Stochastics and Dynamics

From July 9 to 11, 2001, 50 researchers from the fields of climate dynamics and stochastic analysis met in Chorin, Germany, to discuss the idea of stochastic models of climate. The present issue of Stochastics and Dynamics collects several papers from this meeting. In this introduction to the volume, the idea of simple conceptual stochastic climate models is introduced amd recent results in the mathematically rigorous development and analysis of such models are reviewed. As well, a brief overview of the application of ideas from stochastic dynamics to simple models of the climate system is given.

We consider simple stochastic climate models, described by slowly time-dependent Langevin equations. We show that when the noise intensity is not too large, these systems can spend substantial amounts of time in metastable equilibrium, instead of adiabatically following the stationary distribution of the frozen system. This behavior can be characterized by describing the location of typical paths, and bounding the probability of atypical paths. We illustrate this approach by giving a quantitative description of phenomena associated with bistability, for three famous examples of simple climate models: Stochastic resonance in an energy balance model describing the Ice Ages; hysteresis in a box model for the Atlantic thermohaline circulation; and bifurcation delay in the case of the Lorenz model for Rayleigh–Bénard convection.

The investigation of the coupled atmosphere-ocean system is not only scientifically challenging but also practically important.

We consider a coupled atmosphere-ocean model, which involves hydrodynamics, thermodynamics, and random atmospheric dynamics due to short time influences at the air–sea interface. We reformulate this model as a random dynamical system. First, we have shown that the asymptotic dynamics of the coupled atmosphere-ocean model is described by a random climatic attractor. Second, we have estimated the atmospheric temperature evolution under oceanic feedback, in terms of the freshwater flux, heat flux and the external fluctuation at the air–sea interface, as well as the earth's longwave radiation coefficient and the shortwave solar radiation profile. Third, we have demonstrated that this system has finite degree of freedom by presenting a finite set of determining functionals in probability. Finally, we have proved that the coupled atmosphere-ocean model is ergodic under suitable conditions for physical parameters and randomness, and thus for any observable of the coupled atmosphere-ocean flows, its time average approximates the statistical ensemble average, as long as the time interval is sufficiently long.

Data from a recent field campaign in the Himalayan Kali Gandaki Valley are used to construct master equations for the along-valley flow component as observed at a station and the pressure difference between this location and another one in the valley. The inspection of these equations reveals a pronounced difference between the flow dynamics during the day when strong up-valley winds prevail and those during the night when weak down-valley winds are observed. This difference is not captured by a corresponding regression model. Predictions by the master equation cease to be useful after one hour. The relation of master equations to data based Fokker–Planck equations is briefly discussed.

In studies of perturbation dynamics in physical systems, certain specification of the governing perturbation dynamical system is generally lacking, either because the perturbation system is imperfectly known or because its specification is intrinsically uncertain, while a statistical characterization of the perturbation dynamical system is often available. In this report exact and asymptotically valid equations are derived for the ensemble mean and moment dynamics of uncertain systems. These results are used to extend the concept of optimal deterministic perturbation of certain systems to uncertain systems. Remarkably, the optimal perturbation problem has a simple solution: In uncertain systems there is a sure initial condition producing the greatest expected second moment perturbation growth.

Observed near surface air and soil temperature time series reveal a long-term memory, which is associated with a power-law scaling of the frequency spectra, S(ω) ~ ω^{- β} with β ~ 0.6, lying between white and flicker noise, 0 < β < 1. As this power law scaling is not consistent with the Brownian motion concept of climate variability, Fickian diffusion is added to a Newtonian cooling relaxation to provide a more suitable analog of climatic fluctuations: (i) Diffusive plus random heat fluxes parametrise the turbulent mixing by synoptic scale eddy life cycles, affect tropospheric and near surface temperatures and excite a long-term memory regime with a β ~ 0.5 scaling. (ii) Newtonian cooling describes the near surface temperatures relaxing towards a global mean deep soil temperature and stabilises the system to a white noise response at very low frequencies. The long-term memory regime emerges from the high frequency scaling (β ~ 1.5), once temperatures become correlated in space due to diffusion, so that spatially averaged fluctuations correlate for times beyond the diffusion time scale. The long-term memory regime disappears into a white noise plateau (β ~ 0), when low frequencies exceed the damping time scale of Newtonian cooling. This system may be interpreted as a diffusive system relaxing towards the deep soil restoration temperature with an almost infinitely large time scale.

In a two-state Markov chain with time periodic dynamics, we study path properties such as the sojourn time in one state between two consecutive jumps or the distribution of the first jump. This is done in order to exhibit a resonance interval and an optimal tuning rate interpreting the phenomenon of stochastic resonance through quality notions related with interspike intervals. We consider two cases representing the reduced dynamics of particles diffusing in time periodic potentials: Markov chains with piecewise constant periodic infinitesimal generators and Markov chains with time-continuous periodic generators.

A simple model of the thermohaline circulation of the World Ocean is considered, in which fluctuations in internal oceanic mixing and in freshwater forcing are represented by stochastic processes. The effects on the stationary probability density function of correlations between fluctuations in mixing and freshwater forcing, and of finite autocorrelation time in oceanic mixing, are determined using a mixture of analytical and numerical techniques. The quantitative behaviour of the system is found to depend on the strength and correlation character of the noise processes, quite sensitively so in some regions of parameter space. The results of this analysis suggest the importance of accurately modelling high-frequency variability in nonlinear models of the climate system.