The structure of force-free magnetic fields

Incontrovertible evidence is presented that the force-free magnetic fields exhibit strong stochastic behavior. Arnold's solution is given with the associated first integral of energy. A subset of the solution is shown to be non-ergodic whereas the full solution is shown to be ergodic. The first integral of energy is applied to the study of these fields to prove that the equilibrium points of such magnetic configurations are saddle points. Finally, the potential function of the first integral of energy is shown to be a member of the Helmholtz family of solutions. Numerical results corroborate the theoretical conclusions and demonstrate the robustness of the energy integral, which remains constant for arbitrarily long computing times.     

Cylindrical oscillations of force-free electromagnetic fields

This paper is devoted to Force-Free Electromagnetic Oscillations in a constant magnetic field. A correction is made in the derivation of the basic equation. The paper confirms the predicted spectrum of frequencies, namely _n = _o (n + 1)^1/2;n = 0, 1, 2, .... In addition it is suggested that hybrid frequency _n = ( _n ² + _H ² )^1/2 should be found in observational data.     

Cylindrical oscillations of force-free electromagnetic fields

The force-free electromagnetic field represents a natural generalization of the well-known force-free magnetic field model and allows the magnetic field to maintain electric charge separation.The basic equation for the cylindrical oscillations of the force-free electromagnetic field is obtained and solved for a linear case. The spectrum of possible resonances in a magnetized atmosphere is discussed.     

The influence of fluctuation fields on the force-free field

In Paper I (Hu, 1982), we discussed the the influence of fluctuation fields on the force-free field for the case of conventional turbulence and demonstrated the general relationships. In the present paper, by using the approach of local expansion, the equation of average force-free field is obtained as (1+b)?×B ₀=(α#x002B;a)B ₀#x002B;a ⁽¹⁾×B ₀#x002B;K. The average coefficientsa,a ⁽¹⁾,b, andK show the influence of the fluctuation fields in small scale on the configurations of magnetic field in large scale. As the average magnetic field is no longer parallel to the average electric current, the average configurations of force-free fields are more general and complex than the usual ones. From the view point of physics, the energy and momentum of the turbulent structures should have influence on the equilibrium of the average fields. Several examples are discussed, and they show the basic features of the fluctuation fields and the influence of fluctuation fields on the average configurations of magnetic fields. The astrophysical environments are often in the turbulent state, the results of the present paper may be applied to the turbulent plasma where the magnetic field is strong.     

On the time evolution of force-free fields

Using the adiabatic approximation, it is shown that the problem of the continuous deformation of a force-free (f.f.) field, in general, has no solution. This means that f.f. fields are nonevolutionary and even small perturbations may produce drastic changes in them. By analogy with a special case of f.f. field, the current-free field, we conclude that perturbations of a f.f. field in general produce pinch (current) sheets.     

Linear force-free fields in the lower corona

We have computed the surface Green's function for linear force-free magnetic fields, where × B = B and is a constant, for application to low coronal levels of the solar atmosphere. Boundary conditions are imposed on the normal component of B on two parallel planes which delineate the force-free volume. This procedure ensures that the magnetic field energy remains bounded, and that the field lines have a smooth behavior. A simple bipolar source distribution is treated and representative field line tracings are shown.     

Models of force-free magnetic fields in resistive media

A review is given of some of the basic properties of force-free fields under circumstances when the conductivity of the medium is finite. Then the electric current density is related not only to the magnetic field, but also by Ohm's law to the electric field and plasma velocity, which must be considered in the solutions. It is pointed out that the natural constraint that the electric field and plasma velocity should be finite everywhere is not fulfilled in some previously used models. Models with a constant ratio of the electric current density and magnetic field intensity have been used extensively in the past. They are of some importance since solutions with the plasma at rest are possible only if is constant. However, it is shown that solutions of constant cannot be matched to an external current-free region of finite conductivity since can have no discontinuity in a medium where the conductivity varies continuously. Hence, for most applications models with a varying and a moving plasma must be used. Some new, simple and consistent models, especially of cylindrical symmetry, are derived by prescribing the form of the magnetic field lines, or one electic field component, or one time-dependent electric current component.     

The equilibrium shape of slender flux tubes in a linear force-free magnetic field

In this paper we extend previous work of Browning and Priest (1984, 1986) by studying the equilibrium path of twisted and untwisted thin flux tubes in a stratified, isothermal atmosphere using as the ambient field a linear force-free field. When an untwisted flux tube is considered, we find that shearing the magnetic arcade provides a different form to change the parameter which characterizes the external atmosphere, but at the same time this introduces a limitation in the width allowed for the external arcade. Also, the critical width found for the different analytical cases considered is always greater than one arch of the ambient arcade which prevents an eruption inside the arcade. In the case of twisted flux tubes, an analytical solution can be found for the critical _c , which separates regimes of strong and weak gravity, and the shape of the flux tube is now dependent on , a parameter which represents the magnetic field enhancement of the loop at the photosphere.     

Error analysis regarding the calculation of nonlinear force-free field

Magnetic field extrapolation is an alternative method to study chromospheric and coronal magnetic fields. In this paper, two semi-analytical solutions of force-free fields (Low and Lou in Astrophys. J. 352:343, 1990) have been used to study the errors of nonlinear force-free (NLFF) fields based on force-free factor α. Three NLFF fields are extrapolated by approximate vertical integration (AVI) Song et al. (Astrophys. J. 649:1084, 2006), boundary integral equation (BIE) Yan and Sakurai (Sol. Phys. 195:89, 2000) and optimization (Opt.) Wiegelmann (Sol. Phys. 219:87, 2004) methods. Compared with the first semi-analytical field, it is found that the mean values of absolute relative standard deviations (RSD) of α along field lines are about 0.96–1.19, 0.63–1.07 and 0.43–0.72 for AVI, BIE and Opt. fields, respectively. While for the second semi-analytical field, they are about 0.80–1.02, 0.67–1.34 and 0.33–0.55 for AVI, BIE and Opt. fields, respectively. As for the analytical field, the calculation error of 〈|RSD|〉 is about 0.1∼0.2. It is also found that RSD does not apparently depend on the length of field line. These provide the basic estimation on the deviation of extrapolated field obtained by proposed methods from the real force-free field.     

Solar force-free magnetic fields on and above the photosphere

If the problem of a magnetic field being force-free with = constant ( 0) is solved by some previously published methods, then the field obtained in the whole exterior of the Sun cannot have a finite energy content and the solution cannot be determined uniquely from only one magnetic field component given at the photosphere. A magnetic field in the volume between two parallel planes has been investigated by us (Chen and Wang, 1986).Based on observational data we present in this paper a suitable physical model for a half-space and adopted an integral transform established by us (Chen, 1980, 1983) to solve this problem. We then obtain a unique analytical solution of the problem from only one magnetic field component (longitudinal field observed) given at the photosphere. Not only the uniqueness of the solution has been proved but also the finiteness of magnetic energy content in the half-space considered has been verified. We have demonstrated that there is no singular point in the solution. It enables us to describe analytically the configurations of magnetic fields on and above the photosphere.     

Characteristics of proton flare active regions and force-free magnetic fields

Proton flare active regions have the common characteristic of an S-shaped neutral line caused by relative movements of sunspots. This observational feature suggested to us a method of estimating the force-free parameter. For three typical active regions we have estimated the variation in the force-free parameter, and assuming a constant force-free parameter, we calculated the structure of the force-free field and the magnetic energy of the potential field. Our calculations show that before the occurence of proton flares, the force-free parameter increases and the magnetic energy of the potential field decreases, this decrease may well be the source for the development of the force-free field, and its magnitude is sufficient for the requirement of the proton flare.     

Evolution of emerging force-free flux tube to be a flare loop

Evolution of a filamentary magnetic flux tube emerging from the photosphere is investigated in the assumption that the magnetic field is force-free and unchanged during the evolution.If a characteristic radius of the flux tube is 3 km or less setting the field to 1000G, the temperature increases at first due to Joule heating up to about one million degree keeping the plasma density almost constant, and then the density decreases down to a critical value at which a current instability may occur. Thus, a.strong field-aligned electric field of 200 million volts or more is expected to be produced during the following anomalous Ohmic decay of the magnetic field as already shown by a numerical simulation.     

Bifurcation of force-free solar magnetic fields: A numerical approach

Numerical calculations of two-dimensional force-free fields as models of solar active regions are presented. For a given toroidal component of the photospheric magnetic field two branches of solutions are numerically obtained which merge at the critical point of maximum allowed toroidal magnetic field. Depending on boundary conditions magnetic islands may or may not form. The results are discussed with respect to their relevance to the flare process.     

Evolution of emerging force-free flux tube to be a flare loop

Solar Physics - Evolution of a filamentary magnetic flux tube emerging from the photosphere is investigated in the assumption that the magnetic field is force-free and unchanged during the...     

Non-ideal evolution of non-axisymmetric, force-free magnetic fields in a magnetar

Recent numerical magnetohydrodynamic calculations by Braithwaite and collaborators support the 'fossil field' hypothesis regarding the origin of magnetic fields in compact stars and suggest that the resistive evolution of the fossil field can explain the reorganization and decay of magnetar magnetic fields. Here, these findings are modelled analytically by allowing the stellar magnetic field to relax through a quasi-static sequence of non-axisymmetric, force-free states, by analogy with spheromak relaxation experiments, starting from a random field. Under the hypothesis that the force-free modes approach energy equipartition in the absence of resistivity, the output of the numerical calculations is semiquantitatively recovered: the field settles down to a linked poloidal–toroidal configuration, which inflates and becomes more toroidal as time passes. A qualitatively similar (but not identical) end state is reached if the magnetic field evolves by exchanging helicity between small and large scales according to an α-dynamo-like, mean-field mechanism, arising from the fluctuating electromotive force produced by the initial random field. The impossibility of matching a force-free internal field to a potential exterior field is discussed in the magnetar context.