Sergey
Cherkis
(Physics & Astronomy University of California, Los Angeles)
cherkis@ihes.fr
Solitons
in Hierarchical Systems (an example) Slides
We
apply techniques of conformal field theory and integrable
systems to explore the following problem arising in seismology:
prediction of a strong earthquake by the emergence of particular
patters of seismic activity in a lower energy range. Seismic
activity is known to exhibit, on average, scale invariance
of the form dN(e)~E-c dE, where N is the annual
number of earthquakes with energy E and c is a critical exponent.
We model seismicity by a hierarchical model proposed by Belov.
The model is integrable and displays scale invariance.
Using
Lax formalism we find infinitely many conserved quantities
and find solitonic solutions. A soliton solution is interpreted
as free transfer of the abundance of defects on small scales
to large scales. In other words, the original rise of seismic
activity is a perturbation of N(E) which has a very special
form. Such a perturbation propagates without dispersion from
small to large energies.
In
such a system, monitoring its behavior at small scales for
solitonic excitations can provide criteria for predicting
a large scale event. We search for complete Gelfand-Levitan-Marchenko
transformation to scattering data, which would provide such
quantitative criteria.
Even
though the integrability of the considered system is a fine
feature that is lost with a generic perturbation, by the universality
principle we expect his model to provide a good description
near the conformal point for all other systems in the same
universality class.
Susan
Friedlander
(Mathematics, Statistics & Computer Science, University of
Illinois-Chicago) susan@math.northwestern.edu
A
GOY model for the Navier Stokes equations with nonlinear viscosity
Slides
We
discuss a modified Navier Stokes equation that arises in turbulence
modeling and in modeling the motion of visco-elastic fluids.
We present a shell cascade model for the full PDE and show
that for this "GOY" type model the Hausdorff dimension of
the singular set is bounded by a parameter that depends on
the order of the nonlinear viscosity.
This
is joint work with Natasa Pavlovic.
Andrei
Gabrielov
(Mathematics and Geophysics, Purdue University) agabriel@math.purdue.edu
http://www.math.purdue.edu/~agabriel
Modeling
of seismicity: a mathematician's perspective Slides
Modeling
of seismicity leads to new exciting problems in such areas
of mathematics as differential geometry, dynamical systems,
algebraic geometry, and combinatorics. I will overview these
connections between seismology and mathematics, not requiring
the knowledge of either from the audience.
Agnes
Helmstetter
(Geosciences, University of Grenoble) Agnes.Helmstetter@obs.ujf-grenoble.fr
Sub-critical
and Super-critical Regimes in Epidemic Models of Earthquake
Aftershocks Slides
We
present an analytical solution and numerical tests of the
epidemic type aftershock (ETAS) model for aftershocks, which
describes foreshocks, aftershocks and mainshocks on the same
footing. In this model, each earthquake of magnitude M triggers
aftershocks with a rate proportional to 10(AM).
The occurrence rate of aftershocks decreases with the time
from the mainshock according to the modified Omori law K/(t+c)p
with p=1+
.
The background seismicity rate is modeled by a stationary
Poisson process with a constant occurrence rate. Contrary
to the usual definition, the ETAS model does not impose an
aftershock to have a magnitude smaller than the mainshock.
We find two differents regimes depending on the branching
ratio N, defined as the mean aftershock number triggered per
event. In the sub-critical regime (N<1), we recover and document
the crossover from a power-law decrease of the seismicity
rate with an Omori exponent 1-theta at early times to 1+theta
at large times found previously in [Sornette and Sornette,
Geophys. Res. Lett., 26, 1999] for a special case of the ETAS
model. In the super-critical regime (n>1 and theta>0), we
find a novel transition from an Omori decay law with exponent
1-theta to an explosive exponential increase of the seismicity
rate. These results can rationalize many of the stylized facts
reported for aftershock and foreshock sequences, such as (i)
the suggestion that a small p-value may be a precursor of
a large earthquake, (ii) the relative seismic quiescence sometimes
observed before large aftershocks, and (iii) the increase
of seismic activity preceding large earthquakes.
Raymond
Hide
(Oxford University)
Analysis
and interpretation of the main geomagnetic field: The magnetic
field at the core-mantle boundary: some topological speculations
The
determination of the main geomagnetic field at the core-mantle
boundary (CMB) from observations made at nearly twice the
distance from the geocentre, i.e. at and near the Earth's
surface, is a crucial first step in the use of such observations
in the study of core motions and the testing of geodynamo
models. Important details of CMB field patterns remain controversial,so
it is of interest to investigate whether they can be elucidated
by considering topological characteristics of the patterns
associated with the intersection of lines of force of any
solenoidal vector field V with a general spherical surface
S. Such patterns are characterised by (a) patches bounded
by "null flux lines'' where the component of V normal
to S vanishes and (b) dip poles where V is normal to S, and,when
V is sufficiently complex, by (c) touch points on null flux
lines where the component of V that is tangential to S is
also tangential to the null flux line. At the Earth's surface
there is at present just one pair of dip poles and one null
flux line, but no touch points. The more complex CMB field
has several pairs of dip poles and several null flux lines,
not all of which are nested and upon some of which there are
pairs of touch points.
Leo
Kadanoff (Department of Physics & Mathematics,
University of Chicago)
Making
a Splash, Breaking a Neck: The Development of Complexity in
Physical Systems
Joint
work with Michael Brenner, Peter
Constantin, Todd Dupont,
Albert Libchaber, Sidney
Nagel, Robert Rosner,
and many others.
We
study the motion of fluids, with the aim of developing a fundamental
understanding of fluid flow. Our program is characterized
by close cooperation among experimenters, theoreticians, and
simulators. The world about us exhibits many beautiful and
important fluid flows. Consider clouds and waves, storms,
and earthquakes, sunspots and mountain-building. What can
we learn from all this richness?
Mostly our work involves solving particular problems, e.g.
'how does heat flow in a pot of water heated over a flame'.
But, in following these problems we soon get to broader issues:
predictability and chaos, the likelihood of very extreme outcomes,
and the natural formation of complex 'machines'. In the end,
we try to ask if there is a 'science of complexity' and are
there natural 'laws' of complex things. My answer is 'no',
but I do see important lessons to be learned from studying
such systems.
Leon
Knopoff
( Department of Physics and Astronomy Institute of Geophysics
and Planetary Physics University of California, Los Angeles)
lknopoff@eq.ess.ucla.edu
Are
simple models adequate for the simulation of recurrent seismicity?
Recent
statistical studies indicate that the magnitude-frequency
relation for earthquake mainshocks is not scale-independent.
A number of geophysical observations indicate consistency
of the statistics with a fracture model for larger earthquakes
that involves physics on at least four interactive scales.
Because of computational limitations, it is doubtful that
we will be able to take into account all of the different
issues of the physics of fracture of this rather complicated
model in constructing an appropriate computational model.
We discuss the influence of the dynamics of fracture, the
radiation of elastic waves, tensor stresses, the physics of
nucleation and healing, the geometry of faults, three dimensionality
and fault structure, and the influence of fluids and microfracturing
and granularity on the properties of materials, in modeling
the full problem. We discuss the robustness of simulations
to the inclusion of some of these ingredients into models
for the space-time pattern formation of mainshocks.
Vladimir
G. Kossobokov (International Institute of Earthquake
Prediction, Theory and Mathematical Geophysics, Russian Academy
of Sciences, 79-2 Warshavskoye Shosse, Moscow 113556, Russian
Federation Institute de Physique du Globe de Paris, 4 Place
Jussieu, 75252 Paris, Cedex 05, France) volodya@mitp.ru
or volodya@ipgp.jussieu.fr
Complexity
of inverse and direct cascading of earthquakes Slides:
html
pdf
(982KB)
Earthquakes
evidence consecutive stages of inverse cascading of seismic
activity to main shock and direct cascading of aftershocks.
The first may reflect coalescence of instabilities at the
approach, while the second indicates readjustment in a new
state of a complex system of blocks-and-faults after a catastrophe.
The cascades observed in seismic dynamics are by far more
diverse than a power-law family easily tractable in computer
and mathematical modeling.
George
Molchan (International Institute of Earthquake
Prediction Theory & Math Geoscience) molchan@mitp.ru
Mandelbrot
Cascade Measures Independent of Branching Parameter
Mandelbrot
cascade measures arose from the desire to explain intermittency
in the fully developed turbulence. They are defined by the
scale hierarchy with a fixed branching parameter "c" and by
the distribution of breakdown coefficients which are responsible
for the transport of energy from larger to smaller scales.
We show that the measures corresponding to both conservative
and nonconservative cascades strongly depend on the parameter
c. In particular, only Lebesgue measure can be generated by
a cascade process with an arbitrary integer c. That fact creates
difficulties for those physical inferences which rely on c-independent
cascade measures.
Clément
Narteau (Seismological Laboratory, California Institute
of Technology, 252-21, 1200 E. California Pasadena CA 91125)
narteau@gps.caltech.edu
Strike-slip
fault network evolution in the Scaling Organization of Fracture
Tectonic model pdf
(13MB)
From
laboratory experiments, we are recognising that fractures
are rough and irregular. Field surveys show that faults exhibit
similar geometrical features despite the geological complexities.
Meanwhile, slip distribution and the speed of rupture front
propagation along planar faults have become standard seismological
observables of earthquakes.
We
are interested in the spatial-temporal properties of the stress
dissipation within an active tectonic region. Therefore, in
our approach, fractures play a central role and we focus on
their interrelated evolution at different scales, from the
micro-fractures to continental-scale faults.
We
adopt a binary description of the microscopic scale to distinguish
between two blocks of rock separated by a fracture and a solid
rock. In a multiple scale system, geometric interactions extend
this description at larger scales. In return, we define how
any point in space is affected by the fracturing process from
the distribution of fractures at all scales and from the local
shear stress. By calling any perturbation and numerical structure
from their geophysical counterparts, we study the evolution
of our dynamical systems.
From
the statistical results of a model of seismicity along an
isolated fault segment, we extend our approach to the fault
network scale. We present typical patterns of formation and
evolution of a population of faults. Different phases of development
are described: nucleation, growth, interaction, concentration,
branching and relocation. We show that the geometry of the
networks converges to a configuration in which all the stress
dissipation is accumulated on a megafault aligned with
respect to the orientation of the stress field. We conclude
that the fault networks organize themselves in order to dissipate
more and more efficiently the excess of stress. Different
processes are isolated: localization and homogenization of
the state of the stress along faults at different periods
of time and structural regularization of the fault trace.
We discuss the interrelated evolution of the faults within
the network and relationships between the seismicity and the
geometry of the fault network.
William
I. Newman
(Department of Earth and Space Sciences, Physics and Astronomy,
and Mathematics, University of California-Los Angeles) win@ucla.edu
Complexity
and Spatio-Temporal Chaos in Material Failure: Analysis and
Computation of Fiber Bundle Models Slides
Problems
manifesting complexity and spatio-temporal chaos are endemic
in the physical sciences. These problems are often difficult
to describe from first principles and generally beyond the
reach of computational and analytic methods. For example,
earthquakes show self-similar behavior in space and time and
possess several power-law scalings valid over many orders
of magnitudes, yet our knowledge of continuum mechanics is
sufficiently primitive (and linear) that it offers no insight
into the earthquake mechanism. Parallels are often made between
earthquake activity and fluid turbulence; however, nothing
paralleling the Navier-Stokes equations for earthquakes is
known.
Remarkably,
a variety of "toy models" have provided some important new
insights into the problems. Some of these are reminiscent
of the underlying simplicity (and emergent complexity) inherent
in Feigenbaum's seminal work on deterministic chaos and scaling.
Not only do these toy models deliver an improved understanding
of complicated physical processes, they provide a rich set
of problems that are ripe for mathematicians and computer
scientists.
This
lecture will focus on a class of cellular automata models---referred
to as "fiber bundles"---developed to describe material failure
and widely used in applications ranging from materials science
to theoretical seismology. These models employ a probabilistic
formulation applied to cellular automata organized geometrically
according to the nature of the problem, and result in problems
that have a hierarchical flavor, that is a functional iteration
(in contrast with a simple function iteration).
An
important ingredient in these investigations is the interplay
between computation and analysis. Computation is often important
to establishing the nature of large scale behavior and sometimes
leads to theorems, in keeping with Von Neumann's dictum regarding
computation and analysis in nonlinear problems. Sometimes,
analysis is required to make the computation possible, owing
to the large numbers of elements M required to show physical
scalings---characterized by Avogadro's number or 1023
or greater---and restructuring of the problem, in the same
spirit as the Fast Fourier Transform, can be used to render
it computationally irreducible. [Typically, these problems
require O(M*M) operations but can sometimes be reduced to
some power of log(M).]
Donald
L. Turcotte
(Department of Geological Sciences, Cornell University) Turcotte@geology.geo.cornell.edu
Micro
and macroscopic models for material failure
A simple microscopic model for the failure of a composite
material is the fiber bundle model. The failure of a cylindrical
fiber bundle in tension is considered, a statistical "time-to-failure"
model is considered and global load shearing (mean field)
is assumed. A simple analytical solution is found. A simple
macroscopic (continuum) model for failure is the damage model.
Again the failure of a cylindrical rod in tension is considered.
The analytical solution found is identical to that for the
microscopic model. The models are used to determine the acoustic
emissions during failure. The results re shown to be in good
agreement with experiments carried out on the failure fiber
board.
Misha
Vishik
(Department of Mathematics, University of Texas at Austin
Austin , TX 78712) vishik@mail.ma.utexas.edu
Incompressible
flows of an ideal fluid with unbounded vorticity Slides
We
discuss solutions to the Euler equations of an ideal incompressible
fluid in dimension 2 and higher with special attention to
function classes described in terms of the wavelets coefficients.
In dimension 2 both existence and uniqueness can be proved
for classes of flows that contain essentially unbounded functions.
In dimension 3 where the existence of weak solutions with
bounded vorticity is open the local in time results are proved
for certain classes of flows with vorticity discontinuous
at one point.
David
A. Yuen (Department of Geology and Geophysics and
Minnesota Supercomputing Institute, University of Minneaota,
Minneapolis) davey@krissy.msi.umn.edu
Controlling
Thermal Chaos in the Mantle by Feedback due to Radiative Thermal
Conductivity Slides
The
role of nonlinear aspects of thermal conductivity has been
neglected in studies of mantle convection, even though it
is well known from solid-state physics, that it is temperature-
and pressure-dependent. The temperature equation acquires
a distinct nonlinear character by virtue of the nonlinear
term involving the square of the temperature gradient. We
have employed the recently developed thermal conductivity
model by Hofmeister (Hofmeister, Science, 1999) in both 2-D
and 3-D mantle convection studies. The thermal conductivity
of mantle materials has two components, the lattice component
klat from phonons and the radiative component krad
due to photons. The temperature (T) derivatives of these mechanisms
have different signs, with d klat /d T negative
and d krad /d T positive. This attribute of a positive
temperature derivative on the part of k-rad offers the possibilities
for the actual temperature at the core-mantle boundary (CMB)
to be a stabilizing factor on boundary layer instabilities
at the core-mantle boundary. We have parameterized the weight
factor between krad and klat with a
dimensionless number f , where f =1 corresponds to the reference
conductivity model given by Hofmeister (1999). For this thermal
conductivity model (f = 1 ) we have found that by increasing
the temperature at the CMB, Tcmb , from 3000 to
4200 K, the boundary layer instabilities are quenched more
and become more stabilized. For purely basal heating situations
the time-dependent chaotic flows at Tcmb = 3000K
become stabilized for values of f between 1.5 and 2. As we
increase the Tcmb to 4000 K the critical value
of f, fc, needed for flow stabilization is correspondingly
reduced .These results argue for the possible constraints
on Tcmb from the presence of radiative thermal
conductivity in the deep mantle and the development of secondary
instabilities on the CMB. Too high a Tcmb would
quench the instabilities. This work is the first to address
the important role played by variable thermal conductivity
in controlling chaotic flows in mantle convection, the number
of hotspots and the attendant mixing of geochemical anomalies.
