Stochastic Parameterization: A Rigorous Approach to Stochastic Three-Dimensional Primitive Equations

The atmosphere is a strongly nonlinear and infinite-dimensional dynamical system acting on a multitude of different time and space scales. A possible problem of numerical weather prediction and climate modeling using deterministic parameterization of subscale and unresolved processes is the incomplete consideration of scale interactions. A stochastic treatment of these parameterizations bears the potential to improve the simulations and to provide a better understanding of the scale interactions of the simulated atmospheric variables. The scientific community that is dealing with stochastic meteorological models can be divided into two groups: the first one uses pragmatic approaches to improve existing complex models. The second group pursues a mathematical rigorous way to develop stochastic models, which is currently limited to conceptual models. The overall objective of this work is to narrow the gap between pragmatic approaches and the mathematical rigorous methods. Using conceptual climate models, we point out that a stochastic formulation must not be chosen arbitrarily but has to be derived based on the physics of the system at hand. Equally important is a rigorous numerical implementation of the resulting stochastic model. The dynamics of sub grid and unresolved processes are often described by time continuous stochastic processes, which cannot be treated with deterministic numerical schemes. We show that a stochastic formulation of the three-dimensional primitive equations fits in the mathematical framework of abstract stochastic fluid models. This allows us to utilize recent results regarding existence and uniqueness of solutions of such systems. Based on these theoretical results we propose a Galerkin scheme for the discretization of spatial and stochastic dimensions. Using the framework of mild solutions of stochastic partial differential equations we are able to prove quantitative error bounds and strong mean square convergence. Under additional assumptions we show the convergence of a numerical scheme which combines the Galerkin approximation with a temporal discretization.Stochastische Parametrisierung: Ein Rigoroser Ansatz für die Stochastischen Drei-Dimensionalen Primitiven Gleichungen Die Atmosphäre ist ein von starken Nichtlinearitäten geprägtes, unendlich-linebreak dimensionales dynamisches System, dessen Variablen auf einer Vielzahl verschiedener Raum- und Zeitskalen interagieren. Ein potentielles Problem von Modellen zur numerischen Wettervorhersage und Klimamodellierung, die auf deterministischen Parametrisierungen subskaliger Prozesse beruhen, ist die unzureichende Behandlung der Interaktion zwischen diesen Prozessen und den Modellvariablen. Eine stochastische Beschreibung dieser Parametrisierungen hat das Potential die Qualität der Simulationen zu verbessern und das Verständnis der Skalen-Interaktion atmosphärischer Variablen zu vertiefen. Die wissenschaftlich Gemeinschaft, die sich mit stochastischen meteorologischen Modellen beschäftigt, kann grob in zwei Gruppen unterteilt werden: die erste Gruppe ist bemüht durch pragmatische Ansätze bestehende, komplexe Modelle zu erweitern. Die zweite Gruppe verfolgt einen mathematisch rigorosen Weg, um stochastische Modelle zu entwickeln. Dies ist jedoch aufgrund der mathematischen Komplexität bisher auf konzeptionelle Modelle beschränkt. Das generelle Ziel der vorliegenden Arbeit ist es, die Kluft zwischen den pragmatischen und mathematisch rigorosen Ansätzen zu verringern. Die Diskussion zweier konzeptioneller Klimamodelle verdeutlicht, dass eine stochastische Formulierung nicht willkürlich gewählt werden darf, sondern aus der Physik des betrachteten Systems abgeleitet werden muss. Ebenso unabdingbar ist eine rigorose numerische Implementierung des resultierenden stochastischen Modells. Diesem Aspekt wird besondere Bedeutung zu Teil, da dynamische subskalige Prozesse oftmals durch zeitabhängige stochastische Prozesse beschrieben werden, die sich nicht mit deterministischen numerischen Methoden behandeln lassen. Wir zeigen auf, dass eine stochastische Formulierung der dreidimensionalen primitiven Gleichungen im mathematischen Rahmen abstrakter stochastischer Fluidmodelle behandelt werden kann. Dies ermöglicht die Anwendung kürzlich gewonnener Erkenntnisse bezüglich Existenz und Eindeutigkeit von Lösungen. Wir stellen einen auf dieser theoretischen Grundlage basierenden Galerkin Ansatz zur Diskretisierung der räumlichen und stochastischen Dimensionen vor. Mit Hilfe sogenannter milder Lösungen der stochastischen partiellen Differentialgleichungen leiten wir quantitative Schranken der Diskretisierungsfehler her und zeigen die starke Konvergenz des mittleren quadratischen Fehlers. Unter zusätzlichen Annahmen leiten wir die Konvergenz eines numerischen Verfahrens her, das den Galerkin Ansatz um eine zeitliche Diskretisierung erweitert

Based Upon Repeat Pattern (BURP): an algorithm to characterize the long-term evolution of Staphylococcus aureus populations based on spa polymorphisms

Mellmann A, Weniger T, Berssenbrügge C, et al. Based Upon Repeat Pattern (BURP): an algorithm to characterize the long-term evolution of Staphylococcus aureus populations based on spa polymorphisms. BMC MICROBIOLOGY. 2007;7(1):98.Background: For typing of Staphylococcus aureus, DNA sequencing of the repeat region of the protein A ( spa) gene is a well established discriminatory method for outbreak investigations. Recently, it was hypothesized that this region also reflects long-term epidemiology. However, no automated and objective algorithm existed to cluster different repeat regions. In this study, the Based Upon Repeat Pattern ( BURP) implementation that is a heuristic variant of the newly described EDSI algorithm was investigated to infer the clonal relatedness of different spa types. For calibration of BURP parameters, 400 representative S. aureus strains with different spa types were characterized by MLST and clustered using eBURST as "gold standard" for their phylogeny. Typing concordance analysis between eBURST and BURP clustering ( spa-CC) were performed using all possible BURP parameters to determine their optimal combination. BURP was subsequently evaluated with a strain collection reflecting the breadth of diversity of S. aureus (JCM 2002; 40: 4544). Results: In total, the 400 strains exhibited 122 different MLST types. eBURST grouped them into 23 clonal complexes (CC; 354 isolates) and 33 singletons (46 isolates). BURP clustering of spa types using all possible parameter combinations and subsequent comparison with eBURST CCs resulted in concordances ranging from 8.2 to 96.2%. However, 96.2% concordance was reached only if spa types shorter than 8 repeats were excluded, which resulted in 37% excluded spa types. Therefore, the optimal combination of the BURP parameters was "exclude spa types shorter than 5 repeats" and "cluster spa types into spa-CC if cost distances are less than 4" exhibiting 95.3% concordance to eBURST. This algorithm identified 24 spa-CCs, 40 singletons, and excluded only 7.8% spa types. Analyzing the natural population with these parameters, the comparison of whole-genome micro-array groupings ( at the level of 0.31 Pearson correlation index) and spa-CCs gave a concordance of 87.1%; BURP spa-CCs vs. manually grouped spa types resulted in 95.7% concordance. Conclusion: BURP is the first automated and objective tool to infer clonal relatedness from spa repeat regions. It is able to extract an evolutionary signal rather congruent to MLST and micro-array data

Identifizierung von Schadstoff Hot Spots unterschiedlicher Belastungs-geschichte in Auen der Mittelelbe

Es wurde die zeitliche Belastungsentwicklung von Dioxinen und Metallen untersucht, um prüfen, ob Metalle als Trcer für Dioxine in Oberböden der Elbtalaue geeignet sind. Die unterschiedlichen Belastungsgeschichten machen lassen Metalle als ungeeignet erscheinen. Allerdings konnten im Tiefenprofil Metallmuster identifiziert werden, die auch in Oberböden eine hohe Dioxinbelastung indizieren. Standorte mit hoher Dioxinkonzentration im Oberboden befinden sich in Senkenlage oder Flutrinnen, gekennzeichnet durch geringe, bzw. unterbrochene Sedimentation. Die Standorte weisen geringmächtige Oberböden auf, meist mit rel. geringen pH-Werten sowie mit rel. geringen Anteilen oxalatlöslichen Eisens, d. h. Chartakteristika einer vorangeschrittenen Bodenbildung aus Sedimenten

Implementation of the Combined--Nonlinear Condensation Transformation

We discuss several applications of the recently proposed combined nonlinear-condensation transformation (CNCT) for the evaluation of slowly convergent, nonalternating series. These include certain statistical distributions which are of importance in linguistics, statistical-mechanics theory, and biophysics (statistical analysis of DNA sequences). We also discuss applications of the transformation in experimental mathematics, and we briefly expand on further applications in theoretical physics. Finally, we discuss a related Mathematica program for the computation of Lerch's transcendent.Comment: 23 pages, 1 table, 1 figure (Comput. Phys. Commun., in press

An Off-center Density Peak in the Milky Way's Dark Matter Halo?

We show that the position of the central dark matter density peak may be expected to differ from the dynamical center of the Galaxy by several hundred parsec. In Eris, a high resolution cosmological hydrodynamics simulation of a realistic Milky-Way-analog disk galaxy, this offset is 300 - 400 pc (~3 gravitational softening lengths) after z=1. In its dissipationless dark-matter-only twin simulation ErisDark, as well as in the Via Lactea II and GHalo simulations, the offset remains below one softening length for most of its evolution. The growth of the DM offset coincides with a flattening of the central DM density profile in Eris inwards of ~1 kpc, and the direction from the dynamical center to the point of maximum DM density is correlated with the orientation of the stellar bar, suggesting a bar-halo interaction as a possible explanation. A dark matter density offset of several hundred parsec greatly affects expectations of the dark matter annihilation signals from the Galactic Center. It may also support a dark matter annihilation interpretation of recent reports by Weniger (2012) and Su & Finkbeiner (2012) of highly significant 130 GeV gamma-ray line emission from a region 1.5 degrees (~200 parsec projected) away from Sgr A* in the Galactic plane.Comment: 12 pages, 11 figures, replaced with version accepted for publication in Ap

Lost in spatial translation - A novel tool to objectively assess spatial disorientation in Alzheimer's disease and frontotemporal dementia

Spatial disorientation is a prominent feature of early Alzheimer's disease (AD) attributed to degeneration of medial temporal and parietal brain regions, including the retrosplenial cortex (RSC). By contrast, frontotemporal dementia (FTD) syndromes show generally intact spatial orientation at presentation. However, currently no clinical tasks are routinely administered to objectively assess spatial orientation in these neurodegenerative conditions. In this study we investigated spatial orientation in 58 dementia patients and 23 healthy controls using a novel virtual supermarket task as well as voxel-based morphometry (VBM). We compared performance on this task with visual and verbal memory function, which has traditionally been used to discriminate between AD and FTD. Participants viewed a series of videos from a first person perspective travelling through a virtual supermarket and were required to maintain orientation to a starting location. Analyses revealed significantly impaired spatial orientation in AD, compared to FTD patient groups. Spatial orientation performance was found to discriminate AD and FTD patient groups to a very high degree at presentation. More importantly, integrity of the RSC was identified as a key neural correlate of orientation performance. These findings confirm the notion that i) it is feasible to assess spatial orientation objectively via our novel Supermarket task; ii) impaired orientation is a prominent feature that can be applied clinically to discriminate between AD and FTD and iii) the RSC emerges as a critical biomarker to assess spatial orientation deficits in these neurodegenerative conditions