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Nonlinear softening as a predictive precursor to climate tipping  [PDF]
Jan Sieber,J. Michael T. Thompson
Physics , 2011, DOI: 10.1098/rsta.2011.0372
Abstract: Approaching a dangerous bifurcation, from which a dynamical system such as the Earth's climate will jump (tip) to a different state, the current stable state lies within a shrinking basin of attraction. Persistence of the state becomes increasingly precarious in the presence of noisy disturbances. We consider an underlying potential, as defined theoretically for a saddle-node fold and (via averaging) for a Hopf bifurcation. Close to a stable state, this potential has a parabolic form; but approaching a jump it becomes increasingly dominated by softening nonlinearities. If we have already detected a decrease in the linear decay rate, nonlinear information allows us to estimate the propensity for early tipping due to noise. We argue that one needs to extract information about the nonlinear features (a "softening") of the underlying potential from the time series to judge the probability and timing of tipping. This analysis is the logical next step if one has detected a decrease of the linear decay rate. If there is no discernable trend in the linear analysis, nonlinear softening is even more important in showing the proximity to tipping. After extensive normal form calibration studies, we check two geological time series from paleo-climate tipping events for softening of the underlying well. For the ending of the last ice age, where we find no convincing linear precursor, we identify a statistically significant nonlinear softening towards increasing temperature. The analysis has thus successfully detected a warning of the imminent tipping event.
Tipping points in open systems: bifurcation, noise-induced and rate-dependent examples in the climate system  [PDF]
Peter Ashwin,Sebastian Wieczorek,Renato Vitolo,Peter Cox
Physics , 2011, DOI: 10.1098/rsta.2011.0306
Abstract: Tipping points associated with bifurcations (B-tipping) or induced by noise (N-tipping) are recognized mechanisms that may potentially lead to sudden climate change. We focus here a novel class of tipping points, where a sufficiently rapid change to an input or parameter of a system may cause the system to "tip" or move away from a branch of attractors. Such rate-dependent tipping, or R-tipping, need not be associated with either bifurcations or noise. We present an example of all three types of tipping in a simple global energy balance model of the climate system, illustrating the possibility of dangerous rates of change even in the absence of noise and of bifurcations in the underlying quasi-static system.
Parameter shifts for nonautonomous systems in low dimension: Bifurcation- and Rate-induced tipping  [PDF]
Peter Ashwin,Clare Perryman,Sebastian Wieczorek
Mathematics , 2015,
Abstract: We discuss the nonlinear phenomena of irreversible tipping for nonautonomous systems where time-varying inputs correspond to a smooth "parameter shift" from one asymptotic value to another. We define notions of bifurcation-induced and rate-induced irreversible tipping of the nonautonomous system in terms of local pullback point attractors and present several results on how nontrivial dynamics for nonautonomous systems can be deduced from analysis of the structure of the bifurcation diagram for an associated autonomous system where parameters are fixed at intermediate values. In one-dimension, we give a number of sufficient conditions for the presence or absence of the less understood rate-induced tipping, and we discuss consequences of our results in a conceptual climate model example.
Tipping points near a delayed saddle node bifurcation with periodic forcing  [PDF]
Jielin Zhu,Rachel Kuske,Thomas Erneux
Mathematics , 2014,
Abstract: We consider the effect on tipping from an additive periodic forcing in a canonical model with a saddle node bifurcation and a slowly varying bifurcation parameter. Here tipping refers to the dramatic change in dynamical behavior characterized by a rapid transition away from a previously attracting state. In the absence of the periodic forcing, it is well-known that a slowly varying bifurcation parameter produces a delay in this transition, beyond the bifurcation point for the static case. Using a multiple scales analysis, we consider the effect of amplitude and frequency of the periodic forcing relative to the drifting rate of the slowly varying bifurcation parameter. We show that a high frequency oscillation drives an earlier tipping when the bifurcation parameter varies more slowly, with the advance of the tipping point proportional to the square of the ratio of amplitude to frequency. In the low frequency case the position of the tipping point is affected by the frequency, amplitude and phase of the oscillation. The results are based on an analysis of the local concavity of the trajectory, used for low frequencies both of the same order as the drifting rate of the bifurcation parameter and for low frequencies larger than the drifting rate. The tipping point location is advanced with increased amplitude of the periodic forcing, with critical amplitudes where there are jumps in the location, yielding significant advances in the tipping point. We demonstrate the analysis for two applications with saddle node-type bifurcations.
Dansgaard-Oeschger events: tipping points in the climate system  [PDF]
A. A. Cimatoribus,S. S. Drijfhout,V. Livina,G. van der Schrier
Climate of the Past Discussions , 2012, DOI: 10.5194/cpd-8-4269-2012
Abstract: The largest variability in temperature over the last sixty thousand years is connected with Dansgaard-Oeschger events. Various prototype models have been proposed to explain these rapid climate fluctuations, but until now no observational constraint has been forwarded to choose between different theories. We assess the bimodality of the system reconstructing the topology of the multi-dimensional attractor over which the climate system evolves. Furthermore, we show that Dansgaard-Oeschger events are compatible with the crossing of a tipping point in the climate system. We use high-resolution ice core isotope data to investigate the statistical properties of the climate fluctuations in the period before the onset of the abrupt change. We find that the statistics are consistent with the switches between two different climate equilibrium states in response to a changing external forcing.
Climate bifurcation during the last deglaciation  [PDF]
T. M. Lenton,V. N. Livina,V. Dakos,M. Scheffer
Climate of the Past Discussions , 2012, DOI: 10.5194/cpd-8-321-2012
Abstract: The last deglaciation was characterised by two abrupt warming events, at the start of the B lling-Aller d and at the end of the Younger Dryas, but their underlying causes are unclear. Some abrupt climate changes may involve gradual forcing past a bifurcation point, in which a prevailing climate state loses its stability and the climate tips into an alternative state, providing an early warning signal in the form of slowing responses to perturbations. However, the abrupt Dansgaard-Oeschger (DO) events during the last ice age were probably triggered by stochastic fluctuations without bifurcation or early warning, and whether the onset of the B lling-Aller d (DO event 1) was preceded by slowing down or not is debated. Here we show that the interval from the Last Glacial Maximum to the end of the Younger Dryas, as recorded in three Greenland ice cores with two different climate proxies, was accompanied by a robust slowing down in climate dynamics and an increase in climate variability, consistent with approaching bifurcation. Prior to the B lling warming there was a robust increase in climate variability but no consistent slowing down signal, suggesting this abrupt change was probably triggered by a stochastic fluctuation. The B lling warming marked a distinct destabilisation of the climate system, which excited an internal mode of variability in Atlantic meridional overturning circulation strength, causing multi-centennial climate fluctuations. There is some evidence for slowing down in the transition to and during the Younger Dryas. We infer that a bifurcation point was finally approached at the end of the Younger Dryas, in which the cold climate state, with weak Atlantic overturning circulation, lost its stability, and the climate tipped irreversibly into a warm interglacial state. The lack of a large triggering perturbation at the end of the Younger Dryas, and the fact that subsequent meltwater perturbations did not cause sustained cooling, support the bifurcation hypothesis.
Climate bifurcation during the last deglaciation?
T. M. Lenton, V. N. Livina, V. Dakos,M. Scheffer
Climate of the Past (CP) & Discussions (CPD) , 2012,
Abstract: There were two abrupt warming events during the last deglaciation, at the start of the B lling-Aller d and at the end of the Younger Dryas, but their underlying dynamics are unclear. Some abrupt climate changes may involve gradual forcing past a bifurcation point, in which a prevailing climate state loses its stability and the climate tips into an alternative state, providing an early warning signal in the form of slowing responses to perturbations, which may be accompanied by increasing variability. Alternatively, short-term stochastic variability in the climate system can trigger abrupt climate changes, without early warning. Previous work has found signals consistent with slowing down during the last deglaciation as a whole, and during the Younger Dryas, but with conflicting results in the run-up to the B lling-Aller d. Based on this, we hypothesise that a bifurcation point was approached at the end of the Younger Dryas, in which the cold climate state, with weak Atlantic overturning circulation, lost its stability, and the climate tipped irreversibly into a warm interglacial state. To test the bifurcation hypothesis, we analysed two different climate proxies in three Greenland ice cores, from the Last Glacial Maximum to the end of the Younger Dryas. Prior to the B lling warming, there was a robust increase in climate variability but no consistent slowing down signal, suggesting this abrupt change was probably triggered by a stochastic fluctuation. The transition to the warm B lling-Aller d state was accompanied by a slowing down in climate dynamics and an increase in climate variability. We suggest that the B lling warming excited an internal mode of variability in Atlantic meridional overturning circulation strength, causing multi-centennial climate fluctuations. However, the return to the Younger Dryas cold state increased climate stability. We find no consistent evidence for slowing down during the Younger Dryas, or in a longer spliced record of the cold climate state before and after the B lling-Aller d. Therefore, the end of the Younger Dryas may also have been triggered by a stochastic perturbation.
Dansgaard-Oeschger events: tipping points in the climate system  [PDF]
Andrea A. Cimatoribus,Sybren S. Drijfhout,Valerie Livina,Gerard van der Schrier
Physics , 2011, DOI: 10.5194/cp-9-323-2013
Abstract: Dansgaard-Oeschger events are a prominent mode of variability in the records of the last glacial cycle. Various prototype models have been proposed to explain these rapid climate fluctuations, and no agreement has emerged on which may be the more correct for describing the paleoclimatic signal. In this work, we assess the bimodality of the system reconstructing the topology of the multi--dimensional attractor over which the climate system evolves. We use high-resolution ice core isotope data to investigate the statistical properties of the climate fluctuations in the period before the onset of the abrupt change. We show that Dansgaard-Oeschger events have weak early warning signals if the ensemble of events is considered. We find that the statistics are consistent with the switches between two different climate equilibrium states in response to a changing external forcing (e.g. solar, ice sheets...), either forcing directly the transition or pacing it through stochastic resonance. These findings are most consistent with a model that associates Dansgaard-Oeschger with changing boundary conditions, and with the presence of a bifurcation point.
Controlling systems that drift through a tipping point  [PDF]
Takashi Nishikawa,Edward Ott
Physics , 2014, DOI: 10.1063/1.4887275
Abstract: Slow parameter drift is common in many systems (e.g., the amount of greenhouse gases in the terrestrial atmosphere is increasing). In such situations, the attractor on which the system trajectory lies can be destroyed, and the trajectory will then go to another attractor of the system. We consider the case where there are more than one of these possible final attractors, and we ask whether we can control the outcome (i.e., the attractor that ultimately captures the trajectory) using only small controlling perturbations. Specifically, we consider the problem of controlling a noisy system whose parameter slowly drifts through a saddle-node bifurcation taking place on a fractal boundary between the basins of multiple attractors. We show that, when the noise level is low, a small perturbation of size comparable to the noise amplitude applied at a single point in time can ensure that the system will evolve toward a target attracting state with high probability. For a range of noise levels, we find that the minimum size of perturbation required for control is much smaller within a time period that starts some time after the bifurcation, providing a "window of opportunity" for driving the system toward a desirable state. We refer to this procedure as tipping point control.
Dansgaard–Oeschger events: bifurcation points in the climate system
A. A. Cimatoribus, S. S. Drijfhout, V. Livina,G. van der Schrier
Climate of the Past (CP) & Discussions (CPD) , 2013,
Abstract: Dansgaard–Oeschger events are a prominent mode of variability in the records of the last glacial cycle. Various prototype models have been proposed to explain these rapid climate fluctuations, and no agreement has emerged on which may be the more correct for describing the palaeoclimatic signal. In this work, we assess the bimodality of the system, reconstructing the topology of the multi-dimensional attractor over which the climate system evolves. We use high-resolution ice core isotope data to investigate the statistical properties of the climate fluctuations in the period before the onset of the abrupt change. We show that Dansgaard–Oeschger events have weak early warning signals if the ensemble of events is considered. We find that the statistics are consistent with the switches between two different climate equilibrium states in response to a changing external forcing (e.g. solar, ice sheets), either forcing directly the transition or pacing it through stochastic resonance. These findings are most consistent with a model that associates Dansgaard–Oeschger with changing boundary conditions, and with the presence of a bifurcation point.
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