S.

displacernent of the site t han an equivalent change in longitude. For the site in Laurentia a 10' shift in CO-latitude moves the site approximately three times as far as a 10' shift in longitude; however the effect upon the predicted net relative sea-level variation is approxirnately six times larger for the shift in CO-latitude than the shift in longitude. Thus, the apparent difference in sensit ivity to cdatitude and longitude seen in Figure 5.3 is a combination of the difference in site displacement for equivalent angular changes and a true difference in the sensitivity to t hat displacement. In general, the form of the predicted sea-level changes is unaffected, aithough the magnitude of the sea-level transgressions and regressions varies as does the net sea-level change experienced. Obviously the time history of the IITPW event will also play a prominent role in determining the induced sea-level signai. The magnitude of the total polar displacernent is constrained to - 90° but the duration of the event could vary significantly. Figure 5.5 shows the predicted sea-level trend at the Laurentian site For IITPW durations of 15 Myr. 25 Myr (as in Figure 5.2) and 35 Myr. These values reflect the uncertainty in the elapsed Time (Myr)

Figure 5.5: Predictions of the relative sea-level response at the site in central Laurentia for different durations of the IITPW event. The dashed line corresponds to a duration of 15 Myr. the solid line 25 Mvr. and the dotted line 35 Myr. ..\II calculations ii.wd an Earth model with LT = 100 km, vu, = 1021 Pas and ul, = 3 x loZ2Pa S.

time for the Early Cambrian event; however, t hey also provide a measure of the sensi t ivity of the sea-level response to the duration of a generic IITPW event. The magnitude of the sea-level highstand for the Laurentian site grows monotonically larger as the duration of the IITP W event is shortened. Indeed, the maximum excursion of sea level from the value at the onset of the event increases by - -50% from - 130 m to .- 190 m as the duration is reduced from -35 Myr to 15 MF. As the elapsed tirne for the HTPW event increases, the efficiency by which viscous relaxations compensate for the large instantaneous elastic response to TPW also increases. As has been discussed previously, sea-level fluctuations induced by TPW are nearly zero on an inviscid Earth mode1 and as the duration of the IITPW event is increased the behaviour of the Earth model approaches the inviscid limit. Thus, sea-level fluctuations will be larger for IITPW events of shorter duration. This result was also apparent in Figure 2.3 in which it was found that faster TPW rates induced a greater change in relative sea level for a given displacement of the pole from its initial position. In Figure 5.6 1 explore the sensitivity of the relative sea-level predictions to different evolutions of the the IITPW event. The solid lines in Figure 5.6 correspond to the IITPW Cliapter 5: Earfy Cambrian IITPW and SeaAevel Change

Figure 5.6: Sensitivity of the relative sea-level predictions to changes in the history of the IITPW event. (A) The position of the geographic CO-latitudeof the north rotation pole with respect to time for three different models of the IITPW event. Note that in al1 cases the motion is constrained to lie within the same great circle. (B) The sea-level response at the central Laurentian site to the V~~OUSIlTPW paths shown in (A). Al1 calculations used an Earth mode1 with LT = 100 km, vu, = loZ1 Pa s and VI, = 3 x 10~~Pa S. event characterized by a uniform rate of polar motion (as in the previous predictions of this section); the dotted lines correspond to a case in which the IITPW event begins with a rapid TPW rate t hat gradua1 slows (the position of the pole versus t irne being described by a quartet-wawtength sinasaid): the dashed [ines are generated by assuming that the greatest speed of TPW occurs at the midpoint of the event (the position of the pole versus time being described by a half-wavelengt h sinusoid). In bot h of the non-uniform IITPW histories the rate of TPW is greater t han in the uniform case at some point during the polar motion, and it is therefore not surprising that these IITPW histories lead to relative sea-level peaks of greater magnitude than predicted using the constant rate event. The timing of the peak and the generd form of the sea-level response are also dependent on the adopted IITPW history, however the net sea-level change at the end of the event is roughly the same for al1 cases considered. Since the evolution of the IITPW event is not well constrained, my cornparison between the predicted sea-level trends and the geological observations will focus on the broad character of the sea-level change. Chapter 5: Early Cambrian IITP W a~dSea-Levet Change 67

A final point bo be considemi is the diof the pre&&ms to variations in the physical properties of the Earth model. As has ben discussed in previous chapters, the effective elastic thickness of the lithosphere and the radial viscosity profile of the made are the source of some contention, and t hey are likely functions of the timescale of the applied forcing. In Figure 5.7 1 present results for Laurentia for the case where the lit hospheric elastic t hickness, upper mant le viscosity and lower mantle viscosity are independently varied from the values used to generate Figures 5.2 through 5.6. The predictions in Figure 5.7 show the greatest sensitivity to changes in lithospheric thickness, the sensitivity of the transgressive peak (but not of the net sea-level change at the end of the event) to miations in upper mantle viscosity is nearly as large, and there is relatively lit tle sensitivity to variations in lower mantle viscosity. The predicted peak sea-level fluctuation in Figure 5.7 ranges from -- 75 m to over 200 m. The sensitivities seen in Figure 5.7 are consistent with the results of the previous chapter in which I considered TPW over the last 130 Myr. However Sabadini et al. (1990), who corisiciered the sea-level response over I Myr to a constant TPW rate of I0/Myr, found a strong sensitivity to wiations in the viscosity of the lower mantle. whereas I have found that the magnitude of the sea-level fluctuation over the course of the IITP W event is not a part icularly strong funct ion of lower mantle viscosity. To explore this difference 1 show in the inset to Figure 5.ïC an enlargement of the first 1 Myr of Our sea-Ievel predictions for riou us values of lower mantle viscosity. These results exhibi t the same sensitivity to variations in lower mantle viscosity first described by Sabadini et al. (1990; compare with their Figure 2). (As discussed by Sabadini and Vermeenen (1997), the amplitudes predicted by Sabadini et al. (1990) are also sensitive to the Eart h model discret ization procedure adopted in t hat study.) Over t imescales of 10'-1O6 yr. increasing the viscosity of the lwer mantle reduces the level of viscous relaxation that acts to compensate the direct effect of the TPW,and this reduction Ieads to sea-Ievel predictions of higher amplitude. Over timescales longer than a few million years these differences represent relat ively srnall transients on total sea-level predict ions w hich exceed 100 m. Indeed, as predicted by Sabadini et al. (1990) the sea-level curves for the different values of lower mantle viscosity become essentially parallel by 1 Myr after the onset of TPW (see inset Figure 5.X).Furthermore, the time derivative of the predicted sea- level curve at the 1 Myr mark for the model with a lower mantle viscosity of 10U Pa s is somewhat lower than the other two curves. As the TPW progresses the differences between the predicted relative sea-Ievel change for the high lower mantle viscosity mode1 and the other two modeis is gradually decreased. This difference in the rate of relative Time (lyr)

Figure 5.7: Predictions of IITPW-induced sea-level change for the central Laurentian site in response to a 25 Myr IITPW event for a suite of Earth modeb in which the physicat parameters are varied from their 'standard' values of LT = 100 km. vu, = 102' Pa s and VI, = 3 x 10~~Pa S. (A) LT = 50 km (dotted line), 100 km (solid line), or 150 km (dashed line). (B) vu, = 5 x 102' Pa s (dotted line), 102' Pa s (solid line), or 2 x 102' Pa s (dashed line). (C) zq, = 10~~Pa s (dotted üne), 3 x Pa s (solid line), or 10" Pa s (dashed line). The inset in panel C provides a detailed view of the shaded region. Chapter 5: EdyCambrian IITP W and Seatevef Change 69

sea-kveL thange predicted for diffetenf; toopet madie vkosities was atm seeo by Sabadini et ai. (1990). I conclude that IITPW-induced sea-level trends, in contrast to third-order TP W-induced sea-level cycles, will not be strong functions of the lower mantle viscosity (assurning that IITPW events occur over periods in excess of a few million years). The results presented here indicate that IITPW events are capable of of producing relative sea-level change of up to - 200 m depending upon the site location, evolution of the event, and the viscoelastic structure of the Earth. The position of a site relative to the polar wander path and the effective elastic thickness of the lithosphere are the two factors which have the greatest effect on the net sea-level change experienced over the course of the IITPW event. Factors such as the evolution of the IITPW event and the mantle's viscosity profile may change the peak sea-level departure however they appear to have relatively little effect on the net sea-level change. I conclude t hat IITPW events are capable of producing large sea-level signals with a distinct spatial geometry. Sea- level records may therefore provide an important independent test for an IITPW event proposed upon the basis of paleomagnetic observations. In the next section such a test will be applied to the Early Cambrian IITPW event proposed by Kirschvink et id. (1997).

5.3 A Preliminary Cornparison With Cambrian Sea- level Records

To illustrate the utility of sea-level predictions as a test of any proposed IITPW event. 1 will now present a comparison of the relative sea-level predictions with constraints on Early Cambrian sea-level trends; the sea-level data considered in this section was primarily cornpiled by D. A. D. Evans. The comparison is preliminary for two main reasons. First. an exhaustive compilation of available Cambrian sea-level records (which may, in any case, not yet be sufficient for the ta&) has not ben attempted. Second, the cornparison will not attempt to correct for the effects on sea level of eustasy. long-term subsidence and sediment compaction, and far-field tectonic influences. Furtherrnore, regional patterns of flooded continental interiors can only be converted to absolute sea-level variations via assumpt ion of a given continental hypsomet ry (c. f., -4lgeo and Seslavinsky, 1995). Nevertheless. the focus on regional sea-level trends for the three sites in central Australia. Laurent ia and Balt ica is mot ivated by t heir tectoaically stable Iocat ions near the centre of cratonic regions, and the differences in their position relative to the IITPW and rotation axes. Indeed. the position of these continents is somewhat fortuitous as they sample the range of signds which can aise due to the geornetry of the IITPW-induced sea-level Cliapter 5: MyCambrian IITP W and Sea-Level Change 70

Buetuations; Laurcntir is predicted ta hexperiencect a net sea-ievel rise, Bdtica a net sea-level fall, and Australia virtudly no sea-level change. Australia's proxirnity to the proposed IITPW axis, and the correspondingly insignif- icant IITPW-induced sea-level variation (Figure 5.2), makes it a favourable reference locali ty for determini ng Early Cambrian eustasy. Carnbrian sedimentary rocks are distributed in several regions of Austrdia, predorninantly in the Flinders and Mount Lofty Ranges of South Australia (Gravestock, 1995) and the disrupted 'Centrahan Superbasin' farther north (Walter et al., 1995). The former region was a rapidly subsiding basin which was affected by the Delamerian orogeny, perhaps as early as Early Cambrian time (Chen and Liu, 1996). For that reason, we opt for more northerly regions (Amadeus and Georgina basins ) to cons t rain cratonic Booding. A detailed Cambrian paleogeographic atlas of cratonic Australia (Cook, 1988) shows a shallow (0-20 m) marine transgression over these areas during Early Cambrian time. The low amplitude of sea-level changes in this area is consistent with the results of little IITPW-induced onlap or offlap, and could also suggest t hat Early Cambrian eustasy was relatively constant. The Phanerozoic sea-level curves of Vail et al. (1977) are based primarily on data frorn cratonic sequences in North America. These curves show a marked Cambrian onlap, the so-called Sauk transgression ( Sloss. 1963 ). Algeo and Seslavins ky !1995) argw that this onlap represents - 75-150 m of rnonotonic sea-level rise, depending on the paleo-hypsometry chosen for Laurentia. In contrast, some syntheses of Early Paleozoic Laurentian flooding records include withio this transgression a slight regression near the Early-Middle Cambrian boundary (Wise, 1974; after Schuchert , 1955). The predicted relative sea-level trends for Laurentia indicate a transgression leading to highstands of 7.3-200 m above the pre-IITP W baseline, which is consistent with the observat ional record. The predictions also suggest a sea-level regression toward the end of the IITP W event, alt hough the magnitude of the regression relative to the earlier transgression is model-dependent. For example, some Earth models yield a significant IITPW-induced regression (eg. for the LT = 30 km prediction io Figure 5.7.4, the reps- sion is almost equal in amplitude to the transgression), and thus appear to be inconsistent with the geological record. However, the predicted regression associated wit h the IITP W event is weakened relative to the transgression as the adopted upper made viscosity is reduced or the elastic lithosphere is thickened. For example, v,, = 5 x 1OZ0 Pa s yields a regression roughly t hree times smaller t han the earlier transgression, t hus providing a reasonable match to the observed Sauk flooding of Laurentia during Early-Middle Cam- brian time. Upper mant le viscosit ies of around t hat value have beeu favoured on the basis of near-surface plume deflection (Richards and Griffiths, I988), glacial rebound data (eg. Chapter 5: Early Cambrian IITPW and Sea-Lwel Change 71

Nakada a& Lôrnkk, 1989) and jomt irwersims of hg-waveiength geoid and rebound data (hfitrovica and Forte, 1997). A relative sea-level curve for the Early Cambrian of Baltica may be estimated from flooding records of the Russian cratoaic platform. Unfortunately, t hose available to us (Ronov et al., 1984) are only compiled at epoch-resolution, i.e., Early (544-518 Ma), Middle (518-505 Ma), and Late (505-495 Ma) Cambrian. These maps, as well as a quantified update at the same resolution (Algeo and Seslavinsky, 1995), confirm that central Baltica experienced a marked regression of perhaps 100-200 m during Cambrian time. A bout half of t his sea-level fdl occurred during the interval of proposed IITP W. The results for Baltica in Figure 5.2 indicate a moderate transgression, Followed by a substantial regression. with a net regression of about 80 m. As For Laurentia, this value is dependent upon the effective elastic lit hospheric t hickness, mant le viscosity, and the rate of TP W. Baltica's hypot hesized Cambrian pdeogeographic position is rat her poorly known because of its lack of reliable Cambrian paleomagnetic poles (Kirschvink et al., 1997; Evans et al., 1998) and as seen in Figure 5.4 the initial position of Baltica can greatly influence the size of the predicted regression. Nevertheless, the proposed IITPW event is predicted to produce a net sea-level drop for Baltica, consistent with its observed Cambrian emergence. This preliminary comparison bet ween sea-level predict ions based on the proposed Early Cambrian IITPW event of Kirschvink et al. (1997) and observed Early-Middle Cambrian sea-level records for Australia, Laurentia and Balt ica, shows qualitative agreement. However, any defini t ive judgement on the proposed event will reguire a systernatic, quantitative comparison with a sufficiently accurate database of Cambrian sea-level histories. The results presented in this chapter provide the required framework for such a Future test, in addition to demonstrating that IITPW events will. in general, produce sizeabte changes in regiona1 sea-leveI trends. Chapter 6

True Polar Wander Induced Sea-Level Change: A Test of Rapid True Polar Wander During the Late Cret aceous

6.1 Introduction

[Ljet us enter into sorne examinations jar ourseives. bejore me make up an opinion respect ing the m.

Edgar Man Poe, The !Murders in the Rue Morgue

In previous chapters I have shown that TPW can produce large amplitude sea-level variations which have a distinctive global pattern. TPW events proposed on the bais of paleomagnetic data can be the source of contention as the paleomagnetic signature of TPW is seldom unambiguous. Sealevel records may tkrefore provide a vatuabte mde- pendent means of determining if a proposed TPW event is consistent with the geological record. In this chapter 1 will examine the compatibility of a proposed period of rapid TPW during the Late Cretaceous with published les-level records for that time. Motion of the solid Earth with respect to the rotation axis (ie. TPW) occurs as a consequence of the conservation of angular momentum when rnass is redistributed either within the Earth (for example, due to mantle convection) or upon its surface (for example. due to the growth and ablation of ice sheets). Improvements in both paleomagnetic data and continental reconstructions have yielded est imates of TPW over various t imescales. Besse and Courtillot (1991) have argued for rates of TPW as high as 0.7O/Myr during the last 200 Myr. Their approach involved averaging paleomagnetic data in 20-30 .My Chapter 6: Late Cretaceous TPW and Sea-Level Cbmge 73

Sager and Koppers (2000a) have recently proposed, based on paleomagnetic poles obtained from Pacific seamounts, a TPW event in the Late Cretaceous in which the pole moved 16-21' in 2-5 Myr. The mean age of the paleomagnetic poles used to constrain the proposed TPW event of Sager and Koppers (2000a) is 84f 2 MF, coinciding ( within error) with the end the Cretaceous superchron which terrninated at the Santonian-Campanian boundary - 83.5 Ma (Obradovich, 1993). The Cretaceous superchron was a period of - 40 Myr during which time there were no reversals of the Earth's magnetic field and thus it is one of the most distinctive features in the magnetic reversal time scale. Depending upon the evolution of the proposed TPW event, the instantaneous rates of TPW could be considerably larger than 10°/Myr and average TPW rates in excess of JO/Wyr are iodicated; for comparison the average rate of TPW during the 90' in 25 Myr IITPW event considered in the previous chapter is 3.6*/Myr. The proposal of Sager and Koppen (2000a), which is based on data solely from Pacific seamounts, calls for a dramatic shift in the orientation of the rotation axis and one would expect such a s hift to be recorded in ot her paleomagnetic records. Cot t rell and Tarduno (2000) however have shown that the TPW hypothesis does not appear to be compatible with a set of paleolatitude determinations from an Italian sedimentary sequeoce. On the other hand, Sager and Koppers (2000b) have argued that there is a general agreement between their proposed Late Cretaceous TPW event and records of long-term TPW based upon global synt heses of paleomagnet ic data. The rate of TP W in t hese global TPW pat hs (Livermore et al., 1984; Besse and Courtillot, 1991; Prévot et al., 2000) is not as great as in the Sager and Koppers (2000a) proposal, however since these paths are found by averaging paleomagnetic data within large (10-30 Myr) windows of time one ivould not expect an exact match. It is therefore unclear from the paleomagnetic data. whether or not there has been a period of rapid TPW during the tate Cretaceous; the size and rapidity of the proposed polar offset should drive large regional sea-level changes and thus the observational record of these changes may provide an independent means of testing t his contentious TPW proposal. Material in this chapter is reprinted with permission from Mound et al. (2001). copyright Arnerican Geophysical Union.

6.2 Result s of Sea-Level Modelling

As in Chapter 5, the sea-level predictions presented here will neglect the loading and self-gravitation of the redistributed ocean mass, and 1 will consider predictions for a suite Chapter 6: Late Cretaceous TPW adSa-Level Change 74 of Eartb rnodals wbich va~ybhe ehsbie Lithospheric tkickmss, and average viscosities of the upper and lower made regions. 1 consider four different cases of TPW that reflect ranges representing the uncertainty in both the position and age of the Late Cretaceous paleomagnetic poles (Sager and Koppers, 2000a). Specifically, I assume eit her 16 or 21" of polar motion lasting for a period of either 2 or 5 Myr. Sager and Koppers (2000a) specify only the start and end locations of the pole and 1 simply assume that the displacement versus time is monotonic and smooth (in fact, I mode1 the path as a half-wavelength sinusoid) until it reaches its maximum displacement. Subsequently, the pole is held fixed for an additional 2 Myr, t his allows the sea-level response bot h during and after the TPW event to be studied. The paleomagnetic poles used by Sager and Koppers (2000a) to constrain the pre posed Late Cretaceous TP W event have a mean age of 84 f 2 klyr. Most of the polar motion is believed to have occurred within this dating uncertainty, and thus the start of the event is constrained to approximately 85-86.5 Ma (Sager and Koppers, 2000a). The onset of rapid polar motion therefore fdls either in the Late Coniacian (the Coniacian extends from 89.0 f 0.5 Ma to 85.8 f0.5 Ma) or Early Santonian (the Santonian extends from 55.8 f 0.5 Ma to 83.5 f 0.5 Ma) and the event ends by, at the latest, the Early Campanian (the Campanian extends from 83.5 10.5 Ma to 71.3 f 0.5 Ma). (The timing of the various geologicai stages are taken from Gradstein and Ogg (1996).) Figure 6.1 shows the predicted sea-level response at three geographic sites for the four different models of the TPW event. Site positions are based on reconstructed plate locations for 54 Ma (Hay et al., 1999), corresponding to the mean age of the paleomagnetic poles used to constrain the proposed TPW event. The predictions adopt an Earth mode1 characterized by a lit hospheric t hickness of 70 km, upper made viscosity of 102' Pa s and lower mantle viscosity of 3 x 10~~Fa S. As expected, the predicted sea-level variations are targer, the faster and hirther the poh moves. The genera! form of the sea-lever curve is approximately the same at the different sites in Figure 6.1 regardless of the rnagni t ude, or even sign, of the fluctuations. Table 6.1 lists the peak amplitude of the sea-level response. for the case of 21' of TPW lasting 2 Myr, for a larger set of geographic locations. The shape of the cornplete sea-level curves at these sites is similar to the curves shown in Figure 6.1, adjusting for differences in amplitude. The results shown in Figure 6.1 and Table 6.1 highlight the geometry of the TPLV- induced relative sea-level signal. Regions located at mid-latitudes close to the great circle of polar motion (for example, the sites in Austrdia and North America) experience a large change in their rotationai colatitude over the course of the event and a large relative sea-level signal. Sites which experience relatively litt le change in rotational colatitude Chapter 6: Labe Cretaceous TPW and Sea-Level Change

O 1 2 3 4 5 6 7 Western lnterior US

Western Austraiia

Time (Myr)

Figure 6.1: Relative sea-level response to the proposed TP W event at sites located in (A) the western interior of the United States, (B) South Africa, and (C) western Australia. The calcu1ations assumed a polar motion of either 21" (solid line) or 16" (dotted Iine) over a period of either 2 or 5 Myr (these cases are easily distinguished on the figure). The Earth modei is constraineâ to LT, vu, and VI, values of 70 km, IO2' Pa s and 3 x Pa s respectively. The tirne and relative sea level are set to zero at the onset of the TPW event. Chapter 6: Late Cretaceous TPW and Sea-Level Chmge

Regionai Location Peak Amplitude (m)

US Western Interior US Gulf Coast Venezuela Sweden Russian Platform Spain G reece Oman South .4frica Madagascar Western Aust ralia Eastern Australia

Table 6.1: Maximum amplitude of the relative sea-level perturbation at various sites in response to a TPW history with 21' of polar motion in 2 Myr, and an Earth mode1 characterized by LT = 70 km, vu, = 102' Pa s, and ul, = 3 x 10~~Pa S. Negative amplitudes indicate a relative sea-level faII. over the course of the TPW event (as is the case for Oman) experience virtually no relative sea-level change. Table 6.1 also illustrates the difficulty in distinguishing between a global sea-level signal and a TPW-induced signal (which has a long-ivavelength spatial variation). Despite considering sites on six continents only one region (represented by South Africa and Madagascar) is predicted to experience a significant sea-level fall, and it is possible that one would be unable to distinguish between a global and a TPW-induced regionally varying sea-level signal. The larger the magnitude of the TPW-induced sea-level signal the more likely it will be t hat the signal can be detected wit hin the geologic record. The t hree main factors which cont rol the magnitude of TPW-i nduced sea-level variations are the location of the si te relative to the pole and its direction of motion, the speed and duration of TPW. and the viscoelastic structure of the Eart h. There is uncertainty in the effective elastic t hickness of the lithosphere and the made's viscosity profile, and bot h are likely dependent on the timescale of the applied forcing. Accordingly, as in previous chapters, 1 investigate the sensitivity of the predictions to the values of effective elastic lit hospheric t hickness. upper rnantie viscosity and lower made viscosity adopted for the Eart h model. Figure 6.2 shows the predicted sea-ievel response at the site in the western interior Chapter 6: Late Cretaceous TPW and Sea-Level Change of the United States fo~a suite of khh models in wkkh the efkéive k&cbof the elastic lithosphere is varied from 20 to 150 km. The upper and lower made viscosities were fixed to 1021 and 3 x 102' Pa s, respectively, and al1 curves are based on a polar motion of 21' over a period of 2 Myr. The amplitude of the transgression decreases as the elastic lithosphere is thinned. but it remains above 100 m for the range considered. The appropriate value for the elastic lithospheric thickness is difficult to estimate, both because this value is dependent on the timescale of the forcing and also because the mode1 neglects lateral variations in Eart h structure. The t hickness of the lit hosphere varies substantially over the surface of the Eart h. The lit hosphere is generally thinner in regions of hi& heat flow such as along mid-ocean ridges and at *hotspots7over ascending thermal plumes. Passive continental margins and continental interiors are usually regions with a relatively cold and t hick lit hosphere and these are also the regions in which the geologic record of sea-level change is generally studied. In any case, the lower bound on

Time (Myr)

Figure 6.2: Relative sea-level predictions for the western interior of the United States for a suite of Earth models in which LT = 20 km (dotted), 50 km (dash-dot), 70 km (solid), 100 km (double dash-double dot) or 150 km (dashed). The right ordinate axis gives the relative sea-tevel change normalized by the peak value for the mode1 in which LT = 70 km (-- 140 m). In al1 cases, the values of vu, and VI, were 10'~and 3 x 10~~Pa s, respectively. The calculations used a total polar rnot.ion of 21° over a duration of 2 iMyr. The time and relative sea levet are set to zero at the onset of the TPW event. Chapter 6: Late Cretaceous TP W aod Ses-Level Change

lithospheiic thickness trea6ed in Figwe h2 k &sinIy conservative. The right ordinate axis in Figure 6.2 gives the relative sea-level values normalized by the peak prediction for the model with a lithospheric thickness of 70 km, upper mantle viscosity of 102' Pa s and lower mantle viscosity of 3 x 10Z2 Pa s (ie. -- 140 m. see Table 6.1). The 'normdized' sensitivity evident in Fiyre 6.2 is very similar for all of the sites considered when the çame TP W history (21' in 2 Myr) is adopted. Therefore, the right ordinate axis provides a multiplicative factor that can be applied to the predictions in Table 6.1 to assess how their amplitude would be altered by a change in the assumed lithospheric thickness. For example, a model with a 20 km thick elastic lithosphere would produce a transgressive peak at the western Australian site of - 0.73 x 136 m = 10'2 m. The increase in the predicted relative sea-level signal as the elast ic lit hosphere is t hickened is a result of the increasing departure of the Earth's response from that of an inviscid planet, as has been discussed in previous chapters. In Figure 6.3 1 investigate the sensitivity of the TP W-induced sea-Ievel response at the site in the western interior of the United States to variations in both upper and lovw mantle viscosity for the TPW event in which the pole moved 21' in 2 Myr. .411 of the models used in the construction of this figure had a lithospheric thickness of 70 km. The figure shows contours of the peak sea-level response normalized by the value obtained for the Eart h model wit h an upper mantle viscosity of 102' Pa s and a lower mant le viscosi ty of 3 x 10~~Pa s, the same mode1 that was used to normalize the response in Figure 6.2. The results shown in Figure 6.3 provide a viscosity-mode1 mapping that is relatively robust for the other sites and TPW histories considered. As seen in previous chapters the sensitivity of the predicted peak amplitude to variations in upper mantle viscosity is greater than to variations in lower mantle viscosity. Estimates of mantle viscosity, whet her generated from analyses of glacial isostat ic adjustment data (eg. Lam beck et al., 1998) or a combinat ion of t his data wit h long-wavelengt h mant le convection observables (eg. Forte and Mitrovica, 1996) appear to be converging, and they indicate upper mantle viscosity in the range of roughly 3 x lo2* to 102' Pa s and a lower mantle viscosity greater than 5 x 1021 Pa S. This suggests that the predicted tirne series in Figures 6.1 and 6.2 may be scaled in the range .- 0.5-1.1 to account for uncertainties in the adopted mant le viscosi ty profile. The TPW t ime series considered in the predictions shown in Figure 6.1 correspond to the extrema of displacement and duration proposed for the Late Cretaceous event of Sager and Koppers (2000a). The calcdations summarized by figures 6.2 and 6.3 provide a measure of the sensitivity of the relative sea-level predictions to variations in the adopted viscoelashic structure of the Eart h model. Toget her, these calcdations (and Table 6.1) Chapter 6: Late Cretaceous TP W aod Sea-Level Change

J 19.75 k I I I I I I I 21.00 21.25 21.50 21.75 22.00 22.25 22.50 Log Lower Mantle Viscosity (Pa s)]

Figure 6.3: Contour plot showing the sensitivity of the predicted relative sea-level response to variations in both upper and lower mantle viscosity. The contours represent the maximum amplitude of the transgressive peak predicted for the western interior of the United States normalized by the value obtained for an Earth model with 4, and vim equal to lo2' and 3 x loa2 Pa s, respectively (- 140 m). In all cases LT was equal to iO km. The calculations used a total polar motion oF21° over a duration of 2 Myr. provide a bound on the range of sea-level signals that would be expected as a consequence of the proposed TP W event. The next section focuses on whet her or not such signals are evident within observational records of Late Cretaceous sea-level change.

6.3 Comparison With Observed Late Cretaceous Sea- Level Trends

Constraints on sea-level variations determined from the stratigraphic record are subject to uncertainties in both timing and magnitude. Additionally, there are numerous processes that effect the stratigraphic record on local, regional and global scales. Ac- cordingly, in this section I will be concerned mainly with regions in which the predicted TPW-induced sea-level signal is large and/or where I have been able to find relati-~ely high-resolut ion observat ion4 records in the literature. Therefore, the observational constraints presented here do not include all published Late Cretaceous sea-level records, Chapter 6: Late Cretaceous TPW and S&Level Change

Sub-stage Approximate Western Gulf Coast Western Souteast Scandinavia Rusaian Interval (Ma) Interior US US Aust dia Africa Plat form

E Coniacian 89.0-88.7 S T T T I hi Coniacian 88.7-87.5 S T T T R 1 L Coniacian 87.5-85.8 S T T T T E Santonian 85.H5.0 S T T T T M Santonian 85.0-84.8 S T T T R L Santonian 84.843.5 R T T T T T E Campaniart 83.5-81 .O R T S T R LM Campanian 8 1 .O-76.5 T R R T T L Campanian 76.5-71.3 R-T T R T R

Table 6.2: Observational constraints on Late Cretaceous sea-level change (see text for references). Sea level during each sub-stage is identified as either regressing (R), transgressing (T) or stable (S); a dash (-) indicates that 1 found no data for that tirne.

instead 1 have focused on sites where the size of the predicted signal and the quality of the observations make it most likely that the effect of TPW would be detectable. The observational constraints that will be discussed more fully below are surnmarized in Ta- ble 6.2. A cornparison between Table 6.2 and the predict ions of the previous section (especially Figure 6.1 and Table 6.1) allçivj me to ases the sompatibility ol the proposeci TPW event ivith the observed sea-level record. Recall that the proposed TPW event began approximately 85-86.5 Ma (near the Coniacian-Santonian boundary ) and t hat for al1 regions in Table 6.2, except for sout heast Africa, a transgression is predicted to begin at t hat tirne, slowing and changing to a regression by approximately the Santonian-Campanian boundaxy. For southeast Africa an opposite trend of regression changing to transgression is predicted. Sea level in the western interior of the United States and Canada appears to have been relatively stable from the Coniacian through Santonian until the onset of a large and rapid regression near the Santonian-Campanian boundary (Kadrnan, 1977, 1984: Payenberg et al.. 2000). The region was contemporaneously subject to well-documented epeirogenic motions due to subduction on the western margin of the continent (eg. Mitrovica et al.. 1989) and thus it is not an ideal site to isolate TPW-induced effects. The short-lived regression which begins in the Late Santonian might be linked to TPWo however the absence of a transgression in the observational record prior to the observed regression does oot support such a link. TPW-induced sea-level trends dong the guif coast of the United States would have been similar to those in the western interior although slightly smaller in magnitude (see Chapter 6: Late Cretaceous TPUrand Seéi-Level Change 81

region; however there is a recorded transgression during the Late Santonian through Early Campanian followed by a regression which appears to be faster than the preceding transgression (King Jr., 1994; Mancini et ai., 1996). Although the gap in the stratigraphie record makes it impossible to detennine the start of the transgression. the timing of the transgression-regression cycle may be consistent with a rotational effect if the TPW occurs as recently as allowable within the paleomagnetic constraints. It should be noted that in the predicted sea-level curves the rate of transgression and regession are roughly equal, the transgressive phase being somewhat more rapid. A cornparison of the observations from the gulf coast and western interior of the United States indicates t hat the sea-level trends experienced at t hese sites was ant i-correlated during the Late Santonian t hrough Middle Campanian (see Table 6.2). This anti-correlation is incompatible with the sea- level signal predicted for the proposed Late Cretaceous TPW event and is likely due to the tectonic influences decting the western interior. Due to the near antipodal positions of Australia and North America during the Late Cretaceous (Hay et al., 1999) and the symmetry of the TPW-induced potential load, these continents would have experienced very similar sea-Ievel trends in response to the proposed TPW event (see Figure 6.1 and Table 6.1). In western Australian basins, a transgression beginning in the Coniacian reached a maximum during the Santonian and was followed by a Middle to Late Campanian regression, however the timing of the cycle is unique in each basin around the continental margin (Quilty, 1980; Frakes et al., 1957). Alt hough t hese observations are roughly in agreement with the predict ions for the longest of the proposed TPW events, the transgression begins earlier and the regression occurs later t han predicted. Furt hermore, a TP W-induced signal should occur simultaneously in al1 western Australian basins. Although the predictions indicate t hat al1 of Australia would have experienced a sirnifar TPW-kdnced sea-kvei signal (see Table 6. t ) t here is no evidence of a Late Cretaceous highstand in eastern Australia, possibly because the eastern half of the continent was uptifted relative to the west at this tirne (Frakes et al., 1987: Russell and Gurnis. 1994). A transgression initiated in the Late Turonian (the Turonian extends from 93.5 f 0.2 Ma to 89.0 & 0.5 Ma) and progressing through the Coniacian and Santonian until at Ieast the Late Campanian is recorded along the south and east coasts of Africa. alt hough a Santonian regression is seen in Madagascar (Reyment and Dingle, 1987). The sea-level predictions based upon the proposed TPW event show a drop in sea level for this region beginning near the Coniacian-Santonian boundary, one which should be larger on the mainland than in Madagascar (see Figure 6.1 and Table 6.1) and clearly do not match Chapter 6: Late Cretaceous TP W and Sea-Level Change

Christ ensen ( 1984) st udied seven Scandinavian areas and found a good correlat ion between t heir Middle to Late Cretaceous stratigraphic records. An Early Coniacian transgression in this region peaked in the Middle to Late Santonian and was followed by an Early Campanian regression, however this w.u just one of a number of similar transgressive-regressive phases (Christensen, 1984). The observed regression-transgression cycle could be ascribed to one of the longer duration TPW events, although the transgressive pulse begins earlier t han predicted for the TPW-induced response. Furt hermore, since the observed sea-ievel fluctuation is one of a series of similar cycles any argument for a link to TPW is not cornpelling. A high-resolution Jurassic and Cretaceous sea-level curve, which unfortunately does not extend beyond the Santonian-Campanian boundary, has been constructed based on Russian Platforrn and Siberian stratigraphy (Sabagian et al., 1996). Sea level was found to be - 15 rn higher at the end of the Santonian than at the beginning of the Coniacian and there were relatively short-lived drops in sea level of 20-30 m magnitude during both the Middle Coniacian and Middle Santonian. Although the observed sea-level change does not match the predicted sea-level variations for t his region, the TP W-induced signal may be masked by a larger global or regional signal. The transgressive peak amplitude of 52 m for the Russian Platform listed in Table 6.1 can be reduced to less than 20 m by adopting a weaker viscoelastic profile and a less dramatic TPW event. The proposed Late Cretaceous TPW event of Sager and Koppers (2000a), in which the rotation axis moved 16-21' degrees in 2-5 Myr, would have produced a distinctive pattern of large sea-level variations. There has been debate about the compatibility of the TPW proposa1 wi t h paleomagnet ic data (Cot t rell and Tarduno, 2000; Sager and Koppers. '2000b) and the stratigraphic record thus holds the potential for an important independent test O€the proposed event. A preliminary analysis has faiIed to find consistent evidence for the expected TPW-induced sea-level changes in the observational record. Despite uncertaint ies in the viscoelast ic Eart h mode1 and the proposed TP W history, t his lack of consistency suggests that if there was a period of increased TPW during the Late Cretaceous it was less dramatic than the proposai of Sager and Koppers (2000a). Chapter 7

Discussion and Conclusions

[U]ltimately, science is ideas . . .

Richard Cytowic, The Mun Who Tasted Shapes

7.1 Variations in Rotation - Variations by Rotation

Ultimately this thesis has been concerned with a simple and long-recognized idea in geophysics; the Earth's rotation affects the shape of the planet. Changes in the pot.mt.ial associated wit h the Eart h's rotation will deform bot h the geoid and the solid surface, but in different amounts, and thus relative sea-level change will occur. This idea has been invoked in previous work that considered rotational variations occurring on timescales ranging up to a million years; in this thesis I have extended those studies to consider the affect on sea-level of rotational variations that occur over tens to hundreds of millions of years. Furthermore, 1 have turned the idea around to consider whether observed changes in sea-level may be used to constrain the history of planetary rotation. There is no direct measure of the Eart h's paleorot at ion vector. however its orientation does influence the orientation of the planetary rnagnetic field and it is thus possible, albeit difficult. to infer TPW frorn paleornagnetic data. The idea that long-term changes in rotation can influence sea level therefore links two of the classic observational records in Earth science, and provides a new rneans of test ing TP W proposais. Before invest igating the sea-level signal associated wit h changes in rotation 1 first examined the affect of rotation on the present day shape of the Earth. The present day ellipticity of the geoid has a measured flattening of & whereas the value expected for a fluid body wit h the same density distribution as the Eart h is & (Nakiboglu. 1982): the Earth is therefore more ellipticai (ie. flatter) than expected. This difference implies that the Earth is not a fluid body in hydrmtatic equilibriurn; either the viscoelastic structure C6apter 7: Discussion and Conclusions 84

d the plan&, or a dyoamie prefess, is rnpomibk fur the obsed departme from the hydrostatic form. In Chapter 3 1 calculated the effect that adding a thin elastic shell had on the equilibrium shape of an otherwise fluid Eart h model. Estimates of the long- term effective elast ic lit hospheric t hickness (eg. Watts and Daly, 1981; Turcot te and Schubert, 1982; Pilkington, 1991) vary depending on the tectonic setting. The inclusion of a 45 km thick elastic shell. for example, was found to decrease the magnitude of the spherical harmonic degree two, order zero coefficient of the equilibrium geoid by .- 14 m. Although this perturbation is small with respect to the total flattening, it increases the

discrepancy between the observed and hydrostatic flattening of the geoid by 5 50%. The geodynamic mechanism responsi ble for t his difference, likely the convection of mant le density heterogeneit ies (eg. Forte and Mitrovica, 1997), must t herefore be st ronger t han previously supposed. Part of the excess flattening of the geoid may be due to the delay in the response of the viscous mantle to the gradua1 deceleration of the Earth's rotation. Using geologic constraints on the long-term rate of tidd deceleration I calculated, for a suite of mantle viscosity rnodels, the magnitude of the present day fossil bulge. For the most viscous models I considered the predicted amplitude of of the fossil bulge was - 30% of the effect associated with the presence of a 45 km thick elastic lithosphere (and of opposite si@ to this effect ). Thus, the presence of a fossil bulge explains at most - 10% of the between the observed excess Bat tening of the geoid. Changes in the rotation vector will perturb the Earth from its equilibrium shape and can lead to relative sea-level change. In Chapter 4 1 computed the sea-Ievel effect associated with the inferred rotationai variations for the past 130 Myr of the Earth's history. 1 found that the long-term wander of the rotation pole could be a significant contributor to second-order sea-level variations; TPW-induced differential sea-level trends coutd be as targe as 5 200 m, comparable to the observationaIly inferred sea-IeveI change since the Early Cretaceous (eg. Vail et al., 1977; Haq et al., 1987). The observed signal has been presumed to be globally uniform; the sea-level signal associated wit h TPW is not uniform, however it is of sufficiently long wavelength that it may be mistaken for a global signal. For example, the sites used by Vail et al. (1977), although spread over six continents, are heavily biased towards regions which are predicted to experience similar TPW-induced signals. In general. the results of my study indicate that care must be taken when comparing sea-level data from different regions of the globe; in particular, I argue that the observed Cretaceous-Tertiary sea-level cycle should be re-interpreted as a combination of global and regiondy varying. TPW-induced, signals. The size of the TPW-induced sea-level signal depends on the viscoelastic structure of C6apte.r 7: Discussion and Condusions

will deform equally in response to a given potential perturbation and no relative sea-level change will result. On the other hand, for a completely elastic planet there would be a large difference between the responses of the geoid and solid surface. The Earth is neither perfectly elastic nor perfectly Buid, but instead behaves as a viscoelastic body. On short timescales the response of the Eart h is nearly elastic, but on the very long timescales considered in t his t hesis viscous compensation acts to great ly reduce the sea- level signal. The TPW-induced relative sea-level signal is not zero on these timescales due to the presence of a t hin effectively elastic lit hosphere and very slowly decaying mantle relaxation modes, bot h of which prevent the Earth from reaching its inviscid fluid limit. Furthermore, TPW is a continuous process and the induced sea-level fluctuation at any given time will be a response to both the prior and the ongoing polar motion. The rate and duration of TPW therefore bot h affect the size of the sea-level signal. Proposals of dramatic TPW events based on paleomagnetic data are often contentious. The distinct pattern of large sea-level variations that would accornpany such events provide an independent means of testing those proposais. In Chapters 5 and 6 1 considered two cases of rapid TPW and compared the predicted regional sea-Ievel trends to observa- tionai records. For the proposed Early Carnbrian IITPW event proposed by Kirschvink et al. (1997) good general agreement was found between sea-level records for Laurentia, Baltica and Australia and the predicted TPW-induced signal. For the Late Cretaceous TPW proposa1 of Sager and Koppers (20004 consistent sea-level evidence was not found wi t hi n the geologic record. The comparison bet ween the sea-level predict ions and the stratigraphie record was preliminary, but it should motivate further work on applying the sea-level test. I did not correct for the effects of eustatic sea-level change, long-term subsidence and sediment compaction, or far-field tectonic effects; however, these problems are létrgety avoided by considering the differentiai patteni of sea-levet change between sites which appear to have been tectonicdy stable. The effectiveness of a differential comparison depends in large part on the positions of sites with robust sea-level constraints relative to the TPW path.

7.2 Speculations, F'uture Work and Final Remarks

There are a number of ways that the ideas which have been explored within this thesis may be advanced. TPW is a manifestation of the heterogeneous structure of the Earth, however my models have assumed a spherically symmetnc planet. Incorporating lateral variations Chapter 7: Discussion and Coaclusio~s

into the viscoelaskic sb~uebureOF the modek is on impartant amme far fdnre research. A normal mode theory governing the load induced response of a fully t hree dimensionai, heterogeneous Eart h model has been outlined (eg. Tromp and Mitrovica, 1999ab, 2000) and a number of groups are current ly developing sophisticated numerical models to ad- dress this problem. It is likely that lateral variations in the effective elastic thickness of the lithosphere will have a larger influence on the results of my thesis than will variations in mantle viscosity. The predicted sea-level change induced by TPW on geologically long timescaies was found to be more sensitive to reasonable variations in lit hospheric t hickness than mantle viscosi ty, and the departure of the infinite time response of the model from t hat of a Ruid body is controlled entirely by the lit hosphere. Another obvious example of the lateral variation present in the real Earth is topography. The predictions of sea-level change presented here consider the relative vertical motions of the geoid and solid surface. In order to convert these motions to horizontal shoreline motion or continental flooding percentages it is necessary to include the contemporaneous topography (or hypsometry ) of the solid surface (eg. Algeo and Seslavinsky, 19%). Strat igraphic const raints on local sea-level change can be converted to vertical sea-surface mot ions by using regional paleeshoreline reconstructions; in general, however, only the sign of the sea-level trend can be recovered from the observations. Since the TPW-induced sea-level signal consists of a distinct ive pattern of transgressions and regressions, the sign of the sea-level trends can be used to assess the impact of TP W on the stratigraphic record. The stratigraphic record is influenced by many processes. For example, the advection of mantle heterogeneities that give rise to a TPW event would also produce a sea-level perturbation. 1 do not include this "direct" contribution in my predictions because the nature of advection is generally unknown and it cannot be uniquely reconstructed from a proposed TPW pat h. This direct effect wouId be important to sea-Ievel trends in certain regions but by considering predictions at a globally distributed set of reasonably stable geographic sites it s hould be possible to isolate the TP W-induced signal. Furt hermore, changes in TPW rate are controlled by the relative values of the maximum and intermediate moments of inertia of the planet and do not necessarily correspond to changes in the rate of mass redistribution (Goldreich and Toomre. 1969). Therefore, despite the fact that a rapid change in the location of the rotation pole is likely driven by mantle convectiont the overall convection pattern may remain fairly stable during this time and the direct effect of this advection on regional sea-level trends may be relatively unimportant. To determine the relative importance of the TPW-induced sea-Ievel effect and the direct sea-level effect associated with convection it would be necessary to combine my Cbapter 7: Discussion and Conclusions 87

~esultswibh it eonveckian dei. The cowettion madek mfd ne& to compute bath the resulting change in the rotation vector associated with the redistribution of mass (eg. Richards et al., 1997; Steinberger and O'Connell, 1997; Richards et al.. 1999) as well as the the changes in dynamic topography supported by the mantle ffow (eg. Richter, 1973; McKenzie et al.. 1974; Pysklywec, 1998). For relat ively recent t imes it may be possi bly to match the observed TPW path with a general convection pattern based upon back- advection of present day mant le density discontinuities. It is well known t hat subduction can produce large relative sea-level changes on long wavelengt hs by tilt ing continental interiors (eg. Mitrovica et al., 1989) and that the mantle ffow associated with subducted slabs can produce changes in dynamic topography, and hence sea level, well away Erom the subduction zone (eg. Pysklywec, 1998). The combination of such effects with the predict ions presented here could t hus help to reconcile modelled sea-Ievel signals wit h the stratigraphic record, especidly in areas influenced by subducting slabs. The effect of subduction on the stratigraphic record is of special interest as TPW has been linked to changes in the pattern of subduction (eg. Spada et al., 1992; Ricard et al., 1993) and if a TPW sea-level signal is to be recognized within the geologic record it will be helpful to distinguish the sea-level signal directly associated with convection. The continued collection and andysis of paleomagnetic data may lead to new proposais of TPW and to the refinement of events that have been suggested. If the TPW suggested by new paleomagnetic data is sufficientiy rapid then the associated sea-level signal may be geologically significant. As in Chapters 5 and 6, the stratigraphic record of sea-level change may be used to test the proposed event if there is debate on the interpretation of the paleomagnetic data. Alternatively, if the TPW event is conclusively established on the basis of the paleomagnetic data, then the associated sea-level signal will be valuable in the interpretation of the stratigraphic record. Improvements in the observationar constraints on regiond sea-Ievel change would also be of great value to the study of TPW. The correct distribution of sites is quite important if one is to isolate a TPW-induced signal. For example, the North Atlantic bias seen in the site locations of Vail et al. ( 1977) make it very difficult to use t hat data set, although extensive, to analyze the contribution of TPW to Cretaceous-Tertiary sea-level trends. Of course the suitability of a site for the determination of long-term sea-level trends is controiled by many factors, and it may not be possible to obtain a site distribution that is geographically advantageous for the study of TPW. It may however be possible to consider differential flooding histories wit hin a single continent. Both the timing and magnitude of TPW-induced sea-level oscillations is influenced by the site location. For example, consider an IITPW event in which a continent Chapter 7: Discussion and Conclusions

moves fpom mid-latitudes acmsl, the totehioud eqaator. hmight amsider two sites located on this continent with initid rotational CO-latitudes of 30" and 60'; both sites would initially experience a TPW-induced sea-level rise which changes to a sea-level fa11 over the course of the event. However, the timing of the sea-level trend reversal would be different for each site, and the more northerly site would experience a larger transgression larger than the more sout herly site. As discussed above it would be necessary to include the continental topography to establish the continental inundation associated wit h such a scenario, however the timing of the transgressive-regressive reversa1 should not depend on the to~ographyand thus represents a promising Iine of study. The planet Earth is a dynamic body which is constantly deformed by processes that act on a wide range of timescales. Changes in the planetary rotation vector are one result of planetary deformation, however rotational changes deform the planet in turn. In this thesis 1 have shown that the deformation induced by long timescale changes in rotation may have an important influence on the geologic record, specifically by producing long-term sea-level variations. Constraining the history of TPW would also constrain the geodynamic rnechanisrn which drives the rotat ional variations; on the t imescales considered wit hin t his thesis it is mantle convection, and thus mantle heterogeneity? t hat is ultimately responsible for TPW. By studying records of long-term sea-level change it rnay be ~ossibleto test proposals of large and rapid TPW events based on paleomagnetic evidence. Algeo, T. J., and K. B. Seslavinsky, 1995. The Paleozoic world: continental flooding, hypsometry, and sealevel, .4 m. J. Sei., 295, 787-822.

Andrews. J. A., 1983. True polar wander: An analysis of Cenozoic and Mesozoic pdeo- magnetic poles, J. Geophys. Res., 90, 7737-7750.

Besse, J., and V. Courtillot, 1991. Revised and synthetic apparent polar wander paths of the African, Eurasian, North American and Indian Plates, and true polar wander since 200 Ma, J. Geophys. Res., 96, 4029-4050.

Bills, B. G., and T. S. James, 1996. Late Quaternary variations in relative sea level due to glacial cycle polar wander, Geophys. Res. Lett., 23, 3023-3026.

Bowring, S. A., and D. H. Erwin, 1998. A new look at evolutionary rates in deep time: unit ing paleont ology and high-precision geochronology, GSA Today, 8 (9) , 1-8.

Burgers, J., 1955. Rotational motion of a sphere subject to viscoelastic deformation, 1. 2, 3, Ned. Akad. Wetenschap. Proc., 58. 219-337.

Chen, Y. D., and S. F. Liu, 1996. Precise U-Pb zircon dating of post-D2 rneta-dolerite: constraints for rapid tectonic development of southern Adelaide Fold Belt during the Cambrian, J. Ceol. Soc. London, 153, 53-90.

Christensen, W. K., 1984. The Albian to Maastrichtian of southern Sweden and Born- holm, Denmark: a review, Cret. Res., 5, 313-327.

Clairaut, A. C.. 1743. Theorie de la figure de fa Terre, tirée des principes de f 'hydrostatique (in French), Paris.

Clark, R. A., 1983. Crust and uppermost made structure of the Iceland-Faroes region from Rayleigh wave group veloci ty dispersion, Geophys. J. R. Astron. Soc., 72. 255-264.

Coffin, M. C., L. M. Gahagan, L. A. Lawver, T.-Y. Lee, and E. Rosencrantz. 1992. Atlas of Wesozoic/Cenozoic reconstructions (200 bfa to present day), PLATES progress report No. 1-0192, Tech. Rep. Inst. Geophys. Uniu. Texas, 122.

Cook, P. J., 1988. Palaeogeographic .4tlas of Australin: Volume 1, Canibrian, Australian Government Publishing Service, Canberra. Cobtrell, R. D.,and J. A. Th,2ûM tate Maceam trne potar wander: Not so fast, Science, 288,2283a.

Darwin, G. H., 1877. On the influence of geological changes on the Earth's axis of rotation, Phil. Trans. Roy. Soc. London, A, 167, 271-312.

Debayle. E.. and B. L. N. Kennett, 2000. The Australian continental upper rnantle: Structure and deformation inferred frorn surface waves, J. Geophys. Res., 105,25423- 25450.

Derry, L. A., A. J. Kaufrnan, and S. B. Jacobsen, 1992. Sedimentary cycling and envi- ronmental change in the late Proterozoic: evidence from stable radiogenic isotopes, Geochim. Cosmochim. Acta, 56, 131 7-1329. de Sitter, W., 1924. On the Bat tening and the constitution of the Eart h, Bull. Astr. Inst. Neth., 2, 97-108.

DiVenere, V., and D. V. Kent, 1999. Are the Pacific and IndeAtlantic hotspots fixed? Testing the plate circuit through Antarctica, Earth Planet. Sci. Lett., 170, 105-117.

Dziewonski, A. M, and D. L. Anderson, 1981. Preliminary reference Earth mode1 (PREM), Phys. Earth Planet. Inter., 25, 297-356.

Eardley, A. J., 1964. Polar rise and equatorial fd1 of sea level, Amer. Sci., 42. 488-497.

Evans, D. A., 1998. True polar wander, a supercontinental legacy, Earth Planet. Sci. Lett., 157, 1-8.

Evans, D. A., R. L. Ripperdan, and J. L. Kirschvink. 1998. Polar wander and the Cam- brian (response), Science, 279, 9 (correction p. 307).

Evans, J. R.. and 1. S. Sacks, 1979. Deep structure of the Iceland Plateau. J. Geophys. Res., 84, 6859-6866.

Farrell, W. E., and J. A. Clark. 1976. On postglacial sea Level, Geophys. J. R. ..lstron. SOC..46, 647-667.

Fischer, A. C., 1951. Climate oscillations in the biosphere, in M. H. Nitecki. ed.. Biotic Crises in Ecological and Euolutionary Time, Acadernic Press, New York, 102-131.

Fisher, D., 1974. Some more rernarks on polar wandering, J. Geophys. Res.. 79, 4041- 4045. References 91

Flatté, S. M., 1965. Effects of chairgesirr ttieEmttr'srotation rate on sea tevef, J. Geophys. Res., 70, 5189-5191.

Forsyth, D. W., 1977. The evolution of the upper mantle beneath mid-ocean ridges, Tectonophysics, 38, 89-1 18.

Forte. A. M.. .4. M. Dziewonski, and R. J. O'Connell, 1995. Continent-ocean chernical heterogeneity in the mantle based on seismic tornography, Science, 268, 386-388.

Forte, A. M., A. M. Dziewonski, and R. L. Woodward, 1993. Aspherical structure OF the mantle, tectonic plate motions, nonhydrostatic geoid, and topography of the core- mantle-boundary, in J.-L. Le Mouël, D. E. Smylie, and T. Herring, eds., Dynamics of the Earth's Deep Interior and Earth Rotation, AGU Geophys. Monogr. Ser., 72, 135-166.

Forte, A. M., and J. .Y. Mitrovica, 1996. New inferences of mantle viscosity from joint inversion of long-wavelengt h mant le convection and post-glacial rebound data. Geoph ys. Res. Lett., 23, 1147-1150.

Forte, A. M., and J. X. Mitrovica, 1997. A resonance in the Earth's obliquity and pre- cession ^ver the pa~t20 Myr drken by made convection, :Wurc, 390, 676 630.

Fowler, C. M. R., 1990. The Solid Earth: An introduction to Global Ceophysics, Cam- bridge University Press. Cambridge.

Frakes, L. A., D. Burger, M. Apthorpe, J. Wiseman, M. Dettmann. N. Alley. R. Flint, D. 1. Gravestock. N. Ludbrook, J. Backhouse, S. Skwarko, V. Scheibnerova, A. McMinn, P. S. Moore, B. R. Bolton, J. G. Douglas, R. Christ, hl. Wade, R. E. Molnar. B. McGowran, B. E. Balme, R. A. Day, 1987. Australian Cretaceous shorelines, stage by stage, Palaeogeogr. Pnlaeoclimatol. Palaeoecol., 59, 3 1-45.

Gold, TmT 19.53. Instability of the Earth's axis of rotation, Nature, 175, 526-529.

Goldreich, P., and A. Toomre, 1969. Some remarks on polar wandering, J. Geophys. Res., 74, 2555-2567.

Gordon, R. G., 1987. Polar wandering and paleomagnetism. Ann. Rev. Earth Planet. Sei., 15, 567-593.

Gradstein, F. M., and J. G. Ogg, 1996. A Phanerozoic time scale, Episodes, 19. 3-5.

Grand, S. P., 1987. Tomographie inversion for shear velocity beneath the North American References

plate, J. Geoplrys. Re.,02, 1406544040.

Gravestock. D. I., 1995. Early and Middle Palaeozoic, in Drexel, J. F. and W. V. Preiss, eds., The Geology oj South Australia, Volume 2, The Phanerosoic, Ceol. Surv. S. A ust. Bull., 54, 3-6 1.

Grotzinger. J. P.. S. A. Bowring. B. 2. Saylor. and A. J. Kaufman, 1995. Biostratigraphic and geochronologic constraints on early animal evolut ion, Science, 270, 598-604.

Gurnis, M., 1990. Ridge spreading, subduction, and sea level Auct uations, Science, 250, 970-9'72.

Gurnis, M., 1993a. Depressed continental hypsometry behind ocean t renches, a due to subduction controls on sea-level change, Geology, 21, 29-32.

Gurnis, M., 199313. Phanerozoic marine inundation of continents driven by dynamic topography above subducting slabs, Nature, 364, 589-593.

Hager, B. H., 1980. Eustatic sea level and spreading rate are not simply related, EOS Trans. AGU, 61, 374.

Haner.y B. H., 1984. Siibdiicted slahs and thr g~oirl:constraints on mantle rheol~gyancl flow, J. Geoph ys. Res., 89, 6003-6015.

Hallam, A., 1963. Major epeirogenic and eustatic changes since the Cretaceous and their possible relationship to crustal structure. Amer. J. Science. 261, 397-423.

Hallam, A., 1978. Eustatic cycles in the Jurassic. Palaeogeogr. Palaeoclirnatol. Pafaeoecol., 23, 1-32.

Han, D., and J. Wahr. 1989. Post-glacial rebound analysis for a rotating Earth, in Cohen, S.. and P. Vanicek, eds.? Slow Defornations and Transmission of Stress in the Earth, AGU Geophys. Mono. Ser., 49, 1-6.

Harrison, C. G. A.. and T. Lindh, 1952. Comparison between the hot spot and geomag- net ic field reference frarnes, ivatttre, 300, 251-252.

Haq, B. U., J. Hardenbol, and P. R. Vail, 1987. Chronology of fluctuating sea levels since the Triassic, Science, 235, 1156-1 16%

Haubrich. R. A., and W. Munk, 1959. The pole tide, J. Geophys. Res., 64, 2373-2388.

Hay, W. W., R. DeConto, C. N. Wold, K. M. Wilson, S. Voigt, M. Shulz, -4. Wold-Rossby, Re ferea ces 93

W.-C. Dullo, A. B. Ronov, A. N. Bdttbotrskp, adE. Saedhg, 1999. Ahmative global Cretaceous paleogeography, in Barrera, E., and C. Johnson, eds., The Evolution of Cretaceous Ocean/Climute Systems, Ceol. Soc. Am. Spec. Pap., 332, 1-47.

Hays, J. D., and W. C. Pitman III, 1973. Lithospheric plate motion, sea level changes and climatic and ecological consequences, Nature, 246, 18-32.

Hoffman, P. F., A. J. Kaufman, G. P. Halverson, and D. P. Schrag, 1998. A Neoproterozoic snowball earth, Science, 281, 1342-1346.

Inglis, D., 1957. Shifting of the Earth's axis of rotation: Reu. Mod. Phys., 29, 9-19.

Irving, E., 1983. Fragmentation and assembly of the coutinents, mid-Carboniferous to present , Geophys. Suni., 5?299-333.

Jeffreys, H., 1963. On the hydrostatic theory of the figure of the Earth, Geophys. J. R. ristr. Soc., 8, 196-202.

Jordan, T. H., 1975. The continental tectonosphere, Rev. Geophys. Space Phys., 13, 1-12.

Kauffman? E. C.: 1977. Cwlogiral and hiolopical ouerview: Western Interior Crcteccous basins, Mount. Geol., 14, 75-99.

Kauffman, E. C., 1984. Paleobiogeography and 'evol utionary response dynarnic in the Cretaceous Western Interior seaway of North America, Ceol. rlssoc. Can. Spec. Pap.. 27, 273-306.

King Jr., D. T., 1994. Upper Cretaceous depositional sequences in the Alabama Gulf Coastal Plain: t heir characteristics, origin, and constituent clastic aquifers, J. Sed. Res., B64, 255-265.

Kirschvink, J. L., R. L. Ripperdan, and D. A. Evans, 1997. Evidence for a large-scale reorganization of Early Cambrian continental masses by inertial interchange true polar wander, Science. 277, 54 1-545.

Kopal, Z., 1960. Figures O/ Equilibn'um of Celestial Bodies, The University of Wisconsin Press, Madison.

Krishna, M. R., S. Chaud. and C. Subrahrnanyam, 2000. Gravity anomalies, sediment loading and lithospheric flexure associated with the Krishna-Godavari basin, eastern continental margin of India, Earth Planel. Sci. Lett.. 175, 23-232. References 94

tarnbeek, K.; 1980. The Earth 's Viabte Rotation: Geophysical causes and consequences, Cambridge University Press, Cambridge.

Lambeck, K., C. Smither, and P. Johoston, 1998. Sea-level change, glacial rebound and mantle viscosity for northern Europe, Geophys. J. Int., 134, 102-144.

Landing, E., S. A. Bowring, K. L. Davidek, S. R. Westrop, G. Geyer, and W. Heldmaier, 1998. Duration of the early Carnbrian: U-Pb ages of volcanic ashes from Avalon and Gondwana, Can. J. Earth Sci., 35, 329-338.

Leeds. A. R., 1975. Lithospheric thickness in the Western Pacific, Phys. Earth Planet. Int., 11, 61-64.

Liverrnore, R. A., F. J. Vine, and A. G. Smith, 1984. Plate motions and the geomagnetic field, II, Jurassic to Tertiary, Geophys. J. R. Astron. Soc., 79. 939-961.

Magaritz, M., W. T. Holser, and J. L. Kirschvink, 1986. Carbon-isotope events across the Precambrianfcambrian boundary on the Siberian Platform, Xature, 320, 255-259.

Mancini, E. A., T. M. Puckett, and B. H. Tew, 1996. Integrated biostratigraphic and sequence stratigraphic framework for Upper Cretaceous st rata of the eastern Gulf Coastal Plain, USA, Cret. Res., 17, 643-669.

Marcano, M. C., R. Van der Voo, and C. Mac Niocaill, 1999. True polar wander during the Permo-Triassic, Geodynamics, 28, 75-95.

Markowitz, W., 1970. Sudden changes in rotational acceleration of the Earth and secular motion of the pole, in Mansinha, L., D. E. Smyth, and A. E. Beck. eds., Earthquake Displacernent Fields and the Rotation of the Earth, D. Reidel. Dordrecht. 69-8 1.

McElhinny, M., and J. Lock, 1997. IA C.4 Palae~rna~netieDatabase? Version 3.3.

McKenzie, D. P., J. M. Roberts. and N. 0. Weiss, 1974. Convection in the Earth's mantle:

towards a numerical solut ion, J. O/ Ffuid Mech., 62, 463-538.

McMenamin. M. A. S.. and D. L. S. McMenamin, 1990. The Emergence of Animals: The Cambn'an Breakthrough, Columbia University Press, New York.

Meert. J. G., 1999. ..\ paleornagnetic analysis of Cambrian true polar wander. Earth Pfanet. Sei. Lett., 168: 1.31-144.

MerriIl, R. T.. and LI. W. McElhinny, 1983. The Earth's Magnetic Field: [ts History, Origin and Planetary Perspective, Academic, San Diego. Mi&, A. D., 1490. Ptineiptes of Sehimmtary Bmin Rnatgsis, 2nd editiorq Sprhger- Verlag, New York.

Miller, S. P., and C. Wunsch, 1973. The pole tide, Natctre, 246, 9 1-102.

Milne, G. A., and J. X. Mitrovica, 1996. Postglacial sea-level change on a rotating Eart h: first results from a gravitationally self-consistent sea-level equation, Geoph ys. J. Int., 126, F13-FSO.

Milne, G. A.. and J. X. Mitrovica, 1998. Postglacial sea-level change on a rotating Earth, Geophys. J. Int., 133, 1-19.

Mitrovica, J. X.. 1996. Haskell (19351 revisited, J. Geophys. Res., 101, 55.5-569.

Mitrovica, J. K., C. Beaumont, and G. T. Jarvis. 1989. Tilting of continental interiors by the dynamical effects of subduction, Tectonics, 8, 1079-1094.

Mitrovica, J. X., and A. M. Forte, 1997. Radial profile of mantle viscosity: Rcsults from the joint inversion of convection and postglacial rebound observables, J. Geoph ys. Res., 102, 275 l-?ï69.

Mitrovicat .J. X.: .J. E. kloiind, R. N. Pysklyw~r,and C.A. Milne, 3000. Sea-leuel change on a dynamic Earth, in Baschi, E., G. Ekstrom, and A. Morelli. eds., Problems in Geophysics for the New iWillennium: A Collection of Papers in Honor of Adam !CI. Dtie wonski, Edit rice Compositori, Bologna, 499-529.

Mitrovica, J. X., and W. R. Peltier, 1991. On postglacial geoid subsidence over the equatorial oceans, J. Geophys. Res., 96, 2OO53-2OOï 1.

Mitrovica. J. X.. R. N. Pysklywec. C. Beaumont, and A. Rutty, 1996. The Devonian to Permian tilting of the Russian platform: an example of subduction controlled long- wavelength tilting of continents, J. Geodyn., 22, 79-96.

Morgan. W. J., 1983. Hotspot tracks and the early rifting of the Atlantic, Tectonophysics. 94, 123-139.

Morner, NA., 1981. Revolution in Cretaceous sea-level analysis, Geology. 9, 344-346.

Mound, J. E., and J. X. àlitrovica. 1998. True polar wander as a mechanism for second- order sea-level variations, Science, 279, 534-537.

Mound, J. E., J. X. Mitrovica, D. A. D. Evans, and J. L. Kirschvink, 1999. A sea-level test for inertial interchange true polar wander events, Geophys. J. [nt., 136, F5-F10. Refereaces 96

Mound, J. E., J. X. Miirovica, and 6. A. PirEihe, 2ûûh Sea-kvd and tme polar wander during the Late Cretaceous, Geophys. Res. Lett., 28, 2057-9060.

Müller, R. D., J.-Y. Royer, and L. A. Lawver, 1993. Revised plate motions relative to the hotspots from combined Atlantic and Indian Ocean Lotspot tracks, Geoiogy. 21, 275-278.

Munk, W. H., and G. J. F. MacDonald, 1960. The Rotation of the Earth: .A Geophysicaf Discussion, Cambridge University Press, Cambridge.

Nakada, M., and K. Lambeck, 1989. Late Pleistocene and Holocene sea-level change in the Aust ralian region and mant le rheology, Geophys. J. Int., 96, 197-Ei 17.

Nakiboglu, S. M., 1979. Hydrostatic figure and related properties of the Earth. Geophys. J. R. Astron. Soc.. 57, 639-648.

Nakiboglu, S. M., 1982. Hydrostatic t heory of the Earth and its mechanical implications, Phys. Earth Planet. Int., 28, 302-3 1 1.

Narbonne, G. M., 1998. The Ediacara biota; a terminal Neoproterozoic experiment in the evolution of life, GSA Today, 8(2), 1-6.

Narbonne, G. M., A. J. Kaufman. and A. H. Knoll, 1994. Integrated chemostratigraphy and biost rat igraphy of the Windermere supergroup, Nort hwestern Canada: Implica- tions for Neoproterozoic correlat ions and the early evolut ion of animals, Geol. Soc. .h.Bull., 106, 1281-1292.

Obradovich, J. D., 1993. .A Cretaceous time scale, in Caldwell, W. G. E.. and E. G. Kauffman, eds.. Euofution of the Western lnterior Basin. Ceol. Assoc. Can.. Spec. Pap, 39-3'19-396.

Panasyuk, S. V., and B. H. Hager. 2000. Inversion for made viscosity profiles constrained by dynarnic topography and the geoid, and t heir estimated errors. Geophys. J. Int.. 143, 511-836.

Pannella, G., 1976. Geophysicai inferences from stromatolite laminat ion. in Walter, M. R.. ed., Strornatolites. Elsevier, Amsterdam, 673-655.

Payenberg, T. H. D.. D. R. Braman. D. W. Davis, and A. D. Miall, 2000. Geochronolog- ical andysis of the Pakowki and Claggett marine incursions in southern Alberta and sout hem Montana. Proc. GeoCanada-2000. Peltier, W. R., 1974. The mipnke respmse of a Maxwett Earth, Reu. Ceophys., 12, 649-669.

Peltier, W. R., 1976. Glacial-isostatic adjustment, II, The inverse problem, Geophys. J. R. Astron. Soc., 46, 669-705.

Peltier, W. R., 1998. Postglacial variations in the level of the sea: Implications for climate dynamics and solid-Eart h geophysics, Rev. Geophys., 36, 603-689.

Peltier. W. R., and J. T. Andrews, 1976. Ice age paleotopography, Science? 265, 195-201.

Pilkington, M., 1991. Mapping elastic lit hospheric t hickness variations in Canada, Tectono- ph ysics, 190, 283-297.

Pitman, W. C., III, 1978. Relationship between eustacy and stratigraphie sequences of passive margins, Geol. Soc. Amer. Bull., 89, 1389-1403.

Prévot, M.. E. Mattern, P. Camps, and M. DaigniBres, 2000. Evidence for a 20' tilting of the Earth's rotation axis 110 million years ago, Earth Planet. Sci. Lett., 179. 517-528.

Pyrklywec, R. Y., 1998. Made flou' and the geologicd record: Dynamic mcchanisms for continental epeirogeny, Ph. D. Thesis, University of Toronto.

Quilty, P. G., 1980. Sedimentation cycles in the Cretaceous and Cenozoic of Western Australia, Tectonophysics, 63, 349-366.

Reyment, R. A., and R. V. Dingle, 1987. Palaeogeography of Africa during the Cretaceous period, Palaeogeogr. Palaeoclimatol. Palaeoecol., 59, 69-91.

Riad, S., and H. A. El- Etr, 19S5. Bouguer anomalies and lithosphete-crustal t hickness in Uganda, J. Geodyn., 3, 169-186.

Ricard Y., G. Spada. and R. Sabadini, 1993. Polar wandering of a dynamic Earth, Geophys. J. Int., 1 13, 284-298.

Ricard, Y., and B. Wuming, 1991. Inferring the viscosity and the 3-D density structure of the mantle from geoid, topography and plate velocities. Geophys. J. Int., 105, 561-571.

Richards, M. A., H.-P. Bunge, Y. Ricard, and J. R. Baumgardner, 1999. Polar wandering in made convection models, Geophys. Res. Lett., 26, 1777-1 780. Richards, M. A., and- R. W. Griffith, 1988. Defîectim of plumes by made shear Bow: Experimental results and a simple t heory, Geoph ys. J. Int., 94, 367-376.

Richards, M. A., and B. H. Hager, 1984. Geoid anomalies in a dynamic Earth, J. Geophys. Res., 89,5987-6003.

Richards, M. A., Y. Ricard, C. Lithgow-Bertelloni, G. Spada, and R. Sabadini, 1997. An explanation for Earth's long-term rotational stability, Science, 275, 372-315.

Richter, F. R., 1973. Dynamical models for sea floor spreading, Rev. Ceophys. Space Phys.. 11, 223-387.

Ronov, A. B., V. E. Khain, and K. B. Seslavinsky, 1984. Atlas of Lithological-Paleogeo- graphical Maps of the World: Late Precarnbn'an and Paleozoic of Continents (in Rus- sian), USSR Academy of Sciences, Leningrad.

Rosenberg, G. D., and S. K. Runcorn, eds., 1973. Growth Rhgthms and the History O/ the Earth 's Rotation, Wiley, London.

Royer, J.-Y., R. D. hlüller, L. M. Gahagan, L. A. Lawver, C. L. Mayes, D. Nürmer. and J. G. Sclater, 1992. A global iscxhron chart, Tech. Rep. Inst. Geophys. Univ. Texas, Il?.

Russell M., and M. Gurnis, 1994. The planform of epeirogeny: vertical motions of Aus- tralia during the Cretaceous, Basin Res., 6. 63-76.

Sabadini, R., C. Doglioni, and D. A. Yuen, 1990. Eustatic sea level fluctuations induced by polar wander, Nature. 345, 708-710.

Sabadini, R., and L. L. A. Vermeersen, 1997. Ice age cycles: Earth's rotation instabilities and sea-level changes, Geoph ys. Res. Lett., 24, 304 1-3044.

Sager, W. W. and A. P. Koppers, 2000a. Late Cretaceous polar wander of the Pacific plate: Evidence of a rapid true polar wander event, Science, 287, $55-459.

Sager, W. W. and A. P. Koppers, 3000b. Late Cretaceous true polar wander: ?lot so fast (response), Science, 288. 22S3a.

Sahagian, D., O. Pinous, A. Olferiev, and V. Zakharov, 1996. Eustatic curve for the Middle Jurassic-Cretaceous based on Russian Platfom and Siberian stratigraphy, Am. Assoc. Pet. Geol. Bull., 80, 1433-145s.

Schopf, J. W., C. Klein, and D. Des Marais, eds., 1992. The Proterozoic Biosphere: A Re ferea ces

Schuchert, C., 1955. Atlas of Paleogeographic Maps of North America, Vol. xi, Wiley. New York.

Sloss, L. L., 1963. Sequences in the cratonic interior of North America, Geol. Soc. Amer. Bull., 74, 93-1 14.

Spada. G.,Y. Ricard, and R. Sabadini, 1992. Excitation of true polar cvander by subduc- t ion, iVat ure, 360, 453-454.

Steinberger. B. M., 1996. Motion of hotspots and changes of the Earth's rotation axis caused by a convecting mantle, Ph.D. Thesis, Harvard University.

Steinberger, B., and R. J. O'Connell, 1997. Changes of the Earth's rotation axis owing to advection of mantle density heterogenei ties, Nature, 387, 169-1 73.

Stoyko, A., 1968. Mouvement séculaire du pole et la variation des latitudes des stations du SIL (in French), in Markowitz, W., and B. Guinot, eds., Continental Drift,Secnlar Motion of the Pole and Rotation of the Earth, D. Reidel, Dordrecht, 52-56,

Suess. E.. 1906. The Face of the Enrth, Clar~dnnPress, Oxford.

Tarduno, J. A., and R. D. Cottrell, 1997. Paleomagnetic evidence for motion of the Hawaiian hotspot during Format ion of the Ernperor seamounts. Earth Planet. Sci. Lett., 153, 171-180.

Tarduno, J. A., and J. Gee, 1995. Large-scale motion between Pacific and Atlantic hotspots, Nature, 378, 477-480.

Tarduno, J. A., and A. V. Smirnov, 2001. Stability of the Earth with respect to the spin axis for the last 130 million years, Earth Planet. Sci. Lett., 184, 549-533.

Torsvik, T. H., J. G. Meert, and M. A. Srnethurst, 1998. Polar wander and the Cambrian, Science, 279, 9 (correction p. 307).

Tromp, J.? and J. X. Mitrovica? 1999a. Surface loading of a viscoelastic planet - I. General theory. Geophys. J. [nt.. 137. 847-853.

Tromp, J.. and J. X. Mitrovica, 1999b. Surface loading of a viscoelastic planet - II. Spherical models, Geophys. J. [nt., 137, 856-872.

Tromp, J., and J. X. Mitrovicao 2000. Surface loading of a viscoelastic planet - III. Turcotte, D. L., D. C. McAdoo, and J. G. Caldwell, 1978. An elastic-perfectly plastic analysis of the bending of the Iithosphere at a trench, Tectonophysics. 47, 193-205.

Turcotte, D. L., and G. Schubert. 1982. Geodynamics: Applications of Continuum Physics to Ceologtcal Problems. John Wiley k Sons. New York.

Vail, P. R., R. M. Mitchum, and S. Thompson III, 1977. Seisrnic stratigraphy and global changes of sea level, Part 4: Global cycles of relative changes of sea level, Amer. Assoc. Petr. Geol. !l/lem., 26, 83-97.

Van Andel, T. H., 1994. New Mews on an Old Planet: A Hisiory of Global Change. 2nd edition, Cambridge University Press, Cambridge.

Walter, M. R., J. J. Veevers, C. R. Calver, and K. Grey, 1995. Neoproterozoic stratigraphy of the Centralian Superbasin, Australia, Precambr. Res., 74, 173-195.

Watts, A. B., and K. G. Cox, 1989. The Deccan Traps: an interpretation in terms of progressive lit hospheric Rexure in response to a migrating load, Earih Planet. Sei. Lett., 93, 55-97.

Watts, A. B., and S. F. Daly, 1981. Long wavelength gravity and topography anomalies, Ann. Rev. Earth Planet. Sci., 9, 415-448.

Williams, G. E., 1990. Tidal rhythmites: Key to the history of the Earth's rotation and the lunar orbit, J. Phys. Earth, 38, 475-491.

Williams, G. E., 1997. Precambrian length of day and the didity of tidal rhythmite paleotidal values, Geophzjs. Res. Lett., 24, 421-424.

Wise, D. U., 1974. Continental margins, freeboard and the volumes of continents and oceans through time, in Burk, C. A. and C. L. Drake, eds.. The Geology of Continental iWrrgins, Spri nger-Ver Iag, New York, 45-58.

Woods, M. T., J.-J. Leveque, E. A. Okal, and M. Cara, 1991. Two-station rneasurements of Rayleigh wave group velocity along the Hawaiian Swell, Geophys. Res. Lett.. 18? 105-108.

Wu, P., 1978. The response of the Earth to applied surface rnass loads: Glacial isostat ic adjust ment, M.Sc. Thesis, University of Toronto.

Wu, P., and W. R. Peltier, 1984. Pleistocene deglaciation and the Earth's rotation: a Refereoces

oew sndysis, Geapitys. J. R. Rstrarr. Soc., 76, 753-79L

Yanshin, A. L., 1973. O tak nazyvaemykh t ransgressiyakh i regressiyakh (in Russian), Bulletin ~\.loskovskogo Ob-va Isp ytatelei Prirody, Otdel Geologii. XLVIII, 9-45.

Zorin, Y. A., V. M. Kozhevnikov, M. R. Novoselova, and E. K. Turutanov, 1989. Thick- ness of the lithosphere heneath the Baikal rift zone and adjacent regions, Teetono- ph ysics, 168, 327-337.