Joseph
A. Burns (Theoretical & Applied Mechanics and
Astronomy, Cornell University) jab16@cornell.edu
The
Renaissance in Solar System Dynamics Slides
Four
decades ago, at the dawn of the space age, the solar system
was viewed as static. This talk will show why and how
that view has been overturned. Major advances have been
made in understanding the solar system's structure and
its cause owing to provocative data provided by spacecraft
and ground observatories, new algorithms coupled with
improved and widely accessible computers that have allowed
extraordinarily long orbit integrations, and new paradigms
from dynamical systems.
No
longer is the solar system thought to be a fixed, deterministic
entity. Dissipation (by tides, drags and anelasticity)
is now known to have profoundly modified the solar system:
rotations damp into pure spins while orbits evolve significantly,
occasionally leading to the loss of past objects through
collisions. Resonances (two- or three-body mean-motion
as well as secular resonances) play a major role in the
solar system's current structure. Objects that drift into
resonant orbits may become trapped at these positions,
sometimes protecting themselves, but may also suffer substantial
jumps in orbital eccentricity and inclinations, which
allow orbits to cross at high speeds. Many orbits were--and
are--chaotic. In total, dissipation, close-interactions
and chaos have reconfigured the solar system, ejecting
many objects to interstellar space, causing some to collide
and transmuting yet others from "comets" to "asteroids."
Rotations
have also evolved over the eons: Mercury is locked in
a 3:2 spin-orbit resonance while Venus' spin may not be
tidally damped; Mars undergoes substantial obliquity oscillations
that may have markedly influenced its climatic history;
Saturn's satellite Hyperion tumbles chaotically and other
irregular moons may have, with important thermal consequences.
The rotations of asteroids are intimately tied to the
collisional histories of the minor planets; some spin
extraordinarily slowly, others remarkably fast, even a
few wobble noticeably like comet Halley. Planetary rings
form a rich dynamical laboratory, exhibiting a bewildering
array of phenomena: bending and spiral density waves at
resonances; satellite perturbations, including shepherding
and clumping; angular momentum transfer through collisions;
spokes; and electromagnetic interactions.
The
behavior of planets and disks around other stars is just
beginning to be explored and should produce many interesting
models to be interpreted by the techniques of dynamical
systems.
Jinqiao
Duan (Department of Applied Mathematics, Illinois
Institute of Technology, Chicago, IL 60616, USA) duan@iit.edu
Stochastic
dynamics of coupled ocean-atmosphere models Slides
Uncertainty
or randomness is ubiquitous in climate and geophysical
flows. Examples of such randomness are stochastic forcing,
uncertain parameters, random sources or inputs, and random
initial or random boundary conditions due to an uncertain
environment. Taking stochastic effects into account is
of central importance for the development of mathematical
models of climate and geophysical flows. These mathematical
models are in the form of stochastic ordinary or partial
differential equations, and random ordinary or partial
differential equations. Problems arising in the context
of climate and geophysical flows have inspired interesting
research directions in the fields of stochastic partial
differential equations and random dynamical systems. I
will present recent research on geophysical fluid dynamics
under uncertainty, especially the stochastic dynamics
of some coupled ocean-atmosphere models.
Susan
Friedlander (Department of Mathematics, Statistics
and Computer Science, University of Illinois-Chicago)
susan@math.northwestern.edu
Instabilities in Fluid Motion
We discuss some general results concerning linear and
nonlinear instability for the Euler equations. We consider
specific applications of these results to certain examples
in geophysical fluid dynamics, including systems with
rotation, precession, stratification and magnetic fields.
Michael
Ghil (Atmospheric Sciences and IGPP, UCLA)
ghil@atmos.ucla.edu
http://webster.atmos.ucla.edu/ghil/
Quaternary
glaciations and celestial mechanics Slides
This
talk attempts to bridge the two themes of the present
workshop, climate dynamics and celestial mechanics. To
do so, we focus on the relatively recent paleoclimate
of the last 2 My, the Quaternary era. The evidence for
climatic variability on this time scale is reviewed, based
on proxy records from deep-sea sediments and ice cores.
The climatic subsystems that are active over the Quaternary
are discussed, along with the feedback mechanisms that
couple them. These include the atmosphere, oceans, land
and sea ice among the subsystems, as well as ice-albedo,
precipitation-temperature and load-accumulation feedbacks
between them.
The
simplest mathematical models that capture the recorded
variability are systems of nonlinear ordinary differential
equations (ODEs). Multiple equilibria and limit cycles
arise in these forced-dissipative ODE systems via saddle-node
and Hopf bifurcations, respectively, for constant forcing.
The quasi-periodic changes in insolation that are due
to orbital variations lead to a number of interesting
phenomena: nonlinear resonance, frequency locking, and
transition to chaos. The "music of the spheres" is captured
by the climate system through combination tones that were
predicted by these models and confirmed by the analysis
of higher-resolution proxy records.
References
Ghil,
M., R. Benzi, and G. Parisi (Eds.), l985: Turbulence and
Predictability in Geophysical Fluid Dynamics and Climate
Dynamics, North-Holland Publ. Co., Amsterdam/New York/Oxford/
Tokyo, 449 pp.
Ghil,
M., and S. Childress, 1987: Topics in Geophysical Fluid
Dynamics: Atmospheric Dynamics, Dynamo Theory and
Climate Dynamics, Springer-Verlag, New York/Berlin/London/Paris/
Tokyo, 485 pp.
Ghil,
M., 1994: Cryothermodynamics: The chaotic dynamics of
paleoclimate, Physica D, 77, 130-159.
Ghil,
M., 2001 (PDF file): Hilbert problems for the geosciences
in the 21st century, Nonlin. Proc. Geophys., 8, 211-222.
Ghil,
M., 2001 (PDF file): Natural climate variability, in Encyclopedia
of Global Environmental Change, Vol. 1 (M. MacCracken
& J. Perry, eds.), Wiley & Sons, Chichester/New York,
in press.
N.B.
The latter two references can be found as PDF files at
http://www.atmos.ucla.edu/tcd/
.
James
E. Howard (Center for Integrated Plasma Studies
& Laboratory for Atmospheric and Space Physics, University
of Colorado, Boulder, CO) James.Howard@Colorado.EDU
Asteroidal
Satellites Slides
t
is now well established that many asteroids are actually
binary systems - in some cases of comparable mass, in
others a small body orbiting a much larger mass perhaps
100 km in diameter. The satellite may of course be a spacecraft,
e. g. the NEAR-Shoemaker orbiter which landed on 433 Eros.
Thus the system Sun-Asteroid-Satellite forms a three body
problem, with Msun >>Mast >>m. When the satellite
is relatively distant, but within the Hill sphere of the
asteroid, we may neglect solar gravity and treat the system
as a two body problem, with two interesting complications:
(i) the central mass is highly nonspherical, (ii) it is
also rotating.
Most asteroids rotate about the axis of maximum moment
of inertia (pencil-on-the-table mode) with a period of
5-10 hours. There is also a significant and mysterious
population of "slow rotators" with much longer periods.
In
this talk we will focus on orbital stability about axisymmetric
bodies, such as prolate ellipsoids or peanut-shaped bodies
represented by Cassini ovals. We consider two classes
of orbits, those encircling the axis of symmetry (thus
crossing the plane of rotation) and those lying within
this plane. The motion of the transverse class is governed
by an effective potential for nonrotating bodies, which
becomes an averaged effective potential for slow rotators.
The coplanar orbits may be further classified as prograde
or retrograde and are described by zero-velocity curves
and Poincaré sections. Preliminary conclusions
about the stability of the various orbital classes will
be ventured.
Peter
Imkeller (Institut fuer Mathematik, Humboldt-Universitaet
zu Berlin) imkeller@gmx.de
EBM
and stochastic resonance
Energy
Balance Models (EBM) range at the bottom in the hierarchy
of climate models. They describe just a global, temporally
and spatially averaged radiation balance of the earth.
Mathematically, they consist in nonlinear (stochastic)
differential equations in low space dimensions. Climate
states are obtained as their stable (metastable) equilibria.
The models are able to describe some dynamical aspects
of transitions between them. We give a survey on work
done on this class of models in the last two decades.
One
of the paradigms of this area, a simple model designed
to explain glacial cycles, exhibits the phenomenon of
stochastic resonance which has been encountered
in numerous examples in different areas of natural sciences
besides climate dynamics. Roughly, stochastic resonance
is the optimal tuning of a stochastic system to a periodic
input signal. We give the outlines of a mathematically
rigorous approach of this phenomenon, using ideas of large
deviations and spectral theory.
James
F. Kasting
(Department of Geosciences, Penn State University) kasting@essc.psu.edu
Climate
Stability (and Instability) on Long Time Scales Slides:
html
pdf
On billion-year time scales, the classic question for
climatologists is the so-called "faint young Sun problem":
How did Earth's climate remain warm despite a solar
luminosity decrease of 20-30 percent compared to today?
The likely answer is that the atmospheric greenhouse
effect was much larger as a consequence of higher concentrations
of volcanogenic CO2 and biogenic CH4.
Both of these gases are involved in negative feedback
loops that contribute to long-term climate stability.
That said, the controls appear to have broken down several
times in Earth's history, leading to periods of global,
or near-global, glaciation. The first of these Snowball
Earth episodes occurred at ~2.3 Ga. It coincides precisely
with the initial rise of atmospheric O2 and
was probably triggered by the loss of the methane component
of the atmospheric greenhouse. After an ice-free interval
of more than 1.5 billion years, global glaciation recurred
near the end of the Proterozoic at 750 Ma and 600 Ma.
The cause of these latter two Snowball Earth episodes
is less clear. Possible triggering mechanisms include
drawdown of atmospheric CO2 by weathering
of equatorially-situated continents, or further drawdown
of CH4 caused by rising O2 and
sulfate levels and corresponding increases in methanotrophic
and sulfate-reducing bacteria. All Snowball Earth glaciations
were presumably ended within ~10 million years by the
buildup of volcanic CO2. The question of
how Earth's biota managed to survive these climate catastrophes
will be discussed.
Richard
Kleeman (Courant Institute of Mathematical
Sciences, New York University) kleeman@cims.nyu.edu
Predictability
in dynamical systems relevant to climate and weather
Significant
progress has occured in the past two decades in climate
dynamics and prediction. Particularly simple stochastic
models are able to account for both the predictive properties
and the variability of the dominant mode of climate
variability namely the El Nino phenomenon. Skillful
prediction models now exist for this phenomenon for
time scales of the order of 6-12 months.
In this talk the implications of this new understanding
for the field of dynamical systems predictability are
outlined and a new theoretical framework from information
theory introduced. This framework has some very interesting
possible implications for the relatively mature field
of weather prediction which are briefly outlined.
Wang
Sang Koon (Control and Dynamical Systems,
California Institute of Techolnogy, Pasadena, CA 91125)
koon@cds.caltech.edu
Invariant
Manifolds, the Three-Body Problem and a Petit Grand
Tour of Jovian Moons slides.pdf
paper.pdf
The
invariant manifold structures of the collinear libration
points for the spatial restricted three-body problem
provide the framework for understanding complex dynamical
phenomena from a geometric point of view. In particular,
the stable and unstable invariant manifold ``tubes''
associated to libration point orbits are the phase space
structures that provide a conduit for orbits between
primary bodies for separate three-body systems. These
invariant manifold tubes can be used to construct new
spacecraft trajectories, such as a ``Petit Grand Tour''
of the moons of Jupiter. Previous work focused on the
planar circular restricted three-body problem. The current
work extends the results to the spatial case.
This
is joint work with Gerard Gómez,
Martin Lo, Jerrold
Marsden, Josep Masdemont
and Shane Ross.
William
I. Newman
(William I. Newman Department of Earth and Space Sciences,
Physics and Astronomy, and Mathematics, University of
California-Los Angeles) win@ucla.edu
Numerical
Integration, Lyapunov Exponents, and the Outer Solar
System Slides:
pdf
(5MB)
The
numerical integration of ordinary differential equations
introduces errors that can fundamentally alter the nature
of the computed solution, a point eloquently made by
Mitchell Feigenbaum in his development of the quadratic
map. Hamiltonian systems offer some extraordinary challenges
to our ability to quantify the uncertainty present in
our representation of such systems. In this lecture,
I will review some of the fundamental issues germane
to the numerical integration of ordinary differential
equations, including Hamiltonian systems. In particular,
we will focus on the properties of so-called symplectic
methods, showing their significance to Hamiltonian problems,
as well as their limitations. In particular, we will
explore how computed trajectories will differ from their
exactly calculated counterparts as a function of step
size, and focus on how symplectic systems that utilize
inappropriately chosen step sizes can manifest artificial
chaotic behavior. Our presentation will focus on analytic
examples drawn from classical mechanics---including
the harmonic oscillator and the pendulum---as well as
computational examples drawn from celestial mechanics---including
the behavior of the outer solar system.
Richard
Rand (Theoretical & Applied Mechanics, Cornell
University) rhr2@cornell.edu
A
Nonlinear Quasiperiodic Mathieu Equation pdf
(568KB)
This
talk presents some recent results obtained regarding
the differential equation x'' + (d + e cos t + e cos
wt) x + a x3 = 0. Stability of the origin
is investigated using Lyapunov exponents, regular and
singular perturbations, and harmonic balance. Computer
algebra is utilized to handle the complicated resulting
algebraic expressions. In addition, large amplitude
subharmonic motions are investigated by using Lie transforms
with elliptic functions. The resulting approximations
are used to predict the transition from local to global
chaos by the use of Chirikov's overlap criterion. This
work is joint with Randy Zounes,
Stephanie Mason and Rachel
Hastings.
Daniel
Schertzer
(Laboratoire de Modelisation en Mecanique (CNRS UMR
7607) case 162, Universite P. et M. Curie) schertze@ccr.jussieu.fr
http://www.multifractal.jussieu.fr
Chaos,
ergodic theory and multifractal singularities of stochastic
differential equations Slides:
html
pdf
powerpoint
In
collaboration with S. Lovejoy.
Nonlinear
differential equations in low space dimensions (e.g.
Energy Balance Models) have been popular and helpful
to show the strong limitations of classical methods
in climate dynamics. However, the understanding of dynamics
over a wide range of time and space scale is indispensable.
It has been therefore argued that high dimensional dynamical
systems, e.g. partial differential systems, are required
and could be analyzed in the framework of the ergodic
theory of chaos.
Existence
and uniqueness of physical invariant measures have been
obtained for large-scale approximations (e.g. quasi-geostrophic
approximation). Nevertheless, these derivations depend
on the regularity of the deterministic system, its physical
relevance, as well as that of the pertubative noise,
which is usually considered as gaussian and white in
time.
Nevertheless,
we show that: large-scale approximations are rather
incompatible with:
