HR: 0800h
AN: NG21B-0523    [Abstracts]
TI: A new Differential Equation for Anomalous Diffusion with Potential Applications to Nonlinear Space Plasmas
AU: * Watkins, N W
EM: nww@bas.ac.uk
AF: BAS, Madingley Road, Cambridge, CB3 0ET, United Kingdom
AU: Credgington, D
EM: d.credgington@ucl.ac.uk
AF: BAS, Madingley Road, Cambridge, CB3 0ET, United Kingdom
AU: Credgington, D
EM: d.credgington@ucl.ac.uk
AF: UCL, Gower Street, London, WC1E 6BT, United Kingdom
AU: Sanchez, R
EM: sanchezferlr@ornl.gov
AF: ORNL, Fusion Energy Division, Oak Ridge, TN 37831-6169, United States
AU: Chapman, S C
EM: s.c.chapman@warwick.ac.uk
AF: CFSA, University of Warwick, Coventry, CV4 7AL, United Kingdom
AB: Since the 1960s Mandelbrot has advocated the use of fractals for the description of the non-Euclidean geometry of many aspects of nature. In particular he proposed two kinds of model to capture persistence in time (his Joseph effect, common in hydrology and with fractional Brownian motion as the prototpe) and/or prone to heavy tailed jumps (the Noah effect, typical of economic indices, for which he proposed Lévy flights as an exemplar). Both effects are now well demonstrated in space plasmas, notably in indices quantifying Earth's auroral currents and in the turbulent solar wind. Models have, however, typically emphasised one of the Noah and Joseph parameters (the Lévy exponent μ and the temporal exponent β) at the other's expense. I will describe recent work [1] in which we studied a simple self-affine stable model-linear fractional stable motion, LFSM, which unifies both effects. I will discuss how this resolves some contradictions seen in earlier work. Such Noah-Joseph hybrid ("ambivalent" [2]) behaviour is highly topical in physics but is typically studied in the paradigm of the continuous time random walk (CTRW) [2,3] rather than LFSM. I will clarify the physical differences between these two pictures and present a recently-derived diffusion equation for LFSM. This replaces the second order spatial derivative in the equation of fBm [4] with a fractional derivative of order μ, but retains a diffusion coefficient with a power law time dependence rather than a fractional derivative in time (c.f. [2,3]). Intriguingly the self-similarity exponent extracted from the CTRW differs from that seen in LFSM. In the CTRW it is the ratio of μ to a temporal exponent, in LFSM it is an additive function of them. I will also show work in progress using an LFSM model and simple analytic scaling arguments to study the problem of the area between an LFSM curve and a threshold-related to the burst size measure introduced by Takalo and Consolini into solar- terrestrial physics and further studied by Freeman et al [5,6]. The extension of our new LFSM results to the related class of multifractals will be discussed. 1. Watkins et al, Space Sci. Rev. 121, 271, 2005.
2. Brockmann et al, Nature 439, 462, 2006.
3. Zaslavsky et al, Physica A, 373, 11, 2007.
4. Wang and Lung, Phys. Lett. A 151, 119, 1990.
5. Freeman et al, Geophys. Res. Lett. 27, 1367, 2000.
6. Freeman et al, Phys. Rev. E 62, 8794, 2000.
DE: 3235 Persistence, memory, correlations, clustering (3265, 7857)
DE: 4468 Probability distributions, heavy and fat-tailed (3265)
DE: 4475 Scaling: spatial and temporal (1872, 3270, 4277)
DE: 7863 Turbulence (4490)
SC: Nonlinear Geophysics [NG]
MN: 2007 Fall Meeting