Explore the fascinating world of planetary dynamos in this comprehensive lecture. Delve into the mechanisms behind planetary magnetospheres and auroras, indicators of magnetic dipole fields. Examine the internal dynamo processes occurring in liquid cores that generate main planetary magnetic fields. Study planetary data and learn about key geodynamo researchers. Investigate the Earth's interior structure, including its inner core, and analyze the geological record of geomagnetic excursions. Understand the basic equations and mathematical models based on magnetohydrodynamic (MHD) equations with buoyancy. Discover numerical modeling techniques for solving nonlinear partial differential equations, including spectral methods for regular geometries. Explore geodynamo parameters, magnetic fields, and turbulent relaxation of ideal magneto-fluid. Learn about spherical harmonics, function expansions for spherical shells, and the transformation of PDEs into coupled ODEs. Examine the statistics of ideal MHD turbulence, probability density functions, and invariants. Investigate cross helicity and its relationship to the expected dipole angle. Gain insights into large-scale numerical simulations of ideal, rotating MHD turbulence in spherical shells and their implications for effective cross helicity.
Overview
Syllabus
Planetary Dynamos
Planetary Magnetospheres
Planetary Auroras are one indication of a magnetic dipole field
Internal Dynamo in a Liquid Core Generates the Main Planetary Magnetic Field
Planetary Data
(Some) Geodynamo Researchers
Interior of the Earth
The Earth’s Inner Core
Geological Record: Geomagnetic Excursions
Basic Equations Mathematical model based on the magnetohydrodynamic (MHD) equations with buoyancy → the 'Boussinesq approximation : compositional variation is not included because of strong mixing.
Numerical Modeling Solve nonlinear partial differential equations (PDEs) on computer. Various methods: finite difference, finite volume,.... For regular geometries, we can choose a spectral method
Geodynamo Parameters
Magnetic Fields Turbulent relaxation of ideal magneto-fluid
Spectral Method Model
Spherical Harmonics
Function Expansions for Spherical Shells
Coefficients = Dynamical Variables
A Dynamical System The spectral method tums a few PDEs into many coupled ODES
Statistics of Ideal MHD Turbulence
Probability Density Function (PDF) The invariants for ideal MHD are
Expectation Values
Cross Helicity, HC He is essentially the cross correlation between velocity and magnetic field in the liquid core.
Expected Dipole Angle 0 vs Cross Helicity Hc The dipole angle can be calculated using the statistical theory, with maximum , i, m = 100 Large-scale numerical simulations of ideal, rotating MHD turbulence in a spherical shell are needed to find effective cross helicity He for different values of magnetic helicity I
Summary: The Ideal MHD Geodynamo
Conclusion
Taught by
College of Science and Engineering
