Magnetic helicity and force-free properties of astrophysical magnetic fields

Abstract

Magnetic fields are ubiquitous in the universe and play an important role in variety of astrophysical phenomenon. It is thus very important to understand the origin, structure and strength of these astrophysical magnetic fields. In

this Thesis, we use the concept of magnetic helicity conservation and prop- erties of force-free magnetic fields to investigate the topological properties

of magnetic fields in the solar corona and the amplification and nonlinear saturation of dynamo generated field in disc galaxies. For the case of solar corona, we solve the linear and nonlinear force-free

field equation using photospheric boundary conditions to obtain simple ax- isymmetric magnetic field configurations in spherical geometry. We show

that the condition of separability of solutions in the radial and angular vari- ables leads to two classes of solutions: linear and nonlinear force-free fields

(NLFF). We extended the set of NLFF solutions with radial power law index n = p/q, for all cases of odd p and cases of q > p for even p. We apply these solutions to simulate photospheric vector magnetograms obtained using the spectro-polarimeter on board Hinode and search for best-fit configurations. The effectiveness of our search strategy is demonstrated on test inputs of dipolar, axisymmetric, and non axisymmetric linear force-free fields. Using the best fit, we build three- dimensional axisymmetric field configurations and calculate the energy and relative helicity with two independent meth-

ods. The magnetic helicity and free energy content of these fields are useful indicators of energy available for release during eruptive events like solar

flares. We analyze five magnetograms for active regions (AR) 10930 span- ning a period of three days during which two X-class flares occurred and

calculate the free energy and relative helicity of the active region before and after the flare. Our analysis indicates a peak in these quantities before the flare events, which is consistent with the previous results. We also analyze single-polarity regions AR 10923 and 10933, which showed very good fits to

potential fields. This method provides useful reconstruction of NLFF and in- put fields for other numerical techniques. We also apply the NLFF solutions

to calculate the amount of braiding in coronal magnetic fields using the con- cept of mean crossing number. This is then used to estimate the free energy

content in solar active regions. We find that the free energy estimates ob- tained from calculation of magnetic braiding is in good agreement with those

obtained by exact calculations of NLFF fields. We then apply the model of self-organized criticality (SOC) to these braided field lines and calculate the distribution of coherent braid sequences and flare energies. We find find good agreement in the flare energy distributions obtained using SOC model and

NLFFF extrapolation. These results provide useful information on the coro- nal loop structure and also imply that the coronal heating can be supplied

by the braiding in the case of the active sun. We provide a new formulation for relative helicity in arbitrary geometries using the toroidal-poloidal representation of the magnetic field iand discuss the special cases of planar and spherical geometry. In a general astrophysical

application, the fields penetrate the generation region and extend to a sur- rounding corona. It is important to develop gauge-free form for Helicity that

can be readily used in different geometries without involving integrals over

external volumes. The further extension of the ideas here can be formalized through use of differential geometry. Magnetic fields correlated on kiloparsec scales are seen in disc galaxies.

The origin could be due to amplification of small scale seed fields by a tur- bulent dynamo. Helicity conservation imposes constraints on dynamo action

and one can study the minimal field strength of the large scale magnetic field

that could arise despite the constraint. The calculation of helicity is tech- nically complicated because of open boundaries and the usual form for the

magneto-hydrodynamic (MHD) invariant needs to be modified to take this into account. We then present a global semi-analytic axisymmetric model for a turbulent dynamo operating in a galaxy with a corona. Here, we show that the supernovae (SNe) and magneto-rotational instability (MRI) driven turbulence parameters have nearly the same radial dependence and can be treated in a common formalism; however we assume the main contribution

from SNe. The general toroidal-poloidal representation is then used to cal- culate the global gauge invariant relative magnetic helicity in cylindrical ge- ometry. We present the analytic steady-state solutions within the disc that

are matched to force-free fields in the corona. A dynamical solution for the dynamo is then obtained by expanding the time-dependent field in the basis obtained using the steady-state solutions. The non-linear quenching of the

dynamo is alleviated by inclusion of small-scale advective and diffusive mag- netic helicity fluxes, which allow the helicity to be transferred outside the

disc and consequently build up a corona during the course of dynamo action. We find quadrupolar solutions for in the galactic disc that extend out into the corona and show oscillations radially. The mean field is found to reach saturation within a timescale of 1 Gyr with a strength which is of the order of equipartition magnetic energy (∼ Beq ).

The following is the arrangement of the Thesis. Chapter 1 gives an overview of astrophysical magnetic fields with special focus on observations of solar and galactic magnetic fields. Chapter 2 outlines the basics concepts of MHD and describes the processes relating to magnetic field generation and dissipation. We also discuss the topological properties of magnetic field using magnetic helicity and provide a novel prescription for calculating magnetic helicity in arbitrary geometries. Chapter 3 presents a description of potential and force-free fields and outlines their important properties. We then discuss analytical and numerical techniques for solving potential and force-free fields equations for determining coronal magnetic fields. In Chapter 4, we present an overview of various coronal heating mechanisms and discuss the statistical properties of solar flares. We then discuss braiding in coronal magnetic fields

and calculate the free energy in these configurations due to braiding. Chap- ter 5 gives an introduction to large-scale turbulent dynamos and discusses

various closure approximations used in mean field MHD. We then present its application to disc galaxies, discuss the basic analytic solutions and give an overview of current problems in dynamo theory. In Chapter 6, we present

new solutions to the nonlinear force-free field equation and discuss its appli- cation for determining the topological properties of coronal magnetic fields,

such as their free-energy and relative helicity. We then apply the solutions to a time sequence of vector magnetograms to estimate the energy released in a solar flare due to change in magnetic field configuration. In Chapter 7, we use the NLFF field solutions obtained in Chapter 6 and estimate the amount of free-energy due to braiding in these configurations. We then apply a model of SOC to this field and calculate the power-law distribution of flare energies which is then compared with observations. In Chapter 8, we present a model of nonlinear turbulent dynamo applied to a disc galaxy having a force-free

corona. We discuss the significance of small-scale magnetic helicity fluxes with regards nonlinear saturation of the dynamo. Chapter 9 then presents a summary of the results from all chapters, highlight the novel aspects of this Thesis with its impact. Then, we present future work which includes papers under preparation.