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.