The stochastic exit problem for dynamical systems

The problem of "exit against a flow" for dynamical systems subject to small Gaussian white noise excitation is studied. Here the word "flow" refers to the behavior in phase space of the unperturbed system's state variables. "Exit against a flow" occurs if a perturbation causes the phase point to leave a phase space region within which it would normally be confined. In particular, there are two components of the problem of exit against a flow:

i) the mean exit time

ii) the phase-space distribution of exit locations.

When the noise perturbing the dynamical systems is small, the solution of each component of the problem of exit against a flow is, in general, the solution of a singularly perturbed, degenerate elliptic-parabolic boundary value problem.

Singular perturbation techniques are used to express the asymptotic solution in terms of an unknown parameter. The unknown parameter is determined using the solution of the adjoint boundary value problem.

The problem of exit against a flow for several dynamical systems of physical interest is considered, and the mean exit times and distributions of exit positions are calculated. The systems are then simulated numerically, using Monte Carlo techniques, in order to determine the validity of the asymptotic solutions.

Detailed properties of high redshift galaxies

Galaxies evolve throughout the history of the universe from the first star-forming sources, through gas-rich asymmetric structures with rapid star formation rates, to the massive symmetrical stellar systems observed at the present day. Determining the physical processes which drive galaxy formation and evolution is one of the most important questions in observational astrophysics. This thesis presents four projects aimed at improving our understanding of galaxy evolution from detailed measurements of star forming galaxies at high redshift.

We use resolved spectroscopy of gravitationally lensed z ≃ 2 - 3 star forming galaxies to measure their kinematic and star formation properties. The combination of lensing with adaptive optics yields physical resolution of ≃ 100 pc, sufficient to resolve giant Hii regions. We find that ~ 70 % of galaxies in our sample display ordered rotation with high local velocity dispersion indicating turbulent thick disks. The rotating galaxies are gravitationally unstable and are expected to fragment into giant clumps. The size and dynamical mass of giant Hii regions are in agreement with predictions for such clumps indicating that gravitational instability drives the rapid star formation. The remainder of our sample is comprised of ongoing major mergers. Merging galaxies display similar star formation rate, morphology, and local velocity dispersion as isolated sources, but their velocity fields are more chaotic with no coherent rotation.

We measure resolved metallicity in four lensed galaxies at z = 2.0 − 2.4 from optical emission line diagnostics. Three rotating galaxies display radial gradients with higher metallicity at smaller radii, while the fourth is undergoing a merger and has an inverted gradient with lower metallicity at the center. Strong gradients in the rotating galaxies indicate that they are growing inside-out with star formation fueled by accretion of metal-poor gas at large radii. By comparing measured gradients with an appropriate comparison sample at z = 0, we demonstrate that metallicity gradients in isolated galaxies must flatten at later times. The amount of size growth inferred by the gradients is in rough agreement with direct measurements of massive galaxies. We develop a chemical evolution model to interpret these data and conclude that metallicity gradients are established by a gradient in the outflow mass loading factor, combined with radial inflow of metal-enriched gas.

We present the first rest-frame optical spectroscopic survey of a large sample of low-luminosity galaxies at high redshift (L < L*, 1.5 < z < 3.5). This population dominates the star formation density of the universe at high redshifts, yet such galaxies are normally too faint to be studied spectroscopically. We take advantage of strong gravitational lensing magnification to compile observations for a sample of 29 galaxies using modest integration times with the Keck and Palomar telescopes. Balmer emission lines confirm that the sample has a median SFR ∼ 10 M_sun yr^−1 and extends to lower SFR than has been probed by other surveys at similar redshift. We derive the metallicity, dust extinction, SFR, ionization parameter, and dynamical mass from the spectroscopic data, providing the first accurate characterization of the star-forming environment in low-luminosity galaxies at high redshift. For the first time, we directly test the proposal that the relation between galaxy stellar mass, star formation rate, and gas phase metallicity does not evolve. We find lower gas phase metallicity in the high redshift galaxies than in local sources with equivalent stellar mass and star formation rate, arguing against a time-invariant relation. While our result is preliminary and may be biased by measurement errors, this represents an important first measurement that will be further constrained by ongoing analysis of the full data set and by future observations.

We present a study of composite rest-frame ultraviolet spectra of Lyman break galaxies at z = 4 and discuss implications for the distribution of neutral outflowing gas in the circumgalactic medium. In general we find similar spectroscopic trends to those found at z = 3 by earlier surveys. In particular, absorption lines which trace neutral gas are weaker in less evolved galaxies with lower stellar masses, smaller radii, lower luminosity, less dust, and stronger Lyα emission. Typical galaxies are thus expected to have stronger Lyα emission and weaker low-ionization absorption at earlier times, and we indeed find somewhat weaker low-ionization absorption at higher redshifts. In conjunction with earlier results, we argue that the reduced low-ionization absorption is likely caused by lower covering fraction and/or velocity range of outflowing neutral gas at earlier epochs. This result has important implications for the hypothesis that early galaxies were responsible for cosmic reionization. We additionally show that fine structure emission lines are sensitive to the spatial extent of neutral gas, and demonstrate that neutral gas is concentrated at smaller galactocentric radii in higher redshift galaxies.

The results of this thesis present a coherent picture of galaxy evolution at high redshifts 2 ≲ z ≲ 4. Roughly 1/3 of massive star forming galaxies at this period are undergoing major mergers, while the rest are growing inside-out with star formation occurring in gravitationally unstable thick disks. Star formation, stellar mass, and metallicity are limited by outflows which create a circumgalactic medium of metal-enriched material. We conclude by describing some remaining open questions and prospects for improving our understanding of galaxy evolution with future observations of gravitationally lensed galaxies.

A perturbation procedure for nonlinear oscillations (The dynamics of two oscillators with weak nonlinear coupling)

This thesis considers in detail the dynamics of two oscillators with weak nonlinear coupling. There are three classes of such problems: non-resonant, where the Poincaré procedure is valid to the order considered; weakly resonant, where the Poincaré procedure breaks down because small divisors appear (but do not affect the O(1) term) and strongly resonant, where small divisors appear and lead to O(1) corrections. A perturbation method based on Cole's two-timing procedure is introduced. It avoids the small divisor problem in a straightforward manner, gives accurate answers which are valid for long times, and appears capable of handling all three types of problems with no change in the basic approach.

One example of each type is studied with the aid of this procedure: for the nonresonant case the answer is equivalent to the Poincaré result; for the weakly resonant case the analytic form of the answer is found to depend (smoothly) on the difference between the initial energies of the two oscillators; for the strongly resonant case we find that the amplitudes of the two oscillators vary slowly with time as elliptic functions of ϵ t, where ϵ is the (small) coupling parameter.

Our results suggest that, as one might expect, the dynamical behavior of such systems varies smoothly with changes in the ratio of the fundamental frequencies of the two oscillators. Thus the pathological behavior of Whittaker's adelphic integrals as the frequency ratio is varied appears to be due to the fact that Whittaker ignored the small divisor problem. The energy sharing properties of these systems appear to depend strongly on the initial conditions, so that the systems not ergodic.

The perturbation procedure appears to be applicable to a wide variety of other problems in addition to those considered here.

A dynamical systems analysis of vortex pinch-off

Vortex rings constitute the main structure in the wakes of a wide class of swimming and flying animals, as well as in cardiac flows and in the jets generated by some moss and fungi. However, there is a physical limit, determined by an energy maximization principle called the Kelvin-Benjamin principle, to the size that axisymmetric vortex rings can achieve. The existence of this limit is known to lead to the separation of a growing vortex ring from the shear layer feeding it, a process known as `vortex pinch-off', and characterized by the dimensionless vortex formation number. The goal of this thesis is to improve our understanding of vortex pinch-off as it relates to biological propulsion, and to provide future researchers with tools to assist in identifying and predicting pinch-off in biological flows.

To this end, we introduce a method for identifying pinch-off in starting jets using the Lagrangian coherent structures in the flow, and apply this criterion to an experimentally generated starting jet. Since most naturally occurring vortex rings are not circular, we extend the definition of the vortex formation number to include non-axisymmetric vortex rings, and find that the formation number for moderately non-axisymmetric vortices is similar to that of circular vortex rings. This suggests that naturally occurring vortex rings may be modeled as axisymmetric vortex rings. Therefore, we consider the perturbation response of the Norbury family of axisymmetric vortex rings. This family is chosen to model vortex rings of increasing thickness and circulation, and their response to prolate shape perturbations is simulated using contour dynamics. Finally, the response of more realistic models for vortex rings, constructed from experimental data using nested contours, to perturbations which resemble those encountered by forming vortices more closely, is simulated using contour dynamics. In both families of models, a change in response analogous to pinch-off is found as members of the family with progressively thicker cores are considered. We posit that this analogy may be exploited to understand and predict pinch-off in complex biological flows, where current methods are not applicable in practice, and criteria based on the properties of vortex rings alone are necessary.

Tools for the study of dynamical spacetimes

This thesis covers a range of topics in numerical and analytical relativity, centered around introducing tools and methodologies for the study of dynamical spacetimes. The scope of the studies is limited to classical (as opposed to quantum) vacuum spacetimes described by Einstein's general theory of relativity. The numerical works presented here are carried out within the Spectral Einstein Code (SpEC) infrastructure, while analytical calculations extensively utilize Wolfram's Mathematica program.

We begin by examining highly dynamical spacetimes such as binary black hole mergers, which can be investigated using numerical simulations. However, there are difficulties in interpreting the output of such simulations. One difficulty stems from the lack of a canonical coordinate system (henceforth referred to as gauge freedom) and tetrad, against which quantities such as Newman-Penrose Psi_4 (usually interpreted as the gravitational wave part of curvature) should be measured. We tackle this problem in Chapter 2 by introducing a set of geometrically motivated coordinates that are independent of the simulation gauge choice, as well as a quasi-Kinnersley tetrad, also invariant under gauge changes in addition to being optimally suited to the task of gravitational wave extraction.

Another difficulty arises from the need to condense the overwhelming amount of data generated by the numerical simulations. In order to extract physical information in a succinct and transparent manner, one may define a version of gravitational field lines and field strength using spatial projections of the Weyl curvature tensor. Introduction, investigation and utilization of these quantities will constitute the main content in Chapters 3 through 6.

For the last two chapters, we turn to the analytical study of a simpler dynamical spacetime, namely a perturbed Kerr black hole. We will introduce in Chapter 7 a new analytical approximation to the quasi-normal mode (QNM) frequencies, and relate various properties of these modes to wave packets traveling on unstable photon orbits around the black hole. In Chapter 8, we study a bifurcation in the QNM spectrum as the spin of the black hole a approaches extremality.

Control of dynamical systems with temporal logic specifications

This thesis is motivated by safety-critical applications involving autonomous air, ground, and space vehicles carrying out complex tasks in uncertain and adversarial environments. We use temporal logic as a language to formally specify complex tasks and system properties. Temporal logic specifications generalize the classical notions of stability and reachability that are studied in the control and hybrid systems communities. Given a system model and a formal task specification, the goal is to automatically synthesize a control policy for the system that ensures that the system satisfies the specification. This thesis presents novel control policy synthesis algorithms for optimal and robust control of dynamical systems with temporal logic specifications. Furthermore, it introduces algorithms that are efficient and extend to high-dimensional dynamical systems.

The first contribution of this thesis is the generalization of a classical linear temporal logic (LTL) control synthesis approach to optimal and robust control. We show how we can extend automata-based synthesis techniques for discrete abstractions of dynamical systems to create optimal and robust controllers that are guaranteed to satisfy an LTL specification. Such optimal and robust controllers can be computed at little extra computational cost compared to computing a feasible controller.

The second contribution of this thesis addresses the scalability of control synthesis with LTL specifications. A major limitation of the standard automaton-based approach for control with LTL specifications is that the automaton might be doubly-exponential in the size of the LTL specification. We introduce a fragment of LTL for which one can compute feasible control policies in time polynomial in the size of the system and specification. Additionally, we show how to compute optimal control policies for a variety of cost functions, and identify interesting cases when this can be done in polynomial time. These techniques are particularly relevant for online control, as one can guarantee that a feasible solution can be found quickly, and then iteratively improve on the quality as time permits.

The final contribution of this thesis is a set of algorithms for computing feasible trajectories for high-dimensional, nonlinear systems with LTL specifications. These algorithms avoid a potentially computationally-expensive process of computing a discrete abstraction, and instead compute directly on the system's continuous state space. The first method uses an automaton representing the specification to directly encode a series of constrained-reachability subproblems, which can be solved in a modular fashion by using standard techniques. The second method encodes an LTL formula as mixed-integer linear programming constraints on the dynamical system. We demonstrate these approaches with numerical experiments on temporal logic motion planning problems with high-dimensional (10+ states) continuous systems.

Dynamical stability of nascent neutron stars

This thesis presents a study of the dynamical stability of nascent neutron stars resulting from the accretion induced collapse of rapidly rotating white dwarfs.

Chapter 2 and part of Chapter 3 study the equilibrium models for these neutron stars. They are constructed by assuming that the neutron stars have the same masses, angular momenta, and specific angular momentum distributions as the pre-collapse white dwarfs. If the pre-collapse white dwarf is rapidly rotating, the collapsed object will contain a high density central core of size about 20 km, surrounded by a massive accretion torus extending to hundreds of kilometers from the rotation axis. The ratio of the rotational kinetic energy to gravitational binding energy, β, of these neutron stars is all found to be less than 0.27.

Chapter 3 studies the dynamical stability of these neutron stars by numerically evolving the linearized hydrodynamical equations. A dynamical bar-mode instability is observed when the β of the star is greater than the critical value β_d ≈ 0.25. It is expected that the unstable mode will persist until a substantial amount of angular momentum is carried away by gravitational radiation. The detectability of these sources is studied and it is estimated that LIGO II is unlikely to detect them unless the event rate is greater than 10^-6/year/galaxy.

All the calculations on the structure and stability of the neutron stars in Chapters 2 and 3 are carried out using Newtonian hydrodynamics and gravity. Chapter 4 studies the relativistic effects on the structure of these neutron stars. New techniques are developed and used to construct neutron star models to the first post-Newtonian (1PN) order. The structures of the 1PN models are qualitatively similar to the corresponding Newtonian models, but the values of β are somewhat smaller. The maximum β for these 1PN neutron stars is found to be 0.24, which is 8% smaller than the Newtonian result (0.26). However, relativistic effects will also change the critical value β_d. A detailed post-Newtonian stability analysis has yet to be carried out to study the relativistic effects on the dynamical stability of these neutron stars.

A probabilistic treatment of uncertainty in nonlinear dynamical systems

In this work, computationally efficient approximate methods are developed for analyzing uncertain dynamical systems. Uncertainties in both the excitation and the modeling are considered and examples are presented illustrating the accuracy of the proposed approximations.

For nonlinear systems under uncertain excitation, methods are developed to approximate the stationary probability density function and statistical quantities of interest. The methods are based on approximating solutions to the Fokker-Planck equation for the system and differ from traditional methods in which approximate solutions to stochastic differential equations are found. The new methods require little computational effort and examples are presented for which the accuracy of the proposed approximations compare favorably to results obtained by existing methods. The most significant improvements are made in approximating quantities related to the extreme values of the response, such as expected outcrossing rates, which are crucial for evaluating the reliability of the system.

Laplace's method of asymptotic approximation is applied to approximate the probability integrals which arise when analyzing systems with modeling uncertainty. The asymptotic approximation reduces the problem of evaluating a multidimensional integral to solving a minimization problem and the results become asymptotically exact as the uncertainty in the modeling goes to zero. The method is found to provide good approximations for the moments and outcrossing rates for systems with uncertain parameters under stochastic excitation, even when there is a large amount of uncertainty in the parameters. The method is also applied to classical reliability integrals, providing approximations in both the transformed (independently, normally distributed) variables and the original variables. In the transformed variables, the asymptotic approximation yields a very simple formula for approximating the value of SORM integrals. In many cases, it may be computationally expensive to transform the variables, and an approximation is also developed in the original variables. Examples are presented illustrating the accuracy of the approximations and results are compared with existing approximations.

Multipole moments in general relativity and dynamical perturbations of black-hole magnetospheres

This thesis consists of two parts. In Part I, we develop a multipole moment formalism in general relativity and use it to analyze the motion and precession of compact bodies. More specifically, the generic, vacuum, dynamical gravitational field of the exterior universe in the vicinity of a freely moving body is expanded in positive powers of the distance r away from the body's spatial origin (i.e., in the distance r from its timelike-geodesic world line). The expansion coefficients, called "external multipole moments,'' are defined covariantly in terms of the Riemann curvature tensor and its spatial derivatives evaluated on the body's central world line. In a carefully chosen class of de Donder coordinates, the expansion of the external field involves only integral powers of r ; no logarithmic terms occur. The expansion is used to derive higher-order corrections to previously known laws of motion and precession for black holes and other bodies. The resulting laws of motion and precession are expressed in terms of couplings of the time derivatives of the body's quadrupole and octopole moments to the external moments, i.e., to the external curvature and its gradient.

In part II, we study the interaction of magnetohydrodynamic (MHD) waves in a black-hole magnetosphere with the "dragging of inertial frames" effect of the hole's rotation - i.e., with the hole's "gravitomagnetic field." More specifically: we first rewrite the laws of perfect general relativistic magnetohydrodynamics (GRMHD) in 3+1 language in a general spacetime, in terms of quantities (magnetic field, flow velocity, ...) that would be measured by the ''fiducial observers” whose world lines are orthogonal to (arbitrarily chosen) hypersurfaces of constant time. We then specialize to a stationary spacetime and MHD flow with one arbitrary spatial symmetry (e.g., the stationary magnetosphere of a Kerr black hole); and for this spacetime we reduce the GRMHD equations to a set of algebraic equations. The general features of the resulting stationary, symmetric GRMHD magnetospheric solutions are discussed, including the Blandford-Znajek effect in which the gravitomagnetic field interacts with the magnetosphere to produce an outflowing jet. Then in a specific model spacetime with two spatial symmetries, which captures the key features of the Kerr geometry, we derive the GRMHD equations which govern weak, linealized perturbations of a stationary magnetosphere with outflowing jet. These perturbation equations are then Fourier analyzed in time t and in the symmetry coordinate x, and subsequently solved numerically. The numerical solutions describe the interaction of MHD waves with the gravitomagnetic field. It is found that, among other features, when an oscillatory external force is applied to the region of the magnetosphere where plasma (e⁺e^-) is being created, the magnetosphere responds especially strongly at a particular, resonant, driving frequency. The resonant frequency is that for which the perturbations appear to be stationary (time independent) in the common rest frame of the freshly created plasma and the rotating magnetic field lines. The magnetosphere of a rotating black hole, when buffeted by nonaxisymmetric magnetic fields anchored in a surrounding accretion disk, might exhibit an analogous resonance. If so then the hole's outflowing jet might be modulated at resonant frequencies ω=(m/2) Ω_H where m is an integer and Ω_H is the hole's angular velocity.

Equivalent differential equations for nonlinear dynamical systems

A technique for obtaining approximate periodic solutions to nonlinear ordinary differential equations is investigated. The approach is based on defining an equivalent differential equation whose exact periodic solution is known. Emphasis is placed on the mathematical justification of the approach. The relationship between the differential equation error and the solution error is investigated, and, under certain conditions, bounds are obtained on the latter. The technique employed is to consider the equation governing the exact solution error as a two point boundary value problem. Among other things, the analysis indicates that if an exact periodic solution to the original system exists, it is always possible to bound the error by selecting an appropriate equivalent system.

Three equivalence criteria for minimizing the differential equation error are compared, namely, minimum mean square error, minimum mean absolute value error, and minimum maximum absolute value error. The problem is analyzed by way of example, and it is concluded that, on the average, the minimum mean square error is the most appropriate criterion to use.

A comparison is made between the use of linear and cubic auxiliary systems for obtaining approximate solutions. In the examples considered, the cubic system provides noticeable improvement over the linear system in describing periodic response.

A comparison of the present approach to some of the more classical techniques is included. It is shown that certain of the standard approaches where a solution form is assumed can yield erroneous qualitative results.

Transmission matrices and lumped parameter models for continuous systems

The use of transmission matrices and lumped parameter models for describing continuous systems is the subject of this study. Non-uniform continuous systems which play important roles in practical vibration problems, e.g., torsional oscillations in bars, transverse bending vibrations of beams, etc., are of primary importance.

A new approach for deriving closed form transmission matrices is applied to several classes of non-uniform continuous segments of one dimensional and beam systems. A power series expansion method is presented for determining approximate transmission matrices of any order for segments of non-uniform systems whose solutions cannot be found in closed form. This direct series method is shown to give results comparable to those of the improved lumped parameter models for one dimensional systems.

Four types of lumped parameter models are evaluated on the basis of the uniform continuous one dimensional system by comparing the behavior of the frequency root errors. The lumped parameter models which are based upon a close fit to the low frequency approximation of the exact transmission matrix, at the segment level, are shown to be superior. On this basis an improved lumped parameter model is recommended for approximating non-uniform segments. This new model is compared to a uniform segment approximation and error curves are presented for systems whose areas very quadratically and linearly. The effect of varying segment lengths is investigated for one dimensional systems and results indicate very little improvement in comparison to the use of equal length segments. For purposes of completeness, a brief summary of various lumped parameter models and other techniques which have previously been used to approximate the uniform Bernoulli-Euler beam is a given.

Effects produced in GaAs by MeV ion bombardment

Presented in the first part of this thesis is work performed on the ionizing energy beam induced adhesion enhancement of thin (~ 500 Angstrom) Au films on GaAs substrates. The ionizing beam, employed in the present thesis, is the MeV ions (i.e., ¹⁶O, ¹⁹F, and ³⁵Cl), with energies between 1 and 20 MeV. Using the "Scratch" test for adhesion measurement, and ESCA for chemical analysis of the film-substrate interface, the native oxide layer at the interface is shown to play an important role in the adhesion enhancement by the ionizing radiation. A model is discussed which explains the experimental data on the the dependence of adhesion enhancement on the energy which was deposited into electronic processes at the interface. The ESCA data indicate that the chemical bonds (or compounds), which are responsible for the increase in the thin film adherence, are hydroxides rather than oxides.

In the second part of the thesis we present a research performed on the radiation damage in GaAs crystals produced by MeV ions. Lattice parameter dilatation in the surface layers of the GaAs crystals becomes saturated after a high dose bombardment at room temperature. The strain produced by nuclear collisions is shown to relax partially due to electronic excitation (with a functional dependence on the nuclear and electronic stopping power of bombarding ions). Data on the GaAs and GaP crystals suggest that low temperature recovery stage defects produce major crystal distortion. The x-ray rocking curve technique with a dynamical diffraction theory analysis provides the depth distribution of the strain and damage in the MeV ion bombarded crystals.

Stability of parametrically excited differential equations

Sufficient stability criteria for classes of parametrically excited differential equations are developed and applied to example problems of a dynamical nature.

Stability requirements are presented in terms of 1) the modulus of the amplitude of the parametric terms, 2) the modulus of the integral of the parametric terms and 3) the modulus of the derivative of the parametric terms.

The methods employed to show stability are Liapunov’s Direct Method and the Gronwall Lemma. The type of stability is generally referred to as asymptotic stability in the sense of Liapunov.

The results indicate that if the equation of the system with the parametric terms set equal to zero exhibits stability and possesses bounded operators, then the system will be stable under sufficiently small modulus of the parametric terms or sufficiently small modulus of the integral of the parametric terms (high frequency). On the other hand, if the equation of the system exhibits individual stability for all values that the parameter assumes in the time interval, then the actual system will be stable under sufficiently small modulus of the derivative of the parametric terms (slowly varying).

Dynamical models for the Earth's geoid

The Earth's largest geoid anomalies occur at the lowest spherical harmonic degrees, or longest wavelengths, and are primarily the result of mantle convection. Thermal density contrasts due to convection are partially compensated by boundary deformations due to viscous flow whose effects must be included in order to obtain a dynamically consistent model for the geoid. These deformations occur rapidly with respect to the timescale for convection, and we have analytically calculated geoid response kernels for steady-state, viscous, incompressible, self-gravitating, layered Earth models which include the deformation of boundaries due to internal loads. Both the sign and magnitude of geoid anomalies depend strongly upon the viscosity structure of the mantle as well as the possible presence of chemical layering.

Correlations of various global geophysical data sets with the observed geoid can be used to construct theoretical geoid models which constrain the dynamics of mantle convection. Surface features such as topography and plate velocities are not obviously related to the low-degree geoid, with the exception of subduction zones which are characterized by geoid highs (degrees 4-9). Recent models for seismic heterogeneity in the mantle provide additional constraints, and much of the low-degree (2-3) geoid can be attributed to seismically inferred density anomalies in the lower mantle. The Earth's largest geoid highs are underlain by low density material in the lower mantle, thus requiring compensating deformations of the Earth's surface. A dynamical model for whole mantle convection with a low viscosity upper mantle can explain these observations and successfully predicts more than 80% of the observed geoid variance.

Temperature variations associated with density anomalies in the man tie cause lateral viscosity variations whose effects are not included in the analytical models. However, perturbation theory and numerical tests show that broad-scale lateral viscosity variations are much less important than radial variations; in this respect, geoid models, which depend upon steady-state surface deformations, may provide more reliable constraints on mantle structure than inferences from transient phenomena such as postglacial rebound. Stronger, smaller-scale viscosity variations associated with mantle plumes and subducting slabs may be more important. On the basis of numerical modelling of low viscosity plumes, we conclude that the global association of geoid highs (after slab effects are removed) with hotspots and, perhaps, mantle plumes, is the result of hot, upwelling material in the lower mantle; this conclusion does not depend strongly upon plume rheology. The global distribution of hotspots and the dominant, low-degree geoid highs may correspond to a dominant mode of convection stabilized by the ancient Pangean continental assemblage.