Inhomogeneous magnetic fields in the solar atmosphere
The magnetic field in the solar atmosphere is highly inhomogeneous. In the photosphere, the field is concentrated into intense flux tubes and the coronal magnetic field consists of many loops and regions of open field. This thesis investigates some of the basic properties of inhomogeneous solar magnetic fields. First of all, the equilibrium properties of untwisted flux tubes, confined by a spatially varying external pressure distribution, are investigated. The behaviour of thick flux tubes, including the effects of a transverse field component and a variation in the field across the tube, is compared with slender flux tube theory. It is shown that slender tube theory is accurate for tubes which are approximately slender, but that completely misleading results can be obtained by applying slender tube theory if the pressure distribution is not slowly varying. Twisted flux tubes are then studied, with the aim of finding how twisting affects a tube confined by an inhomogeneous pressure distribution. It is shown that, in general, a tube expands as it is twisted; this is illustrated both by extensions to slender tube theory and by some exact analytical solutions. A family of linear solutions is used to model the evolution of a finite tube confined by a falling external pressure. It is shown that, if the confining pressure falls too low, the tube may burst, with some dynamic process ensuing. The equilibrium properties of a flux tube with a curved axis are then investigated, with the main aim of modelling coronal loops. Previous theory for the equilibrium of a curved slender flux tube in a gravitationally stratified atmosphere, with a balance between magnetic buoyancy and tension forces, is extended to take into account an external field and the effects of twist. Increasing the magnitude of the external field tends to lower the summit height of the tube. It is found that non-equilibrium sets in if the footpoints are separated more than a certain critical width, which does not depend on the magnitude of the external field. It is found that two possible equilibrium heights can exist for a twisted tube; however, if the tube is twisted too far, or if the footpoints are moved apart, non-equilibrium can set in. The critical width at which non-equilibrium occurs is lower for a twisted tube than for an untwisted one. This is suggested as an explanation for the rise of a filament prior to a two ribbon flare, and as a mechanism for coronal transients. An alternative description of the coronal magnetic field is given, using a perturbation expansion for an almost potential field, with small pressure gradients. The field is assumed to be line-tied at the photospheric base. Then the equilibrium properties of the global magnetic field of a star are investigated. A linear and non-linear family of solutions to the magnetostatic equilibrium equation are found. The linear solutions are used to investigate the twisting up of force-free dipolar and quadrupolar fields, including in a simple manner the effects of a stellar wind. In both cases, it was found that the field becomes physically unreasonable if it is twisted too far, with field lines detached from the star being formed, which would be pulled out by the stellar wind. Thus, if the field is twisted more than a critical amount, non-equilibrium sets in and some catastrophic behaviour takes place. This is suggested as a possible mechanism for stellar flares. Similar results are found in a study of the effects of increasing the pressure gradients at the stellar surface of a magnetostatic dipole-like field. The linear solutions are also used to study the equilibrium of a finite magnetosphere, and multiple equilibria are found. Finally, one aspect of the propagation of waves in an inhomogeneous magnetic field is studied, with particular reference to the problem of heating the solar corona. The mechanism of phase-mixing, which provides a means of dissipating shear Alfven waves that propagate in an inhomogeneous magnetic field, is investigated. The onset of Kelvin-Helmholtz instability, which could disrupt the wave and thus enhance the dissipation, is studied. First, the dispersion relation of the instability is calculated for the case of fully developed phase-mixing. Then, the onset of the instability is investigated, to find out whether the instability can grow before the phase-mixing is fully developed. It is found that instability can set in after only a very few wave periods. It is suggested that the instability triggers off a turbulent cascade which dissipates the wave energy. The heating rates that could be produced by such a process are calculated, and are found to be more than adequate for coronal heating.