Statistical physics approaches to the complex Earth system.

Global warming, extreme climate events, earthquakes and their accompanying socioeconomic disasters pose significant risks to humanity. Yet due to the nonlinear feedbacks, multiple interactions and complex structures of the Earth system, the understanding and, in particular, the prediction of such disruptive events represent formidable challenges to both scientific and policy communities. During the past years, the emergence and evolution of Earth system science has attracted much attention and produced new concepts and frameworks. Especially, novel statistical physics and complex networks-based techniques have been developed and implemented to substantially advance our knowledge of the Earth system, including climate extreme events, earthquakes and geological relief features, leading to substantially improved predictive performances. We present here a comprehensive review on the recent scientific progress in the development and application of how combined statistical physics and complex systems science approaches such as critical phenomena, network theory, percolation, tipping points analysis, and entropy can be applied to complex Earth systems. Notably, these integrating tools and approaches provide new insights and perspectives for understanding the dynamics of the Earth systems. The overall aim of this review is to offer readers the knowledge on how statistical physics concepts and theories can be useful in the field of Earth system science.

Climate network percolation reveals the expansion and weakening of the tropical component under global warming.

Global climate warming poses a significant challenge to humanity; it is associated with, e.g., rising sea level and declining Arctic sea ice. Increasing extreme events are also considered to be a result of climate warming, and they may have widespread and diverse effects on health, agriculture, economics, and political conflicts. Still, the detection and quantification of climate change, both in observations and climate models, constitute a main focus of the scientific community. Here, we develop an approach based on network and percolation frameworks to study the impacts of climate changes in the past decades using historical models and reanalysis records, and we analyze the expected upcoming impacts using various future global warming scenarios. We find an abrupt transition during the evolution of the climate network, indicating a consistent poleward expansion of the largest cluster that corresponds to the tropical area, as well as the weakening of the strength of links in the tropic. This is found both in the reanalysis data and in the Coupled Model Intercomparison Project Phase 5 (CMIP5) 21st century climate change simulations. The analysis is based on high-resolution surface (2 m) air temperature field records. We discuss the underlying mechanism for the observed expansion of the tropical cluster and associate it with changes in atmospheric circulation represented by the weakening and expansion of the Hadley cell. Our framework can also be useful for forecasting the extent of the tropical cluster to detect its influence on different areas in response to global warming.

Network analysis reveals strongly localized impacts of El Niño.

Climatic conditions influence the culture and economy of societies and the performance of economies. Specifically, El Niño as an extreme climate event is known to have notable effects on health, agriculture, industry, and conflict. Here, we construct directed and weighted climate networks based on near-surface air temperature to investigate the global impacts of El Niño and La Niña. We find that regions that are characterized by higher positive/negative network "in"-weighted links are exhibiting stronger correlations with the El Niño basin and are warmer/cooler during El Niño/La Niña periods. In contrast to non-El Niño periods, these stronger in-weighted activities are found to be concentrated in very localized areas, whereas a large fraction of the globe is not influenced by the events. The regions of localized activity vary from one El Niño (La Niña) event to another; still, some El Niño (La Niña) events are more similar to each other. We quantify this similarity using network community structure. The results and methodology reported here may be used to improve the understanding and prediction of El Niño/La Niña events and also may be applied in the investigation of other climate variables.

Dynamics of a Snowball Earth ocean.

Geological evidence suggests that marine ice extended to the Equator at least twice during the Neoproterozoic era (about 750 to 635 million years ago), inspiring the Snowball Earth hypothesis that the Earth was globally ice-covered. In a possible Snowball Earth climate, ocean circulation and mixing processes would have set the melting and freezing rates that determine ice thickness, would have influenced the survival of photosynthetic life, and may provide important constraints for the interpretation of geochemical and sedimentological observations. Here we show that in a Snowball Earth, the ocean would have been well mixed and characterized by a dynamic circulation, with vigorous equatorial meridional overturning circulation, zonal equatorial jets, a well developed eddy field, strong coastal upwelling and convective mixing. This is in contrast to the sluggish ocean often expected in a Snowball Earth scenario owing to the insulation of the ocean from atmospheric forcing by the thick ice cover. As a result of vigorous convective mixing, the ocean temperature, salinity and density were either uniform in the vertical direction or weakly stratified in a few locations. Our results are based on a model that couples ice flow and ocean circulation, and is driven by a weak geothermal heat flux under a global ice cover about a kilometre thick. Compared with the modern ocean, the Snowball Earth ocean had far larger vertical mixing rates, and comparable horizontal mixing by ocean eddies. The strong circulation and coastal upwelling resulted in melting rates near continents as much as ten times larger than previously estimated. Although we cannot resolve the debate over the existence of global ice cover, we discuss the implications for the nutrient supply of photosynthetic activity and for banded iron formations. Our insights and constraints on ocean dynamics may help resolve the Snowball Earth controversy when combined with future geochemical and geological observations.

Percolation framework to describe El Niño conditions.

Complex networks have been used intensively to investigate the flow and dynamics of many natural systems including the climate system. Here, we develop a percolation based measure, the order parameter, to study and quantify climate networks. We find that abrupt transitions of the order parameter usually occur ∼1 year before El Niño events, suggesting that they can be used as early warning precursors of El Niño. Using this method, we analyze several reanalysis datasets and show the potential for good forecasting of El Niño. The percolation based order parameter exhibits discontinuous features, indicating a possible relation to the first order phase transition mechanism.

Teleconnection Paths via Climate Network Direct Link Detection.

Teleconnections describe remote connections (typically thousands of kilometers) of the climate system. These are of great importance in climate dynamics as they reflect the transportation of energy and climate change on global scales (like the El Niño phenomenon). Yet, the path of influence propagation between such remote regions, and weighting associated with different paths, are only partially known. Here we propose a systematic climate network approach to find and quantify the optimal paths between remotely distant interacting locations. Specifically, we separate the correlations between two grid points into direct and indirect components, where the optimal path is found based on a minimal total cost function of the direct links. We demonstrate our method using near surface air temperature reanalysis data, on identifying cross-latitude teleconnections and their corresponding optimal paths. The proposed method may be used to quantify and improve our understanding regarding the emergence of climate patterns on global scales.

Dominant imprint of Rossby waves in the climate network.

The connectivity pattern of networks based on ground level temperature records shows a dense stripe of links in the extra tropics of the southern hemisphere. We show that statistical categorization of these links yields a clear association with the pattern of an atmospheric Rossby wave, one of the major mechanisms associated with the weather system and with planetary scale energy transport. It is shown that alternating densities of negative and positive links are arranged in half Rossby wave distances around 3500, 7000, and 10 000 km and are aligned with the expected direction of energy flow, distribution of time delays, and the seasonality of these waves. In addition, long distance links that are associated with Rossby waves are the most dominant in the climate network.

Dynamic Europa ocean shows transient Taylor columns and convection driven by ice melting and salinity.

The deep (~100 km) ocean of Europa, Jupiter's moon, covered by a thick icy shell, is one of the most probable places in the solar system to find extraterrestrial life. Yet, its ocean dynamics and its interaction with the ice cover have received little attention. Previous studies suggested that Europa's ocean is turbulent using a global model and taking into account non-hydrostatic effects and the full Coriolis force. Here we add critical elements, including consistent top and bottom heating boundary conditions and the effects of icy shell melting and freezing on ocean salinity. We find weak stratification that is dominated by salinity variations. The ocean exhibits strong transient convection, eddies, and zonal jets. Transient motions organize in Taylor columns parallel to Europa's axis of rotation, are static inside of the tangent cylinder and propagate equatorward outside the cylinder. The meridional oceanic heat transport is intense enough to result in a nearly uniform ice thickness, that is expected to be observable in future missions.

The surface temperature of Europa.

Previous estimates of the annual mean surface temperature of Jupiter's moon, Europa, neglected the effect of the eccentricity of Jupiter's orbit around the Sun, the effect of the emissivity and heat capacity of Europa's ice, the effect of the eclipse of Europa (i.e., the relative time that Europa is within the shadow of Jupiter), the effect of Jupiter's radiation, and the effect of Europa's internal heating. Other studies concentrated on the diurnal cycle but neglected some of the above factors. In addition, to our knowledge, the seasonal cycle of the surface temperature of Europa was not estimated. Here we systematically estimate the diurnal, seasonal and annual mean surface temperature of Europa, when Europa's obliquity, emissivity, heat capacity, and eclipse, as well as Jupiter's radiation, internal heating, and eccentricity, are all taken into account. For a typical internal heating rate of 0.05 W m - 2 , the equator, pole, and the global and mean annual mean surface temperatures are 96 K, 46 K, and 90 K, respectively. We found that the temperature at the high latitudes is significantly affected by the internal heating, especially during the winter solstice, suggesting that measurements of high latitude surface temperatures can be used to constrain the internal heating. We also estimate the incoming solar radiation to Enceladus, the moon of Saturn.

Possible origin of memory in earthquakes: Real catalogs and an epidemic-type aftershock sequence model.

Earthquakes are one of the most devastating natural disasters that plague society. Skilled, reliable earthquake forecasting remains the ultimate goal for seismologists. Using the detrended fluctuation analysis (DFA) and conditional probability (CP) methods, we find that memory exists not only in interoccurrence seismic records but also in released energy as well as in the series of the number of events per unit time. Analysis of a standard epidemic-type aftershock sequences (ETAS) earthquake model indicates that the empirically observed earthquake memory can be reproduced only for a narrow range of the model's parameters. This finding therefore provides tight constraints on the model's parameters and can serve as a testbed for existing earthquake forecasting models. Furthermore, we show that by implementing DFA and CP results, the ETAS model can significantly improve the short-term forecasting rate for the real (Italian) earthquake catalog.

Snowball Earth climate dynamics and Cryogenian geology-geobiology.

Geological evidence indicates that grounded ice sheets reached sea level at all latitudes during two long-lived Cryogenian (58 and ≥5 My) glaciations. Combined uranium-lead and rhenium-osmium dating suggests that the older (Sturtian) glacial onset and both terminations were globally synchronous. Geochemical data imply that CO2 was 102 PAL (present atmospheric level) at the younger termination, consistent with a global ice cover. Sturtian glaciation followed breakup of a tropical supercontinent, and its onset coincided with the equatorial emplacement of a large igneous province. Modeling shows that the small thermal inertia of a globally frozen surface reverses the annual mean tropical atmospheric circulation, producing an equatorial desert and net snow and frost accumulation elsewhere. Oceanic ice thickens, forming a sea glacier that flows gravitationally toward the equator, sustained by the hydrologic cycle and by basal freezing and melting. Tropical ice sheets flow faster as CO2 rises but lose mass and become sensitive to orbital changes. Equatorial dust accumulation engenders supraglacial oligotrophic meltwater ecosystems, favorable for cyanobacteria and certain eukaryotes. Meltwater flushing through cracks enables organic burial and submarine deposition of airborne volcanic ash. The subglacial ocean is turbulent and well mixed, in response to geothermal heating and heat loss through the ice cover, increasing with latitude. Terminal carbonate deposits, unique to Cryogenian glaciations, are products of intense weathering and ocean stratification. Whole-ocean warming and collapsing peripheral bulges allow marine coastal flooding to continue long after ice-sheet disappearance. The evolutionary legacy of Snowball Earth is perceptible in fossils and living organisms.

Volatility of linear and nonlinear time series.

Previous studies indicated that nonlinear properties of Gaussian distributed time series with long-range correlations, u(i), can be detected and quantified by studying the correlations in the magnitude series |u(i)|, the "volatility." However, the origin for this empirical observation still remains unclear and the exact relation between the correlations in u(i) and the correlations in |u(i)| is still unknown. Here we develop analytical relations between the scaling exponent of linear series u(i) and its magnitude series |u(i)|. Moreover, we find that nonlinear time series exhibit stronger (or the same) correlations in the magnitude time series compared with linear time series with the same two-point correlations. Based on these results we propose a simple model that generates multifractal time series by explicitly inserting long range correlations in the magnitude series; the nonlinear multifractal time series is generated by multiplying a long-range correlated time series (that represents the magnitude series) with uncorrelated time series [that represents the sign series sgn (u(i))]. We apply our techniques on daily deep ocean temperature records from the equatorial Pacific, the region of the El-Ninõ phenomenon, and find: (i) long-range correlations from several days to several years with 1/f power spectrum, (ii) significant nonlinear behavior as expressed by long-range correlations of the volatility series, and (iii) broad multifractal spectrum.

Model for cardiorespiratory synchronization in humans.

Recent experimental studies suggest that there is evidence for a synchronization between human heartbeat and respiration. We develop a physiologically plausible model for this cardiorespiratory synchronization, and numerically show that the model can exhibit stable synchronization against given perturbations. In our model, in addition to the well-known influence of respiration on heartbeat, the influence of heartbeat (and hence blood pressure) on respiration is also important for cardiorespiratory synchronization.

Nonlinear volatility of river flux fluctuations.

We study the spectral properties of the magnitudes of daily river flux increments, the volatility. The volatility series exhibits (i) strong seasonal periodicity and (ii) power-law correlations for time scales less than 1 yr. We test the nonlinear properties of the river flux increment series by randomizing its Fourier phases and find that the surrogate volatility series (i) has almost no seasonal periodicity and (ii) is weakly correlated for time scales less than 1 yr. We quantify the degree of nonlinearity by measuring (i) the amplitude of the power spectrum at the seasonal peak and (ii) the correlation power-law exponent of the volatility series.

Correlation differences in heartbeat fluctuations during rest and exercise.

We study the heartbeat activity of healthy individuals at rest and during exercise. We focus on correlation properties of the intervals formed by successive peaks in the pulse wave and find significant scaling differences between rest and exercise. For exercise the interval series is anticorrelated at short-time scales and correlated at intermediate-time scales, while for rest we observe the opposite crossover pattern--from strong correlations in the short-time regime to weaker correlations at larger scales. We also suggest a physiologically motivated stochastic scenario to provide an intuitive explanation of the scaling differences between rest and exercise.

Complex patterns of abnormal heartbeats.

Individuals having frequent abnormal heartbeats interspersed with normal heartbeats may be at an increased risk of sudden cardiac death. However, mechanistic understanding of such cardiac arrhythmias is limited. We present a visual and qualitative method to display statistical properties of abnormal heartbeats. We introduce dynamical "heartprints" which reveal characteristic patterns in long clinical records encompassing approximately 10(5) heartbeats and may provide information about underlying mechanisms. We test if these dynamics can be reproduced by model simulations in which abnormal heartbeats are generated (i) randomly, (ii) at a fixed time interval following a preceding normal heartbeat, or (iii) by an independent oscillator that may or may not interact with the normal heartbeat. We compare the results of these three models and test their limitations to comprehensively simulate the statistical features of selected clinical records. This work introduces methods that can be used to test mathematical models of arrhythmogenesis and to develop a new understanding of underlying electrophysiologic mechanisms of cardiac arrhythmia.

Characterization of sleep stages by correlations in the magnitude and sign of heartbeat increments.

We study correlation properties of the magnitude and the sign of the increments in the time intervals between successive heartbeats during light sleep, deep sleep, and rapid eye movement (REM) sleep using the detrended fluctuation analysis method. We find short-range anticorrelations in the sign time series, which are strong during deep sleep, weaker during light sleep, and even weaker during REM sleep. In contrast, we find long-range positive correlations in the magnitude time series, which are strong during REM sleep and weaker during light sleep. We observe uncorrelated behavior for the magnitude during deep sleep. Since the magnitude series relates to the nonlinear properties of the original time series, while the sign series relates to the linear properties, our findings suggest that the nonlinear properties of the heartbeat dynamics are more pronounced during REM sleep. Thus, the sign and the magnitude series provide information which is useful in distinguishing between the sleep stages.

Multifractal chaotic attractors in a system of delay-differential equations modeling road traffic.

We study a system of delay-differential equations modeling single-lane road traffic. The cars move in a closed circuit and the system's variables are each car's velocity and the distance to the car ahead. For low and high values of traffic density the system has a stable equilibrium solution, corresponding to the uniform flow. Gradually decreasing the density from high to intermediate values we observe a sequence of supercritical Hopf bifurcations forming multistable limit cycles, corresponding to flow regimes with periodically moving traffic jams. Using an asymptotic technique we find approximately small limit cycles born at Hopf bifurcations and numerically preform their global continuations with decreasing density. For sufficiently large delay the system passes to chaos following the Ruelle-Takens-Newhouse scenario (limit cycles-two-tori-three-tori-chaotic attractors). We find that chaotic and nonchaotic attractors coexist for the same parameter values and that chaotic attractors have a broad multifractal spectrum. (c) 2002 American Institute of Physics.

Biogenic crust dynamics on sand dunes.

Sand dunes are often covered by vegetation and biogenic crusts. Despite their significant role in dune stabilization, biogenic crusts have rarely been considered in model studies of dune dynamics. Using a simple model, we study the existence and stability ranges of different dune-cover states along gradients of rainfall and wind power. Two ranges of alternative stable states are identified: fixed crusted dunes and fixed vegetated dunes at low wind power; and fixed vegetated dunes and active dunes at high wind power. These results suggest a crossover between two different forms of desertification.

Why do active and stabilized dunes coexist under the same climatic conditions?

Sand dunes can be active (mobile) or stable, mainly as a function of vegetation cover and wind power. However, there exists as yet unexplained evidence for the coexistence of bare mobile dunes and vegetated stabilized dunes under the same climatic conditions. We propose a model for dune vegetation cover driven by wind power that exhibits bistabilty and hysteresis with respect to the wind power. For intermediate wind power, mobile and stabilized dunes can coexist, whereas for low (or high) wind power they can be fixed (or mobile). Climatic change or human intervention can turn active dunes into stable ones and vice versa; our model predicts that prolonged droughts with stronger winds can result in dune reactivation.