Applications of topology in magnetic fields

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

Learning by Arguing About Evidence and Explanations

Collaborative learning with cases characteristically involves discussing and developing shared explanations. We investigated the argumentation scheme which learners use in constructing shared explanations over evidence. We observed medical students attempting to explain how a judge had arrived at his verdict in a case of medical negligence. The students were learning within a virtual learning environment and their communication was computer mediated. We identify the dialogue type that these learners construct and show that their argumentation conforms with an abductive form of argumentation scheme ('inference to the best explanation'). We also assessed the students' learning and propose that it is related to particular features of this argumentation scheme

Action potential duration alternans in mathematical models of excitable cells

Action potential duration alternans has been associated with the onset of one of the most common forms of abnormal heart rhythm, atrial fibrillation (Cherry et al., 2012; Nattel, 2002). This thesis concerns identifying variables and parameters responsible for inducing action potential duration alternans. In order to achieve this, we apply asymptotic reduction methods to models of cardiac electrophysiology described by a system of ordinary differential equations and derive explicit discrete restitution maps which specify the action potential duration as a function of the preceding diastolic interval. The bifurcations of equilibria of these maps are studied to determine regions in the parameter space of the models where normal response and alternans occur. Furthermore, explicit parametric representations of both the normal and the alternans equilibrium branches of the restitution map are found. We also develop a framework formulated in terms of a boundary value problems for studying cardiac restitution. This framework can be used to derive analytically or compute numerically different branches of the action potential duration restitution map from the full excitable models. Our method is validated by comparing the asymptotic restitution map with the boundary value problem formulated restitution curves. The proposed method is applied to investigate the restitution properties of three excitable models: one generic excitable model and two ionic cardiac models. The first model is the McKean (1970) model which is a simplified version of the classical FitzHugh (1961) model. The other two models are the Caricature version of the Noble (1962) model derived by Biktashev et al. (2008) and an asymptotically reduced version of the Courtemanche et al. (1998) model of the atrial cell, reduced by Suckley (2004). After deriving the action potential duration restitution map for each of the mentioned model, the region of the models parameters in which alternans occurs is determined. We conclude that alternans appears if the dynamics in the diastolic stage of an action potential are faster than the dynamics in the systolic stage. Furthermore, we show that the time scale for the slow gating variable is responsible for inducing alternans. We outline that the oscillation in the slow activation of the K+ current and the slow inactivation of the L-type Ca+2 current can induce or suppress alternans

Self-organized braiding in solar coronal loops

In this paper, we investigate the evolution of braided solar coronal loops. We assume that coronal loops consist of several internal strands which twist and braid about each other. Reconnection between the strands leads to small flares and heating of the loop to X-ray temperatures. Using a method of generating and releasing braid structure similar to a forest fire model, we show that the reconnected field lines evolve to a self-organised critical state. In this state, the frequency distributions of coherent braid sequences as well as flare energies follow power law distributions. We demonstrate how the presence of net helicity in the loop alters the distribution laws.Leverhulme TrustAI

Effects of Density Fluctuations on Alfvén Wave Turbulence in a Coronal Hole

We present a study of density fluctuations in coronal holes. We used a reduced magnetohydrodynamic (RMHD) model that incorporated observationally constrained density fluctuations to determine whether density fluctuations in coronal holes can enhance Alfvén wave reflection and dissipation, thereby heating coronal holes and driving the fast solar wind. We show results for two models of the background atmosphere. Each model is a solution of the momentum equation and includes the effects of wave pressure on the solar wind outflow. In the first model, the plasma density and Alfvén speed vary smoothly with height. Wave reflection is relatively weak in the smooth model, resulting in a low energy dissipation rate. In the second model, we include additional density fluctuations along the flux tube based on the observations. We find that density ρ fluctuations on the order of δρ/ρ ∼ 0.24 increase the Alfvén wave turbulence to levels sufficient to heat the open field regions in coronal holes

On the Relationship Between Photospheric Footpoint Motions and Coronal Heating in Solar Active Regions

PublishedArticleCoronal heating theories can be classified as either direct current (DC) or alternating current (AC) mechanisms, depending on whether the coronal magnetic field responds quasi-statically or dynamically to the photospheric footpoint motions. In this paper we investigate whether photospheric footpoint motions with velocities of 1-2 km s–1 can heat the corona in active regions, and whether the corona responds quasi-statically or dynamically to such motions (DC versus AC heating). We construct three-dimensional magnetohydrodynamic models for the Alfvén waves and quasi-static perturbations generated within a coronal loop. We find that in models where the effects of the lower atmosphere are neglected, the corona responds quasi-statically to the footpoint motions (DC heating), but the energy flux into the corona is too low compared to observational requirements. In more realistic models that include the lower atmosphere, the corona responds more dynamically to the footpoint motions (AC heating) and the predicted heating rates due to Alfvén wave turbulence are sufficient to explain the observed hot loops. The higher heating rates are due to the amplification of Alfvén waves in the lower atmosphere. We conclude that magnetic braiding is a highly dynamic process

The heating of solar coronal loops by Alfvèn wave turbulence

This project was supported under contract NNM07AB07C from NASA to the Smithsonian Astrophysical Observatory (SAO) and contract SP02H1701R from Lockheed Martin Space and Astrophysics Laboratory (LMSAL) to SAO.In this paper we further develop a model for the heating of coronal loops by Alfvèn wave turbulence (AWT). The Alfvèn waves are assumed to be launched from a collection of kilogauss flux tubes in the photosphere at the two ends of the loop. Using a three-dimensional magneto-hydrodynamic (MHD) model for an active-region loop, we investigate how the waves from neighboring flux tubes interact in the chromosphere and corona. For a particular combination of model parameters we find that AWT can produce enough heat to maintain a peak temperature of about 2.5 MK, somewhat lower than the temperatures of 3 – 4 MK observed in the cores of active regions. The heating rates vary strongly in space and time, but the simulated heating events have durations less than 1 minute and are unlikely to reproduce the observed broad Differential Emission Measure distributions of active regions. The simulated spectral line non-thermal widths are predicted to be about 27 km s−1, which is high compared to the observed values. Therefore, the present AWT model does not satisfy the observational constraints. An alternative “magnetic braiding” model is considered in which the coronal field lines are subject to slow random footpoint motions, but we find that such long period motions produce much less heating than the shorter period waves launched within the flux tubes. We discuss several possibilities for resolving the problem of producing sufficiently hot loops in active regions.PostprintPeer reviewe