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A Dynamical System The spectral method tums a few PDEs into many coupled ODES
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Classroom Contents
Planetary Dynamos
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- 1 Planetary Dynamos
- 2 Planetary Magnetospheres
- 3 Planetary Auroras are one indication of a magnetic dipole field
- 4 Internal Dynamo in a Liquid Core Generates the Main Planetary Magnetic Field
- 5 Planetary Data
- 6 (Some) Geodynamo Researchers
- 7 Interior of the Earth
- 8 The Earth’s Inner Core
- 9 Geological Record: Geomagnetic Excursions
- 10 Basic Equations Mathematical model based on the magnetohydrodynamic (MHD) equations with buoyancy → the 'Boussinesq approximation : compositional variation is not included because of strong mixing.
- 11 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
- 12 Geodynamo Parameters
- 13 Magnetic Fields Turbulent relaxation of ideal magneto-fluid
- 14 Spectral Method Model
- 15 Spherical Harmonics
- 16 Function Expansions for Spherical Shells
- 17 Coefficients = Dynamical Variables
- 18 A Dynamical System The spectral method tums a few PDEs into many coupled ODES
- 19 Statistics of Ideal MHD Turbulence
- 20 Probability Density Function (PDF) The invariants for ideal MHD are
- 21 Expectation Values
- 22 Cross Helicity, HC He is essentially the cross correlation between velocity and magnetic field in the liquid core.
- 23 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 turbul…
- 24 Summary: The Ideal MHD Geodynamo
- 25 Conclusion
