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.

Stochastic methods are a crucial area in contemporary climate research and are increasingly being used in comprehensive weather and climate prediction models as well as reduced order climate models. Stochastic methods are used as subgrid-scale parameterizations (SSPs) as well as for model error representation, uncertainty quantification, data assimilation, and ensemble prediction. The need to use stochastic approaches in weather and climate models arises because we still cannot resolve all necessary processes and scales in comprehensive numerical weather and climate prediction models. In many practical applications one is mainly interested in the largest and potentially predictable scales and not necessarily in the small and fast scales. For instance, reduced order models can simulate and predict large-scale modes. Statistical mechanics and dynamical systems theory suggest that in reduced order models the impact of unresolved degrees of freedom can be represented by suitable combinations of deterministic and stochastic components and non-Markovian (memory) terms. Stochastic approaches in numerical weather and climate prediction models also lead to the reduction of model biases. Hence, there is a clear need for systematic stochastic approaches in weather and climate modeling. In this review, we present evidence for stochastic effects in laboratory experiments. Then we provide an overview of stochastic climate theory from an applied mathematics perspective. We also survey the current use of stochastic methods in comprehensive weather and climate prediction models and show that stochastic parameterizations have the potential to remedy many of the current biases in these comprehensive models

This review paper summarizes current knowledge available for aviation operations related to meteorology and provides suggestions for necessary improvements in the measurement and prediction of weather-related parameters, new physical methods for numerical weather predictions (NWP), and next-generation integrated systems. Severe weather can disrupt aviation operations on the ground or in-flight. The most important parameters related to aviation meteorology are wind and turbulence, fog visibility (Vis) and ceiling, rain and snow amount and rates, icing, ice microphysical parameters, convection and precipitation intensity, microbursts, hail, and lightning. Measurements of these parameters are also functions of sensor response times and measurement thresholds in extreme weather conditions. In addition to these, airport environments can play an important role leading to intensification of extreme weather conditions or high impact weather events, e.g., anthropogenic ice fog. To observe meteorological parameters, new remote sensing platforms, namely wind LIDAR, sodars, radars, and geostationary satellites, and in-situ observations at the surface and in the cloud, as well as aircraft and Unmanned Aerial Vehicles (UAV) mounted sensors, are becoming more common. Because of prediction issues at smaller time and space scales (e.g., <1 km), meteorological forecasts from NWP models need to be continuously improved. Aviation weather forecasts also need to be developed to provide information that represents both deterministic and statistical approaches. In this review, we present available resources and issues for aviation meteorology and evaluate them for required improvements related to measurements, nowcasting, forecasting, and climate change, and emphasize future challenges.

Finite computing resources limit the spatial resolution of state-of-the-art global climate simulations to hundreds of kilometres. In neither the atmosphere nor the ocean are small-scale processes such as convection, clouds and ocean eddies properly represented. Climate simulations are known to depend, sometimes quite strongly, on the resulting bulk-formula representation of unresolved processes. Stochastic physics schemes within weather and climate models have the potential to represent the dynamical effects of unresolved scales in ways which conventional bulk-formula representations are incapable of so doing. The application of stochastic physics to climate modelling is a rapidly advancing, important and innovative topic. The latest research findings are gathered together in the Theme Issue for which this paper serves as the introduction.

The Robert-Asselin time filter is widely used in numerical models of weather and climate. It successfully suppresses the spurious computational mode associated with the leapfrog time-stepping scheme. Unfortunately, it also weakly suppresses the physical mode and severely degrades the numerical accuracy. These two concomitant problems are shown to occur because the filter does not conserve the mean state, averaged over the three time slices on which it operates. The author proposes a simple modification to the Robert-Asselin filter, which does conserve the three-time-level mean state. When used in conjunction with the leapfrog scheme, the modification vastly reduces the impacts on the physical mode and increases the numerical accuracy for amplitude errors by two orders, yielding third-order accuracy. The modified filter could easily be incorporated into existing general circulation models of the atmosphere and ocean. In principle, it should deliver more faithful simulations at almost no additional computational expense. Alternatively, it may permit the use of longer time steps with no loss of accuracy, reducing the computational expense of a given simulation.

Anthropogenic climate change is expected to strengthen the vertical wind shears at aircraft cruising altitudes within the atmospheric jet streams. Such a strengthening would increase the prevalence of the shear instabilities that generate clear-air turbulence. Climate modelling studies have indicated that the amount of moderate-or-greater clear-air turbulence on transatlantic flight routes in winter will increase significantly in future as the climate changes. However, the individual responses of light, moderate, and severe clear-air turbulence have not previously been studied, despite their importance for aircraft operations. Here, we use climate model simulations to analyse the transatlantic wintertime clear-air turbulence response to climate change in five aviation-relevant turbulence strength categories. We find that the probability distributions for an ensemble of 21 clear-air turbulence diagnostics generally gain probability in their right-hand tails when the atmospheric carbon dioxide concentration is doubled. By converting the diagnostics into eddy dissipation rates, we find that the ensembleaverage airspace volume containing light clear-air turbulence increases by 59% (with an intra-ensemble range of 43%-68%), light-to-moderate by 75% (39%-96%), moderate by 94% (37%-118%), moderate-to-severe by 127% (30%-170%), and severe by 149% (36%-188%). These results suggest that the prevalence of transatlantic wintertime clear-air turbulence will increase significantly in all aviation-relevant strength categories as the climate changes.

Atmospheric turbulence causes most weather-related aircraft incidents. Commercial aircraft encounter moderate-or-greater turbulence tens of thousands of times each year worldwide, injuring probably hundreds of passengers (occasionally fatally), costing airlines tens of millions of dollars and causing structural damage to planes. Clear-air turbulence is especially difficult to avoid, because it cannot be seen by pilots or detected by satellites or on-board radar. Clear-air turbulence is linked to wind shears within atmospheric jet streams, which are projected to be strengthened by anthropogenic climate change. However, the response of clear-air turbulence to projected climate change had not previously been studied. We have shown using climate model simulations that clear-air turbulence changes significantly within the transatlantic flight corridor when the concentration of carbon dioxide in the atmosphere is doubled (Williams & Joshi 2013). At cruise altitudes within 50-75°N and 10-60°W in winter, most clear-air turbulence measures show a 10-40% increase in the median strength of turbulence and a 40-170% increase in the frequency of occurrence of moderate-or-greater turbulence. Our results suggest that climate change will lead to bumpier transatlantic flights by the middle of this century. An intensification of clear-air turbulence could have important consequences for aviation. Flight paths could become more convoluted, possibly lengthening journey times and increasing fuel consumption and emissions. Turbulence limits the lifetime of airframes and wings, which may require more frequent inspections. Aviation is partly responsible for changing the climate, but our findings show for the first time how climate change could affect aviation. Reference PD Williams & MM Joshi (2013) Intensification of winter transatlantic aviation turbulence in response to climate change. Nature Climate Change 3(7), 644-648.

We report on the results of a laboratory investigation using a rotating two-layer annulus experiment, which exhibits both large-scale vortical modes and short-scale divergent modes. A sophisticated visualization method allows us to observe the flow at very high spatial and temporal resolution. The balanced long-wavelength modes appear only when the Froude number is supercritical (i.e. F > F critical ≡ π 2 /2), and are therefore consistent with generation by a baroclinic instability. The unbalanced shortwavelength modes appear locally in every single baroclinically unstable flow, providing perhaps the first direct experimental evidence that all evolving vortical flows will tend to emit freely propagating inertia-gravity waves. The short-wavelength modes also appear in certain baroclinically stable flows. We infer the generation mechanisms of the short-scale waves, both for the baroclinically unstable case in which they co-exist with a large-scale wave, and for the baroclinically stable case in which they exist alone. The two possible mechanisms considered are spontaneous adjustment of the large-scale flow, and Kelvin-Helmholtz shear instability. Short modes in the baroclinically stable regime are generated only when the Richardson number is subcritical (i.e. Ri < Ri critical ≡ 1), and are therefore consistent with generation by a Kelvin-Helmholtz instability. We calculate five indicators of shortwave generation in the baroclinically unstable regime, using data from a quasi-geostrophic numerical model of the annulus. There is excellent agreement between the spatial locations of shortwave emission observed in the laboratory, and regions in which the model Lighthill/Ford inertia-gravity wave source term is large. We infer that the short waves in the baroclinically unstable fluid are freely propagating inertia-gravity waves generated by spontaneous adjustment of the large-scale flow.