On the relevance of rotational and divergent modes of motion to mesoscale dynamics and upscale error growth

The atmospheric mesoscales encompass dynamical and thermodynamical processes that are characterized by length scales between a few and a couple of hundred kilometers and temporal scales of several minutes to one day. These processes are associated with mid-latitudinal weather and their skillful prediction is a major aim of meteorological research. In there, two fundamental issues arise: first, there is no consensus about the principal dynamical agent on the mesoscales that gives rise to the observed kinetic energy spectrum. Second, a dominant scale-interaction mechanism that governs the growth of initially small-scale errors to large scales remains undetermined. This thesis contributes to an improved understanding of these two aspects with an emphasis on the relevance of rotational and divergent modes of motion and their interplay. An important observational test of theoretical studies regarding the horizontal mesoscale kinetic energy spectrum is decoupling its rotational and divergent constituents from one-dimensional atmospheric wind measurements. Such a one-dimensional Helmholtz-decomposition method was recently suggested. The first part of this dissertation addresses the validity of the strong mathematical homogeneity and isotropy assumptions underlying this method. To that end, new high-resolution global atmospheric numerical simulations are employed. Rotational and divergent modes of motion are derived by applying the one-dimensional Helmholtz-decomposition method to one-dimensional transects of the horizontal wind field. The results are then compared to the divergent and rotational components obtained from the unambiguous decomposition of the two-dimensional wind field. The mathematical assumptions are found to be fulfilled such that the mesoscale ratio of divergent to rotational kinetic energy can be derived correctly with the one-dimensional Helmholtz-decomposition method. The results suggest a significant dependence of the horizontal divergent- and rotational kinetic energy spectra on the considered height- and latitude ranges. This finding points to the non-universality of the dynamics governing the mesoscale kinetic energy spectrum. Recent studies suggest that small-scale errors in numerical weather predictions quickly amplify through the convective instability and the release of latent heat of condensation within clouds. These errors then propagate to larger scales, whereby their dynamics transition from significantly divergent to mainly rotational. The second part of this dissertation explores the possibility that geostrophic adjustment following deep moist convection is the dominant dynamical process governing this transition with an analytical- and a numerical approach. An analytical framework for the geostrophic adjustment of an initial point-like pulse of heat (representing the error within the prediction of a cloud) is developed based on the linearized, hydrostatic Boussinesq-equations. The solution includes the Green's function of the problem and contains the full temporal evolution of all transient and balanced flow components. Characteristic spatial and temporal scales of the geostrophic adjustment mechanism are deduced and three diagnostics that can be used to identify this process in numerical simulations are proposed. These predictions are then tested in the framework of error growth experiments in highly idealized numerical simulations of a convective cloud field in a rotating environment. The error growth characteristics feature a high level of agreement with the analytical predictions. The results of this thesis suggest that the geostrophic adjustment following convective heating governs upscale error growth through the atmospheric mesoscales.