Ideal Magnetohydrodynamic Turbulence Research Articles

The Statistical Mechanics of Ideal Magnetohydrodynamic Turbulence and a Solution of the Dynamo Problem

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’.

Transition to Equilibrium and Coherent Structure in Ideal MHD Turbulence, Part 2

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’.

Transition to Equilibrium and Coherent Structure in Ideal MHD Turbulence

Transition of ideal, homogeneous, incompressible, magnetohydrodynamic (MHD) turbulence to near-equilibrium from non-equilibrium initial conditions is examined through new long-time numerical simulations on a 1283 periodic grid. Here, we neglect dissipation because we are primarily concerned with behavior at the largest scale which has been shown to be essentially the same for ideal and real (forced and dissipative) MHD turbulence. A Fourier spectral transform method is used to numerically integrate the dynamical equations forward in time and results from six computer runs are presented with various combinations of imposed rotation and mean magnetic field. There are five separate cases of ideal, homogeneous, incompressible, MHD turbulence: Case I, with no rotation or mean field; Case II, where only rotation is imposed; Case III, which has only a mean magnetic field; Case IV, where rotation vector and mean magnetic field direction are aligned; and Case V, which has nonaligned rotation vector and mean field directions. Dynamic coefficients are predicted by statistical mechanics to be zero-mean random variables, but largest-scale coherent magnetic structures emerge in all cases during transition; this implies dynamo action is inherent in ideal MHD turbulence. These coherent structures are expected to occur in Cases I, II and IV, but not in Cases III and V; future studies will determine whether they persist.

Nonlinear Turbulent Dynamo during Gravitational Collapse

Abstract Via amplification by turbulent dynamo, magnetic fields can be potentially important for the formation of the first stars. To examine the dynamo behavior during the gravitational collapse of primordial gas, we extend the theory of the nonlinear turbulent dynamo to include the effect of gravitational compression. The relative importance between dynamo and compression varies during contraction, with the transition from dynamo- to compression-dominated amplification of magnetic fields with the increase of density. In the nonlinear stage of magnetic field amplification with the scale-by-scale energy equipartition between turbulence and magnetic fields, reconnection diffusion of magnetic fields in ideal magnetohydrodynamic turbulence becomes important. It causes the violation of the flux-freezing condition and accounts for (a) the small growth rate of the nonlinear dynamo, (b) the weak dependence of magnetic energy on density during contraction, (c) the saturated magnetic energy, and (d) the large correlation length of magnetic fields. The resulting magnetic field structure and the scaling of magnetic field strength with density are radically different from the expectations of flux freezing.

Magnetohydrodynamic turbulence and the geodynamo

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.

Advanced Diagnostics for the Study of Linearly Polarized Emission. II. Application to Diffuse Interstellar Radio Synchrotron Emission

Abstract Diagnostics of polarized emission provide us with valuable information on the Galactic magnetic field and the state of turbulence in the interstellar medium, which cannot be obtained from synchrotron intensity alone. In Paper I, we derived polarization diagnostics that are rotationally and translationally invariant in the Q–U plane, similar to the polarization gradient. In this paper, we apply these diagnostics to simulations of ideal magnetohydrodynamic turbulence that have a range of sonic and Alfvénic Mach numbers. We generate synthetic images of Stokes Q and U for these simulations for the cases where the turbulence is illuminated from behind by uniform polarized emission and where the polarized emission originates from within the turbulent volume. From these simulated images, we calculate the polarization diagnostics derived in Paper I for different lines of sight relative to the mean magnetic field and for a range of frequencies. For all of our simulations, we find that the polarization gradient is very similar to the generalized polarization gradient and that both trace spatial variations in the magnetoionic medium for the case where emission originates within the turbulent volume, provided that the medium is not supersonic. We propose a method for distinguishing the cases of emission coming from behind or within a turbulent, Faraday rotating medium and a method to partly map the rotation measure of the observed region. We also speculate on statistics of these diagnostics that may allow us to constrain the physical properties of an observed turbulent region.

Galactic interstellar turbulence across the southern sky seen through spatial gradients of the polarization vector

Radio synchrotron polarization maps of the Galaxy can be used to infer the properties of interstellar turbulence in the diffuse warm ionized medium (WIM). In this paper, we investigate the spatial gradient of linearly polarized synchrotron emission ($|\nabla\textbf{P}|/|\textbf{P}|$) as a tracer of turbulence, the relationship of the gradient to the sonic Mach number of the WIM, and changes in morphology of the gradient as a function of Galactic position in the southern sky. We use data from the S-band Polarization All Sky Survey (S-PASS) to image the spatial gradient of the linearly polarized synchrotron emission ($|\nabla \textbf{P}|/|\textbf{P}|$) of the entire southern sky at $2.3$~GHz. The spatial gradient of linear polarization reveals rapid changes of the density and magnetic fluctuations in the WIM due to magnetic turbulence as a function of Galactic position; we make comparisons of these data to ideal MHD numerical simulations. In order to constrain the sonic Mach number ($M_{s}$), we apply a high order moments analysis to the observations and to the simulated diffuse, isothermal ISM with ideal magneto-hydrodynamic turbulence. We find that polarization gradient maps reveal elongated structures, which we associate with turbulence in the ISM. Our analysis corroborates the view of a turbulent WIM in a transonic regime $M_{s}\lesssim 2$. Filamentary structures with typical widths down to the angular resolution are seen and the observed morphologies match closely with numerical simulations and in some cases H$\alpha$ contours. The $|\nabla \textbf{P}|/|\textbf{P}|$ intensity is found to be approximately log-normal distributed. No systematic variations of the sonic Mach number are observed as a function of Galactic coordinates, which is consistent with turbulence in the WIM inferred by the analysis of H$\alpha$ data.

Temperature and Entropy in Ideal Magnetohydrodynamic Turbulence

Fourier analysis of incompressible, homogeneous magnetohydrodynamic (MHD) turbulence produces a model dynamical system on which to perform numerical experiments. Statistical methods are used to understand the results of ideal (i.e., nondissipative) MHD turbulence simulations, with the goal of finding those aspects that survive the introduction of dissipation. This statistical mechanics is based on a Boltzmannlike probability density function containing three “inverse temperatures,” one associated with each of the three ideal invariants: energy, cross helicity, and magnetic helicity. However, these inverse temperatures are seen to be functions of a single parameter that may defined as the “temperature” in a statistical and thermodynamic sense: the average magnetic energy per Fourier mode. Here, we discuss temperature and entropy in ideal MHD turbulence and their use in understanding numerical experiments and physical observations.

Global invariants in ideal magnetohydrodynamic turbulence

Magnetohydrodynamic (MHD) turbulence is an important though incompletely understood factor affecting the dynamics of many astrophysical, geophysical, and technological plasmas. As an approximation, viscosity and resistivity may be ignored, and ideal MHD turbulence may be investigated by statistical methods. Incompressibility is also assumed and finite Fourier series are used to represent the turbulent velocity and magnetic field. The resulting model dynamical system consists of a set of independent Fourier coefficients that form a canonical ensemble described by a Gaussian probability density function (PDF). This PDF is similar in form to that of Boltzmann, except that its argument may contain not just the energy multiplied by an inverse temperature, but also two other invariant integrals, the cross helicity and magnetic helicity, each multiplied by its own inverse temperature. However, the cross and magnetic helicities, as usually defined, are not invariant in the presence of overall rotation or a mean magnetic field, respectively. Although the generalized form of the magnetic helicity is known, a generalized cross helicity may also be found, by adding terms that are linear in the mean magnetic field and angular rotation vectors, respectively. These general forms are invariant even in the presence of overall rotation and a mean magnetic field. We derive these general forms, explore their properties, examine how they extend the statistical theory of ideal MHD turbulence, and discuss how our results may be affected by dissipation and forcing.

Broken ergodicity in magnetohydrodynamic turbulence

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.

Plasma relaxation and the turbulent dynamo

Ideal magnetohydrodynamic (MHD) turbulence may be represented by finite Fourier series whose independent coefficients form a canonical ensemble described by a Gaussian probability density function containing a Hermitian covariance matrix with positive eigenvalues. When the eigenvalues at lowest wave number are very small, a large-scale coherent structure appears: a turbulent dynamo, which is seen in computations. A theoretical explanation is given and contains Taylor’s theory of force-free states. Numerical effects are examined and it is shown that larger grid sizes and smaller time steps provide for better resolution of coherent structure. Ideal hydrodynamic (HD) turbulence is examined and the results are compared and contrasted with those of ideal MHD turbulence. In particular, coherent structure appears in ideal MHD turbulence at the lowest wave number, but can occur in ideal HD turbulence only at the highest wave numbers in a simulation. In the case of real, i.e., dissipative flows, coherent structure and broken ergodicity are expected to occur in MHD turbulence at the largest scale. However, real HD turbulence at all scales and real MHD turbulence at all scales but the largest are expected to be ergodic.

Ideal homogeneous magnetohydrodynamic turbulence in the presence of rotation and a mean magnetic field

The statistical theory of absolute equilibrium ensembles is extended to describe ideal, three-dimensional, magnetohydrodynamic (MHD) turbulence with and without rotation, and with and without a mean magnetic field. Results from seven long-time numerical simulations of five general cases on a 32 3 grid are presented. One notable result is the discovery of a new ideal invariant, the 'parallel helicity,' which arises when rotation and mean magnetic field vectors are aligned. Although the basic equations and statistical theory are symmetric under parity or charge reversal, the presence of invariant cross, magnetic or parallel helicity dynamically breaks this symmetry. Ideal MHD turbulence is, in general, non-ergodic due to the decomposability of the constant energy surface in phase space. Non-ergodicity can be manifested in the appearance of coherent structure as long as magnetic or parallel helicity is invariant. The fact that MHD turbulence inherently contains coherent structure in certain general cases may have important implications for dynamo theory.

Theory and simulation of real and ideal magnetohydrodynamic turbulence

Incompressible, homogeneous magnetohydrodynamic (MHD) turbulence consists of fluctuating vorticity and magnetic fi elds, which are represented in terms of their Fourier coefficients. Here, a set of fi ve Fourier spectral transform method numerical simulations of two-dimensional (2-D) MHD turbulence on a $512^2$ grid is described. Each simulation is a numerically realized dynamical system consisting of Fourier modes associated with wave vectors $\mathbf{k}$, with integer components, such that $k = |\mathbf{k}| \le k_{max}$. The simulation set consists of one ideal (non-dissipative) case and four real (dissipative) cases. All fi ve runs had equivalent initial conditions. The dimensions of the dynamical systems associated with these cases are the numbers of independent real and imaginary parts of the Fourier modes. The ideal simulation has a dimension of $366104$, while each real simulation has a dimension of $411712$. The real runs vary in magnetic Prandtl number $P_M$, with $P_M \in {0.1, 0.25, 1, 4}$. In the results presented here, all runs have been taken to a simulation time of $t = 25$. Although ideal and real Fourier spectra are quite di fferent at high $k$, they are similar at low values of $k$. Their low $k$ behavior indicates the existence of broken symmetry and coherent structure in real MHD turbulence, similar to what exists in ideal MHD turbulence. The value of $P_M$ strongly affects the ratio of kinetic to magnetic energy and energy dissipation (which is mostly ohmic). The relevance of these results to 3-D Navier-Stokes and MHD turbulence is discussed.

Broken symmetry in ideal magnetohydrodynamic turbulence

A numerical study of the long-time evolution of a number of cases of inviscid, isotropic, incompressible, three-dimensional fluid, and magnetofluid turbulence has been completed. The results confirm that ideal magnetohydrodynamic turbulence is nonergodic if there is no external magnetic field present. This is due essentially to a canonical symmetry being broken in an arbitrary dynamical representation. The broken symmetry manifests itself as a coherent structure, i.e., a nonzero time-averaged part of the turbulent magnetic field. The coherent structure is observed, in one case, to contain about 18% of the total energy.

Pressure structure of solar coronal loops, III

The theory of ideal magnetohydrodynamic turbulence in cylindrical geometry is used to study the steady-state structure of a coronal loop. The pressure profile is derived from MHD equations by representing the velocity and magnetic fields as the superposition of Chandrasekhar-Kendall functions. Such a representation brings out the three-dimensional structure of the pressure in the coronal loop. The radial, azimuthal, and axial variations of the pressure for a constant density loop are discussed in detail. The pressure has an oscillatory behavior for different azimuthal angles at some radial positions. This study predicts more features in pressure than can be compared with the presently available observations.

Three-dimensional magnetohydrodynamic turbulence in cylindrical geometry

Ideal magnetohydrodynamic turbulence is treated using more realistic boundary conditions than rectangular periodic boundary conditions. The dynamical equations of incompressible magnetohydrodynamics and the associated fields are expanded in a set of vector eigenfunctions of the curl. The individual eigenfunctions represent force-free fields, but superpositions of them do not. Three integral invariants have simple quadratic expressions in the expansion coefficients: the total energy, the magnetic helicity, and the cross helicity. The invariants remain temporally constant in the face of a truncation at a large but finite number of coefficients. Boundary conditions imposed are those for a rigid, perfectly-conducting cylindrical boundary, with an arbitrary periodicity length parallel to the axis. Canonical distributions are constructed from the invariants. Mean-square turbulent velocity fields 〈v2〉 have finite values for virtually all initial conditions, including quiescient ones. The stability problem can be reformulated as a search for values of the integral invariants which will minimize 〈v2〉. This leads to a principle of extremal helicity, which requires a magnetic configuration which will minimize 〈v2〉 for a given total energy. The development of helical macroscopic structures in the cylinder as a function of increasing ratio of axial current to axial magnetic flux is predicted.