Abstract
We present theoretical and computational results in magnetohydrodynamic turbulence that we feel are essential to understanding the geodynamo. These results are based on a mathematical model that focuses on magnetohydrodynamic (MHD) turbulence, but ignores compressibility and thermal effects, as well as imposing model-dependent boundary conditions. A principal finding is that when a turbulent magnetofluid is in quasi-equilibrium, the magnetic energy in the internal dipole component is equal to the magnetic helicity multiplied by the dipole wavenumber. In the case of the Earth, measurement of the exterior magnetic field gives us, through boundary conditions, the internal poloidal magnetic field. The connection between magnetic helicity and dipole field in the liquid core then gives us the toroidal part of the internal dipole field and a model value of 3 mT for the average core dipole magnetic field. Here, we present the theoretical analysis and numerical simulations that lead to these conclusions. We also test an earlier assertion that differential oblateness may be related to dipole alignment, and while there is an effect, rotation appears to be far more important. In addition, the relationship between dipole quasi-stationarity, broken ergodicity and broken symmetry is clarified. Lastly, we discuss how inertial waves in a rotating magnetofluid can affect dipole alignment.
Highlights
For the average core dipole magnetic field
We showed that extending the statistical mechanics of ideal MHD turbulence opened the door to understanding the fundamental origin of the geodynamo [6,7] and demonstrated that these ideal results appeared to apply to real MHD turbulence [8,9]
We use a Fourier model as a surrogate for a spherical model because the statistical mechanics of ideal, rotating MHD turbulence are equivalent in the two cases [7]
Summary
Explaining the origin of planetary and stellar magnetic fields has been a scientific quest since Larmor [1] hypothesized that magnetohydrodynamic (MHD) motions within the Sun, and by extension, the Earth, were responsible for the creation and maintenance of global magnetic fields. Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations We find, in both Fourier and spherical cases, that the largest-scale (smallest wavenumber k) magnetic field modes are very energetic and quasi-steady, i.e., random variables with large mean values subject to only very small fluctuations. Instead, basing our work on the MHD equations, we see that a quasi-steady, largest-scale component of the magnetic field emerges dynamically and has an explanation by a statistical mechanics with broken ergodicity and broken symmetry, as will be discussed at length in this paper. We briefly review our numerical method, summarize our new theoretical and computational results, followed by detailed explanations of how they were found
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