This thesis concerns applications of topology in magnetic fields. First, we examine
the influence of writhe in the stretch-twist-fold dynamo. We consider a thin flux
tube distorted by simple stretch, twist, and fold motions and calculate the helicity
and energy spectra. The writhe number assists in the calculations, as it tells us how
much the internal twist changes as the tube is distorted. In addition it provides
a valuable diagnostic for the degree of distortion. Non mirror-symmetric dynamos
typically generate magnetic helicity of one sign on large-scales and of the opposite
sign on small-scales. The calculations presented here confirm the hypothesis that
the large-scale helicity corresponds to writhe and the small-scale corresponds to
twist. In addition, the writhe helicity spectrum exhibits an interesting oscillatory
behaviour.
Second, we examine the effect of reconnection on the structure of a braided magnetic
field. A prominent model for both heating of the solar corona and the source
of small flares involves reconnection of braided magnetic flux elements. Much of
this braiding is thought to occur at as yet unresolved scales, for example braiding of
threads within an EUV or X-ray loop. However, some braiding may be still visible
at scales accessible to Trace or the EIS imager on Hinode. We suggest that attempts
to estimate the amount of braiding at these scales must take into account the degree
of coherence of the braid structure. We demonstrate that simple models of
braided magnetic fields which balance input of topological structure with reconnection
evolve to a self-organized critical state. An initially random braid can become
highly ordered, with coherence lengths obeying power law distributions. The energy
released during reconnection also obeys a power law
