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.

The statistical properties are presented for the direct numerical simulation (DNS) of a self-similar adverse pressure gradient (APG) turbulent boundary layer (TBL) at the verge of separation. The APG TBL has a momentum thickness based Reynolds number range from Re δ2 = 570 to 13800, with a self-similar region from Re δ2 = 10000 to 12300. Within this domain the average non-dimensional pressure gradient parameter β = 39, where for a unit density β = δ 1 P ′ e /τ w , with δ 1 the displacement thickness, τ w the mean shear stress at the wall, and P ′ e the farfield pressure gradient. This flow is compared to previous zero pressure gradient (ZPG) and mild APG TBL (β = 1) results of similar Reynolds number. All flows are generated via the DNS of a TBL on a flat surface with farfield boundary conditions tailored to apply the desired pressure gradient. The conditions for self-similarity, and the appropriate length and velocity scales are derived. The mean and Reynolds stress profiles are shown to collapse when non-dimensionalised on the basis of these length and velocity scales. As the pressure gradient increases, the extent of the wake region in the mean streamwise velocity profiles increases, whilst the extent of the log-layer and viscous sub-layer decreases. The Reynolds stress, production and dissipation profiles of the APG TBL cases exhibit a second outer peak, which becomes more pronounced and more spatially localised with increasing pressure gradient. This outer peak is located at the point of inflection of the mean velocity profiles, and is suggestive of the presence of a shear flow instability. The maximum streamwise velocity variance is located at a wall normal position of δ 1 of spanwise wavelength of 2δ 1 . In summary as the pressure gradient increases the flow has properties less like a ZPG TBL and more akin to a free shear layer.

The statistical scaling properties of a self-similar adverse pressure gradient (APG) turbulent boundary layer (TBL) are presented. The intended flow is generated using the direct numerical simulation (DNS) TBL code of Simens et al. (2009) and Borrell et al. (2013), with a modified farfield boundary condition (BC). The conditions for self-similarity and appropriate scaling are derived, with mean and Reynolds stress profiles presented using this scaling. The APG and ZPG DNS are also compared under the classical viscous scaling.

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