Complexity of the magnetosphere: Dynamical and statistical features from extensive correlated data.
The magnetosphere of Earth is a large scale open system driven by the turbulent solar wind, with the plasma processes ranging from the electron to the magnetohydrodynamic scales. The dynamical features exhibit spatio-temporal variations over four orders of magnitude, with the additional feature of overlapping scales in the solar wind and the magnetosphere. This makes the prediction of space weather (the variable conditions of geospace) from first principles a challenge. The extensive time series data of ground-based magnetic field measurements of the magnetospheric response and the spacecraft measurements of the solar wind plasma and field variables have been used to study the geospace using many techniques of complexity science. A database of geospace substorms consisting of more than million events is used in a study of the inherent statistical characteristics, with the main objective of characterizing the extreme events. The long-term correlations are studied using auto-correlation and mutual-information functions, yielding features represented by two exponents. The break in the exponents reflects the existence of two kinds of behavior, viz. the directly driven and internal magnetospheric features. The auto-correlation functions show stronger long-term correlation than the mutual information functions, which represent correlations of all orders. The return intervals for varying thresholds show long-range correlations with decreasing strength for higher thresholds, similar to multifractal systems. The detrended fluctuation analysis is used to compare the features of the magnetosphere to those of multiplicative random cascade processes.
Hydro-Kansas Research Project: Multi-scale Dynamical Understanding of Statistical Scale Invariance in Floods and Riparian Evapotranspiration in River Networks
The Hydro-Kansas (HK) is a multi-institutional, multi-investigator, multi-disciplinary research project. It represents the first illustrative example of a Natural Laboratory (NL), as described in the Water, Earth, Biota (WEB) report to the National Science Foundation (http://cires.colorado.edu/hydrology). HK combines theoretical analyses, numerical modeling, and an observational field program to understand and predict floods during periods of stationary and non-stationary changes in global hydro-climate. The framework is being generalized to include riparian evapotranspiration (RET). The central observational facility of HK is the 1,100 km2 Whitewater basin 50 km east of Wichita, Kansas. HK is addressing the long-standing problem of predicting spatial statistical scale invariance, or scaling, in floods and RET from bio-physical processes on multiple time scales that range from those of individual rainfall-runoff events to annual and longer. Physical predictions of statistical scaling involve non-linear interactions among hydrologic, geomorphologic, atmospheric, climatic, and ecologic processes in mesoscale basins. They require, (i) multi-scale dynamical formulations, (ii) a new ensemble approach to solve multi-scale dynamical equations on random self-similar (RSN) river networks, (iii) diagnostic analyses of theoretical predictions using observations, and (iv) a framework to generalize the results across global hydroclimates. Progress on these four sets of research challenges will be illustrated through examples.
Phase Transitions in Cluster Dynamics - New Type of a Critical Phenomenon
Critical phenomena are usually associated with control parameters, such as temperature, density, etc., passing a critical value. Here we describe a critical phenomenon of an unusual kind, where the role of a control parameter is taken by the time itself. This is demonstrated numerically for the cluster dynamics in a classical system of statistical physics: a frictionless elastic billiard. Cluster dynamics emerges when the interaction between the particles is short-ranged, and the system can be decomposed onto finite clusters so that during some interval of time each cluster moves independently of other clusters as a finite-dimensional dynamical system. After that, a giant, infinite cluster of interacting particles appears. Recently, it turned out that the concept of cluster dynamics has a wide domain of applications in such fields as plasma physics, river networks, and others. We show that dynamical clusters satisfy Horton-Strahler-Tokunaga scaling statistics, originated in the study of river networks and successfully applied in many models of hierarchical aggregation (inverse cascading), such as forest fires, percolation, biological and social systems. We observe critical change in the cluster size distribution that precedes the formation of the giant cluster. We argue that qualitatively similar behavior should be seen in other systems of interacting elements, independently of their origin.
Temperature Observations at the Mauna Loa Observatory, Hawaii
Observations at the Mauna Loa Observatory, Hawaii, established the systematic increase in anthropogenic CO2 in the atmosphere. In this paper we study the hourly temperature records from this observatory for the 30 year period 1977 to 2006. We determine the trends in the data as a function of the time of day for the period. We obtain statistical averages over a year. For the night time data we find a near uniform warming trend dT/dt ≈ 0.04°C yr-1 from 22:00 to 6:00 hours. During the day the warming trend moderates to a slight cooling trend dT/dt ≈ -0.01°C yr-1 at 12:00 hours. The result is a significant decrease in the diurnal temperature range DTR ≈ -0.05°C yr-1 during the period under consideration. Our results are consistent with a direct impact of an increase in CO2 as a greenhouse gas. The major advantage of studies at the Mauna Loa site is that regional and anthropogenic influences are minimal. Our results suggest that global warming due to greenhouse gases is primarily a night time effect. This may explain why changes appear to be a maximum in polar and continental regions.
From weather to climate: a dimensional transition?
Based on anisotropic space-time turbulence theory and both lidar satellite radiances, in situ spectra of temperature, humidity and wind, and numerical models of the atmosphere, we give an objective basis to the weather/climate distinction. We show that the latter accurately follow the predictions of multiplicative cascade models up to about 7- 10 days. This marks the beginning of a weather/climate transition region which extends up to the cascade outer scale (time T) of about 20- 30 days (depending somewhat on the atmospheric field), after which the climate regime begins. We bolster this interpretation by empirically constructing space-time ("Stommel") diagrammes; we obtain near linear relations between time and (horizontal) space and theoretically predicted power law relations between the vertical and time up until the end of the weather regime (~10000 km in the horizontal, ~ 10 km in the vertical, ~10 days in time). Going beyond the weather regime to time scales >T, we see that the spectra flatten out into a "spectral plateau". Using multiplicative space-time cascade models over scales much longer than T, we find that the flattening is nearly as expected and is caused by the transition from a full space-time weather process at scales below T to an increasingly time only type process at larger scales. Further evidence for this "dimensional transition" is the empirical finding that the same type of statistical variability (i.e. probability distributions) persists up to about a year in in situ temperature, humidity and wind statistics. Eventually however for longer periods, we find that new scaling climate processes begin to dominate.
The bifurcation structure and noise assisted transitions in the Pleistocene glacial cycles
The glacial cycles are attributed to the climatic response of the orbital changes in the irradiance to the Earth. These changes in the forcing are too small to explain the observed climate variations as simple linear responses. Non-linear amplifications of the orbital forcing are necessary to account for the glacial cycles. The underlying climate dynamics is extremely complex. Here an empirical model of the non-linear response is presented. From the model it is possible to assess the role of stochastic noise in comparison to the deterministic orbital forcing of the ice ages. The model is based on the bifurcation structure derived from the climate history. It indicates the dynamical origin of the Mid-Pleistocene transition (MPT) from the '41 kyr world' to the '100 kyr world'. The dominant forcing in the latter is still the 41 kyr obliquity cycle, but the bifurcation structure of the climate system is changed. The model suggests that transitions between glacial and interglacial climate are assisted by internal stochastic noise in the period prior to the last five glacial cycles, while the last five cycles are deterministic responses to the orbital forcing.
Volatility of unevenly sampled fractional Brownian motion: an application to ice core records
The analysis of many natural time series and especially those related to ice core records often suffers from uneven sampling intervals. Here, we introduce a method that allows one to reliably estimate the volatility properties of fractional Brownian motion despite uneven sampling. It is based on the linear correlations of the process which are used to rescale the volatility series. For high-resolution temperature proxy records from Antarctica, we confirm that its volatility properties reveal a strong nonlinear component in the time series for time scales of 5 - 200 kyr. The results suggest that temperature increments appear in clusters of big and small increments --- a big (positive or negative) climate change is most likely followed by a big (positive or negative) climate change and a small climate change is most likely followed by a small climate change.
Uncertainty in Multifractal Estimates : a Bottleneck for Applications and a Stimulating Theoretical Question
A fundamental and attractive feature of multifractal fields is that they are extremely variable over a wide range of scale and have long range dependencies. They are therefore at odd with the usual statistical estimator framework and this may explain the limited achievements in defining robust and accurate multifractal estimators. We believe that the lack of the latter has become a stronger and stronger bottleneck for practical applications of multifractals. For instance, rather inconsistent estimates of multifractal parameters of geophysical fields can be easily pointed out. Nevertheless, we also point out the irrelevance of hasty claims that the uncertainty of multifractal attributes should be definitively large. We recall a few general and non parametric results, e.g. non gaussian convergence of the estimators, limitations induced by the length, number and resolution of samples, which have been obtained by various authors in somewhat different frameworks. We then focus on the parametric case study of the universal multifractals and more particularly the Double Trace Moment technique to estimate their parameters. We illustrate a few possible numerical pitfalls, such as those induced by number representation limitations, which depend on the used programming language. We present optimization procedures to take care of the above mentioned limitations, which otherwise may lead to paradoxical results, e.g. parameters of a weakly multifractal field would be harder to estimate than those of a strongly multifractal field. Finally, we discuss and quantify the intrinsic variability of the "best estimates", in particular in relationship with the notion of equifinality, which has been often used in hydrology.