Limit theorems and statistical inference for solutions of some stochastic (partial) differential equations

The starting point for the thesis is an Ornstein-Uhlenbeck type stochastic differential equation dXt=(L(t)-cXt)dt+dBt with a real number value at time zero. The driving process B is a fractional Brownian motion with Hurst parameter H between 0.5 and 1, L is a bounded periodic deterministic function and c is a real valued parameter. In the thesis the task of estimation of L and c is solved when the solution process X is observed continuously. In the first part L is considered to be a linear combination of known functions with the same period, and the vector of the coefficients together with the parameter c is estimated using the least squares method for c<0, i.e. in the non-ergodic setting. We show strong consistency of the estimator as well as its asymptotic normality in the first p components. For the last component a noncentral limit theorem is proved. In the second part we consider the ergodic setting, i.e. c>0, and construct a nonparametric estimator for the function L. The idea for construction is to decompose L with respect to a known orthonormal basis and to estimate coefficients of a finite number of summands, letting this number tend to infinity afterwards. For this estimator L2 consistency is shown and its rate of convergence is calculated. The next part is dealing with a similar equation, this time driven by a Rosenblatt process, a non-Gaussian process with the covariance structure of a fractional Brownian motion with Hurst parameter H between 0.5 and 1. We consider again the setting in which L is a linear combination of known functions and c is positive, i.e. the ergodic case. We construct three estimators for the vector of parameters and study their first and second order asymptotics. The last part of the thesis deals with a different problem. There the stochastic wave equation is considered driven by a random field that is white in space and fractional in time with Hurst parameter H between 0.5 and 1. The objects we investigate are so-called empirical power variations of the solution, i.e. averages over the weighted differences of the solution process raised to some integer powers, where the weights fulfil certain conditions. We show central and noncentral limit theorems for these averages, determine in some cases the third order asymptotics in terms of the Wasserstein distance and use these results to define strongly consistent and asymptotically normal estimators for H

Impact of Diurnal Warm Layers on Atmospheric Convection

AbstractThis manuscript presents a study of oceanic diurnal warm layers (DWLs) in kilometer‐scale global coupled simulations and their impact on atmospheric convection in the tropics. With the implementation of thin vertical levels in the ocean, DWLs are directly resolved, and sea surface temperature (SST) fluctuations of up to several Kelvin appear in regions with low wind and high solar radiation. The increase of SST during the day causes an abrupt afternoon increase of atmospheric moisture due to enhanced latent heat flux (LHF), followed by an increase in cloud cover (CC) and cloud liquid water (CLW). However, although the diurnal SST amplitude is even exaggerated in comparison to reanalysis, this effect only lasts for 5–6 hr and leads to an absolute difference of 1% for CC and 0.01 kg m−2 for CLW. This can be explained by the fact that the low wind over the SST anomalies dampens their potential effect on the LHF and hence clouds. All in all, the impact of DWLs on convective CC is found to be negligible in the tropical mean.Plain Language Summary: The diurnal fluctuations of sea surface temperature (SST) have been extensively studied for the last decades, but the assessment of the importance of this phenomenon for atmospheric convection on the global scale has come within reach only very recently, thanks to the development of simulations with a horizontal resolution of O(1 km). In this manuscript we show that we can indeed observe an impact of SST fluctuations on moisture in the atmosphere. However, the impact on the amount of clouds in the tropics is found to be short‐lived and its magnitude negligible on average.Key Points: diurnal warm layers (DWLs) increase atmospheric moisture The increase of cloud cover (CC) following the formation of a DWL is immediate and only lasts for several hours The magnitude of the CC increase is small and has no discernible influence on the global mean Deutsche Forschungsgemeinschaft http://dx.doi.org/10.13039/501100001659https://gotm.net/https://hdl.handle.net/21.11116/0000-000C-1447-

ICON-Sapphire: simulating the components of the Earth system and their interactions at kilometer and subkilometer scales

International audienceState-of-the-art Earth system models typically employ grid spacings of O(100 km), which is too coarse to explicitly resolve main drivers of the flow of energy and matter across the Earth system. In this paper, we present the new ICON-Sapphire model configuration, which targets a representation of the components of the Earth system and their interactions with a grid spacing of 10 km and finer. Through the use of selected simulation examples, we demonstrate that ICON-Sapphire can (i) be run coupled globally on seasonal timescales with a grid spacing of 5 km, on monthly timescales with a grid spacing of 2.5 km, and on daily timescales with a grid spacing of 1.25 km; (ii) resolve large eddies in the atmosphere using hectometer grid spacings on limited-area domains in atmosphere-only simulations; (iii) resolve submesoscale ocean eddies by using a global uniform grid of 1.25 km or a telescoping grid with the finest grid spacing at 530 m, the latter coupled to a uniform atmosphere; and (iv) simulate biogeochemistry in an ocean-only simulation integrated for 4 years at 10 km. Comparison of basic features of the climate system to observations reveals no obvious pitfalls, even though some observed aspects remain difficult to capture. The throughput of the coupled 5 km global simulation is 126 simulated days per day employing 21 % of the latest machine of the German Climate Computing Center. Extrapolating from these results, multi-decadal global simulations including interactive carbon are now possible, and short global simulations resolving large eddies in the atmosphere and submesoscale eddies in the ocean are within reach

ICON-Sapphire: simulating the components of the Earth system and their interactions at kilometer and subkilometer scales

International audienceState-of-the-art Earth system models typically employ grid spacings of O(100 km), which is too coarse to explicitly resolve main drivers of the flow of energy and matter across the Earth system. In this paper, we present the new ICON-Sapphire model configuration, which targets a representation of the components of the Earth system and their interactions with a grid spacing of 10 km and finer. Through the use of selected simulation examples, we demonstrate that ICON-Sapphire can (i) be run coupled globally on seasonal timescales with a grid spacing of 5 km, on monthly timescales with a grid spacing of 2.5 km, and on daily timescales with a grid spacing of 1.25 km; (ii) resolve large eddies in the atmosphere using hectometer grid spacings on limited-area domains in atmosphere-only simulations; (iii) resolve submesoscale ocean eddies by using a global uniform grid of 1.25 km or a telescoping grid with the finest grid spacing at 530 m, the latter coupled to a uniform atmosphere; and (iv) simulate biogeochemistry in an ocean-only simulation integrated for 4 years at 10 km. Comparison of basic features of the climate system to observations reveals no obvious pitfalls, even though some observed aspects remain difficult to capture. The throughput of the coupled 5 km global simulation is 126 simulated days per day employing 21 % of the latest machine of the German Climate Computing Center. Extrapolating from these results, multi-decadal global simulations including interactive carbon are now possible, and short global simulations resolving large eddies in the atmosphere and submesoscale eddies in the ocean are within reach