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.

[1] In the middle atmosphere, solar thermal tides cause large variations in the background conditions for gravity-wave propagation. The induced modulation of gravity-wave pseudo-momentum fluxes is responsible for a diurnal force. In past studies, this forcing was derived from gravity-wave parameterizations which neglect time-dependence and horizontal inhomogeneities of the background flow. In our study, we evaluate these assumptions using a highly simplified gravity-wave ensemble. With the help of a global ray-tracing model, a small number of different gravity-wave fields is transported through a time-changing background which is composed of a climatological mean and tidal fields from a general circulation model. Within three off-line experiments, assumptions on horizontal and temporal dependence of the background conditions have been successively omitted. Time-dependence leads to a modulation of gravity-wave observed frequencies and its phase velocities. Transient critical layers disappear. The amplitude of the diurnal forcing is reduced. Horizontal inhomogeneities induce a refraction of the gravity waves into the jet stream cores. Horizontal propagation can lead to large meridional displacements and an inter-hemispheric exchange of gravity-wave energy. With equivalent Rayleigh friction coefficients, it is shown that for the gravity-wave ensemble in use the damping of tidal amplitudes is reduced when horizontal and time dependence of tidal background conditions are taken into account.Citation: Senf, F., and U. Achatz (2011), On the impact of middle-atmosphere thermal tides on the propagation and dissipation of gravity waves,

Motivated by the useful new insights from optimal-perturbation theory into the onset of turbulence in other fields singular vectors ͑SVs͒ in stable and unstable gravity waves have been determined within the framework of the Boussinesq equations on an f plane. The difference between the dynamics of normal modes ͑NMs͒ and SV is characterized by a time invariance in the comparative role of the various possible exchange processes between NM and basic wave, while SV can have a highly time-dependent structure, allowing a more efficient energy exchange over a finite time. Both inertia-gravity waves ͑IGWs͒ and high-frequency gravity waves ͑HGWs͒ have been considered. At Reynolds numbers typical for the middle to upper mesosphere IGW admit rapid nonmodal growth even when no unstable NMs exist. SV energy growth within one Brunt-Vaisala period can cover two orders of magnitude, suggesting the possibility of turbulence onset under conditions where this would not be predicted by a NM analysis. HGWs show a dependence of short-term optimal growth on the direction of propagation of the perturbation with respect to the wave which is, at weak to moderate wave amplitudes, quite different from that of NM but reproduced in ensemble integrations from random initial perturbations. Their SVs are sharply peaked pulses with negligible group velocity which are repeatedly excited as the rapidly propagating wave passes over them. The transition of these to the leading NM, which is not moving with respect to the wave and which is typically broader in structure, is very slow, so that in many cases the turbulence onset via local perturbations of a gravity wave might be more appropriately described using optimal-perturbation theory. This might contribute to a better understanding of the often observed occurrence of thin turbulent layers in the middle atmosphere.

Ogura and Phillips derived the original anelastic model through systematic formal asymptotics using the flow Mach number as the expansion parameter. To arrive at a reduced model that would simultaneously represent internal gravity waves and the effects of advection on the same time scale, they had to adopt a distinguished limit requiring that the dimensionless stability of the background state be on the order of the Mach number squared. For typical flow Mach numbers of M ; 1 /30, this amounts to total variations of potential temperature across the troposphere of less than one Kelvin (i.e., to unrealistically weak stratification). Various generalizations of the original anelastic model have been proposed to remedy this issue. Later, Durran proposed the pseudoincompressible model following the same goals, but via a somewhat different route of argumentation. The present paper provides a scale analysis showing that the regime of validity of two of these extended models covers stratification strengths on the order of (h sc /u)du/dz , M 2/3 , which corresponds to realistic variations of potential temperature u across the pressure scale height h sc of Duj h sc 0 , 30 K. Specifically, it is shown that (i) for (h sc /u)du/dz , M m with 0 , m , 2, the atmosphere features three asymptotically distinct time scales, namely, those of advection, internal gravity waves, and sound waves; (ii) within this range of stratifications, the structures and frequencies of the linearized internal wave modes of the compressible, anelastic, and pseudoincompressible models agree up to the order of M m ; and (iii) if m , 2 /3, the accumulated phase differences of internal waves remain asymptotically small even over the long advective time scale. The argument is completed by observing that the three models agree with respect to the advective nonlinearities and that all other nonlinear terms are of higher order in M.

We consider a quasi-linear theory for the acceleration rates and propagation parameters of charged test particles in weakly turbulent electromagnetic plasmas. The similarity between two recent approaches to modelling of therandom electromagnetic field is demonstrated. It is shown that both the concept of dynamical magnetic turbulence and the concept of superposition of individual plasma modes lead to particle Fokker—Planck coefficients in which the sharp delta functions describing the resonant interaction of the particles have to be replaced by Breit—Wigner-type resonance functions, which are controlled by the dynamical turbulence decay time and the wave-damping time respectively. The resulting resonance broadening will significantly change the evaluation of cosmic-ray transport parameters.

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With the aim of contributing to the improvement of subgrid-scale gravity wave (GW) parameterizations in numerical weather prediction and climate models, the comparative relevance in GW drag of direct GW–mean flow interactions and turbulent wave breakdown are investigated. Of equal interest is how well Wentzel–Kramer–Brillouin (WKB) theory can capture direct wave–mean flow interactions that are excluded by applying the steady-state approximation. WKB is implemented in a very efficient Lagrangian ray-tracing approach that considers wave-action density in phase space, thereby avoiding numerical instabilities due to caustics. It is supplemented by a simple wave-breaking scheme based on a static-instability saturation criterion. Idealized test cases of horizontally homogeneous GW packets are considered where wave-resolving large-eddy simulations (LESs) provide the reference. In all of these cases, the WKB simulations including direct GW–mean flow interactions already reproduce the LES data to a good accuracy without a wave-breaking scheme. The latter scheme provides a next-order correction that is useful for fully capturing the total energy balance between wave and mean flow. Moreover, a steady-state WKB implementation as used in present GW parameterizations where turbulence provides by the noninteraction paradigm, the only possibility to affect the mean flow, is much less able to yield reliable results. The GW energy is damped too strongly and induces an oversimplified mean flow. This argues for WKB approaches to GW parameterization that take wave transience into account.

The dynamics of internal gravity waves is modelled using Wentzel–Kramer–Brillouin (WKB) theory in position–wave number phase space. A transport equation for the phase‐space wave‐action density is derived for describing one‐dimensional wave fields in a background with height‐dependent stratification and height‐ and time‐dependent horizontal‐mean horizontal wind, where the mean wind is coupled to the waves through the divergence of the mean vertical flux of horizontal momentum associated with the waves. The phase‐space approach bypasses the caustics problem that occurs in WKB ray‐tracing models when the wave number becomes a multivalued function of position, such as in the case of a wave packet encountering a reflecting jet or in the presence of a time‐dependent background flow. Two numerical models were developed to solve the coupled equations for the wave‐action density and horizontal mean wind: an Eulerian model using a finite‐volume method and a Lagrangian ‘phase‐space ray tracer’ that transports wave‐action density along phase‐space paths determined by the classical WKB ray equations for position and wave number. The models are used to simulate the upward propagation of a Gaussian wave packet through a variable stratification, a wind jet and the mean flow induced by the waves. Results from the WKB models are in good agreement with simulations using a weakly nonlinear wave‐resolving model, as well as with a fully nonlinear large‐eddy‐simulation model. The work is a step toward more realistic parametrizations of atmospheric gravity waves in weather and climate models.