Abstract
Abstract. The understanding of the statistical properties and of the dynamics of multistable systems is gaining more and more importance in a vast variety of scientific fields. This is especially relevant for the investigation of the tipping points of complex systems. Sometimes, in order to understand the time series of given observables exhibiting bimodal distributions, simple one-dimensional Langevin models are fitted to reproduce the observed statistical properties, and used to investing-ate the projected dynamics of the observable. This is of great relevance for studying potential catastrophic changes in the properties of the underlying system or resonant behaviours like those related to stochastic resonance-like mechanisms. In this paper, we propose a framework for encasing this kind of studies, using simple box models of the oceanic circulation and choosing as observable the strength of the thermohaline circulation. We study the statistical properties of the transitions between the two modes of operation of the thermohaline circulation under symmetric boundary forcings and test their agreement with simplified one-dimensional phenomenological theories. We extend our analysis to include stochastic resonance-like amplification processes. We conclude that fitted one-dimensional Langevin models, when closely scrutinised, may result to be more ad-hoc than they seem, lacking robustness and/or well-posedness. They should be treated with care, more as an empiric descriptive tool than as methodology with predictive power.
Highlights
An interesting property of many physical systems with several degrees of freedom is the presence of multiple equilibria for a given choice of the parameters
Among the many interesting properties of multi-stable systems, we may mention their possibility of featuring hysteretic behaviour: starting from an initial equilibrium x = xin realized for a given value of a parameter P = Pin and increasing adiabatically the value of P so that the system is always at equilibrium following x = x (P ), we may eventually encounter bifurcations leading the system to a new branch of equilibria x = x (P ) such that, if we revert the direction of variation of P, we may end up to a different final stable state xfin = x (Pin) = x(Pin) = xin
The problem which has probably attracted the greatest deal of interest in the last two decades is that of the stability properties of the thermohaline circulation (THC)
Summary
An interesting property of many physical systems with several degrees of freedom is the presence of multiple equilibria (or, more in general, of a disconnected attractor) for a given choice of the parameters. Since the landmark Hasselmann’s (1976) contribution, it has become clearer and clearer in the climate science community that stochastic forcing components can be treated as quite reliable surrogates for high frequency processes not captured by the variables included in the climate model under consideration (Fraedrich, 1978; Saltzman, 2002) This has raised the interest in exploring whether transitions between stable modes of operation of the THC far from the actual tipping points could be triggered by noise, representing high-frequency (with respect to the ocean’s time scale) atmospheric forcings, of sufficient amplitude (Cessi, 1994; Monahan, 2002). In this work we would like to examine critically the effectiveness and robustness of using one-dimensional Langevin equations to represent the dynamical and statistical properties of the THC strength resulting from models which feature more than one degree of freedom This is methodologically relevant, in the context of recent efforts directed at understanding whether the transitions between different steady states associated to the tipping points can be highlighted by early indicators (Scheffer et al, 2008).
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