## Chaotic Processes in the Geological SciencesThis IMA Volume in Mathematics and its Applications CHAOTIC PROCESSES IN THE GEOLOGICAL SCIENCES is based on the proceedings of a workshop which was an integral part of the 1989- 90 IMA program on "Dynamical Systems and their Applications". The workshop was intended to be an arena for scientific exchanges between earth scientists and mathematical researchers, especially with experts in dynamical systems. We thank Shui-Nee Chow, Martin Golubitsky, Richard McGehee, George R. Sell and David Yuen for organizing the meeting. We especially thank David Yuen for editing the proceedings. We also take this opportunity to thank those agencies whose financial support made the workshop possible: the Army Research Office, the Minnesota Supercomputer Institute, the National Science Foundation, and the Office of Naval Research. A vner Friedman Willard Miller, Jr. PREFACE The problems in geological sciences have many nonlinearities from the nature of the complicated physical laws which give rise to strongly chaotic behavior. Foremost and most visible are earthquakes and volcanic eruptions, more subtle are the time dependent variations of the Earth's magnetic fields and motions of the surface plates. |

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### Contents

1 | |

Convection and chaos | 43 |

Strongly chaotic Newtonian and nonNewtonian | 70 |

Routes to chaos in the solid earth | 89 |

Mantle phase transitions layered chaotic convection | 111 |

Vigorous motions in magma chambers and lava lakes 141 | 140 |

The eruption and spreading of lava | 175 |

Convection in porous media with thermal | 225 |

Theory of the geodynamo and coremantle coupling | 281 |

The spatial structure of isotropic turbulence | 304 |

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amplitude analysis anomaly aspect-ratio axisymmetric bifurcation boundary conditions boundary layer buoyancy cells chaotic coefficient composition conductivity constant core country rock density diffusion dimensionless dispersion model dynamics Earth earth's mantle effects eigenvalue energy equations equilibrium eruptions eutectic evolution Figure fixed-point subspaces flow Fluid Mech geodynamo geoid Geophys geophysical hard turbulent heat horizontal infinite Prandtl number initial instabilities interface isotropy kinetic lava lakes linear liquid lower mantle Lyapunov exponent magma chambers mantle convection mantle viscosity mass flux melt motions mush region mushy layer non-Newtonian nonlinear nucleation Nusselt observed parameter pattern Peltier phase transitions Phys plate plumes predicted pressure problem radial Rayleigh number rheology scale seismic shows simulations spherical harmonic spherical shell stable stream function structure subspace surface symmetry temperature theory thermal boundary layer thermal convection thickness time-dependent upwelling variations velocity viscosity volcanic volume vorticity Yuen