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While it is known that trapped lee waves propagating at low levels in a stratified atmosphere exert a drag on the mountains that generate them, the distribution of the corresponding reaction force exerted on the atmospheric mean circulation, defined by the wave momentum flux profiles, has not been established, because for inviscid trapped lee waves these profiles oscillate indefinitely downstream. A framework is developed here for the unambiguous calculation of momentum flux profiles produced by trapped lee waves, which circumvents the difficulties plaguing the inviscid trapped lee wave theory. Using linear theory, and taking Scorer's two-layer atmosphere as an example, the waves are assumed to be subject to a small dissipation, expressed as a Rayleigh damping. The resulting wave pattern decays downstream, so the momentum flux profile integrated over the area occupied by the waves converges to a well-defined form. Remarkably, for weak dissipation, this form is independent of the value of Rayleigh damping coefficient, and the inviscid drag, determined in previous studies, is recovered as the momentum flux at the surface. The divergence of this momentum flux profile accounts for the areally-integrated drag exerted by the waves on the atmosphere. The application of this framework to this and other types of trapped lee waves potentially enables the development of physically-based parametrizations of the effects of trapped lee waves on the atmosphere., 24 pages, 5 figures; submitted to Quarterly Journal of the Royal Meteorological Society

In a recent methodological paper, we showed how to learn chaotic dynamics along with the state trajectory from sequentially acquired observations, using local ensemble Kalman filters. Here, we more systematically investigate the possibility to use a local ensemble Kalman filter with either covariance localisation or local domains, in order to retrieve the state and a mix of key global and local parameters. Global parameters are meant to represent the surrogate dynamical core, for instance through a neural network, which is reminiscent of data-driven machine learning of dynamics, while the local parameters typically stand for the forcings of the model. Aiming at joint state and parameter estimation, a family of algorithms for covariance and local domain localisation is proposed. In particular, we show how to rigorously update global parameters using a local domain ensemble Kalman filter (EnKF) such as the local ensemble transform Kalman filter (LETKF), an inherently local method. The approach is tested with success on the 40-variable Lorenz model using several of the local EnKF flavors. A two-dimensional illustration based on a multi-layer Lorenz model is finally provided. It uses radiance-like non-local observations. It features both local domains and covariance localisation in order to learn the chaotic dynamics and the local forcings. This paper more generally addresses the key question of online estimation of both global and local model parameters.

Extreme Precipitation Events (EPEs) can have devastating consequences such as floods and landslides, posing a great threat to society and the economy. Predicting such events long in advance can support the mitigation of negative impacts. Here, we focus on EPEs over the Mediterranean, a region that is frequently affected by such hazards. Previous work identified strong connections between localized EPEs and large-scale atmospheric flow patterns, affecting the weather over the entire Mediterranean. We analyze the predictive skill of these patterns in the ECMWF extended-range forecasts and assess if and where these patterns can be used for indirect predictions of EPEs, using the Brier Skill Score. The results show that the ECMWF model provides skillful predictions of the Mediterranean patterns up to 2 weeks in advance. Moreover, using the forecasted patterns for indirect predictability of EPEs outperforms the reference score up to about 10 days lead time for many locations. Especially for high orography locations or coastal areas, like parts of western Turkey, western Balkans, Iberian Peninsula and Morocco this limit extends from 11 to 14 days lead time. This study demonstrates that connections between localized EPEs and large-scale patterns over the Mediterranean extend the forecasting horizon of the model by over 3 days in many locations, in comparison to forecasting based on the predicted precipitation. Thus, it is beneficial to use the predicted patterns rather than the predicted precipitation at longer lead times for EPEs forecasting. The model's performance is also assessed from a user perspective, showing that the EPEs forecasting based on the patterns increases the economic benefits at medium and extended range lead times. Such information could support higher confidence in the decision-making of various users, e.g., the agricultural sector and (re)insurance companies., Manuscript submitted for review to the Quarterly Journal of the Royal Meteorological Society

The distribution of liquid water in ice-free clouds determines their radiative properties, a significant source of uncertainty in weather and climate models. Evaporation and turbulent mixing cause a cloud to display large variations in droplet-number density, but quite small variations in droplet size [Beals et al. (2015)]. Yet direct numerical simulations of the joint effect of evaporation and mixing near the cloud edge predict quite different behaviors, and it remains an open question how to reconcile these results with the experimental findings. To infer the history of mixing and evaporation from observational snapshots of droplets in clouds is challenging because clouds are transient systems. We formulated a statistical model that provides a reliable description of the evaporation-mixing process as seen in direct numerical simulations, and allows to infer important aspects of the history of observed droplet populations, highlighting the key mechanisms at work, and explaining the differences between observations and simulations., Comment: 18 pages, 5 figures, 2 tables, supplementary material

This data assimilation study exploits infrasound from explosions to probe an atmospheric wind component from the ground up to stratospheric altitudes. Planned explosions of old ammunition in Finland generate transient infrasound waves that travel through the atmosphere. These waves are partially reflected back towards the ground from stratospheric levels, and are detected at a receiver station located in northern Norway at 178 km almost due North from the explosion site. The difference between the true horizontal direction towards the source and the backazimuth direction (the horizontal direction of arrival) of the incoming infrasound wave-fronts, in combination with the pulse propagation time, are exploited to provide an estimate of the average cross-wind component in the penetrated atmosphere. We perform offline assimilation experiments with an ensemble Kalman filter and these observations, using the ERA5 ensemble reanalysis atmospheric product as background (prior) for the wind at different vertical levels. We demonstrate that information from both sources can be combined to obtain analysis (posterior) estimates of cross-winds at different vertical levels of the atmospheric slice between the explosion site and the recording station. The assimilation makes greatest impact at the 12-60 km levels, with some changes with respect to the prior of the order of 0.1-1.0 m/s, which is a magnitude larger than the typical standard deviation of the ERA5 background. The reduction of background variance in the higher levels often reached 2-5%. This is the first published study demonstrating techniques to implement assimilation of infrasound data into atmospheric models. It paves the way for further exploration in the use of infrasound observations - especially natural and continuous sources - to probe the middle atmospheric dynamics and to assimilate these data into atmospheric model products., Comment: Manuscript submitted to the Quarterly Journal of the Royal Meteorological Society. The current document is an e-print typeset by the authors

We describe and illustrate a mechanism whereby convective aggregation and eastward propagating equatorial disturbances, similar in some respects to the Madden--Julian oscillation, arise. We construct a simple, explicit system consisting only of the shallow water equations plus a humidity variable; moisture enters via evaporation from a wet surface, is transported by the flow and removed by condensation, so providing a mass source to the height field. For a broad range of parameters the system is excitable and self-sustaining, even if linearly stable, with condensation producing convergence and gravity waves that, acting together, trigger more condensation. On the equatorial beta-plane the convection first aggregates near the equator, generating patterns related to those in the Matsuno--Gill problem. However, the pattern is unsteady and more convection is triggered on its eastern edge, leading to a precipitating disturbance that progresses eastward. The effect is enhanced by westward prevailing winds that increase the evaporation east of the disturbance. The pattern is confined to a region within a few deformation radii of equator because here the convection can best create the convergence needed to organize into a self-sustaining pattern. Formation of the disturbance preferentially occurs where the surface is warmer and sufficient time (a few tens of days) must pass before conditions arise that enable the disturbance to reform, as is characteristic both of excitable systems and the MJO itself. The speed of the disturbance depends on the efficiency of evaporation and the heat released by condensation, and is typically a few meters per second, much less than the Kelvin wave speed., Comment: 20 pages, 7 figure

Incremental 4D-Var is a data assimilation algorithm used routinely at operational numerical weather predictions centers worldwide.This paper implements a new method for parallelizing incremental 4D-Var, the Randomized Incremental Optimal Technique (RIOT), which replaces the traditional sequential conjugate gradient (CG) iterations in the inner-loop of the minimization with fully parallel randomized singular value decomposition (RSVD) of the preconditioned Hessian of the cost function. RIOT is tested using the standard Lorenz-96 model (L-96) as well as two realistic high-dimensional atmospheric source inversion problems based on aircraft observations of black carbon concentrations. A new outer-loop preconditioning technique tailored to RSVD was introduced to improve convergence stability and performance. Results obtained with the L-96 system show that the performance improvement from RIOT compared to standard CG algorithms increases significantly with non-linearities. Overall, in the realistic black carbon source inversion experiments, RIOT reduces the wall-time of the 4D-Var minimization by a factor 2-3, at the cost of a factor 4-10 increase in energy cost due to the large number of parallel cores used. Furthermore, RIOT enables reduction of the wall-time computation of the analysis error covariance matrix by a factor 40 compared to a standard iterative Lanczos approach. Finally, as evidenced in this study, implementation of RIOT in an operational numerical weather prediction system will require a better understanding of its convergence properties as a function of the Hessian characteristics and, in particular, the degree of freedom for signal (DOFs) of the inverse problem.

This study corrects the hypsometric equation by restoring the nontraditional terms to relax the hydrostatic approximation. The nontraditional terms include one Coriolis term and two metric terms in the vertical momentum equation. The hypsometric equations with and without the nontraditional correction are used to calculate the geopotential height of pressure levels using more than 300,000 selected tropical rawinsonde profiles. With westerlies between two pressure levels, the thickness of the layer increases, which reduces the upward pressure gradient forces to balance the upward nontraditional Coriolis forces; the opposite is true for easterlies. Hence, zonal winds are negatively correlated with traditional geopotential height biases aloft. At 500 hPa, for example, traditional geopotential height error in the tropical troposphere is on the order of at least 0.5 m, which is considerable with respect to geopotential height variability in tropical large-scale flow, on the order of 10 m to 15 m., 10 pages, 3 figures, This is the accepted version of an article that has been published in final form

\citet{turner1957} proposed that dry thermals entrain because of buoyancy (via a constraint which requires an increase in the radius $a$). This however, runs counter to the scaling arguments commonly used to derive the entrainment rate, which rely on either the self-similarity of \citet{scorer1957} or the turbulent entrainment hypothesis of \citet{morton1956}. The assumption of turbulence-driven entrainment was investigated by \citet{lecoanet2018}, who found that the entrainment efficiency $e$ varies by less than $20\%$ between laminar (Re = 630) and turbulent (Re = 6300) thermals. This motivated us to utilize Turner's argument of buoyancy-controlled entrainment in addition to the thermal's vertical momentum equation to build a model for thermal dynamics which does not invoke turbulence or self-similarity. We derive simple expressions for the thermals' kinematic properties and their fractional entrainment rate $\epsilon$ and find close quantitative agreement with the values in direct numerical simulations. In particular, our expression for entrainment rate is consistent with the parameterization $\epsilon \sim B/w^2$, for Archimedean buoyancy $B$ and vertical velocity $w$. We also directly validate the role of buoyancy-driven entrainment by running simulations where gravity is turned off midway through a thermal's rise. The entrainment efficiency $e$ is observed to drop to less than 1/3 of its original value in both the laminar and turbulent cases when $g=0$, affirming the central role of buoyancy in entrainment in dry thermals., Comment: 12 pages, 8 figures

Charging of upper and lower horizontal boundaries of extensive layer clouds results from current flow in the global electric circuit. Layer-cloud charge accumulation has previously been considered a solely electrostatic phenomenon, but it does not occur in isolation from meteorological processes, which can transport charge. Thin layer clouds provide special circumstances for investigating this dynamical charge transport, as disruption at the cloud-top may reach the cloud base, observable from the surface. Here, a thin (~300 m) persistent layer-cloud with base at 300 m and strong wind shear at cloud-top was observed to generate strongly correlated fluctuations in cloud base height, optical thickness and surface electric Potential Gradient (PG) beneath. PG changes are identified to precede the cloud base fluctuations by 2 minutes, consistent with shear-induced cloud-top electrical changes followed by cloud base changes. These observations demonstrate, for the first time, dynamically driven modification of charge within a layer-cloud. Even in weakly charged layer-clouds, redistribution of charge will modify local electric fields within the cloud and the collisional behaviour of interacting charged cloud droplets. Local field intensification may also explain previously observed electrostatic discharges in warm clouds.

Studies conducted by the UK Met Office reported significant skill at predicting the winter NAO index with their seasonal prediction system. At the same time, a very low signal-to-noise ratio was observed, as measured using the `ratio of predictable components' (RPC) metric. We analyse both the skill and signal-to-noise ratio using a new statistical toy-model which assumes NAO predictability is driven by regime dynamics. It is shown that if the system is approximately bimodal in nature, with the model consistently underestimating the level of regime persistence each season, then both the high skill and high RPC value of the Met Office hindcasts can easily be reproduced. Underestimation of regime persistence could be attributable to any number of sources of model error, including imperfect regime structure or errors in the propagation of teleconnections. In particular, a high RPC value for a seasonal mean prediction may be expected even if the models internal level of noise is realistic., Comment: Published in the Quarterly Journal of the Royal Meteorological Society (2019)

The impact of uncertainties in surface layer physics on the atmospheric general circulation is comparatively unexplored. Here the sensitivity of the zonal-mean circulation to reduced air-sea momentum roughness ($Z_{0m}$) at low flow speed is investigated with the Community Atmosphere Model (CAM3). In an aquaplanet framework with prescribed sea surface temperatures, the response to reduced $Z_{0m}$ resembles the La Ni$\tilde{\text{n}}$a minus El Ni$\tilde{\text{n}}$o response to El Ni$\tilde{\text{n}}$o Southern Oscillation variability with: i) a poleward shift of the mid-latitude westerlies extending all the way to the surface; ii) a weak poleward shift of the subtropical descent region; and iii) a weakening of the Hadley circulation, which is generally also accompanied by a poleward shift of the inter-tropical convergence zone (ITCZ) and the tropical surface easterlies. Mechanism-denial experiments show this response to be initiated by the reduction of tropical latent and sensible heat fluxes, effected by reducing $Z_{0m}$. The circulation response is elucidated by considering the effect of the tropical energy fluxes on the Hadley circulation strength, the upper tropospheric critical layer latitudes, and the lower-tropospheric baroclinic eddy forcing. The ITCZ shift is understood via moist static energy budget analysis in the tropics. The circulation response to reduced $Z_{0m}$ carries over to more complex setups with seasonal cycle, full complexity of atmosphere-ice-land-ocean interaction, and a slab ocean lower boundary condition. Hence, relatively small changes in the surface parameterization parameters can lead to a significant circulation response., 13 pages, 15 figures, accepted for publication in QJRMS

The derivation of a quasi-geostrophic (QG) system from the rotating shallow water equations on a midlatitude beta-plane coupled with moisture is presented. Condensation is prescribed to occur whenever the moisture at a point exceeds a prescribed saturation value. It is seen that a slow condensation time scale is required to obtain a consistent set of equations at leading order. Further, since the advecting wind fields are geostrophic, changes in moisture (and hence, precipitation) occur only via non-divergent mechanisms. Following observations, a saturation profile with gradients in the zonal and meridional directions is prescribed. A purely meridional gradient has the effect of slowing down the dry Rossby waves, through a reduction in the "equivalent gradient" of the background potential vorticity. A large scale unstable moist mode results on the inclusion of a zonal gradient by itself, or in conjunction with a meridional moisture gradient. For gradients that are are representative of the atmosphere, the most unstable moist mode propagates zonally in the direction of increasing moisture, matures over an intraseasonal timescale and has small phase speed., 9 pages, 8 figures, Quarterly Journal of the Royal Meteorological Society, DOI:10.1002/qj.2644, 2015

The classical approach for studying atmospheric variability is based on defining a background state and studying the linear stability of the small fluctuations around such a state. Weakly non-linear theories can be constructed using higher order expansions terms. While these methods have undoubtedly great value for elucidating the relevant physical processes, they are unable to follow the dynamics of a turbulent atmosphere. We provide a first example of extension of the classical stability analysis to a non-linearly evolving quasi-geostrophic flow. The so-called covariant Lyapunov vectors (CLVs) provide a covariant basis describing the directions of exponential expansion and decay of perturbations to the non-linear trajectory of the flow. We use such a formalism to re-examine the basic barotropic and baroclinic processes of the atmosphere with a quasi-geostrophic beta-plane two-layer model in a periodic channel driven by a forced meridional temperature gradient $\Delta T$. We explore three settings of $\Delta T$, representative of relatively weak turbulence, well-developed turbulence, and intermediate conditions. We construct the Lorenz energy cycle for each CLV describing the energy exchanges with the background state. A positive baroclinic conversion rate is a necessary but not sufficient condition of instability. Barotropic instability is present only for few very unstable CLVs for large values of $\Delta T$. Slowly growing and decaying hydrodynamic Lyapunov modes closely mirror the properties of the background flow. Following classical necessary conditions for barotropic/baroclinic instability, we find a clear relationship between the properties of the eddy fluxes of a CLV and its instability. CLVs with positive baroclinic conversion seem to form a set of modes for constructing a reduced model of the atmosphere dynamics., Comment: 50 pages, 10 figures

Variational data assimilation in continuous time is revisited. The central techniques applied in this paper are in part adopted from the theory of optimal nonlinear control. Alternatively, the investigated approach can be considered as a continuous time generalisation of what is known as weakly constrained four dimensional variational assimilation (WC--4DVAR) in the geosciences. The technique allows to assimilate trajectories in the case of partial observations and in the presence of model error. Several mathematical aspects of the approach are studied. Computationally, it amounts to solving a two point boundary value problem. For imperfect models, the trade off between small dynamical error (i.e. the trajectory obeys the model dynamics) and small observational error (i.e. the trajectory closely follows the observations) is investigated. For (nearly) perfect models, this trade off turns out to be (nearly) trivial in some sense, yet allowing for some dynamical error is shown to have positive effects even in this situation. The presented formalism is dynamical in character; no assumptions need to be made about the presence (or absence) of dynamical or observational noise, let alone about their statistics., 28 Pages, 12 Figures

Scoring rules are an important tool for evaluating the performance of probabilistic forecasting schemes. In the binary case, scoring rules (which are strictly proper) allow for a decomposition into terms related to the resolution and to the reliability of the forecast. This fact is particularly well known for the Brier Score. In this paper, this result is extended to forecasts for finite--valued targets. Both resolution and reliability are shown to have a positive effect on the score. It is demonstrated that resolution and reliability are directly related to forecast attributes which are desirable on grounds independent of the notion of scores. This finding can be considered an epistemological justification of measuring forecast quality by proper scores. A link is provided to the original work of DeGroot et al (1982), extending their concepts of sufficiency and refinement. The relation to the conjectured sharpness principle of Gneiting et al (2005a) is elucidated., v1: 9 pages; submitted to International Journal of Forecasting v2: 12 pages; Significant change of contents; stronger focus on decomposition; Extensive comments on and extensions of earlier work, in particular sufficiency

Traditionally, turbulence energetics is characterized by turbulent kinetic energy (TKE) and modelled using solely the TKE budget equation. In stable stratification, TKE is generated by the velocity shear and expended through viscous dissipation and work against buoyancy forces. The effect of stratification is characterized by the ratio of the buoyancy gradient to squared shear, called Richardson number, Ri. It is widely believed that at Ri exceeding a critical value, Ric, local shear cannot maintain turbulence, and the flow becomes laminar. We revise this concept by extending the energy analysis to turbulent potential and total energies (TPE and TTE = TKE + TPE), consider their budget equations, and conclude that TTE is a conservative parameter maintained by shear in any stratification. Hence there is no "energetics Ric", in contrast to the hydrodynamic-instability threshold, Ric-instability, whose typical values vary from 0.25 to 1. We demonstrate that this interval, 0.25>1, clarify principal difference between turbulent boundary layers and free flows, and provide basis for improving operational turbulence closure models., Comment: 23 pages, 4 figures, Quarterly Journal of Royal Meteorological Society, in press

The radar scattering properties of realistic aggregate snowflakes have been calculated using the Rayleigh-Gans theory. We find that the effect of the snowflake geometry on the scattering may be described in terms of a single universal function, which depends only on the overall shape of the aggregate and not the geometry or size of the pristine ice crystals which compose the flake. This function is well approximated by a simple analytic expression at small sizes; for larger snowflakes we fit a curve to our numerical data. We then demonstrate how this allows a characteristic snowflake radius to be derived from dual-wavelength radar measurements without knowledge of the pristine crystal size or habit, while at the same time showing that this detail is crucial to using such data to estimate ice water content. We also show that the `effective radius', characterising the ratio of particle volume to projected area, cannot be inferred from dual-wavelength radar data for aggregates. Finally, we consider the errors involved in approximating snowflakes by `air-ice spheres', and show that for small enough aggregates the predicted dual wavelength ratio typically agrees to within a few percent, provided some care is taken in choosing the radius of the sphere and the dielectric constant of the air-ice mixture; at larger sizes the radar becomes more sensitive to particle shape, and the errors associated with the sphere model are found to increase accordingly., Comment: 17 pages 9 figs submitted to Q. J. Roy. Met. Soc