Stochastic physical models for wind fields and precipitation extremes

Abstract

A major goal of this thesis is to introduce stochastic, physically consistent models for precipitation extremes based on the moisture budget. The moisture budget describes the moisture flux convergence and is essential for the generation of precipitation and in particular extreme precipitation. The introduced models are used to extensively study to which extent the budget equation can account for characteristics of precipitation extremes. An important question in this respect is under which conditions the budget equation generates a heavy-tailed behavior. A further point is to understand whether the spatial structure of the humidity transport is essential in generating precipitation extremes. It is demonstrated that the humidity budget equation does not allow for the emergence of heavy-tailed precipitation distributions from light-tailed distribution for wind and humidity. At the same time finite sample approximations of the models suggest that asymptotic properties may be of very limited practical relevance. The models considered here show a remarkable stability to the correlation of wind and humidity. We prove the convergence of a precipitation model to its max-stable limit, which yields asymptotic spatial independence of precipitation extremes. Further, there is no prominent difference between precipitation extremes in purely rotational or purely divergent flow. The budget equation reveals a strong sensitivity to the marginal distributions of wind and humidity and further assumptions, which shows the need for well-established distributional assumptions for these variables.<br /> In order to model moisture flux convergence spatially consistent a multivariate Gaussian random field formulation is introduced. It represents the differential relations of a wind field and related variables such as the streamfunction, velocity potential, vorticity, and divergence. The covariance model of the Gaussian random field is based on a flexible bivariate Mat´ern covariance function for the streamfunction and velocity potential. It allows for different variances in the potentials, nonzero correlations between them, anisotropy, and a flexible smoothness parameter. The joint covariance function of the related variables is derived analytically. Further, it is shown that a consistent model with nonzero correlations between the potentials and positive definite covariance function is possible, rebutting a claim of Obukhov (1954). The statistical model is fitted to forecasts of the horizontal wind fields of a mesoscale numerical weather prediction system. Parameter uncertainty is assessed by a parametric bootstrap method. The estimates reveal only physically negligible correlations between the potentials. The covariance model provides opportunity for a wealth of applications in data assimilation.