Turbulence, Complexity, and Solar Flares

The issue of predicting solar flares is one of the most fundamental in physics, addressing issues of plasma physics, high-energy physics, and modelling of complex systems. It also poses societal consequences, with our ever-increasing need for accurate space weather forecasts. Solar flares arise naturally as a competition between an input (flux emergence and rearrangement) in the photosphere and an output (electrical current build up and resistive dissipation) in the corona. Although initially localised, this redistribution affects neighbouring regions and an avalanche occurs resulting in large scale eruptions of plasma, particles, and magnetic field. As flares are powered from the stressed field rooted in the photosphere, a study of the photospheric magnetic complexity can be used to both predict activity and understand the physics of the magnetic field. The magnetic energy spectrum and multifractal spectrum are highlighted as two possible approaches to this.Comment: 2 figure

Convolution of multifractals and the local magnetization in a random field Ising chain

The local magnetization in the one-dimensional random-field Ising model is essentially the sum of two effective fields with multifractal probability measure. The probability measure of the local magnetization is thus the convolution of two multifractals. In this paper we prove relations between the multifractal properties of two measures and the multifractal properties of their convolution. The pointwise dimension at the boundary of the support of the convolution is the sum of the pointwise dimensions at the boundary of the support of the convoluted measures and the generalized box dimensions of the convolution are bounded from above by the sum of the generalized box dimensions of the convoluted measures. The generalized box dimensions of the convolution of Cantor sets with weights can be calculated analytically for certain parameter ranges and illustrate effects we also encounter in the case of the measure of the local magnetization. Returning to the study of this measure we apply the general inequalities and present numerical approximations of the D_q-spectrum. For the first time we are able to obtain results on multifractal properties of a physical quantity in the one-dimensional random-field Ising model which in principle could be measured experimentally. The numerically generated probability densities for the local magnetization show impressively the gradual transition from a monomodal to a bimodal distribution for growing random field strength h.Comment: An error in figure 1 was corrected, small additions were made to the introduction and the conclusions, some typos were corrected, 24 pages, LaTeX2e, 9 figure