Using Self-affine and Multifractal Testbeds to Calibrate Tests for "SOC" and "IT".

Details Using Self-affine and Multifractal Testbeds to Calibrate Tests for "SOC" and "IT". Using Self-affine and Multifractal Testbeds to Calibrate Tests for "SOC" and "IT".

Dec 2007

adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=2007agufmsm51b0532e&link_type=abstract

American Geophysical Union, Fall Meeting 2007, abstract #SM51B-0532

Mathematics

Probability

3235 Persistence, Memory, Correlations, Clustering (3265, 7857), 4468 Probability Distributions, Heavy And Fat-Tailed (3265), 4475 Scaling: Spatial And Temporal (1872, 3270, 4277), 7863 Turbulence (4490)

Scientific paper

There is by now abundant evidence for scaling in many fluctuating quantities in the coupled solar-terrestrial system (solar wind, magnetosphere and ionosphere). Physical explanations have thus naturally been sought (see e.g. the reviews [1-3]) in descriptions such as low dimensional chaos, intermittent turbulence (IT) and self-organised criticality (SOC). These latter two descriptions differ, however: SOC was directly inspired by a wish to unify spatial (fractal) and temporal ("1/f") scaling; whereas, although the study of turbulence has of course placed increasing emphasis on scaling and multiscaling phenomenology since the seminal work of Kolmogorov [1941], scaling is just one aspect of the subject. In this presentation we discuss a complementary approach ([4,5]), the use of deliberately oversimplified mathematical "testbeds" that allow examination of the logical inferences made in constructing hypotheses such as SOC. The model we will dwell on is Linear Fractional Stable Motion (LFSM). This unites the long range dependence exemplified by fractional Brownian motion (fBm) with the heavy tailed jumps seen in Lévy flights. LFSM is not purely a toy but has known links to extremal dynamics. Intriguingly, LFSM exhibits the appearance of multiaffine behaviour, like IT, while giving (at least in 1D) "avalanche" phenomenology in the sense of power law- tailed pdfs for burst "sizes" and "durations", like SOC. We will discuss numerical simulations, some analytical scaling arguments and our newly found diffusion-like equation for LFSM. We will also present early results from simulations of multifractal processes derived from fBm in order to give insight into how properties of LFSM may carry over into the multifractal domain. The implications for the inference of IT and SOC behaviour from uncontrolled natural datasets will be discussed. 1. Chapman and Watkins, Space Science Reviews, 2001 2. Freeman and Watkins, Science, 2002 3. Vassiliadis, Reviews of Geophysics, 2006 4. Watkins, Nonlinear Processes in Geophysics, 2002 5. Watkins et al., Space Science Reviews, 2005 [Stimulating suggestions from Vadim Uritsky, Mervyn Freeman and Shaun Lovejoy are acknowledged].

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