Stable non-Gaussian random processes , by G. Samorodnitsky and M. S. Taqqu. Pp. 632. £49.50. 1994. ISBN 0-412-05171-0 (Chapman and Hall).

Tipping points in complex systems may imply risks of unwanted collapse, but also opportunities for positive change. Our capacity to navigate such risks and opportunities can be boosted by combining emerging insights from two unconnected fields of research. One line of work is revealing fundamental architectural features that may cause ecological networks, financial markets, and other complex systems to have tipping points. Another field of research is uncovering generic empirical indicators of the proximity to such critical thresholds. Although sudden shifts in complex systems will inevitably continue to surprise us, work at the crossroads of these emerging fields offers new approaches for anticipating critical transitions.

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Anticipating Critical Transitions

A copy of Guangbo jiemu bao [Broadcast Program Report] was being passed from hand to hand among a group of young people eager to be the first to read the article introducing the program "What Is Revolutionary Love?" It said: "… Young friends, you are certainly very concerned about this problem'. So, we would like you to meet the young women workers Meng Xiaoyu and Meng Yamei and the older cadre Miss Feng. They are the three leading characters in the short story ‘The Place of Love.’ Through the description of the love lives of these three, the story induces us to think deeply about two questions that merit further examination.

This is How the Discussion Started

Polar temperatures over the last several million years have, at times, been slightly warmer than today, yet global mean sea level has been 6-9 metres higher as recently as the Last Interglacial (130,000 to 115,000 years ago) and possibly higher during the Pliocene epoch (about three million years ago). In both cases the Antarctic ice sheet has been implicated as the primary contributor, hinting at its future vulnerability. Here we use a model coupling ice sheet and climate dynamics-including previously underappreciated processes linking atmospheric warming with hydrofracturing of buttressing ice shelves and structural collapse of marine-terminating ice cliffs-that is calibrated against Pliocene and Last Interglacial sea-level estimates and applied to future greenhouse gas emission scenarios. Antarctica has the potential to contribute more than a metre of sea-level rise by 2100 and more than 15 metres by 2500, if emissions continue unabated. In this case atmospheric warming will soon become the dominant driver of ice loss, but prolonged ocean warming will delay its recovery for thousands of years.

Contribution of Antarctica to past and future sea-level rise

https://boris.unibe.ch/62575/1/1-s2.0-S0277379114003485-main.pdf

A stratigraphic framework for abrupt climatic changes during the Last Glacial period based on three synchronized Greenland ice-core records: refining and extending the INTIMATE event stratigraphy

Large-amplitude (10–15 K) millennial-duration warming events, the Dansgaard–Oeschger (DO) events, repeatedly occurred in the North Atlantic region during ice ages. So far, the trigger of these events is not known. To explain their recurrence pattern, a ghost stochastic resonance (GSR) scenario has been suggested, i.e., a dynamical scenario in which the events represent the subharmonic response to centennial-scale solar forcing plus noise. According to this hypothesis a multimodal phase distribution of the events is expected, which should be tested on the basis of climate records by means of time series analysis. A major obstacle in these tests, however, is the need of a statistical measure of regularity that can distinguish between a random occurrence of DO events and a GSR scenario. Here we construct and compare three new measures of phase multimodality. In a Monte Carlo simulation with a simple conceptual model of DO events we simulate probability distributions of the measures under both scenarios for r...

https://www.gfy.ku.dk/%7Epditlev/papers/CHAOEH194043132_1.pdf

New measures of multimodality for the detection of a ghost stochastic resonance

Characterizing the multiscale nature of fluctuations from nonlinear and nonstationary time series is one of the most intensively studied contemporary problems in nonlinear sciences. In this work, we address this problem by combining two established concepts—empirical mode decomposition (EMD) and generalized fractal dimensions—into a unified analysis framework. Specifically, we demonstrate that the intrinsic mode functions derived by EMD can be used as a source of local (in terms of scales) information about the properties of the phase-space trajectory of the system under study, allowing us to derive multiscale measures when looking at the behavior of the generalized fractal dimensions at different scales. This formalism is applied to three well-known low-dimensional deterministic dynamical systems (the Henon map, the Lorenz ’63 system, and the standard map), three realizations of fractional Brownian motion with different Hurst exponents, and two somewhat higher-dimensional deterministic dynamical systems (the Lorenz ’96 model and the on–off intermittency model). These examples allow us to assess the performance of our formalism with respect to practically relevant aspects like additive noise, different initial conditions, the length of the time series under study, low- vs high-dimensional dynamics, and bursting effects. Finally, by taking advantage of two real-world systems whose multiscale features have been widely investigated (a marine stack record providing a proxy of the global ice volume variability of the past 5 × 10 6 years and the SYM-H geomagnetic index), we also illustrate the applicability of this formalism to real-world time series.

Multiscale measures of phase-space trajectories.

Few rigorous results are derived for fully developed turbulence. By applying the scaling properties of the Navier–Stokes equation we have derived a relation for the energy spectrum valid for unforced or decaying isotropic turbulence. We find the existence of a scaling function ψ. The energy spectrum can at any time by a suitable rescaling be mapped onto this function. This indicates that the initial (primordial) energy spectrum is in principle retained in the energy spectrum observed at any later time, and the principle of permanence of large eddies is derived. The result can be seen as a restoration of the determinism of the Navier–Stokes equation in the mean. We compare our results with a windtunnel experiment and find good agreement.

Scaling and the prediction of energy spectra in decaying hydrodynamic turbulence

In climatic time series abrupt changes are observed. We hypothesize that these abrupt changes are due to nonlinear responses inherent in the climate system, specifically, so-called tipping points. This behaviour result from non-linear climate response to either external forcing, internal stochastic fluctuations or a combination of both. At some point the forcing will cause the climate to jump from one stable state to another. This scenario is termed a tipping point. The concept of a tipping point is quite broad, but here we shall refrain from any general definitions and consider the following more restricted framework: We consider the climate or some components of the climate as a dynamical system depending on a set of parameters. Factors, not included in the system interacting with components of the system, can then be considered external forcing or stochastic fluctuations. Two common, and often competing, hypotheses are: The climate system’s steady state loses its stability and disappears as an external system (control-)parameter slowly changes, so-called b-tipping, b for bifurcation-induced; or fluctuations spontaneously push the climate system from one stable state to another, so-called n-tipping, n for noise-induced. The cause of the tipping can be very different in the two cases, and especially the possibility of predicting a tipping will be different. In the case that the underlying dynamics or the control-parameter are not completely known, there could still be early warning signals in the statistics of the observed fluctuations prior to a tipping point. The Pleistocene climate is documented in a variety of geological records, most promi-

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Tipping Points in the Climate System

Three-dimensional (3D) turbulence is characterized by a dual forward cascade of both kinetic energy and helicity, a second inviscid flow invariant, from the integral scale of motion to the viscous dissipative scale. In helical flows, however, such as strongly rotating flows with broken mirror symmetry, an inverse energy cascade can be observed analogous to that of two-dimensional turbulence (2D) where a second positive-definite flow invariant, enstrophy, unlike helicity in 3D, effectively blocks the forward cascade of energy. In the spectral-helical decomposition of the Navier-Stokes equation it has previously been show that a subset of three-wave (triad) interactions conserve helicity in 3D in a fashion similar to enstrophy in 2D, thus leading to a 2D-like inverse energy cascade in 3D. In this work, we show both theoretically and numerically that an additional subset of interactions exist conserving a new pseudo-invariant in addition to energy and helicity, which contributes either to a forward or inverse energy cascade depending on the specific triad interaction geometry.

Peter D. Ditlevsen

Papers

New measures of multimodality for the detection of a ghost stochastic resonance

Multiscale measures of phase-space trajectories.

Scaling and the prediction of energy spectra in decaying hydrodynamic turbulence

Tipping Points in the Climate System

Pseudo-invariants causing inverse energy cascades in three-dimensional turbulence