Kinetic energy spectra derived from commercial aircraft observations of horizontal wind velocities exhibit a k −5/3 wavenumber dependence on the mesoscale that merges into a k −3 dependence on the macroscale. In this study, spectral analysis is applied to evaluate the mesoscale ensemble prediction system using the convection-permitting NWP model COSMO-DE (COSMO-DE-EPS). One-dimensional wavenumber spectra of the kinetic energy are derived from zonal and meridional wind velocities, as well as from vertical velocities. Besides a general evaluation, the model spectra reveal important information about spin-up effects and effective resolution. The COSMO-DE-EPS well reproduces the spectral k −5/3 dependence of the mesoscale horizontal kinetic energy spectrum. Due to the assimilation of high-resolution meteorological observations (mainly rain radar), there is no signif cant spin-up of the model simulations within the f rst few hours after initialization. COSMO-DE-EPS features an effective resolution of a factor of about 4 to 5 of the horizontal grid spacing. This is slightly higher in comparison to other limited area models. Kinetic energy spectra derived from vertical velocities exhibit a much f atter wavenumber dependence leading to relatively large spectral energy on smaller scales. This is in good agreement with similar models and also suggested by observations of temporal variance spectra of the vertical velocity.

Under assumptions of horizontal homogeneity and isotropy, one may derive relations between rotational or divergent kinetic energy spectra and velocities along one-dimensional tracks, such as might be measured by aircraft. Two recent studies, differing in details of their implementation, have applied these relations to the Measurement of Ozone and Water Vapor by Airbus In-Service Aircraft (MOZAIC) dataset and reached different conclusions with regard to the mesoscale ratio of divergent to rotational kinetic energy. In this study the accuracy of the method is assessed using global atmospheric simulations performed with the Model for Prediction Across Scales, where the exact decomposition of the horizontal winds into divergent and rotational components may be easily computed. For data from the global simulations, the two approaches yield similar and very accurate results. Errors are largest for the divergent component on synoptic scales, which is shown to be related to a very dominant rotational mode. The errors are, in particular, sufficiently small so that the mesoscale ratio of divergent to rotational kinetic energy can be derived correctly. The proposed technique thus provides a strong observational check of model results with existing large commercial aircraft datasets. The results do, however, show a significant dependence on the height and latitude ranges considered, and the disparate conclusions drawn from previous applications to MOZAIC data may result from the use of different subsets of the data.

Recent numerical studies suggest that convective instability and latent heat release quickly amplify errors in numerical weather predictions and lead to a complete loss of predictability on scales below 100 km within a few hours. These errors then move further upscale, eventually contaminating the balanced flow and projecting on to synoptic‐scale instabilities. According to this picture, the errors have to transition from geostrophically unbalanced to balanced motion while propagating through the mesoscale. Geostrophic adjustment was suggested as the dynamical process of this transition, but so far has not been clearly identified. In the current study, an analytical framework for the geostrophic adjustment of an initial point‐like pulse of heat is developed on the basis of the linearized, hydrostatic Boussinesq equations. The heat pulse is thought to model a convective cloud or an error within the prediction of a cloud. A time‐dependent solution for both transient and balanced flow components is derived from the analytical model. The solution includes the Green's function of the problem, which enables the extension of the model to arbitrary heat sources by linear superposition. Spatial and temporal scales of the geostrophic adjustment mechanism are deduced and diagnostics are proposed that could be used to demonstrate the geostrophic adjustment process in complex numerical simulations of midlatitude convection and upscale error growth.

Research on the mesoscale kinetic energy spectrum over the past few decades has focused on finding a dynamical mechanism that gives rise to a universal spectral slope. Here we investigate the variability of the spectrum using 3 years of kilometer-resolution analyses from COSMO configured for Germany (COSMO-DE). It is shown that the mesoscale kinetic energy spectrum is highly variable in time but that a minimum in variability is found on scales around 100 km. The high variability found on the small-scale end of the spectrum (around 10 km) is positively correlated with the precipitation rate where convection is a strong source of variance. On the other hand, variability on the large-scale end (around 1000 km) is correlated with the potential vorticity, as expected for geostrophically balanced flows. Accordingly, precipitation at small scales is more highly correlated with divergent kinetic energy, and potential vorticity at large scales is more highly correlated with rotational kinetic energy. The presented findings suggest that the spectral slope and amplitude on the mesoscale range are governed by an ever-changing combination of the upscale and downscale impacts of these large- and small-scale dynamical processes rather than by a universal, intrinsically mesoscale dynamical mechanism.

The mechanism for the upscale growth of small errors through the atmospheric mesoscales has not been conclusively identified, but geostrophic adjustment in response to diabatically generated motions such as cumulus convection is a plausible candidate. In a companion paper, an analytic solution of the linearized, hydrostatic Boussinesq equations to an impulsive, localized heat source that mimics the effect of latent heating within a convective cloud on an unperturbed, rotating environment is found. Three characteristics of the solution are shown to be potentially useful for identifying the geostrophic adjustment process in numerical simulations. The predictions relate to the horizontal gravity wave speed, the Rossby number and the quantitative relationship between a precipitation anomaly and the balanced flow response (i.e. large‐scale vorticity). Here these predictions are tested in the framework of error growth experiments in idealized numerical simulations of a convective cloud field. Three different rotation rates are employed in order to identify the geostrophic adjustment mechanism and allow a quantitative comparison with the predictions of the analytic model. The gravity wave speed estimated from the simulations resembles the theoretical value and is independent of the Coriolis parameter, as predicted. The Rossby number resulting from the proposed scaling of temporal and spatial coordinates features a unique shape and the vorticity diagnostic agrees quantitatively with the analytical predictions. Based on these findings, it is proposed that upscale error growth through the atmospheric mesoscales is governed by the geostrophic adjustment process.