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George C. Craig

Abstract

The three dimensional semigeostrophic equations are derived by a formal asymptotic analysis. The equations are scaled with typical values of the gradients across absolute momentum contours along isentropic surfaces and are accurate to first order in a small parameter that can be regarded as a modified Rossby number.

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George C. Craig and Andreas Dörnbrack

Abstract

Systematic numerical experiments were conducted to determine the spatial resolution required to resolve a moist thermal show convergence at a scale proportional to the smaller of the initial thermal diameter D 0 and a buoyancy length scale L buoy. The buoyancy length scale L buoy = ΔT 0/ΔΓ (ΔT 0 is the initial buoyancy excess of the thermal and ΔΓ is the ambient stratification) describes the maximum vertical displacement that can be induced against the stratification in the environment by buoyancy-driven pressure perturbations in the cloud and, thus, the maximum scale of eddies that cross the cloud boundary. For typical atmospheric conditions in which the cloud size D 0 is larger than L buoy, numerical simulations of the mixing processes in cumulus clouds must resolve L buoy.

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Heiner Lange and George C. Craig

Abstract

An idealized convective test bed for the local ensemble transform Kalman filter (LETKF) is set up to perform storm-scale data assimilation of simulated Doppler radar observations. Convective systems with lifetimes exceeding 6 h are triggered in a doubly periodic domain. Perfect-model experiments are used to investigate the limited predictability in precipitation forecasts by comparing analysis schemes that resolve different length scales. Starting from a high-resolution reference scheme with 8-km covariance localization and observations with 2-km resolution on a 5-min cycle, an experimental hierarchy is set up by successively choosing a larger covariance localization radius of 32 km, observations that are horizontally averaged by a factor of 4, a coarser resolution in the calculation of the analysis weights, and a cycling interval of 20 min. After 3 h of assimilation, the high-resolution analysis scheme is clearly superior to the configurations with coarser scales in terms of RMS error and field-oriented measures. The difference is associated with the observation resolution and a larger localization radius required for filter convergence with coarse observations. The high-resolution analysis leads to better forecasts for the first hour, but after 3 hours, the forecast quality of the schemes is indistinguishable. The more rapid error growth in forecasts from the high-resolution analysis appears to be associated with a limited predictability of the small scales, but also with gravity wave noise and spurious convective cells. The latter suggests that the field is in some sense less balanced, or less consistent with the model dynamics, than in the coarser-resolution analysis.

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John Thuburn and George C. Craig

Abstract

Earlier theoretical and modeling work introduced the concept of a radiative constraint relating tropopause height to tropospheric lapse rate and other factors such as surface temperature. Here a minimal quantitative model for the radiative constraint is presented and used to illustrate the essential physics underlying the radiative constraint, which involves the approximate balance between absorption and emission of thermal infrared (IR) radiation determining tropopause temperature.

The results of the minimal model are then extended in two ways. First, the effects of including a more realistic treatment of IR radiation are quantified. Second, the radiative constraint model is extended to take into account non-IR warming processes such as solar heating and dynamical warming near the tropopause. The sensitivity of tropopause height to non-IR warming is estimated to be a few kilometers per K day−1, with positive warming leading to a lower tropopause. Sensitivities comparable to this are found in GCM experiments in which imposed changes in the ozone distribution or in the driving of the stratospheric residual mean meridional circulation lead to changes in tropopause height. In the Tropics the influence of the stratospheric circulation is found to extend down at least as far as the main convective outflow level, some 5 km below the temperature minimum.

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John Thuburn and George C. Craig

Abstract

The sensitivity of the tropopause height to various external parameters has been investigated using a global circulation model (GCM). The tropopause height was found to be strongly sensitive to the temperature at the earth’s surface, less sensitive to the ozone distribution, and hardly sensitive at all to moderate changes in the earth’s rotation rate. The strong sensitivity to surface temperature occurs through changes in the atmospheric moisture distribution and its resulting radiative effects. The radiative and dynamical mechanisms thought to maintain the tropopause height have been investigated in some detail. The assumption that the lower stratosphere is close to radiative equilibrium leads to an easily computed relationship between tropospheric lapse rate and tropopause height. This relationship was found to hold well in the GCM in the extratropics away from the winter pole. Possible reasons for the breakdown of the relationship in the Tropics and near the winter pole are discussed. Simple relationships predicted by two different baroclinic adjustment theories, between parameters such as potential temperature gradients, the Coriolis parameter, and tropopause height, were examined. When some of these parameters were changed explicitly in GCM experiments, the remaining parameters, determined internally by the GCM, did not respond in the predicted way. These results cast doubt on the relevance of baroclinic adjustment to the height of the tropopause.

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Christian Keil and George C. Craig

Abstract

A field verification measure for precipitation forecasts is presented that combines distance and amplitude errors. It is based on an optical flow algorithm that defines a vector field that deforms, or morphs, one image to match another. When the forecast field is morphed to match the observation field, then for any point in the observation field, the magnitude of the displacement vector gives the distance to the corresponding forecast object (if any), while the difference between the observation and the morphed forecast is the amplitude error. Similarly, morphing the observation field onto the forecast field gives displacement and amplitude errors for forecast features. If observed and forecast features are separated by more than a prescribed maximum search distance, they are not matched to each other, but they are considered to be two separate amplitude errors: a missed event and a false alarm. The displacement and amplitude error components are combined to produce a displacement and amplitude score (DAS). The two components are weighted according to the principle that a displacement error equal to the maximum search distance is equivalent to the amplitude error that would be obtained by a forecast and an observed feature that are too far apart to be matched. The new score, DAS, is applied to the idealized and observed test cases of the Spatial Verification Methods Intercomparison Project (ICP) and is found to accurately measure displacement errors and quantify combined displacement and amplitude errors reasonably well, although with some limitations due to the inability of the image matcher to perfectly match complex fields.

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Kirstin Kober and George C. Craig

Abstract

Stochastic perturbations allow for the representation of small-scale variability due to unresolved physical processes. However, the properties of this variability depend on model resolution and weather regime. A physically based method is presented for introducing stochastic perturbations into kilometer-scale atmospheric models that explicitly account for these dependencies. The amplitude of the perturbations is based on information obtained from the model’s subgrid turbulence parameterization, while the spatial and temporal correlations are based on physical length and time scales of the turbulent motions. The stochastic perturbations lead to triggering of additional convective cells and improved precipitation amounts in simulations of two days with weak synoptic forcing of convection but different amounts of precipitation. The perturbations had little impact in a third case study, where precipitation was mainly associated with a cold front. In contrast, an unphysical version of the scheme with constant perturbation amplitude performed poorly since there was no perturbation amplitude that would give improved amounts of precipitation during the day without generating spurious convection at other times.

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Tobias Selz and George C. Craig

Abstract

The growth of small-amplitude, spatially uncorrelated perturbations has been studied in a weather forecast of a 4-day period in the summer of 2007, using a large domain covering Europe and the eastern Atlantic and with explicitly resolved deep convection. The error growth follows the three-stage conceptual model of Zhang et al., with rapid initial growth (e-folding time about 0.5 h) on all scales, relaxing over about 20 h to a slow growth of the large-scale perturbations (e-folding time 12 h). The initial growth was confined to precipitating regions, with a faster growth rate where conditional instability was large. Growth in these regions saturated within 3–10 h, continuing for the longest where the precipitation rate was large. While the initial growth was mainly in the divergent part of the flow, the eventual slow growth on large scales was more in the rotational component.

Spectral decomposition of the disturbance energy showed that the rapid growth in precipitating regions projected onto all Fourier components; however, the amplitude at saturation was too small to initiate the subsequent large-scale growth. Visualization of the disturbance energy showed it to expand outward from the precipitating regions at a speed corresponding to a deep tropospheric gravity wave. These results suggest a physical picture of error growth with a rapidly growing disturbance to the vertical mass transport in precipitating regions that spreads to the radius of deformation while undergoing geostrophic adjustment, eventually creating a balanced perturbation that continues to grow through baroclinic instability.

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Christian Keil and George C. Craig

Abstract

Errors in regional forecasts often take the form of phase errors, where a forecasted weather system is displaced in space or time. For such errors, a direct measure of the displacement is likely to be more valuable than traditional measures. A novel forecast quality measure is proposed that is based on a comparison of observed and forecast satellite imagery from the Meteosat-7 geostationary satellite. The measure combines the magnitude of a displacement vector calculated with a pyramid matching algorithm and the local squared difference of observed and morphed forecast brightness temperature fields. Following the description of the method and its application for a simplified case, the measure is applied to regional ensemble forecasts for an episode of prefrontal summertime convection in Bavaria. It is shown that this new method provides a plausible measure of forecast error, which is consistent with a subjective ranking of ensemble members for a sample forecast. The measure is then applied to hourly images over a 36-h forecast period and compared with the bias and equitable threat score. The two conventional measures fail to provide any systematic distinction between different ensemble members, while the new measure identifies ensemble members of differing skill levels with a strong degree of temporal consistency. Using the displacement-based error measure, individual ensemble members are found to compare better with observations than either a short-term deterministic forecast or the ensemble mean throughout the convective period.

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Stephan Rasp, Tobias Selz, and George C. Craig

Abstract

The statistical theory of convective variability developed by Craig and Cohen in 2006 has provided a promising foundation for the design of stochastic parameterizations. The simplifying assumptions of this theory, however, were made with tropical equilibrium convection in mind. This study investigates the predictions of the statistical theory in real-weather case studies of nonequilibrium summertime convection over land. For this purpose, a convection-permitting ensemble is used in which all members share the same large-scale weather conditions but the convection is displaced using stochastic boundary layer perturbations. The results show that the standard deviation of the domain-integrated mass flux is proportional to the square root of its mean over a wide range of scales. This confirms the general applicability and scale adaptivity of the Craig and Cohen theory for complex weather. However, clouds tend to cluster on scales of around 100 km, particularly in the morning and evening. This strongly impacts the theoretical predictions of the variability, which does not include clustering. Furthermore, the mass flux per cloud closely follows an exponential distribution if all clouds are considered together and if overlapping cloud objects are separated. The nonseparated cloud mass flux distribution resembles a power law. These findings support the use of the theory for stochastic parameterizations but also highlight areas for improvement.

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