|
Mathematics
in Geosciences, September 2001 - June 2002
Talk
Abstracts:
March
18-22, 2002
Material
from Talks
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
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powerpoint
Video: p20_m10.avi
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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
Video:
conus.avi
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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=1012
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
Movies: Ra1e6_2e4c_i2.mov
Ra1e6_6e4c_i2.mov
tAR16_2e5_7.mpg
coolrc2om_t.mov
hom_c.mov
hom_t.mov
lyr_c.mov
lyr_t.mov
meanTdiff_c.mov
meanTdiff_t.mov
meanTiso_t.mov
Images: Deformation2x.jpg
m6_varyYS_for_PR.jpg
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.
Material
from Talks
Mathematics
in Geosciences, September 2001 - June 2002
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