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## Über dieses Buch

This thesis presents a study of strong stratification and turbulence collapse in the planetary boundary layer, opening a new avenue in this field. It is the first work to study all regimes of stratified turbulence in a unified simulation framework without a break in the paradigms for representation of turbulence.

To date, advances in our understanding and the parameterization of turbulence in the stable boundary layer have been hampered by difficulties simulating the strongly stratified regime, and the analysis has primarily been based on field measurements. The content presented here changes that paradigm by demonstrating the ability of direct numerical simulation to address this problem, and by doing so to remove the uncertainty of turbulence models from the analysis. Employing a stably stratified Ekman layer as a simplified physical model of the stable boundary layer, the three stratification regimes observed in nature— weakly, intermediately and strongly stratified—are reproduced, and the data is subsequently used to answer key, long-standing questions.

The main part of the book is organized in three sections, namely a comprehensive introduction, numerics, and physics. The thesis ends with a clear and concise conclusion that distills specific implications for the study of the stable boundary layer. This structure emphasizes the physical results, but at the same time gives relevance to the technical aspects of numerical schemes and post-processing tools. The selection of the relevant literature during the introduction, and its use along the work appropriately combines literature from two research communities: fluid dynamics, and boundary-layer meteorology.

## Inhaltsverzeichnis

### Chapter 1. Introduction

Abstract
The SBL still poses a challenge in terms of both its modelling and fundamental understanding; problems are particularly pertinent where turbulence is globally intermittent and assumptions underlying common turbulence closures break down. This thesis, for the first time systematically, employs DNS, a widely-used tool to study canonical problems in fluid mechanics, to study stratified Ekman flow. Particular emphasis is on very strong stability where other approaches have problems and many open questions remain. As a simplified set-up, turbulent Ekman flow over a smooth flat plate is chosen and this work complements existing studies of stably stratified flows relevant to the atmospheric boundary layer.
Cedrick Ansorge

### Chapter 2. Problem Formulation and Tools

Abstract
In this chapter, the physical problem is formulated using the governing equations. The Navier-Stokes equations in the Boussinesq limit are identified as the appropriate set of equations to study the flow. These equations are introduced along some common assumptions and simplifications, and their simplified version is subsequently non-dimensionalized. The non-dimensional parameter space of the problem is introduced and qualitatively characterized, and the main analysis methods are presented.
Cedrick Ansorge

### Chapter 3. Discretization

Abstract
This chapter briefly lays out the general aspects of incompressible flow simulations, and thereafter focuses on the time integration. Time stepping schemes available for DNS codes are reviewed, and the selection of an appropriate schemes is detailed. The linear stability region of the selected semi-implicit third-order Runge-Kutta scheme is calculated and its actual implementation is discussed.
Cedrick Ansorge

### Chapter 4. Overlapping Communication and Computation

Abstract
From a computational perspective direct numerical simulation (DNS) of turbulent flows is enormously challenging. A utilization of current high-performance computing (HPC) resources demands a distribution of the problem to a large number of independently acting computing entities which requires communication in between those entities. I present here a suitable approach to overlap communication and computation for fluid mechanics simulations along with a comprehensive scaling study.
Cedrick Ansorge

### Chapter 5. A Test Bed for the Numerical Tool

Abstract
When simulating flows numerically, a validation of the tools is a necessary step. To ensure a correct implementation of the flow solver implemented here, it has been tested using problems at various stages of complexity ranging from an ordinary differential equation via one- and two-dimensional laminar flow solutions to fully three-dimensionally turbulent flows.
Cedrick Ansorge

### Chapter 6. The Neutrally Stratified Ekman Layer

Abstract
In this chapter, the neutrally stratified Ekman flow is discussed. For the Reynolds numbers $$\mathscr {R} \in \left.500, 750, 1000 \right.$$, the set-up is well within the turbulent regime, and the scale separation is large enough for a logarithmic layer to develop. The impact of the averaging period on flux measurements based on single-point probes is quantified. For averaging intervals of the order of the eddy-turnover time, fluxes are underestimated by about 1 %, an acceptable error given the precision achieved by boundary-layer measurements in the field. The analogy of the surface layer of Ekman flow with that of channel flow is investigated. In agreement with previous work, a quasi-logarithmic layer above $$z^{+} \simeq 20$$ is found that extends up to $$z^{+} \simeq 100$$ at $$Re = 1000$$. This well-established analogy of the mean-flow profiles is extended here, and shown to also apply to the turbulence-energy budget. External intermittency is quantified by means of the enstrophy allowing to partition the flow to turbulent and non-turbulent regions. Using conditional statistics the impact of external intermittency on the logarithmic law for the mean velocities is quantified: A prominent dip in $${U^{+}}/(ln {z^+}\,+\,\mathscr {A}_{0})$$ in the upper part of the logarithmic layer that was observed before, is shown to be a consequence of external intermittency. This dip is largely reduced by considering only the turbulent subvolumes of the flow, and the data fit then the formulation $$U_{turb}^{+} = \kappa ^{-1} ln z^{+} + \mathscr {A}_{0}$$ with $$\kappa = 0.413$$ and $$\mathscr {A}_{0} = 4.46$$.
Cedrick Ansorge

### Chapter 7. Turbulence Regimes and Stability

Abstract
This chapter demonstrates that the DNS set-up before and used throughout this work is suitable to study all regimes of stratified turbulence without the need to tweak underlying assumptions when stratification is increased to the extreme limit. While the qualitative behavior of the flow agrees well with theory and observations, the present approach allows insight into the dynamics of turbulence based on fundamental principles only, and it evades uncertainties related to the application of turbulence closures. A finding of particular relevance is that global intermittency—in both the time and space dimensions—occurs in absence of external triggers.
Cedrick Ansorge

### Chapter 8. Flow Organization and Global Intermittency Under Strong Stratification

Abstract
In this chapter, the partial collapse of turbulence in the form of global intermittency is studied. The flow’s large-scale organization is shown to scale with the outer scale, and not with the wall unit. The localized collapse of turbulence provoking the presence of these large-scale structures in the buffer and surface layers of the flow is, however, a surface-layer process and as such governed by inner scalings, namely the Obukhov length expressed in wall units L $$_{O}^{+}$$. A new partitioning method developed here allows to partition globally intermittent flow to turbulent and non-turbulent regions and yields new insight into the dynamics of turbulence under strong stratification: The main impact of stratification on turbulence is not a change in the dynamics of turbulence in the regions where the flow is turbulent, but it is rather a confinement of the turbulent area fraction which is quantified by the intermittency factor that can now be determined with the above partitioning method. This finding is consistent with the analysis of flow visualizations, spectra and probability density functions.
Cedrick Ansorge

### Chapter 9. Implications for the Study of the Atmospheric Boundary Layer

Abstract
This work is motivated by a lack of process-level understanding in the stable boundary layer—a meteorological problem at its core. The direct numerical simulation of a turbulent flow is relatively new in a meteorological context, and the technical framing of this work is borrowed from an engineering context where DNS as a tool in turbulence research is widely used since decades. In fact, part III of this book studies Ekman flow as a fluid-mechanics problem. In the present chapter, the key findings of this work are outlined in a more applied context, and their implications for work on the SBL are discussed.
Cedrick Ansorge

### Chapter 10. Résumé

Abstract
The stably stratified boundary layer was early recognized as a fluid-mechanics problem of fundamental interest. Nonetheless, the study of stratified turbulence and the stable boundary layer have taken different paths: Stratified turbulence is mostly studied as a canonical-flow problem in fluid mechanical engineering. On the contrary, the stably stratified boundary layer is commonly investigated as a parameterization problem—motivated by the need for a working turbulence closure in general circulation and numerical weather prediction models. This gave rise to a number of fixes in turbulence closures which are motivated from a large-scale point of view. Still recently, a lack of fundamental understanding in stratified turbulence is identified as a pertinent challenge in understanding the boundary layer. These two paths are consolidated here applying systematically the direct numerical simulation of a turbulent flow, a tool widely used in fluid mechanical engineering, to study the stably stratified boundary layer.
Cedrick Ansorge

### Backmatter

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