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

The existence of partially conserved enstrophy-like quantities is conjectured to cause inverse energy transfers to develop embedded in magnetohydrodynamical (MHD) turbulence, in analogy to the influence of enstrophy in two-dimensional nonconducting turbulence. By decomposing the velocity and magnetic fields in spectral space onto helical modes, we identify subsets of three-wave (triad) interactions conserving two new enstrophy-like quantities which can be mapped to triad interactions recently identified with facilitating large-scale $\alpha$-type dynamo action and the inverse transfer of magnetic helicity. Due to their dependence on interaction scale locality, the invariants suggest that the inverse transfer of magnetic helicity might be facilitated by both local- and nonlocal-scale interactions, and is a process more local than the $\alpha$-dynamo. We test the predicted embedded (partial) energy fluxes by constructing a shell model (reduced wave-space model) of the minimal set of triad interactions (MTI) required to conserve the ideal MHD invariants. Numerically simulated MTIs demonstrate that, for a range of forcing configurations, the partial invariants are, with some exceptions, indeed useful for understanding the embedded contributions to the total spectral energy flux. Furthermore, we demonstrate that strictly inverse energy transfers may develop if enstrophy-like conserving interactions are favoured, a mechanism recently attributed to the energy cascade reversals found in nonconducting three-dimensional turbulence subject to strong rotation or confinement. The presented results have implications for the understanding of the physical mechanisms behind large-scale dynamo action and the inverse transfer of magnetic helicity, processes thought to be central to large-scale magnetic structure formation.

Partial invariants, large-scale dynamo action, and the inverse transfer of magnetic helicity.

The mean surface temperature on Earth and other planets with atmospheres is determined by the radiative balance between the non-reflected incoming solar radiation and the outgoing long-wave black-body radiation from the atmosphere. The surface temperature is higher than the black-body temperature due to the greenhouse warming. Balancing the ice-albedo cooling and the greenhouse warming gives rise to two stable climate states. A cold climate state with a completelyice-covered planet, called Snowball Earth, and a warm state similar to our present climate where greenhouse warming prevents the total glacition. The warm state has dominated Earth in most of its geological history despite a 30 % fainter young Sun. The warming could have been controlled by a greenhouse thermostat operating by temperature control of the weathering process depleting the atmosphere from $CO_2$. This temperature control has permitted life to evolve as early as the end of the heavy bombartment 4 billion years ago.

A climatic thermostat making Earth habitable

ABSTRACT The Pleistocene glacial cycles are the most prominent climate variations over the past three million years. They are the climatic response to variations in the incoming solar radiation, insolation, due to the deviations of the Earth's orbital configuration from the perfect Keplerian orbit is caused by the influence of the other planets in the solar system. This climatic response to astronomical forcing is highly non-linear, which is most pronounced expressed in the changing duration of the glacial cycles through the Middle Pleistocene Transition around a million years ago, where the duration of glacial cycles changed from 40 kyr to approximately 100 kyr without any corresponding changes in the astronomical forcing. In the late Pleistocene glaciations, the Northern Hemisphere ice sheets have grown larger than before, causing an increased cooling through the ice-albedo feedback, which makes it harder with increased insolation to cause deglaciation. The cold climate with extended glaciations has also made the climate more unstable, where strong millennial-scale oscillations related to changes in the Atlantic Meridional Overturning Circulation are observed in the paleoclimatic records.

The Pleistocene Glacial Cycles and Millennial-Scale Climate Variability

Since the beginning of satellite observations, the Arctic sea ice extent has shown a downward trend. The decline has been weaker in the March maximum than in the September minimum and masked by inter-annual fluctuations. One of the less understood aspects of the sea ice response is the persistence times for fluctuations, which could indicate the dominant physical processes behind the sea ice decline. To determine the fluctuation persistence times, however, it is necessary to first filter out the dominant effect of the seasonal cycle. In the current study, we thus develop a statistical model, which accurately decomposes the ice area changes into: (1) a variable seasonal cycle component with a constant shape and (2) a residual (short term) fluctuation. 
We find the persistence time of fluctuations to be only about three weeks, independently from season, which is substantially shorter than previously reported. Such short time scale points to the dominance of atmospheric forcing. The shape of the seasonal cycle is surprisingly constant for the whole observational record despite the rapid decline. This is in agreement with the suggestion that the asymmetry of the seasonal cycle is an effect of Arctic land-sea geography, which has not changed with climate change. 
The analysis suggest a jump in the annual sea ice area amplitude occurring in 2007, from which it has not yet recovered, possibly revealing a permanent amplitude shift. In physical sense, this could imply a shift towards the younger, thinner and more susceptible ice cover commencing after the immense 2007 multi-year ice loss.

Fluctuations and seasonality in the Arctic sea ice area: A sudden regime shift in 2007?

We consider point particle that collides with a periodic array of hard-core elastic scatterers where the length of the free flights is unbounded (the infinite-horizon Lorentz gas, LG). The Bleher central limit theorem (CLT) states that the distribution of the particle displacement divided by $\sqrt{t\ln t}$ is Gaussian in the limit of infinite time $t$. However it was stressed recently that the slow convergence makes this result unobservable. Using a Levy walk model (LW) of the LG, it was proposed that the use of a rescaled Lambert function instead of $\sqrt{t\ln t}$ provides a fast convergent, observable CLT, which was confirmed by the LG simulations. We demonstrate here that this result can simplified to a mixed CLT where the scaling factor combines normal and anomalous diffusions. For narrow infinite corridors the particle for long time obeys the usual normal diffusion, which explains the previous numerical observations. In the opposite limit of small scatterers the Bleher CLT gives a good guiding. In the intermediate cases the mixed CLT applies. The Gaussian peak determines moments of order smaller than two. In contrast, the CLT gives only half the coordinate dispersion. The missing half of the dispersion and also moments of order higher than two are described by the distribution's tail (the infinite density) which we derive here. The tail is supported along the infinite corridors and formed by anomalously long flights whose duration is comparable with the whole time of observation. The moments' calculation from the tail is confirmed by direct calculation of the fourth moment from the statistics of the backward recurrence time defined as time that elapsed since the last collision. This completes the solution of the LW model allowing full comparison with the LG.

Peter D. Ditlevsen

Papers

Partial invariants, large-scale dynamo action, and the inverse transfer of magnetic helicity.

A climatic thermostat making Earth habitable

The Pleistocene Glacial Cycles and Millennial-Scale Climate Variability

Fluctuations and seasonality in the Arctic sea ice area: A sudden regime shift in 2007?

Refined central limit theorem and infinite density tail of the Lorentz gas from Levy walk