Force-free State Research Articles

Approximate model and analysis of flux-core-spheromaks in force-free state.

Approximate model and analysis of flux-core-spheromaks in force-free state.

Structure and Equilibrium of Coronal Magnetic Fields

In “closed” magnetic structures (i.e., coronal loops) the random shuffling of magnetic footpoints in the photosphere causes twisting and braiding of field lines in the corona. If the motions are sufficiently slow, the coronal field evolves through a sequence of force-free equilibrium states. Numerical simulations are presented for a simplified model in which the overall curvature of the coronal loop is neglected. It is shown that magnetic fine structures develop on spatial scales significantly smaller than those of the imposed “photospheric” velocity field.

Expanding force-free magnetized plasmoid

Some properties of a plasmoid in which magnetic field dominates are discussed. Linear expansion, which have been proposed, provides a simple model describing slow and topologically self-similar expansion of a low-beta plasmoid. An initially force-free magnetized plasmoid expanding linearly may evolve toward consecutive force-free states. The magnetic field energy of such an expansion is studied by applying the virial equation. The result indicates that the decrease of the magnetic energy of the plasmoid is exactly equal to work done expelling its external pressure. This means that there is no residue of magnetic energy if a force-free magnetized plasmoid expands linearly.

Expansion of solar-terrestrial low-beta plasmoid

A magnetized plasma system confined inside a finite volume may expand (or contract) when its surrounding pressure changes. In this paper, two possible expanding processes of a low-beta plasmoid are discussed: (1) linear expansion and (2) expansion with fixed footpoints of field lines. The constant-zeta expansion, discussed in this paper, may maintain a low-beta plasmoid in a nearly force-free state during the expanding process. This simple model is applied to the magnetic clouds observed in the interplanetary space. The estimated scale size and expanding speed of the magnetic cloud are found to be in reasonable agreement with the observation. The energy conversion process is investigated: the free 'magnetic potential energy' stored in a magnetized system, which is pressed by external pressure, may be released to accelerate the plasma. The concept of a 'magnetic burst' is briefly discussed.

The stability of the force-free states in the light of the entropy principle

A functional of the quantities describing the macroscopic equilibrium of a magnetized plasma is defined, which has the meaning of the entropy and is constructed by a direct calculation of the probability of the distributions of the current density in a coarse-grained configuration space, under suitable constraints. The entropy law is applied to the derivation of the stability properties of the force-free states, with respect both to the resistive and to the ideal MHD disturbances.

Quasi-Static Evolution of a Force-Free Magnetic Field and Conditions for the Onset of a Stellar Flare

Magnetic fields in the solar corona are braught into an endless evolution by the never-ceasing motions of the subphotospheric plasma in which the feet of their lines are anchored. It is generally thought that this evolution is essentially quasi-static, the field passing through a sequence of force-free equilibrium states. Sporadically, however, the equilibrium is broken in a region of limited extent, and during a relatively short interval of time a catastrophic highly dynamic evolution takes place, giving rise to such wellknown phenomena as flares or coronal transients. Understanding the factors which determine if a magnetohydrostatic coronal equilibrium is maintained or, on the contrary, destroyed, when boundary conditions change at the photospheric level, then appears as a central theoretical problem of solar physics. In this Communication, we report some recent results which shed some new light onto this old problem.

Relaxation phenomena in the high-temperature S-1 spheromak.

Operation of the S-1 device in a high-current-density ($\frac{j}{{n}_{e}}\ensuremath{\ge}2\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}14}$ A m) regime has created high-electron-temperature spheromaks ($50 \mathrm{eV}\ensuremath{\le}{T}_{e}\ensuremath{\le}130 \mathrm{eV}$). The mechanisms and causes of the periodic relaxation events often observed in these hotter plasmas were made clear. A relationship between the MHD relaxation cycle and confinement was revealed for the first time. Resistive loss at the outer edge causes a departure from the initial force-free minimum-energy Taylor state to an MHD profile unstable to low-$n$ ideal MHD modes; a relaxation event then returns the configuration to a Taylor state.

Spontaneous formation of electric current sheets and the origin of solar flares

It is demonstrated that the continuous boundary motion of a sheared magnetic field in a tenuous plasma with an infinite electrical conductivity can induce the formation of multiple electric current sheets in the interior plasma. In response to specific footpoint displacements, the quadrupolar magnetic field considered is shown to require the formation of multiple electric current sheets as it achieves a force-free state. Some of the current sheets are found to be of finite length, running along separatrix lines of force which separate lobes of magnetic flux. It is suggested that current sheets in the form of infinitely thin magnetic shear layers may be unstable to resistive tearing, a process which may have application to solar flares.

Turbulent relaxation of a confined magnetofluid to a force-free state

Three-dimensional, pseudo-spectral computation is used to follow the evolution of a resistive, incompressible magnetofluid. The magnetofluid is confined by rigid, free-slip, perfectly-conducting square boundaries in the x, y directions (‘poloidal’ boundaries), and periodic boundary conditions are assumed in the z direction (‘toroidal’ direction). A constant, uniform d.c. magnetic field B0 is assumed in the z direction and a non-uniform current density j flows along it initially. Starting from a non-equilibrium hollow current profile, the evolution is followed for several tens of Alfvén transit times. Considerable small-scale turbulence develops, which causes energy to decay more rapidly than magnetic helicity. The average toroidal magnetic field at the (x, y) boundary reverses sign spontaneously. The near spatial constancy of the ratio jB/(jB) ≡ cos θ, in the relaxed state at late times, suggests that the state is nearly force-free. However, the ratio j. B/B2 ≡ α is considerably less uniform than is cos θ suggesting more residual disorder than a pure minimum-energy state would display.

Self-organization process in three-dimensional compressible magnetohydrodynamics

A three-dimensional self-organization process in a compressible dissipative plasma is investigated in detail by means of a variational method and a magnetohydrodynamic (MHD) simulation. In the closed system where the total energy and total mass are conserved, the compressible MHD plasma relaxes to a maximum entropy state under the constraint that the total magnetic helicity is constant. This state is equivalent to a force-free equilibrium in which the pressure distribution is uniform. Magnetic reconnection converts the magnetic energy into the thermal energy in the vicinity of the reconnection points where the selective dissipation of magnetic energy effectively takes place owing to the generation of magnetic bubbles [R. Horiuchi and T. Sato, Phys. Fluids 29, 1161 (1986)]. The enchanced thermal energy is carried away all over the system by the fast MHD waves. The swift expansion of the thermal energy leads to a uniform distribution of the pressure profile in the transit time scale of the fast MHD wave. Numerical simulation shows that the system tends to self-organize to a force-free state where the physical quantities are almost exactly the same as those of the zero pressure case. It is also observed that the magnetic energy spectrum cascades toward the high wavenumber region while the helicity spectrum inversely cascades toward the low wavenumber region.

Force-free magnetic fields - The magneto-frictional method

view Abstract Citations (251) References (9) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS Force-free Magnetic Fields: The Magneto-frictional Method Yang, W. H. ; Sturrock, P. A. ; Antiochos, S. K. Abstract The problem under discussion is that of calculating magnetic field configurations in which the Lorentz force j x B is everywhere zero, subject to specified boundary conditions. We choose to represent the magnetic field in terms of Clebsch variables in the form B = grad alpha x grad beta. These variables are constant on any field line so that each field line is labeled by the corresponding values of alpha and beta. When the field is described in this way, the most appropriate choice of boundary conditions is to specify the values of alpha and beta on the bounding surface. We show that such field configurations may be calculated by a magneto-frictional method. We imagine that the field lines move through a stationary medium, and that each element of magnetic field is subject to a frictional force parallel to and opposing the velocity of the field line. This concept leads to an iteration procedure for modifying the variables alpha and beta, that tends asymptotically towards the force-free state. We apply the method first to a simple problem in two rectangular dimensions, and then to a problem of cylindrical symmetry that was previously discussed by Barnes and Sturrock (1972). In one important respect, our new results differ from the earlier results of Barnes and Sturrock, and we conclude that the earlier article was in error. Publication: The Astrophysical Journal Pub Date: October 1986 DOI: 10.1086/164610 Bibcode: 1986ApJ...309..383Y Keywords: Boundary Value Problems; Force-Free Magnetic Fields; Lorentz Force; Magnetic Field Configurations; Aerospace Environments; Boundary Conditions; Friction; Iteration; Velocity; Astrophysics; HYDROMAGNETICS full text sources ADS |

Blowup of force-free magnetic fields in the infinite region of space

view Abstract Citations (54) References (28) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS Blowup of Force-free Magnetic Fields in the Infinite Region of Space Low, B. C. Abstract The model for the quasi-static evolution of a magnetic field through force-free states in response to slow motions of its footpoints at the boundary is considered using an axisymmetric model for a bipolar magnetic field in the infinite space outside a unit sphere. It is shown that, as has been suggested, footpoint displacements exist which can take a magnetic field in further evolution beyond a critical point, through force-free states with a suitably modified form of the alpha scalar function. However, not all footpoint displacements have this property. By direct example, it is shown that other footpoint displacements exist, in response to which quasi-static evolution breaks down and the magnetic field undergoes a catastrophic opening of part of its flux. Publication: The Astrophysical Journal Pub Date: August 1986 DOI: 10.1086/164407 Bibcode: 1986ApJ...307..205L Keywords: Astrophysics; Force-Free Magnetic Fields; Magnetohydrodynamics; Lines Of Force; Solar Atmosphere; Solar Flares; Astrophysics; HYDROMAGNETICS full text sources ADS |

Magnetic islands at the field reversal surface in reversed field pinches

In the reversed field pinch (RFP), magnetic field perturbations having zero poloidal mode number and any toroidal mode number are resonant at the field reversal surface. Such perturbations are a particular threat to the RFP because of their weak radial dependence at low toroidal mode number, and because the toroidal field ripple is essentially of this type. The widths of the resulting islands are calculated in this paper. The self-consistent plasma response is included through the assumption that the plasma relaxes to a Taylor force-free state. The connection with linear tearing mode theory is established for those limits where arbitrarily large islands result from infinitesimal perturbations. Toroidal effects are considered, and application of the theory to RFP experiments is discussed.

A probable initial configuration of a flaring loop

A clue to the preflare configuration of a magnetic loop is obtained based on Taylor's hypothesis that the decay of energy to a minimum with magnetic helicity remaining constant leads to a force free state. The numerical investigation (Riyopoulos et al. 1982) of a MHD turbulent plasma indicates that a high energy and nearly force free state of the plasma decays to a minimum energy state with a release of energy via magnetic reconnection processes.

Accretion disk flares

Flare activity on the sun may be attributed to the distortion of coronal magnetic field caused primarily by vortical (Shearing) photospheric motion. If an accretion disk is permeated by a magnetic field, the strong differential rotation of the disk will lead to similar stressing of the magnetosphere, and one may expect similar flare activity to occur. It is likely that the density of plasma in the maganetosphere is sufficiently large that the “frozen flux” condition is satisfied, and sufficiently small that the magnetic field will be substantially in the force-free state.

Critical currents and flux cutting in thin superconducting films

Experiments have been performed on thin films with the magnetic field applied parallel to the sample current (the force-free state). Plots of the flux-flow critical current versus magnetic field show a pronounced temperature independent structure which is sensitive to the film thickness. This structure is interpreted in terms of flux vortex arrangements that are matched to the sample geometry. A flux-cutting model for the dynamic behavior of vortices in this system is proposed and shown to be in qualitative agreement with the results.

On an Enhancement of a Critical Current of Superconductors in s Longitudinal Magnetic Field

An expression of the torque density on torsionally sheared fluxoids is derived and the torque balance is investigated for the force-free state in superconductors placed in a longitudinal magnetic field. The critical current in the force-free geometry is determined by the moment of pinning forces which prevent fluxoids from rotating to reduce the torsional shear. The ratio of this critical current density to that in the transverse field geometry is estimated for specimens pinned strongly. And the enhancement for weak pinning is also discussed qualitatively.

Reversed Field Configuration in Force-Free State

Two reversed field configurations in force-free state of rot B =α B have been considered; one is a straight cylindrical configuration with α( r ) dependent on radial coordinate r and the other is a toroidal one. The cylindrical configuration is obtained by choosing functional form of α( r ) in such a way that it closely match experimental results. The toroidal configuration is analyzed by using an expansion procedure. A new description of F-\( \varTheta \) diagram has been introduced in order to explain experiments.

Fluxoid motion and electromagnetic phenomena of a superconductor in a longitudinal magnetic field

The fluxoid motion during the process of applying the transport current to a superconductor in the longitudinal field configuration is visualized by solving the continuity equation of the fluxoid number. In this configuration, the force-free state described by rot B=αB is assumed. The obtained fluxoid velocity v contains the torsionally shearing component. The well-known relation E=B×v does not hold in the present field configuration. This fluxoid motion and the related phenomena are compared with those in the rotating transverse field configuration.

Most probable magnetohydrodynamic tokamak and reversed field pinch equilibria

The statistical theory of Montgomery, Turner, and Vahala, which determines the most probable ideal magnetohydrodynamic equilibrium compatible with given information on only a few global parameters (e.g., energy E, magnetic helicity H, flux Φ, current I, ⋅⋅⋅) is extended and investigated for both the tokamak regime (in which experimentally Φ≫μ0aI, with a being the plasma radius) and the reversed field pinch regime (Φ≪μ0aI). One obtains typical experimentally relevant profiles in the appropriate regimes. Most probable equilibria sequences are investigated as the energy/magnetic helicity ratio is decreased at fixed flux and current: In the tokamak regime (flux≫current) the diamagnetic toroidal field Bz becomes less diamagnetic and tends to a uniform field, while in the reversed field pinch regime (flux≪current), field reversal sets in Bz with the radial reversal position moving farther into the plasma and the eventual appearance of hollow pressure profiles. It appears that, in both regimes, the most probable equilibria are becoming more stable as μ0aE/H decreases. Linearized analytic force-free states can also be constructed for certain regimes of the global parameters together with their nonlinear quasi-force-free numerical counterparts.