NG44A-01 INVITED
Linear Fractional Stable Motion: can we use it to Model the Noah and Joseph Effects in Space Physics and Elsewhere ?
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 surveys of Chapman and Watkins [2001]; Freeman and Watkins [2002] and Vassiliadis [2006]) in descriptions such as low dimensional chaos, turbulence and SOC. These latter two models differ, in that SOC was directly inspired by a wish to unify spatial (fractal) and temporal (1/f) scaling, while the study of turbulence has over time placed increasing emphasis on scaling and multiscaling phenomenology since the seminal work of Kolmogorov [1941]. I here discuss a complementary approach (Watkins [2002]; Watkins et al. [2005]) - the use of deliberately oversimplified mathematical testbeds that may capture relevant phenomenology and/or give insight. The model I will discuss is Linear Fractional Stable Motion (LFSM), which unites long range dependence-the Joseph effect-exemplified by fractional Brownian motion with the heavy tailed jumps-the Noah effect-of Levy flights. LFSM is in fact not purely a toy model but has known links to extremal dynamics. Intriguingly, LFSM exhibits the appearance of multiaffinity while giving (at least in 1D) avalanche phenomenology in the sense of power law-tailed pdfs for burst sizes and durations. I will discuss numerical simulations, some analytical scaling arguments and a diffusion-like equation for LFSM.
NG44A-02
DETERMINISTIC GEOMETRIC MODELING OF NATURAL COMPLEXITY
A geometric procedure producing a host of complex distributions over one, two or three dimensions, as transformations of multifractal distributions via fractal functions, is reviewed. Then, it is shown how such notions may be extended in order to produce yet richer sets of patterns that may be used to represent the geometry of a variety of geophysical sets. The ideas are illustrated via various examples that include evolutions of patterns, and their related statistics, based on suitable parameter changes.
NG44A-03
Specification and Forecast of Energetic magnetospheric Particles and Geomagnetic Activity
Prompt and accurate assessments of enhancements in near-Earth energetic particles are crucial to understanding causes of space system operational problems. For this purpose, it is necessary to know the state of the present (and recent past) space environment. Ideally, this means that one should be able to specify the temporal behavior of energetic particles at all relevant altitudes, latitudes, and local times over the entire energy range of interest to space system operators. Through the use of data from a variety of scientific and operational spacecraft, it has been possible in recent times to develop "dynamic" radiation models and predictions of geomagnetic conditions. Our present work in this regard uses a variety of data modeling techniques. With such modeling, we are generally able to achieve reasonable accuracies of energetic particle flux specification throughout the outer magnetosphere. We are also able to forecast geomagnetic indices (Ap) and particle fluxes for some 3-4 days based upon analog modeling techniques. Future work employing these methods should allow even more accurate, reliable specification for magnetospheric conditions.
NG44A-04 INVITED
Model Assessment With Lagrangian Metrics and Data
As geophysical predictive models typically are Eulerian, it seems natural to evaluate their performance with Eulerian metrics and observations. For example, meteorological predictions are assessed by how well they predict precipitation or temperature at specific locations while oceanographic models often are evaluated by comparison of predicted currents with current meter moorings and predicted water mass properties with CTD casts. In the oceanographic case, assessment often is problematic since most such models are exercised in the forward mode. A true test of model skill requires data assimilation since it is then possible to compare model results directly with independent observations of specific ‘events'. Predictive models, judged successful by Eulerian metrics, often fail to perform as well when required to predict Lagrangian properties such as the paths of hurricanes, the motion of ocean eddies, the dispersion of drifting sensor arrays, and the movement of contaminants in the environment. Model assessment with Lagrangian data and metrics, which are inherently nonlinear, is an emergent issue in oceanography. Here, two types of metrics, along with appropriate data are used to assess a data-assimilating model of ocean currents. The first type, the prediction of individual trajectories, generally show poor performance when compared with observations. On the other hand, methods adapted from dynamical systems theory show remarkable ability to account for the outbreak of chlorophyll plumes and the dispersion of drifting sensor arrays. We also compare model predictions of eddy formation, breakup and the movement of cyclones around large anticyclones with conventional Eulerian observations.
NG44A-05
Characterizing the Multiscale Phenomena of the Magnetosphere
The multiscale nature of the magnetospheric response to the driving by the turbulent solar wind is prototypical of open natural systems. The distribution of scales in the magnetosphere are studied using data from space- borne and ground-based measurements. The burst lifetime distribution of the AL index, which is a characteristic response of the magnetosphere, has been found to be a combination of a power law with an exponential cutoff and a log-normal function [Freeman et al., GRL, 27, 1087, 2000]. The distribution of waiting times between substorms above a certain threshold have also been shown to have a similiar behavior [Freeman et.al, Phys. Rev. E 2000]. An important feature of both these distributions is the clear deviation from the power law behavior for longer time scales. The distribution of scales in the AL index (1 min resolution) from January 1978 to June 1988 and from January 1990 to December 1995, consisting of approximately 8.6 million data points, is studied to determine the distribution of scales. There is a clear deviation of the burst duration distribution at lower thresholds from a stretched exponential and the same occurs for the waiting time distribution at higher thresholds. The stretched exponential distributions of waiting times are recognized as a universal behavior for long term correlated data, and the deviations from such a distribution in the case of the magnetosphere indicate the presence of both long and short range correlations. This implies that the magnetospheric dynamics exhibit both global and multiscale phenomena.
NG44A-06 INVITED
Mutltifractal Predictability, Predictions and Forecasts
Multifractals are widely recognized as powerful tools to analyze spatial heterogeneities or temporal variability of
complex fields. They are in fact extremely powerful to analyze together space and time fluctuations, in particular
their scaling anisotropy. Furthermore, we argue that multifractals are not limited to analyze: their capacity to
predict and forecast has to be better investigated and exploited.
It was first necessary to clarify the intrinsic predictability limits of space time scaling systems, e.g. dynamics of
the atmosphere coupled with various fields such as the water content. They are quite different from those of
systems that are only complex in time. Indeed, space time scaling systems do not yield characteristic times of
predictability: a limited uncertainty on initial and/or boundary conditions on a given range of time and space
scales rapidly grows across the scales and yields power-law decays of the predictability, not exponential decays.
Furthermore, the predictability decay is highly intermittent: the loss of information occurs by intermittent puffs. The
predictability itself is multifractal: an infinite hierarchy of power-law exponents is required to characterize the
predictability decay from average to extreme events. In particular, we will discuss the multifractal behaviour of the
error flux.
The second step was the recognition that multifractals can lead to statistical predictions, e.g. that the extremes of
a field can be predicted from its rather average behavior. More recently, we have been interested to predict its
maxima at different scales.
The third step is to proceed to multifractal forecasts in a dynamical manner. We were first interested by stochastic
forecasts, i.e. simulating a given number of possible future realizations and comparing their relative dispersion
to their multifractal predictability. More recently we have been developing probability forecasts.
http:www.enpc.fr/multifractal/
NG44A-07
An Information-Theoretical Approach to Identify Nonlinearity in Magnetospheric Activity
We discuss a method to detect nonlinear dependencies in multivariate time series using mutual information and
cumulant-based cost as discriminating statistics. The method is applied to the historical data stream of
geomagnetic indices (which are constructed to characterize the magnetospheric state) spanning six solar cycles.
The discriminating statistics of the historical data set are compared with the discriminating statistics of surrogate
data streams that share the same linear properties as the historical data set. Both discriminating measures are
significantly different from the surrogates a few years prior to solar minima, while no differences are apparent at
the time of solar maximum. The result suggests that statistically the dynamics of the magnetosphere tend to be
more linear at solar maximum than at solar minimum. We discuss how this behavior of magnetospheric
dynamics may be related to the strength of the solar wind driver as well as its sensitivity to the changing dynamics
of the solar wind over the course of the solar cycle. Because the strong nonlinear dependencies tend to peak on
a timescale around 40-50 hours and are statistically significant up to one week, the analysis may also imply what
physical processes are responsible for the nonlinear behavior. Finally, we discuss how information-theoretical
methods may be used to improve predictive modeling, and we discuss the relative merits of mutual information
and cumulant-based cost as discriminating statistics in the contex of limited or noisy datasets.
http:w3.pppl.gov/~jrj/cumulant.html
NG44A-08
Signatures of Self-Organized Criticality in Low-Latitude Magnetosphere
In a series of previous publications (see e.g. Consolini, 1997; Uritsky and Pudovkin, 1998; Uritsky et al., 2002, 2006; Lui et al., 2000; Chapman and Watkins, 2001) it has been shown that spatiotemporal activity in high-latitude magnetosphere exhibits signatures of self-organized criticality (SOC) - a robust multiscale stochastic regime observed in driven nonlinear systems with many couple degrees of freedom. This regime has been identified by a set of mutually consistent scaling laws describing dynamical and statistical properties of magnetospheric substorms. Here, we report the existence of SOC in the dynamics of SYM-H index, whish is a marker of space storms and other types of low-latitude geomagnetic disturbances. The ensemble average dynamics of activity bursts in the SYM-H index are scale-free and are characterized by spreading critical scaling exponents whose values are consistent with the shape of the scaling of burst sizes versus bust lifetimes. The probability distributions for the lifetime and the size of the SYM-H bursts show robust power laws which are essentially independent of the lower activity threshold used to detect the bursts, and they extend over many orders of magnitude. The avalanche distribution exponents are also in agreement with theoretical predictions for SOC systems. All of these results, together with previous studies (Wanliss and Weygand, [2007]), begin to paint a coherent picture of critical state in the inner magnetosphere possibly associated with SOC dynamics of the ring current and other constituents of low-latitude activity.