Skill prediction of local weather forecasts based on the ECMWF ensemble

Ensemble Prediction has become an essential part of numerical weather forecasting. In this paper we investigate the ability of ensemble forecasts to provide an a priori estimate of the expected forecast skill. Several quantities derived from the local ensemble distribution are investigated for a two year data set of European Centre for Medium-Range Weather Forecasts (ECMWF) temperature and wind speed ensemble forecasts at 30 German stations. The results indicate that the population of the ensemble mode provides useful information for the uncertainty in temperature forecasts. The ensemble entropy is a similar good measure. This is not true for the spread if it is simply calculated as the variance of the ensemble members with respect to the ensemble mean. The number of clusters in the C regions is almost unrelated to the local skill. For wind forecasts, the results are less promising

Statistics of finite-time Lyapunov exponents in the Ulam map

The statistical properties of finite-time Lyapunov exponents at the Ulam point of the logistic map are investigated. The exact analytical expression for the autocorrelation function of one-step Lyapunov exponents is obtained, allowing the calculation of the variance of exponents computed over time intervals of length $n$. The variance anomalously decays as $1/n^2$. The probability density of finite-time exponents noticeably deviates from the Gaussian shape, decaying with exponential tails and presenting $2^{n-1}$ spikes that narrow and accumulate close to the mean value with increasing $n$. The asymptotic expression for this probability distribution function is derived. It provides an adequate smooth approximation to describe numerical histograms built for not too small $n$, where the finiteness of bin size trimmes the sharp peaks.Comment: 6 pages, 4 figures, to appear in Phys. Rev.

Analysing Lyapunov spectra of chaotic dynamical systems

It is shown that the asymptotic spectra of finite-time Lyapunov exponents of a variety of fully chaotic dynamical systems can be understood in terms of a statistical analysis. Using random matrix theory we derive numerical and in particular analytical results which provide insights into the overall behaviour of the Lyapunov exponents particularly for strange attractors. The corresponding distributions for the unstable periodic orbits are investigated for comparison.Comment: 4 pages, 4 figure

Quantification of the performance of chaotic micromixers on the basis of finite time Lyapunov exponents

Chaotic micromixers such as the staggered herringbone mixer developed by Stroock et al. allow efficient mixing of fluids even at low Reynolds number by repeated stretching and folding of the fluid interfaces. The ability of the fluid to mix well depends on the rate at which "chaotic advection" occurs in the mixer. An optimization of mixer geometries is a non trivial task which is often performed by time consuming and expensive trial and error experiments. In this paper an algorithm is presented that applies the concept of finite-time Lyapunov exponents to obtain a quantitative measure of the chaotic advection of the flow and hence the performance of micromixers. By performing lattice Boltzmann simulations of the flow inside a mixer geometry, introducing massless and non-interacting tracer particles and following their trajectories the finite time Lyapunov exponents can be calculated. The applicability of the method is demonstrated by a comparison of the improved geometrical structure of the staggered herringbone mixer with available literature data.Comment: 9 pages, 8 figure

Classification of Possible Finite-Time Singularities by Functional Renormalization

Starting from a representation of the early time evolution of a dynamical system in terms of the polynomial expression of some observable f (t) as a function of the time variable in some interval 0 < t < T, we investigate how to extrapolate/forecast in some optimal stability sense the future evolution of f(t) for time t>T. Using the functional renormalization of Yukalov and Gluzman, we offer a general classification of the possible regimes that can be defined based on the sole knowledge of the coefficients of a second-order polynomial representation of the dynamics. In particular, we investigate the conditions for the occurence of finite-time singularities from the structure of the time series, and quantify the critical time and the functional nature of the singularity when present. We also describe the regimes when a smooth extremum replaces the singularity and determine its position and amplitude. This extends previous works by (1) quantifying the stability of the functional renormalization method more accurately, (2) introducing new global constraints in terms of moments and (3) going beyond the ``mean-field'' approximation.Comment: Latex document of 18 pages + 7 ps figure

Uncertainty dynamics and predictability in chaotic systems

An initial uncertainty in the state of a chaotic system is expected to grow even under a perfect model; the dynamics of this uncertainty during the early stages of its evolution are investigated. A variety of error growth statistics are contrasted, illustrating their relative strengths when applied to chaotic systems, all within a perfect-model scenario. A procedure is introduced which can establish the existence of regions of a strange attractor within which all infinitesimal uncertainties decrease with time. It is proven that such regions exist in the Lorenz attractor, and a number of previous numerical observations are interpreted in the light of this result; similar regions of decreasing uncertainty exist in the Ikeda attractor. It is proven that no such regions exist in either the Rössler system or the Moore-Spiegel system. Numerically, strange attractors in each of these systems are observed to sample regions of state space where the Jacobians have eigenvalues with negative real parts, yet when the Jacobians are not normal matrices this does not guarantee that uncertainties will decrease. Discussions of predictability often focus on the evolution of infinitesimal uncertainties; clearly, as long as an uncertainty remains infinitesimal it cannot pose a limit to predictability. to reflect realistic boundaries, any proposed limit of predictability must be defined with respect to the nonlinear behaviour of perfect ensembles. Such limits may vary significantly with the initial state of the system, the accuracy of the observations, and the aim of the forecaster. Perfect-model analogues of operational weather forecasting ensemble schemes with finite initial uncertainties are contrasted both with perfect ensembles and uncertainty statistics based upon the dynamics infinitesimal uncertainties

Ein internes Vorhersagbarkeitsexperiment im Lorenz-Modell

The predictability of the atmosphere usually is quantified by the error doubling time or the limit of predictability. These measures are times when small errors double on average or reach a certain threshold value respectively. In this article we analyse the error doubling time in the Lorenz Model. We want to demonstrate, that averaged measures of predictability are not necessarily typical measures for the system. Predictability is a variable and it is an important task to identify which initial states lead to enhanced, or reduced, predictabilit