A Minimal Self-affine Stable Model for Space Physics Time Series

Direct inspiration for the measurement of scaling in space physics has come from inherently multiscale physics such as self-organised criticality and turbulence. Additionally, the ability to assess the likelihood of a fluctuation of a given size has "space weather" implications. However, we have elsewhere argued that it is also useful to capture the "stylised facts" of the scaling behaviour of auroral indices and solar wind quantities by much simpler, purely phenomenological models. To make this idea more concrete we here illustrate it by studying the use of fractional Lévy motion as a model for solar wind and auroral index time series, although this example could be taken as a prototype for other possible models. In fLm there are only three exponents, the Lévy exponent μ, the persistence exponent β and the selfsimilarity exponent H which depends additively on the other two. By postulating an fLm description we explore how the previously experimentally measured scaling exponents for quantities like superposed epoch averaged activity, or the probability distribution of the differenced time series, would depend on the model's parameters. We can then also derive predictions for the exponents of the more complicated measurements which have also been made, such as size and duration of bursts above a threshold, or the survival probability of a burst. Comparison of these predictions with data is then used to assess the usefulness of fLm as a toy model for space physics time series.