Comprehensive analysis of tornado statistics in comparison to earthquakes: intensity and temporal behaviour

Tornadoes and earthquakes are characterised by a high variability in their properties concerning intensity, geometric properties and temporal behaviour. Earthquakes are known for power-law behaviour in their intensity (Gutenberg–Richter law) and temporal statistics (e.g. Omori law and interevent waiting times). The observed similarity of high variability of these two phenomena motivated us to compare the statistical behaviour of tornadoes using seismological methods and quest for power-law behaviour. In general, the statistics of tornadoes show power-law behaviour partly coextensive with characteristic scales when the temporal resolution is high (10 to 60 min). These characteristic scales match with the typical diurnal behaviour of tornadoes, which is characterised by a maximum of tornado occurrences in the late afternoon hours. Furthermore, the distributions support the observation that tornadoes cluster in time. Finally, we shortly discuss a possible similar underlying structure composed of heterogeneous, coupled, interactive threshold oscillators that possibly explains the observed behaviour

Scale Dependent Analytical Investigation of the Dynamic State Index Concerning the Quasi-Geostrophic Theory

The Dynamic State Index (DSI) is a scalar diagnostic field that quantifies local deviations from a steady and adiabatic wind solution and thus indicates non-stationarity aswell as diabaticity. The DSI-concept has originally been developed through the Energy-Vorticity Theory based on the full compressible flow equations without regard to the characteristic scale-dependence of many atmospheric processes. But such scaledependent information is often of importance, and particularly so in the context of precipitation modeling: Small scale convective events are often organized in storms, clusters up to “Großwetterlagen” on the synoptic scale. Therefore, a DSI index for the quasi-geostrophic model is developed using (i) the Energy-Vorticity Theory and (ii) showing that it is asymptotically consistent with the original index for the primitive equations. In the last part, using meteorological reanalysis data it is demonstrated on a case study that both indices capture systematically different scale-dependent precipitation information. A spin-off of the asymptotic analysis is a novel non-equilibrium time scale combining potential vorticity and the DSI indices

The Dynamic State Index with Moisture and Phase Changes

The dynamic state index (DSI) is a scalar field that combines variational information on the total energy and enstrophy of a flow field with the second law of thermodynamics. Its magnitude is a combined local measure for non-stationarity, diabaticity, and dissipation in the flow, and it has been shown to provide good qualitative indications for the onset and presence of precipitation and the organization of storms. The index has been derived thus far for ideal fluid models only, however, so that one may expect improved and quantitative insights from a revised definition of the quantity that includes more complex aerothermodynamics. The present paper suggests definitions of the DSI for flows of moist air with phase changes and precipitation

Process-oriented statistical-dynamical evaluation of LM precipitation forecasts

International audienceThe objective of this study is the scale dependent evaluation of precipitation forecasts of the Lokal-Modell (LM) from the German Weather Service in relation to dynamical and cloud parameters. For this purpose the newly designed Dynamic State Index (DSI) is correlated with clouds and precipitation. The DSI quantitatively describes the deviation and relative distance from a stationary and adiabatic solution of the primitive equations. A case study and statistical analysis of clouds and precipitation demonstrates the availability of the DSI as a dynamical threshold parameter. This confirms the importance of imbalances of the atmospheric flow field, which dynamically induce the generation of rainfall

From Metastable to Coherent Sets – time-discretization schemes

Given a time-dependent stochastic process with trajectories x(t) in a space \Omega, there may be sets such that the corresponding trajectories only very rarely cross the boundaries of these sets. We can analyze such a process in terms of metastability or coherence. Metastable sets M are defined in space M \subset \Omega, coherent sets M(t) \subset \Omega are defined in space and time. Hence, if we extend the space \Omega by the time-variable t, coherent sets are metastable sets in \Omega \times [0,\infty). This relation can be exploited, because there already exist spectral algorithms for the identification of metastable sets. In this article we show that these well-established spectral algorithms (like PCCA+) also identify coherent sets of non-autonomous dynamical systems. For the identification of coherent sets, one has to compute a discretization (a matrix T) of the transfer operator of the process using a space-time-discretization scheme. The article gives an overview about different time-discretization schemes and shows their applicability in two different fields of application

Symmetry Analysis of Barotropic Potential Vorticity Equation

Recently F. Huang [Commun. Theor. Phys. V.42 (2004) 903] and X. Tang and P.K. Shukla [Commun. Theor. Phys. V.49 (2008) 229] investigated symmetry properties of the barotropic potential vorticity equation without forcing and dissipation on the beta-plane. This equation is governed by two dimensionless parameters, $F$ and $\beta$, representing the ratio of the characteristic length scale to the Rossby radius of deformation and the variation of earth' angular rotation, respectively. In the present paper it is shown that in the case $F\ne 0$ there exists a well-defined point transformation to set $\beta = 0$. The classification of one- and two-dimensional Lie subalgebras of the Lie symmetry algebra of the potential vorticity equation is given for the parameter combination $F\ne 0$ and $\beta = 0$. Based upon this classification, distinct classes of group-invariant solutions is obtained and extended to the case $\beta \ne 0$.Comment: 6 pages, release version, added reference for section

Statistical and dynamical analyses of atmospheric blocking with an idealized point vortex model

We investigate a reduced point vortex model for a statistical and dynamical analysis of atmospheric blocking phenomena. Thereby, we consider high-over-low and omega blocking as equilibria of two and three point vortices. Based on fields of the kinematic vorticity number, two novel methods, the contour and the trapezoid method, are introduced in order to identify the vortices that form the blocking pattern as well as their local positions and circulation magnitudes. Using an instantaneous blocking index a total number of 347 blocking periods were identified in NCEP-NCAR Reanalysis data for the Euro-Atlantic region during the time period 1990-2012. This procedure provides the basis to corroborate the applicability of the point vortex model to atmospheric blocking in a statistical framework. The calculated translation speed of the point vortex systems associated with the atmospheric blocking appears to match the zonal mean velocity reasonably well. This model explains the stationary behaviour of blocking patterns. A comparison between the theoretical and a statistical model further reveals that the circulation of the blocking high follows the principles of the point vortex model to a large extent. However, the low-pressure-systems behave more variable. Moreover, the stability of point vortex equilibria is analysed regarding the relative distances by considering linear stability analysis and simulations. This reveals, that the point vortex blocking model corresponds to an unstable saddle point. Also, a possible transition between high-overlow and omega blocking situations is indicated. Furthermore, we take viscosity and a Brownian motion into account to simulate the influence of the smaller, subgrid-scale disturbances. As a result a clustering near the equilibrium state emerges indicating the persistence of the atmospheric blocking pattern

Low-order point vortex models of atmospheric blocking

Conceptual models of blocking structures are constructed by reducing the two-dimensional atmospheric vorticity ®eld to a few point vortices. The ¯ow is assumed to be barotropic and divergence-free, and a blocking event is represented by a point vortex dipole. The focus is here on the motion of the blocking dipole under the in¯uence of the zonal mean ¯ow. This is modelled in three different ways: A dipole embedded in a latitude-dependent zonal mean ¯ow exhibits neutrally stable oscillations; their period is estimated analytically. A cyclonic point vortex approaching from upstream can either pass the dipole or break it up, so that an -shaped pattern of three vortices emerges. The stationarity of a blocking between two troughs is modelled by four point vortices. These low-order point vortex models are compared with the dynamics of real blockings in case studies. Despite their high degree of simpli®cation, those models reproduce the kinematics of blocking events properly. This results from the discretization of the ¯ow to its actual physical states, the vortices, in contrast to the common, purely mathematical discretization to grid points. Thus, point vortex dynamics are proposed to be a powerful completion of continuous ¯uid dynamics in explaining blocking events

Large-scale secondary circulations in a limited area model - the impact of lateral boundaries and resolution

Within their domain, regional climate and weather forecasting models deviate from the driving data. Small-scale deviations are a desired effect of adding regional details. There are, however, also deviations of the large-scale circulation, which can be caused by orographic effects and depend on the large-scale flow condition. These ‘secondary circulations’ (SCs) are confined to the model domain due to the prescribed boundary conditions. Here, the impact of different regional model configurations on the SC is analysed in a case study for the European region using an ensemble approach. It is shown that at 500 hPa, vortices of the SC have diameters on the order of several thousand kilometres and are related to wind speed anomalies of more than 5 m/s and geopotential height anomalies of more than 5 dam. The spatial structure and the amplitude of the SC strongly depend on the location of the lateral boundaries. The impact of the boundary location on the anomalies is on the same order of magnitude as the anomalies themselves. The resolution of the regional model, as well as the application of spectral nudging and a smoothed topography, affects mainly the amplitude of the SC, but not the spatial structure