How to cope with climate's complexity

Climate exhibits a vast range of dissipative structures. Some have characteristic times of a few days; others evolve on thousands of years. All these structures are interdependent; in other words, they communicate. It is often considered that the only way to cope with climate complexity is to integrate the equations of atmospheric and oceanic motion with the finer possible mesh. Is this the sole strategy? Aren't we missing another characteristic of the climate system: its ability to destroy and generate information at the macroscopic scale? Paleoclimatologists consider that much of this information is present in palaeoclimate archives. It is therefore natural to build climate models such as to get the most of these archives. The strategy proposed here is based on Bayesian statistics and low-order non-linear dynamical systems, in a modelling approach that explicitly includes the effects of uncertainties. Its practical interest is illustrated through the problem of the timing of the next great glaciation. Is glacial inception overdue, or do we need to wait for another 50,000 years before ice caps grow again? Our results indicate a glaciation inception in 50,000 years.Comment: proceedings of a talk given at the "Complexity Workshop", Academia Europeae, Heidelberg, May 2008, to be submitted to European Review

Effects of additive noise on the stability of glacial cycles

It is well acknowledged that the sequence of glacial-interglacial cycles is paced by the astronomical forcing. However, how much is the sequence robust against natural fluctuations associated, for example, with the chaotic motions of atmosphere and oceans? In this article, the stability of the glacial-interglacial cycles is investigated on the basis of simple conceptual models. Specifically, we study the influence of additive white Gaussian noise on the sequence of the glacial cycles generated by stochastic versions of several low-order dynamical system models proposed in the literature. In the original deterministic case, the models exhibit different types of attractors: a quasiperiodic attractor, a piecewise continuous attractor, strange nonchaotic attractors, and a chaotic attractor. We show that the combination of the quasiperiodic astronomical forcing and additive fluctuations induce a form of temporarily quantised instability. More precisely, climate trajectories corresponding to different noise realizations generally cluster around a small number of stable or transiently stable trajectories present in the deterministic system. Furthermore, these stochastic trajectories may show sensitive dependence on very small amounts of perturbations at key times. Consistently with the complexity of each attractor, the number of trajectories leaking from the clusters may range from almost zero (the model with a quasiperiodic attractor) to a significant fraction of the total (the model with a chaotic attractor), the models with strange nonchaotic attractors being intermediate. Finally, we discuss the implications of this investigation for research programmes based on numerical simulators. }Comment: Parlty based on a lecture given by M. Crucifix at workshop held in Rome in 2013 as a part of Mathematics of Planet Earth 201

Uncertainty in climate science and climate policy

This essay, written by a statistician and a climate scientist, describes our view of the gap that exists between current practice in mainstream climate science, and the practical needs of policymakers charged with exploring possible interventions in the context of climate change. By `mainstream' we mean the type of climate science that dominates in universities and research centres, which we will term `academic' climate science, in contrast to `policy' climate science; aspects of this distinction will become clearer in what follows. In a nutshell, we do not think that academic climate science equips climate scientists to be as helpful as they might be, when involved in climate policy assessment. Partly, we attribute this to an over-investment in high resolution climate simulators, and partly to a culture that is uncomfortable with the inherently subjective nature of climate uncertainty.Comment: submitted as contribution to Conceptual Foundations of ClimateModeling, Winsberg, E. and Lloyd, E., eds., The University of Chicago Pres

On the use of simple dynamical systems for climate predictions: A Bayesian prediction of the next glacial inception

Over the last few decades, climate scientists have devoted much effort to the development of large numerical models of the atmosphere and the ocean. While there is no question that such models provide important and useful information on complicated aspects of atmosphere and ocean dynamics, skillful prediction also requires a phenomenological approach, particularly for very slow processes, such as glacial-interglacial cycles. Phenomenological models are often represented as low-order dynamical systems. These are tractable, and a rich source of insights about climate dynamics, but they also ignore large bodies of information on the climate system, and their parameters are generally not operationally defined. Consequently, if they are to be used to predict actual climate system behaviour, then we must take very careful account of the uncertainty introduced by their limitations. In this paper we consider the problem of the timing of the next glacial inception, about which there is on-going debate. Our model is the three-dimensional stochastic system of Saltzman and Maasch (1991), and our inference takes place within a Bayesian framework that allows both for the limitations of the model as a description of the propagation of the climate state vector, and for parametric uncertainty. Our inference takes the form of a data assimilation with unknown static parameters, which we perform with a variant on a Sequential Monte Carlo technique (`particle filter'). Provisional results indicate peak glacial conditions in 60,000 years.Comment: superseeds the arXiv:0809.0632 (which was published in European Reviews). The Bayesian section has been significantly expanded. The present version has gone scientific peer review and has been published in European Physics Special Topics. (typo in DOI and in Table 1 (psi -> theta) corrected on 25th August 2009

Why could ice ages be unpredictable?

It is commonly accepted that the variations of Earth's orbit and obliquity control the timing of Pleistocene glacial-interglacial cycles. Evidence comes from power spectrum analysis of palaeoclimate records and from inspection of the timing of glacial and deglacial transitions. However, we do not know how tight this control is. Is it, for example, conceivable that random climatic fluctuations could cause a delay in deglaciation, bad enough to skip a full precession or obliquity cycle and subsequently modify the sequence of ice ages? To address this question, seven previously published conceptual models of ice ages are analysed by reference to the notion of generalised synchronisation. Insight is being gained by comparing the effects of the astronomical forcing with idealised forcings composed of only one or two periodic components. In general, the richness of the astronomical forcing allows for synchronisation over a wider range of parameters, compared to periodic forcing. Hence, glacial cycles may conceivably have remained paced by the astronomical forcing throughout the Pleistocene. However, all the models examined here also show a range of parameters for which the structural stability of the ice age dynamics is weak. This means that small variations in parameters or random fluctuations may cause significant shifts in the succession of ice ages if the system were effectively in that parameter range. Whether or not the system has strong structural stability depends on the amplitude of the effects associated with the astronomical forcing, which significantly differ across the different models studied here. The possibility of synchronisation on eccentricity is also discussed and it is shown that a high Rayleigh number on eccentricity, as recently found in observations, is no guarantee of reliable synchronisation.Comment: article submitted to Climate of the Past, and will appear first in Climate of the Past Discussion. keywords: astronomical theory, synchronisation, strange non-chaotic attractors, ice ages, Milankovitc

Bifurcations and strange nonchaotic attractors in a phase oscillator model of glacial-interglacial cycles

Glacial-interglacial cycles are large variations in continental ice mass and greenhouse gases, which have dominated climate variability over the Quaternary. The dominant periodicity of the cycles is $\sim$40 kyr before the so-called middle Pleistocene transition between $\sim$1.2 and $\sim$0.7 Myr ago, and it is $\sim$100 kyr after the transition. In this paper, the dynamics of glacial-interglacial cycles are investigated using a phase oscillator model forced by the time-varying incoming solar radiation (insolation). We analyze the bifurcations of the system and show that strange nonchaotic attractors appear through nonsmooth saddle-node bifurcations of tori. The bifurcation analysis indicates that mode-locking is likely to occur for the 41 kyr glacial cycles but not likely for the 100 kyr glacial cycles. The sequence of mode-locked 41 kyr cycles is robust to small parameter changes. However, the sequence of 100 kyr glacial cycles can be sensitive to parameter changes when the system has a strange nonchaotic attractor.Comment: 25 pages, 9 figure

Emulation of the MBM-MEDUSA model: exploring the sea level and the basin-to-shelf transfer influence on the system dynamics

Complex climate models require high computational burden. However, computational limitations may be avoided by using emulators. In this work we present several approaches for dynamical emulation (also called metamodelling) of the Multi-Box Model (MBM) coupled to the Model of Early Diagenesis in the Upper Sediment A (MEDUSA) that simulates the carbon cycle of the ocean and atmosphere [1]. We consider two experiments performed on the MBM-MEDUSA that explore the Basin-to-Shelf Transfer (BST) dynamics. In both experiments the sea level is varied according to a paleo sea level reconstruction. Such experiments are interesting because the BST is an important cause of the CO2 variation and the dynamics is potentially nonlinear. The output that we are interested in is the variation of the carbon dioxide partial pressure in the atmosphere over the Pleistocene. The first experiment considers that the BST is fixed constant during the simulation. In the second experiment the BST is interactively adjusted according to the sea level, since the sea level is the primary control of the growth and decay of coral reefs and other shelf carbon reservoirs. The main aim of the present contribution is to create a metamodel of the MBM-MEDUSA using the Dynamic Emulation Modelling methodology [2] and compare the results obtained using linear and non-linear methods. The first step in the emulation methodology used in this work is to identify the structure of the metamodel. In order to select an optimal approach for emulation we compare the results of identification obtained by the simple linear and more complex nonlinear models. In order to identify the metamodel in the first experiment the simple linear regression and the least-squares method is sufficient to obtain a 99,9% fit between the temporal outputs of the model and the metamodel. For the second experiment the MBM’s output is highly nonlinear. In this case we apply nonlinear models, such as, NARX, Hammerstein model, and an ’ad-hoc’ switching model. After the identification we perform the parameter mapping using spline interpolation and validate the emulator on a new set of parameters

Crossover and peaks in the Pleistocene climate spectrum; understanding from simple ice age models

The power spectrum provides a compact representation of the scale dependence of the variability in time series. At multi-millennial time scales the spectrum of the Pleistocene climate is composed of a set of narrow band spectral modes attributed to the regularly varying changes in insolation from the astronomical change in Earth’s orbit and rotation superimposed on a continuous background generally attributed to stochastic variations. Quantitative analyses of paleoclimatic records indicate that the continuous part comprises a dominant part of the variance. It exhibits a power-law dependency typical of stochastic, self-similar processes, but with a scale break at the frequency of glacial-interglacial cycles. Here we discuss possible origins of this scale break, the connection between the continuous background and the narrow bands, and the apparently modest spectral power above the continuum at these scales. We demonstrate that the observed scale break around 100 ka can have a variety of different origins and does not imply an internal time scale of correlation as implied by the simplest linear stochastic model