Magnetohydrodynamic turbulence and the geodynamo

Abstract

The ARES Directorate at JSC has researched the physical processes that create planetary magnetic fields through dynamo action since 2007. The "dynamo problem" has existed since 1600, when William Gilbert, physician to Queen Elizabeth I, recognized that the Earth was a giant magnet. In 1919, Joseph Larmor proposed that solar (and by implication, planetary) magnetism was due to magnetohydrodynamics (MHD), but full acceptance did not occur until Glatzmaier and Roberts solved the MHD equations numerically and simulated a geomagnetic reversal in 1995. JSC research produced a unique theoretical model in 2012 that provided a novel explanation of these physical observations and computational results as an essential manifestation of broken ergodicity in MHD turbulence. Research is ongoing, and future work is aimed at understanding quantitative details of magnetic dipole alignment in the Earth as well as in Mercury, Jupiter and its moon Ganymede, Saturn, Uranus, Neptune, and the Sun and other stars. Current computational research results showing effective dipole angle θD with respect to normalized rotation vector Ωo are given in figure 1. (The dotted line is what would be expected if Ωo = 0. Also, note that saturation occurs.

Mantle Electrical Conductivity and the Magnetic Field at the Core–Mantle Boundary

“…(Conduction in the mantle starts at r s = 5371 km and finishes at the CMB at r o = 3480 km; its effects are modeled as due to a spherical surface current of conductance σ c at r = r c .) With regard to the magnetic spectra due to magnetohydrodynamic (MHD) turbulence, consider Figure 2, where we use, as an example, the spectrum R o n (r 3 ) from Figure 1, and then scale and overlay on R o n (r 3 ) a turbulent spectrum from simulation NM06 in [26]. This simulation was of rotating, dissipative, forced MHD turbulence run on a 64 3 grid for a very long time; the kinetic and magnetic Reynolds numbers were 285 and 200, respectively; the wavenumber range was 1 ≤ k < 32; and the forcing wave number was k f = 9, which pushed the spectrum up for k∼k f .…”

“…Figure 2 demonstrates that the presence of mantle electrical conductivity is needed to produce a magnetic spectrum on the CMB that is not flat and, at least at low-k, behaves as if it originated out of a turbulent outer core. 1 along with a scaled, overlaid turbulent spectrum from simulation NM06 discussed in [26]; NM06 was forced at wavenumber k f = 9 (please see text for more details). The Mauersberger-Lowes spectrum R n (r o ) on the CMB and a representative Kolmogorov spectrum k −5/3 are also shown.…”

Magnetic Helicity and the Solar Dynamo

“…If the expressions for E and are placed into ( 6 ), it is straightforward to determine the partition function and from this the expectation values of the means and variances for the variables and . The means are expected to be zero, but in numerical simulations of both ideal and real MHD turbulence in a periodic box, it has been found that the dynamical time-averages of the magnetic field coefficients with the smallest wavenumber can be very large compared to their standard deviations [ 15 , 16 , 22 , 23 , 24 , 25 ]. This is an example of broken ergodicity [ 26 ].…”

“…In periodic box models, this dipole moment vector tends to align itself with a rotation axis, if one is present in ideal MHD turbulence [ 22 ]. However, in dissipative, driven MHD turbulence, this alignment can be affected by the manner in which the system is forced [ 15 , 16 ]. If the angular rotation vector is in the z -direction, alignment in the spherical shell model occurs because the component becomes large dynamically, while the components become much smaller, i.e., we have broken symmetry.…”

“…That the statistical mechanics of ideal MHD turbulence is applicable to real systems in quasi-equilibrium was shown by [ 15 , 16 ]. It should also be applicable to stars such as the Sun because the kinetic and magnetic Reynolds numbers are so high [ 17 ].…”