The influence of a flow field on the stability of the force-free magnetic field

In order to analyse the convective instability of the force-free magnetic field, an exact solution of the MHD equation for the magnetic field (1) together with the flow field (2) of constant speed V₀ making an angle θ with the magnetic field, was chosen as the unperturbed state. The stability of the fields between two parallel conducting walls of seperation d was studied by a linear perturbation method, which led to the eigenvalue problem (12), X being given by (13). It was shown by an approximate variational method that instability will set in by the flow field if V₀ is greater than 1/ $\text{3}$ times Alfven velocity V_A. For , the stability of the force-free field (1) is not influenced by the flow field, which may still be significant in other respects. Perturbations transverse to the magnetic field were found to be the most unstable modes.     

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.     

The flow field in a force-free magnetic field

We have obtained a complete set of the zeroth-order equations for a force-free field at large magnetic Reynolds numbers. One of the equations has often been overlooked in previous works. We discuss the question of uniqueness of solution of the Cauchy problem and outline a general method of solution in the plane and axisymmetric cases.     

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.     

Influence of fluctuating field on the configuration of force-free field

The consistent equations for random force-free magnetic fields are obtained from the fluctuation and average equations of both momentum and induction ones based on the statistical approach. Some features of average force-free fields are discussed. This idea may be applied to explain many configurations of magnetic field in the astronomical environment.     

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.     

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.     

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.     

The calculation of force-free fields from discrete flux distributions

This paper presents particularly simple mathematical formulas for the calculation of force-free fields of constant α from the distribution of discrete sources on a flat surface. The advantage of these formulas lies in their physical simplicity and the fact that they can be easily used in practice to calculate the fields. The disadvantage is that they are limited to fields of ‘sufficiently small α’. These formulas may be useful in the study of chromospheric magnetic fields by the comparison of high-resolution Hα photographs and photospheric magnetograms.     

The topology of force-free magnetic field near bipolar sunspots

The topological character of a new type of solution representing a force-free magnetic field near bipolar sunspots is examined. It is shown that some of the observed topological features of chromospheric fibrils and filaments in H can be interpreted in terms of the configuration of the magnetic lines of force of the present solution. In particular, by the examples considered the observed twisted S-shape topology of lower lying fibrils and the orientation of prominences (higher filaments) associated with sunspots are successfully reproduced.Visiting scientist at the High Altitude Observatory.The National Center for Atmospheric Research is sponsored by the National Science Foundation.     

Equilibrium and stability of the force-free magnetic field

The exact limit of stability of force-free fields for the case α=const and spherical region of the disturbances is obtained. The stable fields of active regions must be close to the harmonical ones for length scales which are less than the linear dimensions of spots.     

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.     

Equilibrium and stability of force-free magnetic field

Force-free magnetic fields (f.f.f) are considered as the first approximation of magnetic hydrodynamic equations in the case when the energy of the field exceeds the thermal energy of the medium. Such a relation of energies takes place in the upper atmosphere of the Sun in active regions.The consequence of the virial theorem obtained shows that for any solution of the corresponding non-linear system of equations only two cases are possible: either the total energy of the field is given by a divergent integral, or in some regions the force-free character of field is destroyed. This permits the conclusion that it is impossible to build f.f. current systems everywhere, and therefore boundary problems for this type of fields are of the same importance as for harmonic fields.Integral relations are obtained which are the necessary conditions for the solution of boundary problems. According to the classical principle of Thompson the harmonic fields are always stable, while f.f.f. may be stable or unstable.It is shown that: (1) arbitrary f.f.f. are stable to small changes of boundary conditions; (2) among f.f.f. the hydrodynamically stable configurations exist.The hydrodynamic stability condition restricts the size of force-free currents in such configurations.     

Singular lines of one-dimensional force-free magnetic field

The behavior of adiabatically slow deformations of the force-free field is investigated. Using the linear approximation it is shown that for a rather wide class of boundary perturbations of one-dimensional force-free field there appear singular magnetic force lines or surfaces. Hence the problem of quasi-steady deformation of frozen-in magnetic field has no solution. Relating to the problem of magnetic field in the solar corona it means that there will appear discontinuities (current sheets), when the magnetic field is deformed, for example, due to photospheric motion.     

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.