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Mathematics in the Geosciences, September 2001 - June 2002
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 M_{sun} >>M_{ast} >>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 CO_{2} and biogenic CH_{4}. 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 O_{2} 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 CO_{2} by weathering of equatorially-situated continents, or further drawdown of CH_{4} caused by rising O_{2} 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 CO_{2}. 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 x^{3} = 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:
- a scaling anisotropic regime from planetary scale down to km scale,
- colored and strongly non-gaussian noises.
We show that these properties, which rather correspond to the appearance of multifractal singularities, are rather well empirically supported. We discuss the corresponding (stochastic) models.
James A. Sethian (Department of Mathematics, University of California, Berkeley)
Ordered Upwind Methods: Computing Viscosity Solutions to Optimal Control and Non-Viscosity Solutions to Wave Propagation
Ordered Upwind Methods are techniques for computing solutions of static Hamilton-Jacobi equations; they use partial information about characteristic directions as the computation unfolds to greatly reduce the computational labor. The methods are O(N log N) where N is the number of points in the computational domain. In this talk, we show how to design and build these schemes to find viscosity solutions to problems in optimal control, and non-viscosity multiple arrival solutions to problems in wave propagation. Applications include anisotropic front propagation in semiconductor manufacturing, computing multiple arrivals in seismic imaging, and finding local geodesic paths on complex manifolds. The work on optimal control is joint with A. Vladimirsky, and the work on multiple arrivals is joint with S. Fomel.
Steve Shkoller (University of California, Davis) shkoller@math.ucdavis.edu
A Variational Level-Set Approach for Two-Phase Incompressible Fluids
Using a simple variational principle, we derive a coupled Navier-Stokes phase-field model of two-phase incompressible fluids with a moving interface. The zero set of the phase-field is precisely the interface between the two fluids. Solutions of our model converge to the Navier-Stokes equations with the traditional kinetic and kinematic interface conditions, whenever the interface can initially be characterized by a distance function (when the interface is not a breaking wave). After discussing analytic properties of the solutions, we shall present results of a linear stability analysis of an idealized ocean-air interface problem. This is joint work with Chun Liu and Glenn Ierley.
Glen R. Stewart (Laboratory for Atmospheric and Space Physics, University of Colorado) glen@artemis.colorado.edu
Resonant Planet-Disk Interactions in the Solar System Slides
During the formation of the solar system, the planets grew from a disk of gas and dust in orbit around the newly formed sun. When the embryonic planets grew large enough, they began to resonantly excite spiral density waves and spiral bending waves in the surrounding disk of material. These planet-disk interactions resulted in substantial changes in the orbital eccentricities, inclinations and semimajor axes of the planetary embryos, and therefore likely determined the final number of planets and their orbital spacing in the solar system. The discovery of many extrasolar planets with very different orbital parameters has stimulated many recent efforts to model planet-disk interactions in large-scale fluid dynamic simulations.
Idealized mathematical models of planet-disk interactions were first developed in the context of observed satellite interactions with planetary rings. Inelastic collisions between ring particles damp their relative velocities to such low levels that pressure forces can often be neglected in Saturn's rings. Hence, satellites of Saturn launch spiral density waves in the rings that can be modeled by a self-gravitating, pressureless fluid. These models have also been applied to resonant interactions between Neptune and the Kuiper belt. The spatial dependence of the disk waves can be interpreted as the time evolution of a pendulum that is subject to a period force with a slowly varying frequency. Nonlinear extensions of the model lead to new phenomena, such as spatial autoresonance, which may help explain why strong satellite resonances open up wide gaps in Saturn's rings.
The idealized models of planet-disk interactions can be reformulated as variational principles. The variational principle allows one to derive conservation laws by Noether's method and provides a convenient starting point for deriving discrete approximations to the disk dynamics. Although lagrangian variational principles are relatively easy to derive, it is also possible to derive eulerian variational principles that may be be better suited for treating gaseous disks in the early solar system where pressure forces dominated wave propagation rather than self-gravity.
Roger Temam (Institute for Scientific Computing & Applied Mathematics, Indiana University) temam@indiana.edu
Some recent developments on the Primitive Equations
In this lecture we will present some recent developments concerning the Primitive Equations of the atmosphere, the ocean and the coupled atmosphere ocean. Results of existence and uniqueness of solutions will be addressed as well as properties of the dynamical systems that they generate.
Edriss S. Titi (Department of Mathematics, University of California, Irvine) etiti@math.uci.edu http://www.math.uci.edu/~etiti/
Mathematical Study of Certain Geophysical Models Slides
The basic problem faced in geophysical fluid dynamics is that a mathematical description based only on fundamental physical principles, which are called the ``Primitive Equations,'' is often prohibitively expensive computationally, and hard to study analytically. In this talk I will present a formal derivation of more manageable shallow water approximate models for the three dimensional Euler equations in a basin with slowly spatially varying topography, the so-called "Lake Equation" and "Great Lake Equation," which should represent the behavior of the physical system on time and length scales of interest. These approximate models will be shown to be globally well-posed. I will also show that the Charney-Stommel model of the gulf-stream, which is a two-dimensional damped driven shallow-water model for ocean circulation, has a global attractor. Whether this attractor is finite- or infinite-dimensional is still an open question. Other results concerning the global well-posedess of three dimensional viscous planetary geostrophic models will be presented.
Ferenc Varadi (Institute of Geophysics and Planetary Physics, University of California-Los Angeles)
Problems in solar system dynamics Slides
Several related topics are discussed, from large-scale perturbation computations to periodic orbits and chaos. Secular perturbation theory and its implementation for large-scale computations are demonstrated for the case of the Jovian planets. The theory ultimately fails due to the 2:5 orbital near-resonance between Jupiter and Saturn, which is elucidated through numerical simulations. Next another resonance, the well-known and somewhat misunderstood 3:1 case, is discussed, from the point of view classical Hamiltonian chaos and periodic orbits. The properties of periodic orbits and their continuation is also illustrated for 2:3 case. Finally, the latest results of solar system simulations are presented, showing chaotic jumps between different dynamical regimes in the motion of the major planets. The dynamics of asteroid orbits exhibits a surprisingly large variety of behaviors, ranging from the very regular to the chaotic, accompanied with intermittent capture into resonances and drift in phase space.
Shouhong Wang (Department of Mathematics, Indiana University) showang@indiana.edu http://php.indiana.edu/~showang
Topology of 2D Incompressible Flows
First I shall present a geometrical/topological theory of the structure and its transitions of 2D incompressible flows. This includes in particular a). structural and block stabilities, and b). structural bifurcation and boundary layer separations of 2D incompressible flows.
Then I shall discuss two applications. The first application is on the characterization and numerical simulation of boundary layer separations in driven cavity flows. The second application is on wind driven ocean circulation.
Dynamical Systems in Celestial Mechanics and Climate Dynamics
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