Force-free Equilibrium Research Articles

Magnetohydrodynamics dynamical relaxation of coronal magnetic fields

Context. For the last thirty years, most of the studies on the relaxation of stressed magnetic fields in the solar environment have onlyconsidered the Lorentz force, neglecting plasma contributions, and therefore, limiting every equilibrium to that of a force-free field. Aims. Here we begin a study of the non-resistive evolution of finite beta plasmas and their relaxation to magnetohydrostatic states, where magnetic forces are balanced by plasma-pressure gradients, by using a simple 2D scenario involving a hydromagnetic disturbance to a uniform magnetic field. The final equilibrium state is predicted as a function of the initial disturbances, with aims to demonstrate what happens to the plasma during the relaxation process and to see what effects it has on the final equilibrium state. Methods. A set of numerical experiments are run using a full MHD code, with the relaxation driven by magnetoacoustic waves damped by viscous effects. The numerical results are compared with analytical calculations made within the linear regime, in which the whole process must remain adiabatic. Particular attention is paid to the thermodynamic behaviour of the plasma during the relaxation. Results. The analytical predictions for the final non force-free equilibrium depend only on the initial perturbations and the total pressure of the system. It is found that these predictions hold surprisingly well even for amplitudes of the perturbation far outside the linear regime. Conclusions. Including the effects of a finite plasma beta in relaxation experiments leads to significant differences from the force-free case.

Trapped particle effect on the velocity of circularly polarized electromagnetic waves in an isotropic plasma

An exact solution of Maxwell–Vlasov equations describing a stationary transverse wave with trapped charged particles is analyzed. In the absence of a background plasma, the phase velocity of such a wave is always less than speed of light. Physical structure of the wave looks most simply in the wave frame of reference. In this frame, the constructed solution corresponds to one of the simplest force-free equilibrium plasma configurations of the type of uniformly sheared current sheets and self-consistent magnetostatic field. In the presence of the background plasma, a simple condition for existence of the waves with sublight velocities is established.

Magnetohydrodynamic stability of plasmas with ideal and relaxed regions

AbstractA unified energy principle approach is presented for analysing the magnetohydrodynamic (MHD) stability of plasmas consisting of multiple ideal and relaxed regions. The gauge a = ξ × B for the vector potential, a, of linearized perturbations is used, with the equilibrium magnetic field B obeying a Beltrami equation, ∇ × B = αB, in relaxed regions. In a region with such a force-free equilibrium Beltrami field we show that ξ obeys the same Euler–Lagrange equation whether ideal or relaxed MHD is used for perturbations, except in the neighbourhood of the magnetic surfaces where B · ∇ is singular. The difference at singular surfaces is analysed in cylindrical geometry: in ideal MHD only Newcomb's small solutions are allowed, whereas in relaxed MHD only the odd-parity large solution and even-parity small solution are allowed. A procedure for constructing global multi-region solutions in cylindrical geometry is presented. Focusing on the limit where the two interfaces approach each other arbitrarily closely, it is shown that the singular-limit problem encountered previously by Hole et al. in multi-region relaxed MHD is stabilized if the relaxed-MHD region between the coalescing interfaces is replaced by an ideal-MHD region. We then present a stable (k, pressure) phase-space plot, which allows us to determine the form a stable pressure and field profile must take in the region between the interfaces. From this knowledge, we conclude that there exists a class of single-interface plasmas that were found to be stable by Kaiser and Uecker, but are shown to be unstable when the interface is resolved.

The correlation of fractal structures in the photospheric and the coronal magnetic field

This work examines the relation between the fractal properties of the photospheric magnetic patterns and those of the coronal magnetic fields in solar active regions.Our results show no correlation between the fractal dimensions of the observed 2d photospheric structures and the extrapolated unstable volumes in the corona, when nonlinear force-free extrapolation is used. This result is independent of efforts to (1) bring the photospheric magnetic fields closer to a nonlinear force-free equilibrium and (2) omit the lower part of the modeled magnetic field volume that is almost completely filled by unstable volumes. A significant correlation between the fractal dimensions of the photospheric and coronal magnetic features is only observed at the zero level (lower limit) of approximation of a current-free (potential) magnetic field extrapolation. We conclude that the complicated transition from photospheric non-force-free fields to coronal force-free ones hampers any direct correlation between the fractal dimensions of the 2d photospheric patterns and their 3d counterparts in the corona at the nonlinear force-free limit.

THREE-DIMENSIONAL RELATIVISTIC MAGNETOHYDRODYNAMIC SIMULATIONS OF CURRENT-DRIVEN INSTABILITY. I. INSTABILITY OF A STATIC COLUMN

We have investigated the development of current-driven (CD) kink instability through three-dimensional relativistic magnetohydrodynamic simulations. A static force-free equilibrium helical magnetic configuration is considered in order to study the influence of the initial configuration on the linear and nonlinear evolution of the instability. We found that the initial configuration is strongly distorted but not disrupted by the kink instability. The instability develops as predicted by linear theory. In the nonlinear regime, the kink amplitude continues to increase up to the terminal simulation time, albeit at different rates, for all but one simulation. The growth rate and nonlinear evolution of the CD kink instability depend moderately on the density profile and strongly on the magnetic pitch profile. The growth rate of the kink mode is reduced in the linear regime by an increase in the magnetic pitch with radius and reaches the nonlinear regime at a later time than the case with constant helical pitch. On the other hand, the growth rate of the kink mode is increased in the linear regime by a decrease in the magnetic pitch with radius and reaches the nonlinear regime sooner than the case with constant magnetic pitch. Kink amplitude growth in the nonlinear regime for decreasing magnetic pitch leads to a slender helically twisted column wrapped by magnetic field. On the other hand, kink amplitude growth in the nonlinear regime nearly ceases for increasing magnetic pitch.

MAGNETIC BRAIDING AND PARALLEL ELECTRIC FIELDS

The braiding of the solar coronal magnetic field via photospheric motions - with subsequent relaxation and magnetic reconnection -- is one of the most widely debated ideas of solar physics. We readdress the theory in the light of developments in three-dimensional magnetic reconnection theory. It is known that the integrated parallel electric field along field lines is the key quantity determining the rate of reconnection, in contrast with the two-dimensional case where the electric field itself is the important quantity. We demonstrate that this difference becomes crucial for sufficiently complex magnetic field structures. A numerical method is used to relax a braided magnetic field to an ideal force-free equilibrium; that equilibrium is found to be smooth, with only large- scale current structures. However, the equilibrium is shown to have a highly filamentary integrated parallel current structure with extremely short length- scales. An analytical model is developed to show that, in a coronal situation, the length scales associated with the integrated parallel current structures will rapidly decrease with increasing complexity, or degree of braiding, of the magnetic field. Analysis shows the decrease in these length scales will, for any finite resistivity, eventually become inconsistent with the stability of a force- free field. Thus the inevitable consequence of the magnetic braiding process is shown to be a loss of equilibrium of the coronal field, probably via magnetic reconnection events.

One-Dimensional Vlasov-Maxwell Equilibrium for the Force-Free Harris Sheet

In this Letter, the first nonlinear force-free Vlasov-Maxwell equilibrium is presented. One component of the equilibrium magnetic field has the same spatial structure as the Harris sheet, but whereas the Harris sheet is kept in force balance by pressure gradients, in the force-free solution presented here force balance is maintained by magnetic shear. Magnetic pressure, plasma pressure and plasma density are constant. The method used to find the equilibrium is based on the analogy of the one-dimensional Vlasov-Maxwell equilibrium problem to the motion of a pseudoparticle in a two-dimensional conservative potential. The force-free solution can be generalized to a complete family of equilibria that describe the transition between the purely pressure-balanced Harris sheet to the force-free Harris sheet.

M = 1 ideal kink modes in a line-tied screw pinch with finite plasma pressure

A new method for computing ideal magnetohydrodynamic linear eigenmodes in a cylindrical screw pinch with line-tying boundary conditions at the axial ends is presented. In this method, plasma volume is reflected over one of the end planes, and equations and field components are continued into the extended volume with the continuation rules prescribed by the line-tying boundary conditions. Field components in the combined volume are expanded in Fourier series in the axial coordinate. The resulting set of coupled differential equations is solved numerically in the radial coordinate by a finite difference method yielding growth rates and eigenmodes for the system. An example of an m=1 (m is the poloidal wave number) internal kink instability in a force-free plasma equilibrium with uniform pressure is considered. In contrast to a periodic screw pinch, marginally stable perturbations are essentially compressible in the line-tied geometry. Finite compressibility makes the mode more stable in addition to the usual line-tying stabilization in zero pressure plasma. The critical length corresponding to the marginal stability increases with the increase of plasma beta. A universal axial dependence for marginally stable density perturbations ρ(r,z)=ρ(r)exp[−izμ(r)] is predicted analytically and confirmed numerically, where μ(r) depends on the equilibrium magnetic field components as μ(r)=B¯θ∕rB¯z.

On the Expansion of Quasi‐Static, Twisted Coronal Loops in Uniform Gravity

Coronal loop emission profiles are often of remarkably constant width along their entire lengths, contradicting expectations based on model coronal magnetic field strengths decreasing with height. Meanwhile P. Bellan has produced a theoretical model in which an initially empty, twisted force-free loop, on being filled with plasma via upflow at each footpoint, in the absence of significant gravitational effects, forms a narrow, filamentary loop of constant cross section. In this paper, we focus on equilibrium states that include stratification by uniform gravity while retaining the effects of magnetic field twist. Comparing these with related force-free equilibria, it is found that injection of low-? plasma under coronal conditions is not likely to change the shape of a loop significantly. These linear equilibria apply to the interiors and boundaries of loops only, with external influences modeled by boundary total pressures. The effects of total pressure balance with surroundings and of gravitational stratification are to inhibit the pinching of a loop to a constant cross section. Only if the plasma ? were high enough for the plasma to reconfigure the external field and the hydrostatic scale height much greater than the loop size could the final state have nearly constant cross section. We do not expect this to occur in the corona.

Saturation of tearing modes in reversed field pinches with locally linear force-free magnetic fields and its application to quasi-single-helicity states

A simple formula for predicting the width of a saturated island, formed as a consequence of tearing perturbation of linear force-free fields in cylindrical geometry, is derived. The formula makes it possible to calculate the saturated island width in terms of the values of parameters characterizing the initial force-free equilibrium and can be applied to equilibria of interest for reversed field pinches. In particular it is applied, in this paper, to force-free equilibria with piecewise constant radial profile of the pinch parameter, which have been recently suggested to be relevant for the formation of quasi-single-helicity states. The main result is that the island width becomes larger as a parameter, that quantifies the departure from a relaxed Taylor state, increases.

Three-dimensional simulations of compressible tearing instability

Three-dimensional numerical simulations of the tearing instability in the framework of compressible and resistive magnetohydrodynamics are presented. Simulations have been performed with a novel Eulerian conservative high order code, including an explicit resistivity, which uses implicit high order numerical schemes having higher spectral resolution than classical schemes. The linear and non linear evolution of the tearing instability has been followed for force-free and pressure-balanced initial equilibrium configurations. Pressure equilibrium configurations are subject to a secondary instability which drives the system toward a quasi two dimensional structure oriented perpendicularly to the initial configuration. The development of secondary instabilities is suppressed by a guide field allowing the coalescence instability to fully develop in the system. Force-free initial configurations follow an intermediate path with respect the previous cases: Strong coalescence of magnetic islands, due to the non linear evolution of the tearing instability, is observed before the system enters in a phase dominated by 3D modes. The histories of the differing initial current-sheet equilibria have counterparts in the energy spectra that, for all three cases, are observed to be strongly anisotropic.

Linear stability analysis of force-free equilibria close to Taylor relaxed states

The linear stability of a class of force-free equilibria in cylindrical geometry is investigated. The class consists of cylindrically symmetric force-free equilibria for which the ratio μ between the parallel current density and the magnetic field is a step function of the radius. It is suggested that plasmas in reversed field pinches could be roughly represented by such equilibria as a consequence of a small departure from an initial force-free state with constant μ, the latter being reached after a relaxation process according to the classical theory proposed by Taylor [Phys. Rev. Lett. 33, 1139 (1974)]. A fully analytical derivation of the tearing stability parameter Δ′ for such class of equilibria is given. It is then shown with one explicit example how the presence of a downward step of relatively small height can destabilize the innermost resonant mode, which would otherwise be stable if μ were constant. A possible implication of this mechanism for the formation of cyclic quasisingle helicity states observed in reversed field pinches is proposed. Considerations on the ideal stability of the class of equilibria under investigation are also given.

Engineering Improvements and Physics Studies of the Accelerated Compact Toroid in the CTIX Injector

The main goals of the experimental studies on the CTIX compact toroid accelerator are to improve CT parameters through engineering modification, and to quantify CT dynamics by the comparison of experimental measurements to MHD theories. Over this research period, the final CT kinetic energy has been increased by about 30%, with increased efficiency of first-cycle capacitor energy utilization. The efficiency increase is obtained through a combination of engineering modification to passive magnetic switching (saturable Metglas cores), and a fast gas system in the acceleration section of CTIX. Using the new switching system and gas puffing, the final CT density has been increased up to a factor of 5, compared to operation without puffing. The second task of the CTIX project is to compare the experimental data with the existing theoretical models. The most relevant models are that of force-free equilibrium, and a Hall-MHD model. From analysis of magnetic field data, it is determined that the Hall-MHD model gives better agreement, if effects of fluid vorticity are also included. This analysis has been reported in a recent paper (S. J. Howard et al., Submitted). Finally, a new plasma density measurement technique, deflectometry, has been compared with conventional interferometry method. A significant data base has been produced to show that the deflection measurements correlate well with the HeNe laser interferometry.

Alfvén wave dissipation and topological properties of 3D coronal force-free magnetic fields

We study the propagation and dissipation of Alfvenic perturbation in a 3D equilibrium structure within a WKB model. We assume small amplitude and small wavelength of the perturbation. The generation of small scales in the perturbation is related to the property that nearby magnetic lines move apart from each other locally. This property is quantified by means of the Kolmogorov entropy H of magnetic lines. We numerically calculate the distribution of H for a 3D complex force-free equilibrium, which models the magnetic field above a quiet-Sun region, both for nonvanishing current and for a potential field. It is found that H decreases slightly with the altitude due to the decreasing complexity of the field, but it is relatively uniform except for the presence of sharp peaks, where H reaches much higher values than the average. These locations are those where Alfven waves are preferentially dissipated. By analyzing the magnetic topology at these locations, we find that they correspond to separator lines which are intersections of separatrix surfaces. Then, in a high-Reynolds number plasma, such as in the solar corona, heating due to Alfven wave dissipation takes place mainly at magnetic separatrices.

Resolution of the 180o Ambiguity for Inverse Horizontal Magnetic Field Configurations

A well-known problem in solar physics is that solutions for the transverse magnetic direction are ambiguous with respect to a 180° reversal in the direction. In this paper we focus on three methods for the removal of the 180° ambiguity applied to three MHD models. These methods are (1) the reference method, (2) the method of magnetic pressure gradient, and (3) the magnetic divergence-free method. All three methods are noniterative, and methods 2 and 3 are analytical and fast. We apply these methods to three MHD equilibrium model fields: (1) an analytical solution of a nonlinear force-free magnetic equilibrium from Low, (2) a simulation of an emerging twisted flux tube from Fan & Gibson, and (3) a pre-eruptive twisted magnetic flux rope equilibrium reached by relaxation from Amari et al. We measure the success of methods within horizontal field regions in the boundary, which are mathematically defined by B⊥ ∇⊥Bz > 0. When such regions overlap with the magnetic neutral lines, they are known as bald patches (BPs) or inverse topology. Our most important conclusion is that the magnetic divergence-free method is far more successful than the other two methods within BPs. This method requires a second level of measurements of the vertical magnetic field. As high-quality multilevel magnetograms will come online in the near future, our work shows that multilayer magnetic measurements will be highly desirable to objectively and successfully tackle the 180° ambiguity problem.

Structure of Magnetic Tower Jets in Stratified Atmospheres

Based on a new approach on modeling the magnetically dominated outflows from AGNs (Li et al. 2006), we study the propagation of magnetic tower jets in gravitationally stratified atmospheres (such as a galaxy cluster environment) in large scales (> tens of kpc) by performing three-dimensional magnetohydrodynamic (MHD) simulations. We present the detailed analysis of the MHD waves, the cylindrical radial force balance, and the collimation of magnetic tower jets. As magnetic energy is injected into a small central volume over a finite amount of time, the magnetic fields expand down the backgroun d density gradient, forming a collimated jet and an expanded “lobe” due to the gradually decreasing background density and pressure. Both the jet and lobes are magnetically dominated. In addition, the injection and expansion produce a hydrodynamic shock wave that is moving ahead of and enclosing the magnetic tower jet. This shock can eventually break the hydrostatic equilibrium in the ambient medium and cause a global gravitational contraction. This contraction produces a strong compression at the head of the magnetic tower front and helps to collimate radially to produce a slendershaped jet. At the outer edge of the jet, the magnetic pressur e is balanced by the background (modified) gas pressure, without any significant contribution from the hoo p stress. On the other hand, along the central axis of the jet, hoop stress is the dominant force in shaping the cent ral collimation of the poloidal current. The system, which possesses a highly wound helical magnetic configurati on, never quite reaches a force-free equilibrium state though the evolution becomes much slower at late stages. The simulations were performed without any initial perturbations so the overall structures of the jet r emain mostly axisymmetric. Subject headings:magnetic fields — galaxies: active — galaxies: jets — methods : numerical — magnetohydrodynamics (MHD)

Tearing and Kelvin-Helmholtz instabilities in the heliospheric plasma

We used 2.5D simulations to analyze the magnetohydrodynamic instabilities arising from an initial equilibrium configuration consisting of a plasma jet or wake in the presence of a magnetic field with strong transverse gradients, such as those arising in the solar wind. Our analysis extends previous results by considering both a force-free equilibrium and a pressure-balance condition for a jet in a plasma sheet, along with arbitrary angles between the magnetic field and velocity field. In the force-free case, the jet/wake does not contain a neutral sheet but the field rotates through the flow to invert its polarity. The presence of a magnetic field component aligned with the jet/wake destroys the symmetric nature of the fastest growing modes, leading to asymmetrical wake acceleration (or, equivalently, jet deceleration). In the case of a jet, the instability properties depend both on the magnetic field and flow gradients, as well as on the length of the jet. The results are applied to the post-termination shock jet recently found in 3D global heliospheric simulations, where our analysis confirms and explains the stability properties found in such simulations.

Dissipation of Alfvén waves in complex 3D coronal force-free structures

The propagation and dissipation of Alfven waves in a three-dimensional inhomogeneous force-free equilibrium structure is considered. Assuming small wavelengths the wave propagation is described within the framework of a WKB theory. The equilibrium structure is a complex magnetic field, resulting from a superposition of several harmonies, which represents a model of the coronal magnetic field above a quiet-Sun region. The properties of wave dissipation are related to the topological complexity of the background magnetic field, and the typical scale law of fast dissipation is recovered. A partial reflection of waves is included in the model by calculating reflection and transmission coefficients of both Alfven and magnelosonic waves. An energy balance is derived between the wave energy dissipated inside the coronal structure and the energy lost through the lower boundary. We find that even for large values of the Reynolds number, a non-negligible fraction of the Alfven wave energy is dissipated inside the corona.

Simulation of spheromak evolution and energy confinement

Simulation results are presented that illustrate the formation and decay of a spheromak plasma driven by a coaxial electrostatic plasma gun, and model the plasma energy confinement. The physics of magnetic reconnection during formation is also illuminated. The simulations are performed with the three-dimensional, time-dependent, resistive magnetohydrodynamic NIMROD code [C. R. Sovinec, A. H. Glasser, T. A. Gianakon, D. C. Barnes, R. A. Nebel, S. E. Kruger, D. D. Schnack, S. J. Plimpton, A. Tarditi, and M. S. Chu, J. Comput. Phys. 195, 355 (2004)]. The simulation results are compared to data from the Sustained Spheromak Physics Experiment (SSPX) [E. B. Hooper, L. D. Pearlstein, and R. H. Bulmer, Nucl. Fusion 39, 863 (1999)]. The simulation results are tracking SSPX with increasing fidelity (e.g., improved agreement with measured magnetic fields, fluctuation amplitudes, and electron temperature) as the simulation has been improved in its representations of the experimental geometry, the magnetic bias coils, and the detailed time dependence of the current source driving the plasma gun, and uses realistic parameters. The simulations confirm that controlling the magnetic fluctuations is influenced by the current drive history and by matching the gun current in sustainment approximately to the value corresponding to the eigenvalue in the flux-conserver for the parallel current in a force-free equilibrium.

Extrapolation of a nonlinear force-free field containing a highly twisted magnetic loop

The stress-and-relax method for the extrapolation of nonlinear force-free coronal magnetic fields from photospheric vector magnetograms is formulated and implemented in a manner analogous to the evolutionary extrapolation method. The technique is applied to a numerically constructed force-free equilibrium that has a simple bipolar structure of the normal field component in the bottom (magnetogram) plane but contains a highly twisted loop and a shear (current) layer, with a smooth but strong variation of the force-free parameter α in the magnetogram. A standard linear force-free extrapolation of this magnetogram, using the so-called αbest value, is found to fail in reproducing the twisted loop (or flux rope) and the shear layer; it yields a loop pair instead and the shear is not concentrated in a layer. With the nonlinear extrapolation technique, the given equilibrium is readily reconstructed to a high degree of accuracy if the magnetogram is sufficiently resolved. A parametric study quantifies the requirements on the resolution for a successful nonlinear extrapolation. Permitting magnetic reconnection by a controlled use of resistivity improved the extrapolation at a resolution comparable to the smallest structures in the magnetogram.