Frozen-in Fractals All Around: Inferring the Large-Scale Effects of Small-Scale Magnetic Structure

The large-scale structure of the magnetic field in the solar corona provides the energy to power large-scale solar eruptive events. Our physical understanding of this structure, and hence our ability to predict these events, is limited by the type of data currently available. It is shown that the multifractal spectrum is a powerful tool to study this structure, by providing a physical connection between the details of photospheric magnetic gradients and current density at all size scales. This uses concepts associated with geometric measure theory and the theory of weakly differentiable functions to compare Ampère's law to the wavelet-transform modulus maximum method. The Hölder exponent provides a direct measure of the rate of change of current density across spatial size scales. As this measure is independent of many features of the data (pixel resolution, data size, data type, presence of quiet-Sun data), it provides a unique approach to studying magnetic-field complexity and hence a potentially powerful tool for a statistical prediction of solar-flare activity. Three specific predictions are provided to test this theory: the multifractal spectra will not be dependent on the data type or quality; quiet-Sun gradients will not persist with time; structures with high current densities at large size scales will be the source of energy storage for solar eruptive events.