| Summary: | Metastability in simple climate models: pathwise analysis of slowly driven Langevin equations. (English summary) Special issue on stochastic climate models. Stoch. Dyn. 2 (2002), no. 3, 327–356. The stochastic differential equation (Langevin equation) dxt = f(xt, εt) dt + σG(εt) dWt describes the evolution of a variable in the presence of noise. If f = −∂V/∂x, then the motion generally evolves on a rapid time scale to an equilibrium in a well of the potential V. For small ε, the locations of such equilibria will themselves evolve on the longer time scale 1/ε, and on an exponentially long (Kramers) time scale exp[O(1/σ 2)], small noise will allow migration between adjacent potential wells. These ideas are applied to three examples of environmental application. The first is the periodic occurrence of ice ages. The idea is that the 100 000 year periodicity is associated with the weak forcing by variations of solar radiation at this period, but the transitions are effected by noise (i.e., weather). The second model is Stommel’s box model for switch on and off of the North Atlantic oceanic circulation, in which two stable dynamic states are possible. The third example is the onset of “convection ” in the Lorenz equations (where two steady states bifurcate from the origin as the forcing parameter r increases through r = 1). It is assumed that r varies slowly, and near the onset of “convection”, the dynamics is governed approximately by a first order amplitude equation. This is converted to a stochastic differential equation if noise is added.
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