The last decade has seen the success of stochastic parameterizations in short-term, medium-range, and seasonal forecasts: operational weather centers now routinely use stochastic parameterization schemes to represent model inadequacy better and to improve the quantification of forecast uncertainty. Developed initially for numerical weather prediction, the inclusion of stochastic parameterizations not only provides better estimates of uncertainty, but it is also extremely promising for reducing long-standing climate biases and is relevant for determining the climate response to external forcing. This article highlights recent developments from different research groups that show that the stochastic representation of unresolved processes in the atmosphere, oceans, land surface, and cryosphere of comprehensive weather and climate models 1) gives rise to more reliable probabilistic forecasts of weather and climate and 2) reduces systematic model bias. We make a case that the use of mathematically stringent methods for the derivation of stochastic dynamic equations will lead to substantial improvements in our ability to accurately simulate weather and climate at all scales. Recent work in mathematics, statistical mechanics, and turbulence is reviewed; its relevance for the climate problem is demonstrated; and future research directions are outlined.

We study a system of particles which jump on the sites of the interval of . The density at the boundaries is kept fixed to simulate the action of mass reservoirs. The evolution depends on two parameters and which are the strength of an external potential and respectively of an attractive potential among the particles. When the system behaves diffusively and the density profile of the final stationary state is linear, Fick's law is satisfied. In this paper we show that when 0$ " src="http://cdn.iopscience.com/images/1751-8121/50/43/435002/aaa8c68ieqn006.gif"> and the system models the diffusion of carbon in the presence of silicon as in the Darken experiment: the final state of the system is in qualitative agreement with the experimental one and uphill diffusion is present at the weld. Finally if and 0$ " src="http://cdn.iopscience.com/images/1751-8121/50/43/435002/aaa8c68ieqn009.gif"> is suitably large, the system simulates a vapor-liquid phase transition and we have a surprising phenomenon, as studied in Colangeli et al (2016 Phys. Lett. A 380 1710–3) and Colangeli et al (2017 J. Stat. Phys. 167 1081–111). Namely when the densities in the reservoirs correspond respectively to metastable vapor and metastable liquid we find a final stationary current which goes uphill from the reservoir with smaller density (vapor) to that with larger density (liquid). Our results are mainly numerical, we have theoretical explanations yet we miss a complete mathematical proof.

In dissipative dynamical systems phase space volumes contract, on average. Therefore, the invariant measure on the attractor is singular with respect to the Lebesgue measure. As noted by Ruelle, a generic perturbation pushes the state out of the attractor, hence the statistical features of the perturbation and, in particular, of the relaxation cannot be understood solely in terms of the unperturbed dynamics on the attractor. This remark seems to seriously limit the applicability of the standard fluctuation-dissipation procedure in the statistical mechanics of nonequilibrium (dissipative) systems. In this letter we show that the singular character of the steady state does not constitute a serious limitation in the case of systems with many degrees of freedom. The reason is that one typically deals with projected dynamics, and these are associated with regular probability distributions in the corresponding lower dimensional spaces.