We review and extend the theory of ideal, homogeneous, incompressible, magnetohydrodynamic (MHD) turbulence. The theory contains a solution to the ‘dynamo problem’, i.e., the problem of determining ...how a planetary or stellar body produces a global dipole magnetic field. We extend the theory to the case of ideal MHD turbulence with a mean magnetic field that is aligned with a rotation axis. The existing theory is also extended by developing the thermodynamics of ideal MHD turbulence based on entropy. A mathematical model is created by Fourier transforming the MHD equations and dynamical variables, resulting in a dynamical system consisting of the independent Fourier coefficients of the velocity and magnetic fields. This dynamical system has a large but finite-dimensional phase space in which the phase flow is divergenceless in the ideal case. There may be several constants of the motion, in addition to energy, which depend on the presence, or lack thereof, of a mean magnetic field or system rotation or both imposed on the magnetofluid; this leads to five different cases of MHD turbulence that must be considered. The constants of the motion (ideal invariants)—the most important being energy and magnetic helicity—are used to construct canonical probability densities and partition functions that enable ensemble predictions to be made. These predictions are compared with time averages from numerical simulations to test whether or not the system is ergodic. In the cases most pertinent to planets and stars, nonergodicity is observed at the largest length-scales and occurs when the components of the dipole field become quasi-stationary and dipole energy is directly proportional to magnetic helicity. This nonergodicity is evident in the thermodynamics, while dipole alignment with a rotation axis may be seen as the result of dynamical symmetry breaking, i.e., ‘broken ergodicity’. The relevance of ideal theoretical results to real (forced, dissipative) MHD turbulence is shown through numerical simulation. Again, an important result is a statistical solution of the ‘dynamo problem’.
We continue our study of the transition of ideal, homogeneous, incompressible, magnetohydrodynamic (MHD) turbulence from non-equilibrium initial conditions to equilibrium using long-time numerical ...simulations on a 1283 periodic grid. A Fourier spectral transform method is used to numerically integrate the dynamical equations forward in time. The six runs that previously went to near equilibrium are here extended into equilibrium. As before, we neglect dissipation as we are primarily concerned with behavior at the largest scale where this behavior has been shown to be essentially the same for ideal and real (forced and dissipative) MHD turbulence. These six runs have various combinations of imposed rotation and mean magnetic field and represent the five cases of ideal, homogeneous, incompressible, and MHD turbulence: Case I (Run 1), with no rotation or mean field; Case II (Runs 2a and 2b), where only rotation is imposed; Case III (Run 3), which has only a mean magnetic field; Case IV (Run 4), where rotation vector and mean magnetic field direction are aligned; and Case V (Run 5), which has non-aligned rotation vector and mean field directions. Statistical mechanics predicts that dynamic Fourier coefficients are zero-mean random variables, but largest-scale coherent magnetic structures emerge and manifest themselves as Fourier coefficients with very large, quasi-steady, mean values compared to their standard deviations, i.e., there is ‘broken ergodicity.’ These magnetic coherent structures appeared in all cases during transition to near equilibrium. Here, we report that, as the runs were continued, these coherent structures remained quasi-steady and energetic only in Cases I and II, while Case IV maintained its coherent structure but at comparatively low energy. The coherent structures that appeared in transition in Cases III and V were seen to collapse as their associated runs extended into equilibrium. The creation of largest-scale, coherent magnetic structure appears to be a dynamo process inherent in ideal MHD turbulence, particularly in Cases I and II, i.e., those cases most pertinent to planets and stars. Furthermore, the statistical theory of ideal MHD turbulence has proven to apply at the largest scale, even when dissipation and forcing are included. This, along with the discovery and explanation of dynamically broken ergodicity, is essentially a solution to the ‘dynamo problem’.
Solar magnetism is believed to originate through dynamo action in the tachocline. Statistical mechanics, in turn, tells us that dynamo action is an inherent property of magnetohydrodynamic (MHD) ...turbulence, depending essentially on magnetic helicity. Here, we model the tachocline as a rotating, thin spherical shell containing MHD turbulence. Using this model, we find an expression for the entropy and from this develop the thermodynamics of MHD turbulence. This allows us to introduce the macroscopic parameters that affect magnetic self-organization and dynamo action, parameters that include magnetic helicity, as well as tachocline thickness and turbulent energy.
We find the analytical form of inertial waves in an incompressible, rotating fluid constrained by concentric inner and outer spherical surfaces with homogeneous boundary conditions on the normal ...components of velocity and vorticity. These fields are represented by Galerkin expansions whose basis consists of toroidal and poloidal vector functions, i.e., products and curls of products of spherical Bessel functions and vector spherical harmonics. These vector basis functions also satisfy the Helmholtz equation and this has the benefit of providing each basis function with a well-defined wavenumber. Eigenmodes and associated eigenfrequencies are determined for both the ideal and dissipative cases. These eigenmodes are formed from linear combinations of the Galerkin expansion basis functions. The system is truncated to numerically study inertial wave structure, varying the number of eigenmodes. The largest system considered in detail is a 25 eigenmode system and a graphical depiction is presented of the five lowest dissipation eigenmodes, all of which are non-oscillatory. These results may be useful in understanding data produced by numerical simulations of fluid and magnetofluid turbulence in a spherical shell that use a Galerkin, toroidal–poloidal basis as well as qualitative features of liquids confined by a spherical shell.
The Earth’s magnetic field is measured on and above the crust, while the turbulent dynamo in the outer core produces magnetic field values at the core–mantle boundary (CMB). The connection between ...the two sets of values is usually assumed to be independent of the electrical conductivity in the mantle. However, the turbulent magnetofluid in the Earth’s outer core produces a time-varying magnetic field that must induce currents in the lower mantle as it emerges, since the mantle is observed to be electrically conductive. Here, we develop a model to assess the possible effects of mantle electrical conductivity on the magnetic field values at the CMB. This model uses a new method for mapping the geomagnetic field from the Earth’s surface to the CMB. Since numerical and theoretical results suggest that the turbulent magnetic field in the outer core as it approaches the CMB is mostly parallel to this boundary, we assume that this property exists and set the normal component of the model magnetic field to zero at the CMB. This leads to a modification of the Mauersberger–Lowes spectrum at the CMB so that it is no longer flat, i.e., the modified spectrum depends on mantle conductance. We examined several cases in which mantle conductance ranges from low to high in order to gauge how CMB magnetic field strength and mantle ohmic heat generation may vary.
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.
•Ideal MHD turbulence contains an intrinsic dynamo mechanism (broken ergodicity).•Real MHD turbulence forced at mid-wavenumber k behaves like ideal MHD at low k.•Periodic box (Fourier) simulations ...are surrogates for spherical shell simulations.•Quasi-stationary or slowly translating largest-scale magnetic structures appear.•If Fourier methods are good surrogates, then Geodynamo – MHD turbulence.
Ideal magnetohydrodynamic (MHD) turbulence contains an intrinsic statistical mechanism called ‘broken ergodicity’ for producing an energetic, large-scale, quasi-stationary coherent structure. Since MHD turbulence occurs in the Earth’s liquid outer core, the implication is that the Earth’s dipole magnetic field may be due to MHD turbulence, per se. The purpose of this work is to examine how ideal MHD results apply to the real (i.e., dissipative) MHD turbulence found in the outer core. Prior theoretical and computational results will be briefly reviewed, after which new results will be presented. The significant effects caused by variations in the time-dependence of the forcing will be discussed. It will be also shown that dissipative MHD turbulence driven with isotropic forcing produces large-scale, coherent structures similar to those appearing in ideal MHD turbulence. A connection between spherical harmonic degree and turbulence wave number will be made, allowing for a qualitative comparison of numerical magnetic energy spectra with the International Geomagnetic Reference Field. In addition, a mechanism for explaining the dominance of the Earth’s axial dipole component over its azimuthal ones will also be described. These results indicate that MHD turbulence, per se, may underlie the geodynamo.
A technique for the kinetic simulation of plasma flow in strong external magnetic fields was developed which captures the compression and expansion of plasma bound to a magnetic flux tube as well as ...forces on magnetized particles within the flux tube. This quasi-one-dimensional (Q1D) method resolves a single spatial dimension while modeling two-dimensional effects. The implementation of this method in a Particle-In-Cell (PIC) code was verified with newly formulated test cases which include two-particle motion and particle dynamics in a magnetic mirror. Results from the Q1D method and fully two dimensional simulations were compared and error analyses performed verifying that the Q1D model reproduces the fully 2D results in the correct regimes. The Q1D method was found to be valid when the hybrid Larmor radius was less than 10% of the magnetic field scale length for magnetic field guided plasma expansions and less than 1% of the magnetic field scale length for a plasma in a converging–diverging magnetic field. The simple and general Q1D method can readily be incorporated in standard 1D PIC codes to capture multi-dimensional effects for plasma flow along magnetic fields in parameter spaces currently inaccessible by fully kinetic methods.
Turbulent magnetofluids appear in various geophysical and astrophysical contexts, in phenomena associated with planets, stars, galaxies and the universe itself. In many cases, large-scale magnetic ...fields are observed, though a better knowledge of magnetofluid turbulence is needed to more fully understand the dynamo processes that produce them. One approach is to develop the statistical mechanics of ideal (i.e. non-dissipative), incompressible, homogeneous magnetohydrodynamic (MHD) turbulence, known as "absolute equilibrium ensemble" theory, as far as possible by studying model systems with the goal of finding those aspects that survive the introduction of viscosity and resistivity. Here, we review the progress that has been made in this direction. We examine both three-dimensional (3-D) and two-dimensional (2-D) model systems based on discrete Fourier representations. The basic equations are those of incompressible MHD and may include the effects of rotation and/or a mean magnetic field
B
o
. Statistical predictions are that Fourier coefficients of the velocity and magnetic field are zero-mean random variables. However, this is not the case, in general, for we observe non-ergodic behavior in very long time computer simulations of ideal turbulence: low wavenumber Fourier modes that have relatively large means and small standard deviations, i.e. coherent structure. In particular, ergodicity appears strongly broken when
B
o
= 0 and weakly broken when
B
o
≠ 0. Broken ergodicity in MHD turbulence is explained by an eigenanalysis of modal covariance matrices. This produces a set of modal eigenvalues inversely proportional to the expected energy of their associated eigenvariables. A large disparity in eigenvalues within the same mode (identified by wavevector
k
) can occur at low values of wavenumber k = |
k
|, especially when
B
o
= 0. This disparity breaks the ergodicity of eigenvariables with smallest eigenvalues (largest energies). This leads to coherent structure in models of ideal homogeneous MHD turbulence, which can occur at lowest values of wavenumber k for 3-D cases, and at either lowest or highest k for ideal 2-D magnetofluids. These ideal results appear relevant for unforced, decaying MHD turbulence, so that broken ergodicity effects in MHD turbulence survive dissipation. In comparison, we will also examine ideal hydrodynamic (HD) turbulence, which, in the 3-D case, will be seen to differ fundamentally from ideal MHD turbulence in that coherent structure due to broken ergodicity can only occur at maximum k in numerical simulations. However, a nonzero viscosity eliminates this ideal 3-D HD structure, so that unforced, decaying 3-D HD turbulence is expected to be ergodic. In summary, broken ergodicity in MHD turbulence leads to energetic, large-scale, quasistationary magnetic fields (coherent structures) in numerical models of bounded, turbulent magnetofluids. Thus, broken ergodicity provides a large-scale dynamo mechanism within computer models of homogeneous MHD turbulence. These results may help us to better understand the origin of global magnetic fields in astrophysical and geophysical objects.
We consider an unforced, incompressible, turbulent magnetofluid constrained by concentric inner and outer spherical surfaces. We define a model system in which normal components of the velocity, ...magnetic field, vorticity, and electric current are zero on the boundaries. This choice allows us to find a set of Galerkin expansion functions that are common to both velocity and magnetic field, as well as vorticity and current. The model dynamical system represents magnetohydrodynamic (MHD) turbulence in a spherical domain and is analyzed by the methods similar to those applied to homogeneous MHD turbulence. We find a statistical theory of ideal (i.e. no dissipation) MHD turbulence analogous to that found in the homogeneous case, including the prediction of coherent structure in the form of a large-scale quasistationary magnetic field. This MHD dynamo depends on broken ergodicity, an effect that is enhanced when total magnetic helicity is increased relative to total energy. When dissipation is added and large scales are only weakly damped, quasiequilibrium may occur for long periods of time, so that the ideal theory is still pertinent on a global scale. Over longer periods of time, the selective decay of energy over magnetic helicity further enhances the effects of broken ergodicity. Thus, broken ergodicity is an essential mechanism and relative magnetic helicity is a critical parameter in this model MHD dynamo theory.
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