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Advances in Nonlinear Geosciences is a set of contributions from the participants of “30 Years of Nonlinear Dynamics” held July 3-8, 2016 in Rhodes, Greece as part of the Aegean Conferences, as well as from several other experts in the field who could not attend the meeting. The volume brings together up-to-date research from the atmospheric sciences, hydrology, geology, and other areas of geosciences and presents the new advances made in the last 10 years. Topics include chaos synchronization, topological data analysis, new insights on fractals, multifractals and stochasticity, climate dynamics, extreme events, complexity, and causality, among other topics.

Pullback Attractor Crisis in a Delay Differential ENSO Model

We study the pullback attractor (PBA) of a seasonally forced delay differential model for the El Niño–Southern Oscillation (ENSO); the model has two delays, associated with a positive and a negative feedback. The control parameter is the intensity of the positive feedback and the PBA undergoes a crisis that consists of a chaos-to-chaos transition. Since the PBA is dominated by chaotic behavior, we refer to it as a strange PBA. Both chaotic regimes correspond to an overlapping of resonances but the two differ by the properties of this overlapping. The crisis manifests itself by a brutal change not only in the size but also in the shape of the PBA. The change is associated with the sudden disappearance of the most extreme warm (El Niño) and cold (La Niña) events, as one crosses the critical parameter value from below. The analysis reveals that regions of the strange PBA that survive the crisis are those populated by the most probable states of the system. These regions are those that exhibit robust foldings with respect to perturbations. The effect of noise on this phase-and-parameter space behavior is then discussed. It is shown that the chaos-to-chaos crisis may or may not survive the addition of small noise to the evolution equation, depending on how the noise enters the latter.

Mickaël D. Chekroun, Michael Ghil, J. David Neelin

Shear-Wave Splitting Indicates Non-Linear Dynamic Deformation in the Crust and Upper Mantle

We demonstrate that non-linear dynamic deformation exists throughout the crust and upper mantle of the Earth. Stress-aligned shear-wave splitting, seismic birefringence, is widely observed in the Earth’s upper crust, lower-crust, and uppermost ∼400 km of the mantle. Attributed to the effects of pervasive distributions of stress-aligned fluid-saturated microcracks in the crust (and controversially intergranular films of hydrated melt in the mantle), the degree splitting indicates that ‘microcracks’ are so closely spaced that they verge on failure in fracturing and earthquakes if there is any disturbance. Phenomena that verge on failure are critical systems with non-linear dynamics that impose a range of new properties on conventional sub-critical geophysics that we suggest is a New Geophysics. Consequently, shear-wave splitting provides directly interpretable information about the progress of non-linear dynamic deformation in the deep otherwise-inaccessible interior of the microcracked Earth. Possibly uniquely for non-linear dynamic phenomena, observation of shear-wave splitting allows the progress towards singularities to be monitored in deep in situ rock, so that earthquakes and volcanic eruptions can be predicted (we prefer stress-forecast). The response to other processes, such as hydraulic fracking, can be monitored, and in some cases calculated and effects predicted. Here, we review shear-wave splitting and demonstrate the prevalence of non-linear dynamic deformation of the New Geophysics in the crust and uppermost ∼400 km of the mantle.

Stuart Crampin, Gulten Polat, Yuan Gao, David B. Taylor, Nurcan Meral Ozel

Stochastic Parameterization of Subgrid-Scale Processes: A Review of Recent Physically Based Approaches

We review some recent methods of subgrid-scale parameterization used in the context of climate modeling. These methods are developed to take into account (subgrid) processes playing an important role in the correct representation of the atmospheric and climate variability. We illustrate these methods on a simple stochastic triad system relevant for the atmospheric and climate dynamics, and we show in particular that the stability properties of the underlying dynamics of the subgrid processes have a considerable impact on their performances.

Jonathan Demaeyer, Stéphane Vannitsem

Large-Scale Atmospheric Phenomena Under the Lens of Ordinal Time-Series Analysis and Information Theory Measures

This review presents a synthesis of our work done in the framework of the European project Learning about Interacting Networks in Climate (LINC, climatelinc.eu). We have applied tools of information theory and ordinal time series analysis to investigate large scale atmospheric phenomena from climatological datasets. Specifically, we considered monthly and daily Surface Air Temperature (SAT) time series (NCEP reanalysis) and used the climate network approach to represent statistical similarities and interdependencies between SAT time series in different geographical regions. Ordinal analysis uncovers how the structure of the climate network changes in different time scales (intra-season, intra-annual, and longer). We have also analyzed the directionally of the links of the network, and we have proposed novel approaches for uncovering communities formed by geographical regions with similar SAT properties.

J. I. Deza, G. Tirabassi, M. Barreiro, C. Masoller

Supermodeling: Synchronization of Alternative Dynamical Models of a Single Objective Process

Imperfect models of the same objective process give an improved representation of that process, from which they assimilate data, if they are also coupled to one another. Inter-model coupling, through nudging, or more strongly through averaging of dynamical tendencies, typically gives synchronization or partial synchronization of models and hence formation of consensus. Previous studies of supermodels of interest for weather and climate prediction are here reviewed. The scheme has been applied to a hierarchy of models, ranging from simple systems of ordinary differential equations, to models based on the quasigeostrophic approximation to geophysical fluid dynamics, to primitive-equation fluid dynamical models, and finally to state-of-the-art climate models. Evidence is reviewed to test the claim that, in nonlinear systems, the synchronized-model scheme surpasses the usual procedure of averaging model outputs.

Gregory S. Duane, Wim Wiegerinck, Frank Selten, Mao-Lin Shen, Noel Keenlyside

Are We Measuring the Right Things for Climate?

If one could exist on climate scales would it make any more sense to measure laboratory-scale quantities to capture climate conditions than it does for us on the laboratory scale to compute wave functions to understand the weather? Clearly the quantum mechanical and the laboratory regime are constructed in terms of different physical variables. Why do we presume, then, that laboratory regime quantities like temperature continue to be the appropriate physical variables to measure in a climate regime? This paper suggests why we may not be measuring the right things and it will broach some alternatives in the context of a reformulation for relevant physics more natural to long timescales: slow time. Specifically it shows that fluctuating velocities can be “thermalized” in suitable averages suggesting that one might imagine climate in terms of a generalization of wind which may include persistent meteorological winds, or none at all. But it also shows that temperature cannot be “thermalized” on long time and space scales, making the notion of local equilibrium and simple generalizations of temperature problematic for climate.

Christopher Essex, Bjarne Andresen

What Have Complex Network Approaches Learned Us About El Niño?

A short overview is given of recent work on the application of network techniques to the El Niño/Southern Oscillation phenomenon in the Tropical Pacific. Although several new and useful diagnostics have been developed, progress regarding the understanding of El Niño dynamics has been rather limited. Success has been claimed to forecast El Niño events 1 year ahead using network-based predictors, but tests are limited and the reason for this skill is still unclear.

Qing Yi Feng, Henk A. Dijkstra

Late Quaternary Climate Response at 100 kyr: A Noise-Induced Cycle Suppression Mechanism

Late quaternary climate proxies suggest the presence of a strong cycle at a period of about 100 kyr. It is thought that this cycle could be due to variations in the eccentricity of the Earth’s orbit, as part of the Milankovitch forcing. However, based on simple energy balance arguments, the eccentricity variations are too small to explain the strength of the climatic response. Some amplification mechanisms based on ice sheet dynamics or ocean circulation models have been suggested to explain this paradox. But recently (Wallmann 2014), a different explanation was proposed. There, a non-linear biogeochemical model coupling seawater alkalinity, dissolved phosphate, dissolved inorganic carbon, and atmospheric carbon dioxide without any orbital forcing was developed. As the parameters vary, the system may undergo a Hopf bifurcation and exhibits self-organized oscillations with a period that has the appropriate order of magnitude but remains larger than 100 kyr. In this contribution, I revisit Wallmann’s model by adding a weak stochastic periodic Milankovitch forcing at 100 kyr in the spirit of stochastic resonance phenomena. It is seen that for sufficiently high noise intensity, a noise-induced cycle suppression occurs, whereby the self-sustained oscillation of biogeochemical origin is destroyed and a strong signal persists at 100 kyr. This mechanism could thus provide an amplification mechanism for the presence of a strong response under the influence of a weak Milankovitch forcing.

Ivan L’Heureux

Role of Nonlinear Eddy Forcing in the Dynamics of Multiple Zonal Jets

Turbulent oceanic and atmospheric flows are characterized by persistent nearly zonal alternating currents, referred to as multiple zonal jets. These jets emerge from self-organization of the flow, and are maintained by persistent action of transient fluctuations (“eddies”). The action of these eddies takes a form of internally generated eddy forcing that can either resist the jets or support them against dissipation and other processes. This review chapter is concerned with the role of the eddy forcing in the dynamics of the multiple zonal jets in the ocean, but the results are also applicable to atmospheric flows and some isolated jets in the ocean.

Igor Kamenkovich, Pavel Berloff

Data-Adaptive Harmonic Decomposition and Stochastic Modeling of Arctic Sea Ice

We present and apply a novel method of describing and modeling complex multivariate datasets in the geosciences and elsewhere. Data-adaptive harmonic (DAH) decomposition identifies narrow-banded, spatio-temporal modes (DAHMs) whose frequencies are not necessarily integer multiples of each other. The evolution in time of the DAH coefficients (DAHCs) of these modes can be modeled using a set of coupled Stuart-Landau stochastic differential equations that capture the modes’ frequencies and amplitude modulation in time and space. This methodology is applied first to a challenging synthetic dataset and then to Arctic sea ice concentration (SIC) data from the US National Snow and Ice Data Center (NSIDC). The 36-year (1979–2014) dataset is parsimoniously and accurately described by our DAHMs. Preliminary results indicate that simulations using our multilayer Stuart-Landau model (MSLM) of SICs are stable for much longer time intervals, beyond the end of the twenty-first century, and exhibit interdecadal variability consistent with past historical records. Preliminary results indicate that this MSLM is quite skillful in predicting September sea ice extent.

Dmitri Kondrashov, Mickaël D. Chekroun, Xiaojun Yuan, Michael Ghil

Cautionary Remarks on the Auto-Correlation Analysis of Self-Similar Time Series

As the time-domain analysis of non-linear time series in geosciences, the auto-correlations of the self-similar time series are examined to identify spurious decorrelation structures in terms of the number of independent pulses and the shape of decay patterns. The self-similar time series is defined as a continuous time series having similar shapes of disturbance or amplitudes of which statistics is non-Gaussian, such as records of river flows, rainfall, wind speed, concentration of Chlorophyll, and inertial amplitudes in geosciences. In this chapter, the auto-correlations of the modeled self-similar time series are evaluated and the relevant cautionary remarks are discussed.

Sung Yong Kim

Emergence of Coherent Clusters in the Ocean

Why does material tend to congregate in long coherent clusters at the surface of the ocean when it is well known that the ocean is dispersive? Here we review some recent research that addresses this question. A standard diagnostic for discerning transport pathways in incompressible 2D flows is the finite time Lyapunov exponent (FTLE). The FTLE can be expressed as the average of two rarely evaluated Lagrangian objects: the dilation and stretch rates. The stretch rate accounts for the ability of fluid shear to change the shape of fluid blobs, and for incompressible fluids it is the FTLE. However, in the real ocean and especially at submesoscales, the horizontal divergence is not negligible. This is quantified by the dilation rate, which is identically zero in 2D incompressible flow. Our analysis demonstrates that the combination of fluid dilation and stretch enhances accumulation of buoyant material along thin clusters in an otherwise dispersing ocean.

A. D. Kirwan, H. S. Huntley, H. Chang

The Rise and Fall of Thermodynamic Complexity and the Arrow of Time

Complexity Theory is an eclectic collection of theoretical approaches to a wide variety of nonlinear problems that typically involve many degrees of freedom. Despite numerous claims, there does not appear to be a universal basis for the various approaches. Here we report on recent attempts to provide such a basis. Our approach is based on the partial order of Boltzmann states under majorization and thus is grounded in the Second Law of Thermodynamics. However, here we do not appeal to any energetic constraints. By majorizing the Boltzmann states we identify a new statistical mechanical entity, namely a multivalued function that maps Boltzmann entropy to the size or order of sets of incomparable Boltzmann entropy states. We call this thermodynamic complexity. This is a concave function of entropy, peaking near mid-entropy and falling to zero at maximum and minimum entropies. It remains to be seen if this approach can be rigorously applied to other areas, but heuristic arguments given here indicate broad applicability.

A. D. Kirwan, William Seitz

From Fractals to Stochastics: Seeking Theoretical Consistency in Analysis of Geophysical Data

Fractal-based techniques have opened new avenues in the analysis of geophysical data. On the other hand, there is often a lack of appreciation of both the statistical uncertainty in the results and the theoretical properties of the stochastic concepts associated with these techniques. Several examples are presented which illustrate suspect results of fractal techniques. It is proposed that concepts used in fractal analyses are stochastic concepts and the fractal techniques can readily be incorporated into the theory of stochastic processes. This would be beneficial in studying biases and uncertainties of results in a theoretically consistent framework, and in avoiding unfounded conclusions. In this respect, a general methodology for theoretically justified stochastic processes, which evolve in continuous time and stem from maximum entropy production considerations, is proposed. Some important modelling issues are discussed with focus on model identification and fitting often made using inappropriate methods. The theoretical framework is applied to several processes, including turbulent velocities measured every several microseconds, and wind and temperature measurements. The applications show that several peculiar behaviours observed in these processes are easily explained and reproduced by stochastic techniques.

Demetris Koutsoyiannis, Panayiotis Dimitriadis, Federico Lombardo, Spencer Stevens

Role of Nonlinear Dynamics in Accelerated Warming of Great Lakes

In recent decades, the Laurentian Great Lakes have undergone rapid surface warming with the summertime trends substantially exceeding the warming rates of surrounding land. Warming of the deepest Lake Superior was the strongest, and that of the shallowest Lake Erie—the weakest of all lakes. We investigate the dynamics of accelerated lake warming in idealized coupled thermodynamic lake–ice–atmosphere models. These models are shown to exhibit, under identical seasonally varying forcing, multiple possible stable equilibrium cycles, or regimes, with different maximum summertime temperatures and varying degrees of wintertime ice cover. The simulated lake response to linear climate change in the presence of the atmospheric noise rationalizes the observed accelerated warming of the lakes, the correlation between wintertime ice cover and next summer’s lake-surface temperature, as well as higher warming trends of the (occasionally wintertime ice-covered) deep-lake vs. shallow-lake regions, in terms of the corresponding characteristics of the forced transitions between colder and warmer lake regimes. Since the regime behavior in the models considered arises due to nonlinear dynamics rooted in the ice–albedo feedback, this feedback is also the root cause of the accelerated lake warming simulated by these models.

Sergey Kravtsov, Noriyuki Sugiyama, Paul Roebber

The Prediction of Nonlinear Polar Motion Based on Artificial Neural Network (ANN) and Fuzzy Inference System (FIS)

The Earth rotation movement characterizes the situation of the whole Earth movement, as well as the interaction between the Earth’s various layers such as the Earth’s core, mantle, crust, and atmosphere. Prediction of the Earth rotation parameters (ERPs) is important for near real-time applications including navigation, precise positioning, remote sensing and landslide monitoring, etc. In such studies, the analysis of time series is also important for highly accurate and reliable predictions. Therefore, prediction of ERPs at least over a few days in the future is necessary. At present, there are two major forecasting methods for ERP: linear and nonlinear models. The nonlinear models include: sequence of artificial neural network (ANN), fuzzy inference system, and other methods. Fuzzy inference system (FIS) and traditional artificial neural networks (ANN) provide good predictions of polar motion (PM). In this study, for the prediction of Earth rotation parameters, International Earth Rotation and Reference System Service (IERS) C04 daily time series data from 1990 to 2015 was used for training. From 1 to 120 days in future of ERPs values were predicted by using the data of 5, 15, and 25 years in ANN. The results of ANN and ANFIS were compared with observed values. The results indicate that the longer training data are used in ANN and ANFIS, the more accurate prediction can be obtained.

Ramazan Alper Kuçak, Raşit Uluğ, Orhan Akyılmaz

Harnessing Butterflies: Theory and Practice of the Stochastic Seasonal to Interannual Prediction System (StocSIPS)

The atmosphere is governed by continuum mechanics and thermodynamics yet simultaneously obeys statistical turbulence laws. Up until its deterministic predictability limit (τ w ≈ 10 days), only general circulation models (GCMs) have been used for prediction; the turbulent laws being still too difficult to exploit. However, beyond τ w —in macroweather—the GCMs effectively become stochastic with internal variability fluctuating about the model—not the real world—climate and their predictions are poor. In contrast, the turbulent macroweather laws become advantageously notable due to (a) low macroweather intermittency that allows for a Gaussian approximation, and (b) thanks to a statistical space-time factorization symmetry that (for predictions) allows much decoupling of the strongly correlated spatial degrees of freedom. The laws imply new stochastic predictability limits. We show that pure macroweather—such as in GCMs without external forcings (control runs)—can be forecast nearly to these limits by the ScaLIng Macroweather Model (SLIMM) that exploits huge system memory that forces the forecasts to converge to the real world climate.To apply SLIMM to the real world requires pre-processing to take into account anthropogenic and other low frequency external forcings. We compare the overall Stochastic Seasonal to Interannual Prediction System (StocSIPS, operational since April 2016) with a classical GCM (CanSIPS) showing that StocSIPS is superior for forecasts 2 months and further in the future, particularly over land. In addition, the relative advantage of StocSIPS increases with forecast lead time.In this chapter we review the science behind StocSIPS and give some details of its implementation and we evaluate its skill both absolute and relative to CanSIPS.

S. Lovejoy, L. Del Rio Amador, R. Hébert

Regime Change Detection in Irregularly Sampled Time Series

Irregular sampling is a common problem in palaeoclimate studies. We propose a method that provides regularly sampled time series and at the same time a difference filtering of the data. The differences between successive time instances are derived by a transformation costs procedure. A subsequent recurrence analysis is used to investigate regime transitions. This approach is applied on speleothem-based palaeoclimate proxy data from the Indonesian–Australian monsoon region. We can clearly identify Heinrich events in the palaeoclimate as characteristic changes in dynamics.

Norbert Marwan, Deniz Eroglu, Ibrahim Ozken, Thomas Stemler, Karl-Heinz Wyrwoll, Jürgen Kurths

Topological Data Analysis: Developments and Applications

Topological Data Analysis (TDA) and its mainstay computational device, persistent homology (PH), has established a strong track record of providing researchers across the data-driven sciences with new insights and methodologies by characterizing low-dimensional geometric structures in high-dimensional data. When combined with machine learning (ML) methods, PH is valued as a discriminating-feature extraction tool. This work highlights many of the recent successes at the intersection of TDA and ML, introduces some of the foundational mathematics underpinning TDA, and summarizes the efforts to strengthen the bridge between TDA and ML. Thus, this document is a launching point for experimentalists and theoreticians to consider what can be learned from the shape of their data.

Francis C. Motta

Nonlinear Dynamical Approach to Atmospheric Predictability

The principal properties of initial condition and of model errors along with their repercussions on atmospheric predictability are reviewed. A general nonlinear dynamics-inspired approach is developed, from which generic trends are derived. The main ideas are illustrated on selected low-order models capturing the principal qualitative aspects of the phenomena of interest.

C. Nicolis

Linked by Dynamics: Wavelet-Based Mutual Information Rate as a Connectivity Measure and Scale-Specific Networks

Experimentally observed networks of interacting dynamical systems are inferred from recorded multivariate time series by evaluating a statistical measure of dependence, usually the cross-correlation coefficient, or mutual information. These measures reflect dependence in static probability distributions, generated by systems’ evolution, rather than coherence of systems’ dynamics. Moreover, these “static” measures of dependence can be biased due to properties of dynamics underlying the analyzed time series. Consequently, properties of local dynamics can be misinterpreted as properties of connectivity or long-range interactions. We propose the mutual information rate as a measure reflecting coherence or synchronization of dynamics of two systems and not suffering by the bias typical for the “static” measures. We demonstrate that a computationally accessible estimation method, derived for Gaussian processes and adapted by using the wavelet transform, can be effective for nonlinear, nonstationary, and multiscale processes. The discussed problem and the proposed method are illustrated using numerically generated data of coupled dynamical systems as well as gridded reanalysis data of surface air temperature as the source for the construction of climate networks. In particular, scale-specific climate networks are introduced.

Milan Paluš

Non-Extensive Statistical Mechanics: Overview of Theory and Applications in Seismogenesis, Climate, and Space Plasma

In this small review, the theoretical framework of non-extensive statistical theory, introduced by Constantino Tsallis in 1988, is presented in relation with the q-triplet estimation concerning experimental time series from climate, seismogenesis, and space plasmas systems. These physical systems reveal common dynamical, geometrical, or statistical characteristics. Such characteristics are low dimensionality, typical intermittent turbulence multifractality, temporal or spatial multiscale correlations, power law scale invariance, non-Gaussian statistics, and others. The aforementioned phenomenology has been attributed in the past to chaotic or self-organized critical (SOC) universal dynamics. However, after two or three decades of theoretical development of the complexity theory, a more compact theoretical description can be given for the underlying universal physical processes which produce the experimental time series complexity. In this picture, the old reductionist view of universality of particles and forces is extended to the modern universality of multiscale complex processes from the microscopic to the macroscopic level of different physical systems. In addition, it can be stated that a basic and universal organizing principle exists creating complex spatio-temporal and multiscale different physical structures or different dynamical scenarios at every physical scale level. The best physical representation of the underline universal organizing principle is the well-known entropy principle. Tsallis introduced a q-entropy (S q ) as a non-extensive (q-extension) of the Boltzmann–Gibbs (BG) entropy (for q = 1, the BG entropy is restored) and statistics in order to describe efficiently the rich phenomenology that complex systems exhibit. Tsallis q-entropy could be a strong candidate for entropy principle according to which nature creates complex structures everywhere, from the microscopic to the macroscopic level, trying to succeed the extremization of the Tsallis entropy. In addition, this Sq entropy principle is harmonized with the q extension of the classic and Gaussian central limit theorem (q-CLT). The q-extension of CLT corresponds to the Levy a-stable extension of the Gaussian attractor of the classic statistical theory. The q-CLT is related to the Tsallis q-triplet theory of random time series with non-Gaussian statistical profile. Moreover Tsallis q-extended entropy principle can be used as the theoretical framework for the unification of some new dynamical characteristics of complex systems such as the spatio-temporal fractional dynamics, the anomalous diffusion processes and the strange dynamics of Hamiltonian and dissipative dynamical systems, the intermittent turbulence theory, the fractional topological and percolation phase transition processes according to Zelenyi and Milovanov non-equilibrium and non-stationary states (NESS) theory, as well as the non-equilibrium renormalization group theory(RNGT) of distributed dynamics and the reduction of dynamical degrees of freedom.

G. P. Pavlos, L. P. Karakatsanis, A. C. Iliopoulos, E. G. Pavlos, A. A. Tsonis

Spatial Patterns of Peak Flow Quantiles Based on Power-Law Scaling in the Mississippi River Basin

This study explores the spatial variability of peak flows for different drainage area sizes in the Mississippi River Basin (MRB) based on the power-law relation between flood quantiles (Q p ) and drainage areas (A) expressed as Qp=αpA??p $${Q}_p={\upalpha}_p{A}^{\uptheta_p}$$ . The aim is to reveal consistent regional flood patterns within the MRB. The authors use 5137 streamflow gauges with peak flow records and the USGS Hydrologic Unit Code (HUC) catchment organization framework to estimate the scaling parameters (α p and θ p ) at multiple spatial disaggregation levels, including the complete Mississippi River Basin (MRB), six major MRB sub-regions (HUC-2), and finally 84 medium-scale catchments (HUC-4). The analysis at the HUC-4 level exposes remarkable regional flood patterns in θ p and α p , which are used to estimate peak flows at 2.33 and 100 years of return periods at multiple spatial scales including 1, 100, 1000, and 10,000 km2 drainage areas. The results expose a peak flow quantile relation that varies as a function of region and drainage area, demonstrating that the regions with the higher peak flows quantiles are varying with respect to the watershed size along the MRB. Mainly, we found that the cluster of higher floods extends from the center to the eastern MRB for drainage areas from 1 to 10,000 km2. Conversely, the clusters of lower 2.33-year floods are preserved in the western MRB for the same range of drainage areas. The results presented in this study demonstrate that the flood-producing mechanisms are varying with respect to the drainage area size and regions, providing a starting point for a quantitative description of physical processes that dominate the variability of flood-producing mechanisms, a critical step in the design of parsimonious continental scale hydrological models.

Gabriel Perez, Ricardo Mantilla, Witold F. Krajewski

Studying the Complexity of Rainfall Within California Via a Fractal Geometric Method

A deterministic geometric approach, the fractal–multifractal (FM) method, useful in modeling storm events and recently adapted in order to encode highly intermittent daily rainfall records, is employed to study the complexity of rainfall sets within California. Specifically, sets—from south to north—at Cherry Valley, Merced, Sacramento and Shasta Dam and containing, respectively 59, 116, 115, and 72 years, all ending at water year 2015, are studied. The analysis reveals that: (a) the FM approach provides faithful encodings of all records, by years, with mean square and maximum errors in accumulated rain that are less than a mere 2 and 10%, respectively; (b) the evolution of the corresponding “best” FM parameters, allowing visualization of the inter-annual rainfall dynamics from a reduced vantage point, exhibit a highly entropic variation that prevents discriminating between sites and extrapolating to the future; and (c) the rain signals at all sites may be termed “equally complex,” as usage of k-means clustering and conventional phase-space analysis of FM parameters yields comparable results for all sites.

Carlos E. Puente, Mahesh L. Maskey, Bellie Sivakumar

Pandora Box of Multifractals: Barely Open?

Three decades ago, multifractals were a major breakthrough in nonlinear geophysics by providing a general framework to understand, analyze, and simulate fields that are extremely inhomogeneous over a wide range of space-time scales. They have remained on the forefront of nonlinear methodologies, but they are still far from being used or even developed to their full extent. Indeed, they have been too often limited to scalar-valued fields, whereas the relevant geophysical fields are vector fields. This chapter therefore gives new insights on current developments to overcome this limitation. This is done in an inductive manner. For instance, it takes hold on simple considerations on “spherical” and “hyperbolic” rotations to introduce step by step the Clifford algebra of Lévy stable generators of multifractal vectors that have both universal statistical and robust algebraic properties.

Daniel Schertzer, Ioulia Tchiguirinskaia

Complex Networks and Hydrologic Applications

Connections are ubiquitous in hydrology. However, understanding the nature and extent of connections in hydrologic systems has and continues to be a tremendous challenge. In recent years, applications of the concepts of complex networks to study connections in hydrologic systems have started to emerge. This chapter aims to offer an overview of the science of complex networks and its applications in hydrology. First, the basic concept of a network, the history of development of network theory, and some important measures of network properties are presented. Next, applications of complex networks in hydrology are reviewed, including studies on spatial connections, temporal connections, and catchment classification. Finally, some remarks on future directions are made.

Bellie Sivakumar, Carlos E. Puente, Mahesh L. Maskey

Convergent Cross Mapping: Theory and an Example

In this review paper we present the basic principles behind convergent cross mapping, a new causality detection method, as well as an example to demonstrate it.

Anastasios A. Tsonis, Ethan R. Deyle, Hao Ye, George Sugihara

Randomnicity: Randomness as a Property of the Universe

This paper is a concept paper, which discusses the definition of randomness, and the sources of randomness in the mathematical system as well as in the physical system (the Universe). We document that randomness is an inherited property of mathematics and of the physical world, shaping all observed forms and structures, and we discuss its role.

Anastasios A. Tsonis

Insights in Climate Dynamics from Climate Networks

This review is a synthesis of work spanning the last 25 years. It is largely based on the use of climate networks to identify climate subsystems/major modes and to subsequently study how their collective behavior explains decadal variability. The central point is that a network of coupled nonlinear subsystems may at times begin to synchronize. If during synchronization the coupling between the subsystems increases the synchronous state may, at some coupling strength threshold, be destroyed shifting climate to a new regime. This climate shift manifests itself as a change in global temperature trend. This mechanism, which is consistent with the theory of synchronized chaos, appears to be a very robust mechanism of the climate system. It is found in the instrumental records, in forced and unforced climate simulations, as well as in proxy records spanning several centuries.

Anastasios A. Tsonis

On the Range of Frequencies of Intrinsic Climate Oscillations

The purpose of this work is to establish the limits of natural oscillations in the climate system, i.e., not attributed to alleged anthropogenic effects. To this end we considered many proxy climate records representing the state of climate in the past when human activity was not a factor.

Anastasios A. Tsonis, Michael D. Madsen

The Prediction of Nonstationary Climate Series by Incorporating External Forces

Almost all climate time series have some degree of nonstationarity due to external forces of the observed system. Therefore, these external forces should be taken into account when reconstructing the climate dynamics. This paper presents a novel technique in predicting nonstationary time series. The main difference of this new technique from some previous methods is that it incorporates the driving forces in the prediction model. To appraise its effectiveness, some prediction experiments were carried out using the data generated from some known classical dynamical models and climate data. Experimental results indicate that this technique is able to improve the prediction skill effectively.

Geli Wang, Peicai Yang, Anastasios A. Tsonis

The Impact of Nonlinearity on the Targeted Observations for Tropical Cyclone Prediction

This study examines the impact of nonlinearity on the targeted observations for tropical cyclone prediction. The nonlinearity of the typhoon is determined by comparing the first singular vector (FSV) and the conditional nonlinear optimal perturbation (CNOP), which is the nonlinear extension of FSV. If the similarity between the CNOP and FSV is larger than 0.5, then the typhoon is categorized as weak nonlinearity, otherwise, the typhoon is categorized as strong nonlinearity. First, the impact of nonlinearity on the typhoon targeted observations due to different resolutions is studied. Two typhoons, Meari (2004) and Matsa (2005), with 24 h forecast length are chosen, with 120-, 60-, and 30-km resolutions, respectively. It is found that the nonlinearity of both cases becomes stronger as the resolution increases. However, the sensitive areas identified with lower resolutions are more similar to each other than those identified with finer resolutions. This means that when the motion of typhoon has been described as linear or weakly nonlinear, the sensitive area may be easier to determine. Then, the impact of nonlinearity on the typhoon targeted observations due to different forecast length is investigated. In this part, typhoons Meari (2004) and Matsa (2005) with 60 km resolution are considered with 12-, 24-, and 36-h forecast lengths. We further studied two issues. In the first the initial time is fixed, while in the second the forecast time is fixed. Results show that no matter which issue is considered, typhoon Matsa exhibits stronger nonlinearity than typhoon Meari. Accordingly, Meari is categorized as a linear case, while Matsa as a nonlinear case. In the linear case, the sensitive areas identified for special forecast times (when the initial time is fixed) resemble those identified for other forecast times. Targeted observations deployed to improve a specific time forecast would thus also benefit forecasts at other times. In the nonlinear case, the similarities among the sensitive areas identified for different forecast times were more limited. The deployment of targeted observations in the nonlinear case would therefore need to be adapted to achieve large improvements for different targeted forecasts. For both cases, the closer the forecast time, the higher the similarities of the sensitive areas. When the forecast time is fixed, the sensitive areas in the linear case diverge continuously from the verification area as the forecast period lengthens due to the determination of the subtropical high in the movement of the typhoon, while those in the nonlinear case are always located around the initial cyclone indicating that the main factors affecting the typhoon movements are located within the typhoon. The deployment of targeted observations to improve a special forecast depends strongly on the time of deployment. Generally, it seems that the sensitive areas are easy to be determined in the linear case and more beneficial for the forecast. In the nonlinear case, the identification of sensitive areas is more difficult, which results in harder deployments in targeted observations.

Feifan Zhou, He Zhang