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Talk abstract

Nonlinear Continuum Mechanics, Rheology and the Dynamo

Nonlinear Continuum Mechanics, Rheology and the Dynamo

March 18-22, 2002

Mathematics in Geosciences, September 2001 - June 2002

**Bruce
Buffett**
(Department of Earth and Ocean Sciences, University of British
Columbia, 2219 Main Mall, Vancouver, BC, V6T 1Z4 tel: (604)
822-2267) buffett@geop.ubc.ca

**Large-Eddy
Simulations of Convection in the Earth's Core **

Large-eddy simulations (LES) provide a strategy for dealing with flows in which the smallest scales cannot be resolved in numerical calculations. The approach is based on spatial filtering to eliminate the scales that are smaller than the grid spacing. The influence of the subgrid scales must be modeled and several schemes have been proposed, including the Smagorinsky, the multiscale and the similarity methods. We apply each of these methods to the problem of convection in the Earth's core and test the predictions using a direct numerical simulation (DNS) on a finer grid. In order to resolve the smallest dissipative scales in the DNS we are forced to consider only a small volume of the core and assume periodic boundary conditions. The grid in the DNS calculation is a cube with 128x64x32 nodes, oriented so that the z-coordinate is aligned with the rotation axis and the y-coordinate is parallel to an imposed magnetic field. The direction of gravity may be oriented arbitrarily in the x-z plane and several representative cases are considered. Output from the DNS is filtered on to a coarser 32^3 grid for the purpose of testing the LES models. Estimates of the subgrid heat and momentum flux are calculated explicitly using the solution on the finer grid. The results reveal a high degree of anisotropy due to the influences of rotation and the large-scale magnetic field. Comparisons with LES models on the coarser grid are poor when the model is based on a scalar diffusivity or viscosity; this includes the eddy viscosity, Smagorinsky and multiscale models. These (scalar) models are incapable of reproducing the strong anisotropy in the subgrid fluxes. The similarity method is much more succesful in reproducing the anisotropy of the subgrid fluxes. Spatial correlations between the similarity model and the exact subgrid fluxes in the three coordinate directions are typically in excess of 0.8. We incorporate the similarity model into our simulation to extend these calculations to larger scales and discus the implementation.

**Friedrich
H. Busse** (Theoretische Physik IV, Universitaet Bayreuth)
Friedrich.Busse@uni-bayreuth.de

**Convection
Driven Dynamos in Rotating Spherical Shells**

Numerical simulations of the generation of magnetic fields by convection flows in rotating spherical shells have been carried out in collaboration with R. Simitev for the parameter space spanned by the Rayleigh number, Taylor number, Prandtl number, and magnetic Prandtl number. A wide variety of dynamos have been found and their areas of predominance have been mapped in the parameter space. Since the structure of the magnetic field often reflects the character of the convection flow considerable efforts have been expanded to understand the properties of turbulent convection in the absence of a magnetic field. The numerical simulations exhibit coherent structures such as localized convection and relaxation oscillations. Of particular interest are regimes in the parameter space where the magnetostrophic approximation is approximately valid and scaling relationships can be obtained. The difficulty of reaching low magnetic Prandtl numbers casts some doubts on the extrapolation of presently available dynamo models to the case of the Earth´s core.

**Charles
R. Carrigan**
(Flow & Transport Group Leader(L-204) Lawrence Livermore National
Laboratory) carrigan1@llnl.gov

**Viscous
Encapsulation: A Potentially Important Mechanism to Explain
the Occurrence of Effusive Volcanism **

A commonly held view among volcanologists is that volcanic systems involving more than one magma type often erupt lower-silica magmas as a precursor to more silicic ones. A basis for this view is the observed chemical zoning of volcanic rock with the outer layer in a volcanic conduit being composed of the lower silica rock and the central core being made up of the higher silica material. This notion of lower-silica magmas erupting first complicates any associated models of crustal magmatic storage since it is problematic to erupt the more dense, lower-silica magmas before their higher silica counterparts which overlie them in typical models of magma withdrawal. An alternative interpretation of these zoning observations is that both magmas flowed in the dike or conduit together and that viscous segregation or encapsulation occurred within the conduit causing the lower-viscosity, lower-silica magma to viscously decouple the higher-silica magma from the boundary of the conduit. This encapsulation process has been hypothesized to explain chemical zoning observations in volcanic conduits. Both geologic and laboratory evidence is presented to support this interpretation of zoning along with suggestions for more realistic models of magma storage and transport. Through its lubricating effect, the viscous encapsulation mechanism may well be responsible in many instances for allowing the transport of particularly viscous magma across a significant fraction of the cool crust followed by eruption in effusive events that would otherwise not be permitted to occur.

**Anne
Davaille**
(Laboratoire de Dynamique des Systemes Geologiques, IPG Paris)
davaille@ipgp.jussieu.fr

**Thermal
convection in a mantle heterogeneous in viscosity and in density**

Mounting evidence indicates that the Earth's mantle is chemically heterogeneous. To understand the forms that convection might take in such a mantle, we have conducted laboratory experiments on thermochemical convection in a fluid with stratified density and viscosity. Depending on the buoyancy ratio B, two regimes prevail: at high B, convection remains stratified while at low B, hot domes oscillate vertically through the whole tank.

Applied to mantle convection, our experimental results can explain a number of observations on Earth such as hot spots, superswells or the survival of several reservoirs in the mantle. The scaling laws derived from the experimental data base allow now to predict a number of characteristics of those features, such as their geometry, size, time and chemical evolution. In particular, we shall see that 1) density heterogeneities are an efficient way to anchor plumes, and therefore to create relatively fixed hot spots, 2) pulses of activity with characteristic time scale of a 50-500 Myr can be produced by thermochemical convection in the mantle, and 3) because of mixing, no "primitive" reservoir can have survived untouched up to now.

**Andrew
C. Fowler**
(Mathematical Institute, Oxford University) fowler@maths.ox.ac.uk

**Lithospheric
Failure on Venus **

We develop a predictive model which has the ability to explain a postulated style of episodic plate tectonics on Venus, through the periodic occurrence of lithospheric subduction events. Present day incipient subduction zones are associated with the existence of arcuate trenches on the Venusian lithosphere. These trenches resemble terrestrial subduction zones, and occur at the rim of coronae, uplift features thought to be due to deep mantle convective plumes. The model we adopt represents the lithosphere as the thermal boundary layer which lies above a convective plume. We assume a temperature dependent non-linear viscoelastic rheology, and we assume a stress based criterion for plastic yield. In developing this latter criterion, we are led to a re-interpretation of the strength envelope which is commonly used in analysing lithospheric stress, and we propose that the plastic yield strength has meaning (and is finite) below the lithosphere, using behaviour in the Earth as our `laboratory' justification for this view. An inferred yield stress on the Earth is about 300 bars (30 MPa). Our model then shows that a thickening lithosphere becomes progressively more fluid as the stresses induced by the buoyant convective plume become large. Failure occurs when the effective lithosphere viscosity becomes equal to that of the underlying mantle. We show that reasonable expected values of yield stress in the range 100-200 bars (10-20 MPa) for Venusian mantle rocks are consistent within the framework of the model with radii of coronal trenches in the range 100-1200 km, and with the approximate time (200-800 Ma) which they may take to develop.

**Gary
A Glatzmaier** (Earth Sciences Department, University
of California, Santa Cruz) glatz@es.ucsc.edu
http://es.ucsc.edu/~glatz

**Current
Challenges in Dynamo Modeling**

Three-dimensional, dynamically self-consistent, numerical simulations have been used for two decades to study the generation of global magnetic fields in the deep fluid interiors of planets and stars. In particular, the number of geodynamo models has increased significantly within the last five years. These simulations of magnetic field generation by laminar convection have provided considerable insight to the geodynamo process and have produced large-scale fields similar to those observed at the Earth's surface. However, no global convective dynamo simulation has yet been able to afford the spatial resolution required to simulate turbulent convection, which surely must exist in the Earth's low-viscosity liquid core. They have all employed greatly enhanced eddy diffusivities to stabilize the low resolution numerical solutions and crudely account for the transport and mixing by the unresolved turbulence. A grand challenge for the next generation of geodynamo models is to produce a simulation with the thermal and viscous (eddy) diffusivities set no larger than the actual magnetic diffusivity of the Earth's fluid core (2 m^2/s), while using the core's dimensions, mass, rotation rate and heat flow. This would correspond to the Ekman and magnetic Ekman numbers both set to 10^-9 and the Rayleigh number being many orders of magnitude greater than critical. Dynamo models for stars and giant planets present an additional complication: the large variation of density with radius. Two-dimensional calculations will be presented that illustrate the significant effects of low viscous, thermal, and magnetic diffusivities on rotating magneto-convection.

**David
Gubbins**
(School of Earth Sciences, University of Leeds) gubbins@earth.leeds.ac.uk

**Pacific
Secular Variation: A result of hot lower mantle**

The Earth's magnetic field is generated by convection in the liquid core, which in turn is driven by internal sources of buoyancy strongly influenced by rotation, magnetic forces, and the boundary conditions. The core-mantle boundary is an isothermal surface, but convection in the mantle causes variations in the heat flux across the boundary, variations that may exceed the average heat flux out of the core. These variations can influence core convection and place an imprint of lower mantle heat flux onto the geomagnetic field itself. Observational evidence for this comes from the modern field, which is concentrated where lower mantle has high heat flux, beneath the Pacific rim, the paleomagnetic time average for the last 5 million years, and persistent patterns in the transition field during polarity reversals, which appear to have poles which track around the Pacific rim.

Geomagnetic secular variation has been low in historical times. Paleomagnetic results from Hawaii show that this anomaly has persisted for 5 thousand years and probably longer, and is therefore likely to be a permanent feature also associated with heat flux anomalies on the core-mantle boundary. Core convection calculations show that a heat flux boundary condition derived from lower mantle seismic velocities causes convection to be suppressed beneath the Pacific but leaves convection rolls drifting around the Atlantic hemisphere. This is very similar to the appearance of secular variation over the last 400 years, where westward drifting features form near the Pacific rim, drift west, and disappear when they reach the west coast of the Americas. Low Pacific secular variation may therefore be one more result of the lower mantle's influence on the dynamo.

**Yves
Gueguen** (Ecole Normale Superieure, Paris) gueguen@mailhost.geologie.ens.fr

**A
Crackling Crust (une croute craquante) **

Fractures in the crust cover a broad range of scales, from microcracks to large fractures. They control both transport and mechanical properties of rocks, which are of major importance in geology and geophysics. An attempt to clarify our understanding of these properties will be presented as follows. At small scales, the homogeneity of the medium is in general sufficient to assume statistical homogeneity (or equivalently Translational Invariance). Effective Medium Theory allows in that case to derive elastic properties and permeability. As the scale increases however, heterogeneity increases also, so that the assumption of Translational Invariance breaks down, clustering effects are important, and critical thresholds are observed. The assumption of Scale Invariance may be more relevant in such situations than that of Translational Invariance, and various methods inspired from percolation theory can be useful. Elastic wave velocities and permeability of rocks will be discussed using these concepts, and strain localization as well. It will be argued that these physical approaches provide a powerful framework to interpret geophysical data on a sound basis.

**Dominique
Jault** (Laboratoire de Géophysique Interne
et Tectonophysique (LGIT), Centre National de la Recherche Scientifique)
http://www-lgit.obs.ujf-grenoble.fr/users/djault

**Experimental
evidence of nonlinear resonance effects between retrograde precession
and the tilt-over mode within a spheroid**

The Poincare flow (also known as the tilt-over mode) in a precessing cavity filled with water is investigated experimentally. Assuming that the flow is mainly a solid-body rotation, we have used three independent techniques to determine the rotation. Rapid changes in the direction of the axis of the rotation of the fluid for critical values of the rates of precession and rotation of the container are pointed out. A torque approach, which can be generalized to other forcings, shows that this effect is due to a nonlinear resonance between the frequencies of the Poincare mode and of precession. As a result, we can determine the validity domain of current theoretical models of nutations and precessions of planets enclosing a fluid core.

**Dazhi
Jiang**
(Department of Geology, University of Maryland) dzjiang@geol.umd.edu http://www.geol.umd.edu

**Numerical
modeling of the development of kink-bands in anisotropic rocks
**Slides: html
pdf
powerpoint

Video: p20_m10.avi

phi20_m2.avi

phi20_m8.avi

One effective way to constrain the rheology of rocks under natural conditions is to examine structures in rocks. Kink-bands are common structures in well-foliated rocks and anisotropic crystals. We have used anisotropic plastic rheology to numerically model the development of kink-bands. It is shown that as the bulk strength and the degree of anisotropy vary, a variety of deformation mechanisms occur. By examining the geometry of kink-bands in rocks, one can constrain the region in the bulk strength- anisotropy space, the deformation has occurred.

**Chris
A Jones**
(Mathematical Sciences, University of Exeter) C.A.Jones@exeter.ac.uk

**The
Dynamical Regime in the Earth's Core **Slides

An outstanding problem with current geodynamo simulations is that the parameters appropriate to the Earth's core cannot be reached because of numerical difficulties. We can, however, analyse the results available to see whether an asymptotic regime has been reached. Much lower Ekman numbers can be achieved if planar geometry rather than spherical geometry is used. Recent results from a plane layer model with rotation and gravity inclined to each other will be discussed. This model sheds light on how Taylor states are achieved in strongly supercritical dynamo models. The power spectrum and the associated ohmic dissipation will also be considered. Various dynamical regimes are possible, and the magnitude of the heat flux passing through the core is shown to be a key parameter in determining the actual dynamical regime achieved.

**Weijia
Kuang** (Research Associate Professor, Joint Center
for Earth Systems Technology, UMBC Geodesy Branch, Code 926,
NASA GSFC) kuang@bowie.gsfc.nasa.gov

**Multidisciplinary
Studies of Deep Earth: From Geodynamo to Geodesy **Slides:
html
pdf
powerpoint

It has long been known that Earth possesses a magnetic field of internal origin (geomagnetic field). This field is generated and maintained by vigorous convection in the Earth's fluid outer core (geodynamo theory). Recent success in numerical geodynamo modeling has made it possible to analyze the details of dynamical processes in the core and its forcing on solid Earth, such as electromagnetic torque driving the solid inner core rotating relative to the solid mantle, and non-hydrostatic pressure acting on the core-mantle boundary. In parallel, observations on global surface geophysical processes, such as Earth's gravity field and large-scale surface deformation, are reaching to unprecedented level in both accuracy and in long-time measurement coverage, as evidenced by recent and up-coming satellite missions. These advances in both science and technology may provide new opportunities in multi-disciplinary studies on interactions between the solid Earth and the liquid outer core. Two research fields are in particular promising: the influence of large-scale mass redistribution in the core on time-variable gravity field variation, and the effect of non-hydrostatic pressure on deformation of the mantle. Studies of these problems could help us not only on identifying responses of the solid Earth to the forces from the fluid outer core, but also on providing further insights on core dynamical processes from non-geomagnetic, surface observations.

**Vladimir
Lyakhovsky** (Geological Survey of Israel, Jerusalem)
vladi@geos.gsi.gov.il
http://geos.gsi.gov.il/vladi

**Nonlinear
Elasticity, Distributed Damage, and Fracture of Rocks **Slides:
html
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powerpoint

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I present a damage rheology model, which holds a potential for providing a framework for understanding processes of rock deformation such as fracture nucleation, development of process zone at rupture tip, and branching from the main rupture plane. The damage mechanics approach is based on the assumption that the density of micro cracks is uniform over a length scale much larger than the length of a typical crack, yet much smaller than the linear size of the volume considered. For any system with a sufficiently large number of cracks, one can define a representative volume in which the crack density is uniform and introduce an intensive damage variable for this volume. The present model treats two aspects of the physics of damage: (1) A mechanical aspect - the sensitivity of the macroscopic elastic moduli to distributed cracks and to the sense of loading, and (2) a kinetic aspect - the evolution of damage (degradation-recovery of elasticity) in response to loading. Several numerical results reproduce the main features of rock behavior including damage self- organization and localization into a narrow zones and kink angle of the fracture front breakdown under mixed mode loading. The damage model includes post-failure behavior (healing) that allows simulating a stick-slip motion along a narrow zone with localized damage. Being averaged in space and time this stick-slip motion fits experimentally observed relations between slip velocity, normal and shear stress components (RS friction).

**Stephen
Morris**
(Department of Mechanical Engineering, University of California,
Berkeley) morris@newton.berkeley.edu

**On
the olivine-spinel transformation as a rheometer **Slides

Kubo et al (Science, 281, 85-87, 1998) show experimentally that during the growth of a rim of new spinel phase on a grain of olivine, the rheology of the spinel can control the transformation rate. In work in press (Morris, J. Mech. Phys. Solids, 2002), I show that those data can be fitted by a model coupling interface kinetics to the viscoelastic creep required to accomodate the transformation--induced volume change. Because those data cover a limited range of strain rate, they can be fitted by a model in which the creep rate is taken as proportional to deviatoric stress. My theory allows the effective viscosiy of the spinel to be inferred for certain of Kubo's experiments. The viscosity so inferred is, of course, valid only for a limited range of strain rate.

In this talk, I will review the study in press, and then describe analysis in progress which incorporates creep by the actual mechanism of low temperature plasticity occurring in the experiments. The purpose of the new work is to determine the zero temperature yield stress for spinel, and also to predict the variation of transformation-rate with excess pressure.

**W.R
Peltier** (Department of Physics, University of Toronto)
peltier@atmosp.physics.utoronto.ca

**The viscosity of Earth's mantle: Newtonian or non-Newtonian**

Knowledge of the viscosity of Earth's iron-magnesium-silicate mantle is vital insofar as understanding tectonophysical processes is concerned. In particular the strength as well as the "style" of the mantle convection process is fundamentally controlled by the magnitude of the momentum diffuesivity for a given temperature difference between the core-mantle boundary and the Earth's surface. A significant issue furthermore remains as to whether the mechanism by which mantle material "flows" is non-Newtonian, as would be expected given the polycrystalline nature of the material involved, or whether it might be effectively Newtonian in consequence of the low differential stress regime in which mobility occurs.

One possible means of probing the "Newtonian-ness" of Earth's mantle is to employ a variety of solid Earth physical phenomena which depend upon the magnitude of the creep resistance to see whether observations over a significant range of phenomenological timescales are satisfied by the same model of the creep resistance. To this end it proves interesting to compare the mantle viscosity inferred on the basis of the relatively fast timescale glacial isostatic adjustment process to that inferred on the basis of analyses of the process of mantle convection, these two processes differing from one-another in characteristic timescale by five orders of magnitude ( the characteristic timescale of glacial isostatic adjustment is O(1000 yrs) whereas the characteristic timescale of the convection process is O(100,000,000 yrs)).

There are several issues that one must address in attempting to carry out a meaningfull intercomparison of this kind, not the least important of which is that little concensus exists as to the model for viscosity that best reconciles the observations of either of these fundamental geodynamic processes! I will first present the results of a new series of analyses of the radial variation of mantle viscosity based upon anlyses of the observables related to the glacial isostatic adjustment process, analyses which directly probe the relative viability of the two models in the current literature that have been suggested as candidates, one of which has a relatively modest increase of viscosity across the 660 km seismic discontinuity and the competing model that has a much larger increase across this same horizon. This analysis will make use of recent space geodetic constraints as well as absolute gravimeter measurements of g-dot on a traverse across the Canadian Shield. These analyses will be shown to entirely rule out the model with high viscosity contrast across the 660 km discontinuity.

It proves useful to enquire as to whether the model of the depth variation of Newtonian viscosity delivered by these analyses of the process of glacial isostasy is also able to deliver accord with observable properties of the mantle convection process. Neglecting all influence of the surface plates, it is found that a priori models of mantle mixing, in which the cmb temperature is fixed to the high value required by high pressure experimental constraints, inevitably predict anomalously high radial heat transfer unless the flow is assumed to be very strongly layered by the influence of the endothermic phase transformation at 660 km depth. Alternatively, one may assume that the viscosity that governs the process of mantle mixing is approximately one order of magnitude higher than that which governs glacial isostasy. Unless the heat transfer inhibiting influence of the surface plates is more significant than is most often assumed, this constitutes a strong argument that Earth's mantle creeps via a mechanism that is non-Newtonian.

**Thomas
J. Pence**
(Department of Metallurgy Mechanics and Materials Science, Michigan
State University) pence@egr.msu.edu

**A
Multi-field Model for Solid-Solid Phase Transformation**

Co-author Davide Bernardini from University of Rome-La Sapienza.

We present a continuum mechanical framework for modeling crystallographic phase transformations. Scalar field variables for the mass fraction of various crystallographic phases are central to the description as are tensor field variables representing Bain strains associated with transformation. Standard balance laws for stress, energy and entropy are then augmented with additional balance principles for these additional field variables. The constitutive theory involves specification of a free energy function and an entropy production functional. We present a treatment for: the basic structure of the theoretical description, handling of the constraints associated with the additional fields, formulation of free energy functions that deliver physically motivated equilibrium configurations, and formulation of entropy production functionals that deliver physically motivated hysteretic response.

**Yanick
Ricard** (Ecole Normale Supérieure de Lyon, Laboratoirede
géologie, Lyon, France) Yanick.Ricard@ens-lyon.fr

**A
two-phase theory for compaction and damage
**Slides: pdf
postscript

A theoretical model for the dynamics of a simple two-phase mixture is presented. A classical averaging approach combined with symmetry arguments is used to derive the mass, momentum and energy equations for the mixture. Rigorous constraints are used to estimate the form of the averaged stress tensor; it does not involve a bulk viscosity which is often assumed necessary to model compaction. The theory accounts for surficial energy at the interface, and thus pressure differences between phases. We discuss various exemples of compaction or compression of mixture with or without the presence of surface tension. This two-phase theory for compaction and damage employs a nonequilibrium relation between interfacial surface energy, pressure, and viscous deformation and also provides a model for damage (void generation and microcracking) and thus a continuum description of weakening, failure, and shear localization.

**Michael
R. Riedel**
(Institute of Geosciences, University of Potsdam, Germany) miker@geo.uni-potsdam.de

**Plastic
Instabilities as a Possible Physical Mechanism Causing Intermediate-Depth
and Deep-Focus Earthquakes**

Joint work with S. Karato (Department of Geology and Geophysics, Yale University) and D.A. Yuen (Department of Geology and Geophysics, University of Minnesota).

It has been suggested that the occurence of plastic instabilities in the deeper portion of subducting slabs is the responsible mechanism for the generation of deep-focus earthquakes. Heat generated during viscous deformation provides a positive feedback to creep and eventually faulting under high pressure. A similar mechanism could be responsible for the occurence of intermediate-depth earthquakes within portions of the mantle lithosphere, where mechanisms involving dehydration or phase transformations do not apply. Recent detailed receiver function images of the structure of the Japan subduction zone seem to provide support for this notion. First, there is no indication of an existing metastable olivine wedge. Second, the intermediate-depth seismicity seems to be located in the strong and colder portions of the downgoing slab, about 30 km below the oceanic Moho. This suggests that instead of dehydration or phase transformation triggered events, ductile faulting is its predominating cause.

We show that, under certain conditions, a general local criterion for plastic instability can be met for nonlinear power-law creep (dislocation creep) of olivine resp. spinel (below 410 km discontinuity), so that the existence of metastable olivine in the deeper portion of a slab (below 500 km) is not a necessary condition for the generation of deep-focus earthquakes.

**Paul
H. Roberts** (Department of Mathematics and Institute
of Gsophysics and Planetary Physics, University of California,
Los Angeles, Los Angeles, California 90095) roberts@math.ucla.edu

**How
can the energy requirements of the Earth's dynamo be met? **Slides

We
start by reviewing two interesting facts: the geodynamo is expensive
to run, but it may be as old as the Earth. More precisely, Roberts,
Jones and Calderwood (2002) estimate that the energy expenditure
of the geodynamo of order 1TW for ohmic dissipation plus about
5TW to maintain the adiabatic gradient. (1TW=10^{12}
W). Kono and Tanaka conclude from paleomagnetic evidence that
the Earth has possessed a magnetic field, having a strength
within a factor of 3 of its present strength, for at least 3.5
Gyr. Doubtless, these estimates will be modified as new information
becomes available, but are unlikely to be fundamentally changed.
Adopting them as working hypotheses, one is led to some questions
that are curiously difficult to answer without doing damage
to some cherished geophysical notions.

An effort is made to understand how the power requirements of the geodynamo can be met. This involves a new discussion of an old question, `How should the gross thermodynamics of the core be understood?' It is a question that was asked by Gubbins and his associates during the late 70's and that has been reconsidered by them very recently. Thanks to their efforts, the situation has become increasingly well understood. Nevertheless, it is a difficult subject; traps and pitfalls abound. Hopefully, the new discussion given here will not cause even greater confusion. The composition and physical properties of the core are not sufficiently well known for definitive answers to be given. Hopefully the situation will improve as these become increasingly well known, through experimental work and first principles calculations. Even at this stage, however, we are led to further questions, the answers to which may lead to some dissention.

**Gerald
Schubert**
(Department of Earth and Space Sciences, Institute of Geophysics
and Planetary Physics, University of California, Los Angeles,
California) schubert@ucla.edu

**A
Numerical Finite Element Approach to the Solution of the Dynamo
Problem**

Joint work with K. Zhang (School of Mathematical Sciences, University of Exeter, Exeter, England) KZhang@maths.ex.ac.uk, K.H. Chan, and J. Zou (Department of Mathematics, Chinese University of Hong Kong, Hong Kong, China).

We are developing the capability to solve the dynamo problem in spherical geometry using a finite element numerical approach. The aim is to exploit the power of parallel processing to reach parameter values that are more relevant to the geodynamo than have been achieved by spectral methods that use spherical harmonic expansions. We have now completed two stages of the problem. Our progress to date is the subject of this presentation.

In
the first stage, we have focused on the kinematic part of the
problem since that contains some difficulties not previously
dealt with by finite element methods while the finite element
technique has been widely applied to the fluid dynamics of convection.
We will summarize the challenges faced in a finite element solution
of the electromagnetic equations and how we overcame these problems.
We have applied our code to the solution of non-linear, 3-D,
spherical ^{2}
dynamos and will present some results for 2-D and 3-D stationary
and time-dependent dynamos. One particular example will illustrate
how an electrically heterogeneous mantle can modulate the core
dynamo, leading to a vacillating dynamo whose amplitude depends
on the relative phases between the generated magnetic field
and the conductivity anomalies in the mantle.

In the second stage, we have proceeded towards a fully dynamic part of the problem by successfully incorporating into our code a momentum equation containing all important dynamical forces based on Proctor's (1987) model. In this nonlinear, three-dimensional, magnetohydrodynamic dynamo, the large-scale magnetic field is generated by the effect, while the large-scale flow is determined by the balance of the pressure gradient and the coriolis, Lorentz, and viscous forces. The magnetohydrodynamic problem is simply characterized by two parameters, the Ekman number E and the magnetic Reynolds number. Because thermal convection is not involved, the magnitude of E plays a less important role in this problem. This allows us to explore various nonlinear balances in the momentum equation. We find that magnetostrophic balance is achieved at about E = 0.001; the nonlinear flow is nearly two dimensional along the direction of the rotation axis and the dynamo solutions are nearly independent of E. Results of the magnetohydrodynamic dynamo calculations will be presented.

The final stage towards a full dynamo will involve the implementation of the temperature equation and the thermal buoyancy term in the momentum equation. This procedure is technically easiest but computationally most demanding. That effort is underway.

**Slava
Solomatov**
(Department of Physics, New Mexico State University) slava@NMSU.Edu

**Mantle dynamics: Grain size does matter**

The lack of anisotropy in the lower mantle of the Earth suggests that deformation is controlled by diffusion creep or superplasticity. This implies that the viscosity depends on the grain size. Analysis of various kinetic processes in the Earth's mantle suggests that the grain size is mainly controlled by Ostwald ripening of Mg-perovskite, Ca-perovskite and magnesiowustite. The coupling between Ostwald ripening, viscosity and convection strongly affects thermal evolution of the Earth. Depending on the grain growth parameters, the Earth can either quickly forget the initial conditions with temperature and heat loss following the decaying radiogenic heat production (Tozer-type evolution) or the initial conditions can essentially determine the temperature and heat loss (Christensen-type evolution).

**David
J. Stevenson** (Department of Geological and Planetary
Sciences, California Institute of Technology, Caltech 150-21)
djs@gps.caltech.edu

**Conditions
for the Excitation and Maintenance of Planetary Dynamos **Slides

The most likely mechanism for maintaining a planetary dynamo is thermal or compositional convection. It is plausible but not certain that the criterion for a dynamo is not much different from the condition for large scale core convection.I will focus on the recent developments for Mars and Ganymede, and discuss the requirements for convection, especially the role of the overlying mantle and the role of an inner core. Both Venus and Mars may have ceased dynamo generation because of a change in the style or vigor of mantle convection (but at very different epochs). Ganymede may be aided by a high sulfur and potassium content relative to the terrestrial planets, but also has different (low pressure) phase diagrams and thermodynamic relationships, and a different (poorly understood) thermal history. By comparison, the presence of giant planet dynamos is easy to understand.

**Paul
J. Tackley** (Department of Earth and Space Sciences,
University of California, Los Angeles) ptackley@ucla.edu

**Modeling
The Thermochemical Evolution of Planets with Plate Tectonics
or Rigid Lids
**Slides: html
pdf
powerpoint

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It is well-established that the strong temperature-dependence of mantle viscosity leads by itself to a rigid lid style of convection, which is probably representative of Mars or Venus but not Earth. Recent numerical studies in both 2-D and 3-D have shown that accounting for the finite strength of the lithosphere by introducing a simple pseudo-plastic yield stress can lead to a plate-tectonic-like style of convection, with self-consistently generated passive spreading centers and "subduction zones". While promising, it is important to note that this physical description is clearly not complete because (i) the plate regime occurs with a yield stress of order 100 MPa, which is several times lower than the strength of rocks measured in laboratory experiments, (ii) "subduction" is double-sided, and (iii) new plate boundaries can spring up anywhere, rather than in previously weakened areas as observed on Earth. Nevertheless, such models provide a useful tool for studying other aspects of terrestrial planetary dynamics such as their thermochemical evolution. Models that incorporate melting and major- and trace-element| differentiation for rigid lid, plate tectonic, or episodic plate tectonic planets have been developed and will be presented and compared to observational constraints (e.g., geochemical reservoir signatures, crustal thicknesses, outgassing histories) for Earth, Venus and Mars.

**David
A. Yuen**
(Department of Geology and Geophysics and Minnesota Supercomputing
Institute, University Minnesota, Minneapolis, MN 55455) davey@krissy.msi.umn.edu

**The
Dynamical Consequences in Mantle Convection from a Nonlinear
Constitutive Relation in the Temperature Equation due to varlable
Thermal Conductivity
**Slides

In mantle convection one normally attributes all of the nonlinearities in the physical properties, such as the rheology, to the momentum equation. But in mantle convection the momentum equation is elliptic. In contrast , for an infinite Prandtl number fluid, which aptly describes the mantle convection, the governing master equation in time is the temperature equation. With constant thermal conductivity, this equation is a parabolic equation with the nonlinear coupling coming from the advection term involving the velocity and the gradient of the temperature. But the presence of variable thermal conductivity , which depends on both temperature and pressure, will introduce three nonlinear terms in the temperature equation from the divergence of the heat-flux vector, which is K (T,P)grad T where K is the thermal conductivity and T and P are the temperature and hydrostatic pressure. Mantle thermal conductivity has two components with vstly different behavior in their temperature-dependence. They are respectively the phonon-assisted and photon-promoted thermal conducitivities. As a consequence of variable thermal conductivity, the nonlinear terms in the energy equation impart a different character to both the convective pattern and the charcteristic timescales of convection.Some outstanding features , which are different from mantle convection with constant conductivity, are (1.) larger plumes are developed in the lower mantle from the radiative component of the conductivity and a high temperature at the core-mantle boundary( CMB). (2.) plumes and convective patterns can be stabilized by a high temperature at the CMB. (3) the timescale for thermal cooling of the mantle is longer with variable thermal conductivity. These results would argue for the important role played by variable thermal conductivity in the thermal coupling between the core and mantle, since the temperature at the CMB would vary with time.

Mathematics in Geosciences, September 2001 - June 2002