A new formulation of time-dependent Relaxed Magnetohydrodynamics (RxMHD) is derived variationally from Hamilton's Action Principle using microscopic conservation of mass, and macroscopic conservation of total magnetic helicity, cross helicity and entropy, as the only constraints on variations of density, pressure, fluid velocity, and magnetic vector potential over a relaxation domain. A novel phase-space version of the MHD Lagrangian is derived, which gives Euler--Lagrange equations consistent with previous work on exact ideal and relaxed MHD equilibria with flow, but generalizes the relaxation concept from statics to dynamics. The application of the new dynamical formalism is illustrated for short-wavelength linear waves, and the interface connection conditions for Multiregion Relaxed MHD (MRxMHD) are derived. The issue of whether $\vec{E} + \vec{u}\times\vec{B} = 0$ should be a constraint is discussed.
We perform direct numerical simulations of magnetohydrodynamic (MHD) turbulence with kinetic energy and cross helicity injections at large scales. We observe that the cross helicity changes sign as we go from large and intermediate scales to small scales. In addition, the magnetic reconnections are strongest at the regions where the cross helicity changes sign and becomes smallest in magnitude. Thus, our simulations provide an important window to explore the regions of magnetic reconnections in nonlinear MHD.
Ideal magnetohydrodynamics (IMHD) is strongly constrained by an infinite number of microscopic constraints expressing mass, entropy and magnetic flux conservation in each infinitesimal fluid element, the latter preventing magnetic reconnection. By contrast, in the Taylor relaxation model for formation of macroscopically self-organized plasma equilibrium states, all these constraints are relaxed save for global magnetic fluxes and helicity. A Lagrangian variational principle is presented that leads to a new, fully dynamical, \emph{relaxed magnetohydrodynamics} (RxMHD), such that all static solu…
Strong incompressible three-dimensional magnetohydrodynamic turbulence is investigated by means of high resolution direct numerical simulations. The simulations show that the configuration space is characterized by regions of positive and negative cross-helicity, corresponding to highly aligned or anti-aligned velocity and magnetic field fluctuations, even when the average cross-helicity is zero. To elucidate the role of cross-helicity, the spectra and structure of turbulence are obtained in imbalanced regions where cross-helicity is non-zero. When averaged over regions of positive and negativ…
Topological implications of the total generalized electron-flow magnetic helicity He in electron magnetohydrodynamics(EMHD) are explored. The invariance of He is shown to imply the invariance of the sum of the linkage of the magnetic field lines, the linkage of electron-flow vorticity field lines and the mutual linkage among these two sets of field lines. This result appears to support a change in the magnetic field topology and hence pave the way for magnetic reconnection in EMHD via a change in the concomitant electron-flow vorticity topology.
We present an extension of the multi-region relaxed magnetohydrodynamics (MRxMHD) equilibrium model that includes pressure anisotropy and general plasma flows. This anisotropic extension to our previous isotropic model is motivated by Sun and Finn's model of relaxed anisotropic magnetohydrodynamic equilibria. We prove that as the number of plasma regions becomes infinite, our anisotropic extension of MRxMHD reduces to anisotropic ideal MHD with flow. The continuously nested flux surface limit of our MRxMHD model is the first variational principle for anisotropic plasma equilibria with general …
The gap between a recently developed dynamical version of relaxed magnetohydrodynamics (RxMHD) and ideal MHD (IMHD) is bridged by approximating the zero-resistivity "Ideal" Ohm's Law (IOL) constraint using an augmented Lagrangian method borrowed from optimization theory. The augmentation combines a pointwise vector Lagrange multiplier method and global penalty function method and can be used either for iterative enforcement of the IOL to arbitrary accuracy, or for constructing a continuous sequence of magnetofluid dynamics models running between RxMHD (no IOL) and weak IMHD (IOL almost everywh…
Cross-helicity dominated distributed chaos (CHDC) for magnetohydrodynamic (large-scale) and kinetic (small-scale) ranges of space plasmas has been studied using results of direct numerical simulations and of measurements made in the Earth's magnetosheath and in free solar wind by the instruments onboard of the Magnetospheric Multiscale Mission and CLUSTER spacecrafts. Existence of two generic types of transition from the magnetohydrodynamic (MHD) to kinetic range has been shown: 1) from the Kolmogorov-like (scaling) MHD turbulence to the kinetic CHDC and 2) from the magnetohydrodynamic CHDC to…
We present an extension of the multi-region relaxed magnetohydrodynamics (MRxMHD) equilibrium model that includes plasma flow. This new model is a generalization of Woltjer's model of relaxed magnetohydrodynamics equilibria with flow. We prove that as the number of plasma regions becomes infinite our extension of MRxMHD reduces to ideal MHD with flow. We also prove that some solutions to MRxMHD with flow are not time-independent in the laboratory frame, and instead have 3D structure which rotates in the toroidal direction with fixed angular velocity. This capability gives MRxMHD potential appl…
The objective of the present paper is to investigate the constancy of the topological invariant denoted non-barotropic generalized cross helicity in the case of non-ideal magnetohydrodynamic (MHD). Existing work considers only ideal barotropic MHD and ideal non-barotropic MHD. The non-ideal MHD case was not explored probably because of its mathematical complexity. Here we consider dissipative processes in the form of thermal conduction, finite electrical conductivity and viscosity and the effect of these processes on the cross helicity conservation. Analytical approach has been adopted to obta…
Magnetohydrodynamics (MHD) provides the simplest description of magnetic plasma turbulence in a variety of astrophysical and laboratory systems. MHD turbulence with nonzero cross helicity is often called imbalanced, as it implies that the energies of Alfvén fluctuations propagating parallel and anti-parallel the background field are not equal. Recent analytical and numerical studies have revealed that at every scale, MHD turbulence consists of regions of positive and negative cross helicity, indicating that such turbulence is inherently locally imbalanced. In this paper, results from high reso…
We describe a magnetohydrodynamic (MHD) constrained energy functional for equilibrium calculations that combines the topological constraints of ideal MHD with elements of Taylor relaxation. Extremizing states allow for partially chaotic magnetic fields and non-trivial pressure profiles supported by a discrete set of ideal interfaces with irrational rotational transforms. Numerical solutions are computed using the Stepped Pressure Equilibrium Code, SPEC, and benchmarks and convergence calculations are presented.
The recent formulations of multi-region relaxed magnetohydrodynamics (MRxMHD) have generalized the famous Woltjer-Taylor states by incorporating a collection of "ideal barriers" that prevent global relaxation, and flow. In this paper, we generalize MRxMHD with flow to include Hall effects (MRxHMHD), and thereby obtain the partially relaxed counterparts of the famous double Beltrami states as a special subset. The physical and mathematical consequences arising from the introduction of the Hall term are also presented. We demonstrate that our results (in the ideal MHD limit) constitute an import…
We study the transition in dimensionality of a three-dimensional magnetohydrodynamic flow forced only mechanically, when the strength of a magnetic guiding field is gradually increased. We use numerical simulations to consider cases in which the mechanical forcing injects (or not) helicity in the flow. As the guiding field is increased, the strength of the magnetic field fluctuations decrease as a power law of the guiding field intensity. We show that for strong enough guiding fields, the helical magnetohydrodynamic flow can become almost two-dimensional. In this case, the mechanical energy ca…
In the presence of large-scale vortical motions and/or magnetic-field strains, the turbulent cross helicity (velocity--magnetic-field correlation in fluctuations) may contribute to the turbulent electromotive force and the Reynolds stress. These effects of cross helicity are considered to balance the primary effects of turbulence such as the turbulent magnetic diffusivity in magnetic-field evolution and the eddy viscosity in the momentum transport. The cross-helicity effects may suppress the enhanced transports due to turbulence. Physical interpretation of the effects is presented with special…
We describe the construction of stepped-pressure equilibria as extrema of a multi-region, relaxed magnetohydrodynamic (MHD) energy functional that combines elements of ideal MHD and Taylor relaxation, and which we call MRXMHD. The model is compatible with Hamiltonian chaos theory and allows the three-dimensional MHD equilibrium problem to be formulated in a well-posed manner suitable for computation. The energy-functional is discretized using a mixed finite-element, Fourier representation for the magnetic vector potential and the equilibrium geometry; and numerical solutions are constructed us…
We report the first measurements of equations of state of a fully relaxed magnetohydrodynamic (MHD) laboratory plasma. Parcels of magnetized plasma, called Taylor states, are formed in a coaxial magnetized plasma gun, and are allowed to relax and drift into a closed flux conserving volume. Density, ion temperature, and magnetic field are measured as a function of time as the Taylor states compress and heat. The theoretically predicted MHD and double adiabatic equations of state are compared to experimental measurements. We find that the MHD equation of state is inconsistent with our data.
Electron magnetohydrodynamics (EMHD) provides a fluid-like description of small-scale magnetized plasmas. An EMHD wave (also known as whistler wave) propagates along magnetic field lines. The direction of propagation can be either parallel or anti-parallel to the magnetic field lines. We numerically study propagation of 3-dimensional (3D) EMHD wave packets moving in one direction. We obtain two major results: 1. Unlike its magnetohydrodynamic (MHD) counterpart, an EMHD wave packet is dispersive. Because of this, EMHD wave packets traveling in one direction create opposite traveling wave packet…
We study the spatio-temporal behavior of the Elsässer variables describing magnetic and velocity field fluctuations, using direct numerical simulations of three-dimensional magnetohydrodynamic turbulence. We consider cases with relatively small, intermediate, and large values of a mean background magnetic field, and with null, small, and high cross-helicity (correlations between the velocity and the magnetic field). Wavenumber-dependent time correlation functions are computed for the different simulations. From these correlation functions, the decorrelation time is computed and compared with d…
Impulse formulations of Hall magnetohydrodynamic (MHD) equations are developed. The Lagrange invariance of a generalized ion magnetic helicity is established for Hall MHD. The physical implications of this Lagrange invariant are discussed. The discussion is then extended to compressible Hall MHD. Compressibility effects are shown not to affect, as to be expected, the physical implications of the generalized ion magnetic helicity Lagrange invariant.
The nonlinear dynamics of magnetic helicity, $H^M$, which is responsible for large-scale magnetic structure formation in electrically conducting turbulent media is investigated in forced and decaying three-dimensional magnetohydrodynamic turbulence. This is done with the help of high resolution direct numerical simulations and statistical closure theory. The numerically observed spectral scaling of $H^M$ is at variance with earlier work using a statistical closure model [Pouquet et al., J. Fluid Mech. \textbf{77} 321 (1976)]. By revisiting this theory a universal dynamical balance relation is …
The dynamics of a dissipationless incompressible Hall magnetohydrodynamic (HMHD) medium is formulated using Lagrangian mechanics on a semidirect product of two volume preserving diffeomorphism groups. In the case of $\mathbb{T}^3$ or $E^3$, the generalized Elsasser variables introduced by Galtier (S. Galtier 2006 J. Plasma Phys. 72 721-769) yield remarkably simple expressions of basic formulas and equations such as the structure constants of Lie algebra, the equation of motion, and the conservation laws. Four constants of motion, where three of the four are independent, are naturally derived f…
Axisymmetric relaxed states of a cylindrical plasma column are found analytically in both standard and Hall magnetohydrodynamics (MHD) by complete minimization of energy with constraints imposed by invariants inherent in corresponding models. It is shown that the relaxed state in Hall MHD is the force-free magnetic field with uniform axial flow and/or rigid azimuthal rotation. The relaxed states in standard MHD are more complex due to the coupling between velocity and magnetic field. Application of these states for reversed-field pinches (RFP) is discussed.
A unified energy principle approach is presented for analysing the magnetohydrodynamic (MHD) stability of plasmas consisting of multiple ideal and relaxed regions. By choosing an appropriate gauge, we show that the plasma displacement satisfies the same Euler-Lagrange equation in ideal and relaxed regions, except in the neighbourhood of magnetic surfaces. The difference at singular surfaces is analysed in cylindrical geometry: in ideal MHD only Newcomb's [W. A. Newcomb (2006) Ann. Phys., 10, 232] small solutions are allowed, whereas in relaxed MHD only the odd-parity large solution and even-pa…
Magnetic helicity is a fundamental quantity of magnetohydrodynamics that carries topological information about the magnetic field. By `topological information', we usually refer to the linkage of magnetic field lines. For domains that are not simply connected, however, helicity also depends on the topology of the domain. In this paper, we expand the standard definition of magnetic helicity in simply connected domains to multiply connected domains in $\mathbb{R}^3$ of arbitrary topology. We also discuss how using the classic Biot-Savart operator simplifies the expression for helicity and how do…
Nonlocal Evolution of Magnetic Helicity across the Magnetohydrodynamic-Kinetic Interface
Technische Universität Braunschweig（布伦瑞克工业大学）；Max Planck Institute for Solar System Research（马克斯·普朗克太阳系研究所）；Istituto Nazionale di Geofisica e Vulcanologia（意大利国家地球物理和火山学研究所）；University of California, Berkeley（加州大学伯克利分校）；KTH Royal Institute of Technology（瑞典皇家理工学院）；Stockholm University（斯德哥尔摩大学）；Ilia State University（伊利亚州立大学）；Carnegie Mellon University（卡内基梅隆大学）；Adelaide University（阿德莱德大学）；National Institute for Space Research (INPE)（国家空间研究所）
The dynamics of an incompressible, dissipationless Hall magnetohydrodynamic medium are investigated from Lagrangian mechanical viewpoint. The hybrid and magnetic helicities are shown to emerge, respectively, from the application of the particle relabeling symmetry for ion and electron flows to Noether's first theorem, while the constant of motion associated with the theorem is generally given by their arbitrary linear combination. Furthermore, integral path variation associated with the invariant action is expressed by the operation of an integro-differential operator on the reference path. Th…
Plasma relaxation in the presence of an initially braided magnetic field can lead to self-organization into relaxed states that retain non-trivial magnetic structure. These relaxed states may be in conflict with the linear force-free fields predicted by the classical Taylor theory, and remain to be fully understood. Here, we study how the individual field line helicities evolve during such a relaxation, and show that they provide new insights into the relaxation process. The line helicities are computed for numerical resistive-magnetohydrodynamic simulations of a relaxing braided magnetic fiel…
A general formulation of the problem of calculating the spectrum of stable and unstable eigenmodes of linearized perturbations about a magnetically confined toroidal plasma is presented. The analysis is based on a new hydromagnetic dynamical model, Multi-region Relaxed Magnetohydrodynamics (MRxMHD), which models the plasma-magnetic field system as consisting of multiple regions, containing compressible Euler fluid and Taylor-relaxed magnetic field, separated by flexible ideal-MHD current sheets. This is illustrated using a first-principles analysis of a two-region slab geometry, with periodic …
Relaxed magnetohydrodynamics with cross-field flow
Mathematical Sciences Institute, Australian National University（澳大利亚国立大学数学科学研究所）；Proxima Fusion（普罗克西玛聚变）；School of Physical and Mathematical Sciences, Nanyang Technological University（南洋理工大学物理与数理科学学院）
Ensemble averaged high resolution direct numerical simulations of reverse spectral transfer are presented, extending on the many single realization numerical studies done up to now. This identifies this type of spectral transfer as a statistical property of magnetohydrodynamic turbulence and thus permits reliable numerical exploration of its dynamics. The magnetic energy decay exponent from these ensemble runs has been determined to be $n_E = (0.47 \pm 0.03) + (13.9 \pm 0.8)/R_λ$ for initially helical magnetic fields. We show for the first time that even after removing the Lorentz force term i…
Clebsch potential gauge field theory for magnetohydrodynamics is developed based in part on the theory of Calkin (1963). It is shown how the polarization vector ${\bf P}$ in Calkin's approach, naturally arises from the Lagrange multiplier constraint equation for Faraday's equation for the magnetic induction ${\bf B}$, or alternatively from the magnetic vector potential form of Faraday's equation. Gauss's equation, (divergence of ${\bf B}$ is zero), is incorporated in the variational principle by means of a Lagrange multiplier constraint. Noether's theorem, coupled with the gauge symmetries is …
With the help of a model of magnetohydrodynamic (MHD) turbulence tested previously, we explore high Reynolds number regimes up to equivalent resolutions of 6000^3 grid points in the absence of forcing and with no imposed uniform magnetic field. For the given initial condition chosen here, with equal kinetic and magnetic energy, the flow ends up being dominated by the magnetic field, and the dynamics leads to an isotropic Iroshnikov-Kraichnan energy spectrum. However, the locally anisotropic magnetic field fluctuations perpendicular to the local mean field follow a Kolmogorov law. We find that …
The adiabatic limit of a recently proposed dynamical extension of Taylor relaxation, \emph{multi-region relaxed magnetohydrodynamics} (MRxMHD) is summarized, with special attention to the appropriate definition of relative magnetic helicity. The formalism is illustrated using a simple two-region, sheared-magnetic-field model similar to the Hahm--Kulsrud--Taylor (HKT) rippled-boundary slab model. In MRxMHD a linear Grad--Shafranov equation applies, even at finite ripple amplitude. The adiabatic switching on of boundary ripple excites a shielding current sheet opposing reconnection at a resonant…
The evolution of the correlation characteristics in three-dimensional isotropic electron magnetohydrodynamic turbulence is investigated. Universal exact relations between the longitudinal and transverse two-point triple correlations of the components of the fluctuational magnetic fields and the rates of dissipation of the magnetic helicity and energy are obtained in the inertial range.
Blackman & Brandenburg argued that magnetic helicity conservation in dynamo theory can in principle be captured by diagrams of mean field dynamos when the magnetic fields are represented by ribbons or tubes, but not by lines. Here we present such a schematic ribbon diagram for the $α^2$ dynamo that tracks magnetic helicity and provides distinct scales of large scale magnetic helicity, small scale magnetic helicity, and kinetic helicity involved in the process. This also motivates our construction of a new "2.5 scale" minimalist generalization of the helicity-evolving equations for the α^2 dyna…
We have explored the thermodynamics of compressed magnetized plasmas in laboratory experiments and we call these studies "magnetothermodynamics". The experiments are carried out in the Swarthmore Spheromak eXperiment device. In this device, a magnetized plasma source is located at one end and at the other end, a closed conducting can is installed. We generate parcels of magnetized plasma and observe their compression against the end wall of the conducting cylinder. The plasma parameters such as plasma density, temperature, and magnetic field are measured during compression using HeNe laser int…
Cross helicity is not conserved in non-barotropic magnetohydrodynamics (MHD) (as opposed to barotropic or incompressible MHD). Here we show that variational analysis suggests a new kind of cross helicity which is conserved in the non barotropic case. The non barotropic cross helicity reduces to the standard cross helicity under barotropic assumptions. The new cross helicity is conserved even for topologies for which the variational principle does not apply.
Recent progress regarding the noncanonical Hamiltonian formulation of extended magnetohydrodynamics (XMHD), a model with Hall drift and electron inertia, is summarized. The advantages of the Hamiltonian approach are invoked to study some general properties of XMHD turbulence, and to compare them against their ideal MHD counterparts. For instance, the helicity flux transfer rates for XMHD are computed, and Liouville's theorem for this model is also verified. The latter is used, in conjunction with the absolute equilibrium states, to arrive at the spectra for the invariants, and to determine the…
Magnetic helicity fluxes are investigated in a family of gauges in which the contribution from ideal magnetohydrodynamics takes the form of a purely advective flux. Numerical simulations of magnetohydrodynamic turbulence in this advective gauge family exhibit instabilities triggered by the build-up of unphysical irrotational contributions to the magnetic vector potential. As a remedy, the vector potential is evolved in a numerically well behaved gauge, from which the advective vector potential is obtained by a gauge transformation. In the kinematic regime, the magnetic helicity density evolves…
We construct finite element methods for the incompressible magnetohydrodynamics (MHD) system that precisely preserve magnetic and cross helicity, the energy law and the magnetic Gauss law at the discrete level. The variables are discretized as discrete differential forms in a de Rham complex. We present numerical tests to show the performance of the algorithm.
We show that oppositely directed fluxes of energy and magnetic helicity coexist in the inertial range in fully developed magnetohydrodynamic (MHD) turbulence with small-scale sources of magnetic helicity. Using a helical shell model of MHD turbulence, we study the high Reynolds number magnetohydrodynamic turbulence for helicity injection at a scale that is much smaller than the scale of energy injection. In a short range of scales larger than the forcing scale of magnetic helicity, a bottleneck-like effect appears, which results in a local reduction of the spectral slope. The slope changes in …
In our conventional understanding, large-scale magnetic fields are thought to originate from an inverse cascade in the presence of magnetic helicity, differential rotation, or a magneto-rotational instability. However, as recent simulations have given strong indications that an inverse cascade (transfer) may occur even in the absence of magnetic helicity, the physical origin of this inverse cascade is still not fully understood. We here present two simulations of freely decaying helical \& non-helical magnetohydrodynamic (MHD) turbulence. We verified the inverse transfer of helical and non-hel…
We revisit the issue of conservation of magnetic helicity and the Woltjer-Taylor relaxation theory in magnetohydrodynamics in the context of weak solutions. We introduce a relaxed system for the ideal MHD system, which decouples the effects of hydrodynamic turbulence such as the appearance of a Reynolds stress term from the magnetic helicity conservation in a manner consistent with observations in plasma turbulence. As by-products we answer two open questions in the field: We show the sharpness of the $L^3$ integrability condition for magnetic helicity conservation and provide turbulent bounde…
Reconnection is a fundamental event in many areas of science, from the interaction of vortices in classical and quantum fluids, and magnetic flux tubes in magnetohydrodynamics and plasma physics, to the recombination in polymer physics and DNA biology. By using fundamental results in topological fluid mechanics, the helicity of a flux tube can be calculated in terms of writhe and twist contributions. Here we show that the writhe is conserved under anti-parallel reconnection. Hence, for a pair of interacting flux tubes of equal flux, if the twist of the reconnected tube is the sum of the origin…
Magnetohydrodynamic (MHD) turbulence on a $β$-plane with an in-plane mean field, a system which serves as a simple model for the solar tachocline, is investigated analytically and computationally. We first derive two useful analytic constraints: we express the mean turbulent cross-helicity in terms of the mean turbulent magnetic energy, and then show that (for weak turbulence) the time-averaged momentum transport in the system can be expressed in terms of the cross-helicity spectrum. We then complete a closure of the system using weak turbulence theory, appropriately extended to a system with …
It is shown that in magnetohydrodynamics, the shapes of the magnetic energy and magnetic helicity spectra can influence whether the magnetic helicity resistive dissipation time scale is longer, equal to, or shorter than that of magnetic energy. The calculations highlight that magnetic helicity need not always dissipate significantly more slowly than magnetic energy. While this may be implicitly understood in detailed studies of magnetic helicity in plasma devices, it is not sufficiently emphasized in elementary discussions comparing magnetic helicity vs. magnetic energy evolution. Measuring ma…
The theorem of Noether dictates that for every continuous symmetry group of an Action the system must possess a conservation law. In this paper we discuss some subgroups of Arnold's labelling symmetry diffeomorphism related to non-barotropic magnetohydrodynamics (MHD) and the conservations laws associated with them. Those include but are not limited to the metage translation group and the associated topological conservations law of non-barotropic cross helicity.
In this paper, we study the constraints imposed by the invariants (generalized helicities and energy) of extended magnetohydrodynamics on some global characteristics of turbulence. We show that the global turbulent kinetic and magnetic energies will approach equipartition only under certain circumstances that depend on the ratio of the generalized helicities. In systems with minimal thermal energy, we demonstrate that the three invariants collectively determine the characteristic length scale associated with Alfvénic turbulence.
