COMPARATIVE STUDY OF DYNAMICAL CRITICAL SCALING IN THE SPACE STORM INDEX VERSUS SOLAR WIND FLUCTUATIONSJames Wanliss, Presbyterian College, Clinton, SC

Thanks: Vadim Uritsky, James Weygand

Isradynamics, 13 April 2010. Ein Bokek, Israel

1. Statistical Physics Concepts In thermodynamics (statistical mechanics), a phase

transition or phase change is the transformation of a thermodynamic system from one phase to another. The distinguishing characteristic of a phase transition is an abrupt sudden change in one or more physical properties, in particular the heat capacity, with a small change in a thermodynamic variable such as the temperature.

Non-equilibrium thermodynamics is a branch of thermodynamics concerned with studying time-dependent thermodynamic systems.

Open system

Definition: Self-Organized Behavior Spontaneous

change in the internal

organization of the

system.

Change appears not to be guided or managed by natural sources

Definition: Critical Behaviour

In standard critical phenomena, there is a control parameter which an experimenter can vary to obtain this radical change in behaviour. In the case of melting, the control parameter is temperature.

Self-organized critical phenomenon, by contrast, is exhibited by driven systems which reach a critical state by their intrinsic dynamics, independently of the value of any control parameter.

If system is critical, results should be robust irrespective of activity levels.

In a physical system the time interval between two "events" is called a waiting-time, for instance, the time interval of a certain activity.

This can give information on whether storms are independent events, and provides a test for models for storm statistics.

Burst lifetime is the time, T, of a burst ofactivity. Total duration is given by θ.

Standard Poisson waiting-time distributions (‘used is good as new’)

2. Testing for Self-Organized or Critical Behavior: Waiting times

( )P e

Wanliss and Weygand, GRL, 2007

Intermittent behaviour, with long-range dependence.

Low-Latitudes: SYM-H Fluctuations

Burst lifetimes of SYM-H, є, VBs (1995-2005)

Power-law slope over several orders of magnitude

Doesn’t vary for different thresholds (SOC-like)

Scaling properties of the low-latitude magnetosphere, whose output is recorded by SYM-H, is not purely a direct response to the scale-free properties of the solar wind!

.exp)(

cT

AP

SYM-H VBs ε

1995-1998

1.24± 0.06 1.30± 0.08 1.32± 0.09

2000-2003

1.29± 0.05 1.54± 0.07 1.59± 0.08

Demonstrate that the temporal dynamics of SYM-H perturbations exhibit non-trivial power-law relations.

The avalanche dynamics are described in terms of the ensemble averaged number of active sites as a function of delay time from the start of each excitation in the ensemble, and the probability that an avalanche survives this time interval,

For a system near a critical point,

As well, for every avalanche with lifetime T there is a relationship between the lifetime and size of the avalanche, S, viz.

3. Testing for Self-Organised or Critical Behavior: Dynamic Critical Scaling

~ ,~ sPN

1~ TS

)(N

)(sP

Wanliss and Uritsky, JGR, 2010

Size vs Lifetime (S vs. T) shows a power law dependence, as does θ vs T. A very slight break in slope occurs near 10,000 seconds.

Slope for S vs T for whole interval gives a slope of 1.705±0.022

SYM-H fit for τ<2 hours gives η=0.263±0.008; δ=0.416±0.004

Thus 1+η+δ=1.679±0.063

(i.) Spreading critical exponents

THEORY

tT=1.40±0.04; tS=1.18±0.03.

MEASUREMENT

StSSP )(TtTTP )(

.02.024.11

12

St

(ii.) Avalanche critical exponents

4.Testing for Self-Organized or Critical Behavior: (iii.) Power SpectraIn addition to the above results, if the bursty dynamics isdue to a critical avalanching process, the exponent ( ) should allow one to predict the power-law slope β of the Fourier power spectrum describing the average burst shape. It has been shown for < 2 (which is the case for SYM-Hbursts and sandpile avalanching models) the integral relating P(f) with the avalanche size probability distribution and the conditional energy spectrum of avalanches of a given size is dominated by a frequency dependent upper cut-off, yielding the simple scaling relation

ststTS ~

st

st

To verify this relation, we performed two semi-independent statistical tests. In the first test, we put together SYM-H bursts with T < 240 min in their natural order by eliminating the quiet periods separating the bursts. The resulting time series is analogous to the time dependence of the number of topplings in an avalanching model studied under infinitely slow driving conditions

In the 2nd test, we overlapped bursts by randomizing their starting times and taking their sum for each time step.

The time series obtained mimics the dynamics of “running” sandpiles with slow but continuous driving, producing no interference between concurrent avalanches.

In both tests, the power spectra have a distinct power-law region for frequencies above (240 minutes)-1 with the exponent β being statistically indistinguishable from the exponent as predicted for critical avalanching systems.

To make sure that the obtained spectra characterize correlations within bursts we also checked the power spectrum of the lifetime dynamics as represented by consecutive T values. The spectrum of the lifetimes has a clear white noise shape indicating that the interburst correlations have no significant effect on the burst shape, in agreement with

the behavior of the

described class of critical

avalanching models.

6. Summary Burst lifetime distribution functions yield clear power-law

exponents of the lifetime probability distributions. Since SYM-H scaling was remarkably robust, irrespective of solar cycle, it could be that the solar wind almost never acts as a direct driver for the SYM-H scaling.

Tests on ensemble averaged dynamics of the bursts of activity in the SYM-H index demonstrated scale-free behavior, and strong correlation between the size S and lifetime T of the activity bursts. These scaling behaviors were consistent with theoretical predictions from nonequilibrium systems near criticality. Our results show what could be the first quantitative evidence for the same universality class as directed percolation in a natural system.

Similar scaling behavior is NOT observed in solar wind fluctuations.

Summary (2)

The second level of results is our demonstration of the possibility that the multiscale dynamics of the ring current system is a result of its cooperative behavior governed by a specific statistical principle. We associate this dynamics with nonlinear interactions of spatially distributed degrees of freedom (e.g., current filaments) keeping the system in the vicinity of a global critical point.

The results can also be used for validating existing and future ring current models in terms of their ability to correctly represent the cross‐scale coupling effects in this system.

References

J. A. Wanliss and J. M. Weygand (2007), Power law burst lifetime distribution of the SYM‐H index, Geophys. Res. Lett., 34, L04107, doi:10.1029/2006GL028235.

Wanliss, J., and V. Uritsky (2010), Understanding bursty behavior in midlatitude geomagnetic activity, J. Geophys. Res., 115, A03215, doi:10.1029/2009JA014642.