Theoretische Physik I – Prof. Dr. Rainer Grauer

Prof. Dr. Rainer Grauer

grauer

Raum, Telefon und Email

Raum
NB 7/123

Telefon
0234/32-23767

Telefon Sekretariat (Alicia Gonzalez)
0234/32-23800

Email

Publikationen


F. Allmann-Rahn, S. Lautenbach, M. Deisenhofer, R. Grauer The muphyII Code: Multiphysics Plasma Simulation on Large HPC Systems Collisionless astrophysical and space plasmas cover regions that typically display a separation of scales that exceeds any code’s capabilities. To help address this problem, the muphyII code utilizes a hierarchy of models with different inherent scales, unified in an adaptive framework that allows stand-alone use of models as well as a model-based dynamic and adaptive domain decomposition. This requires ensuring excellent conservation properties, careful treatment of inner-domain model boundaries for model coupling, and robust time stepping algorithms, especially with the use of electron subcycling. This multi-physics approach is implemented in the muphyII code, tested on different scenarios of space plasma reconnection and evaluated against space probe data and higher-fidelity simulation results from literature. Adaptive model refinement is highlighted in particular, and a hybrid model with kinetic ions, pressure-tensor fluid electrons, and Maxwell fields is appraised. Comp. Phys. Comm. 296 (2024) 109064

Jeremiah Lübke, Frederic Effenberger, Mike Wilbert, Horst Fichtner, Rainer Grauer Towards Synthetic Magnetic Turbulence with Coherent Structures Synthetic turbulence is a relevant tool to study complex astrophysical and space plasma environments inaccessible by direct simulation. However, conventional models lack intermittent coherent structures, which are essential in realistic turbulence. We present a novel method, featuring coherent structures, conditional structure function scaling and fieldline curvature statistics comparable to magnetohydrodynamic turbulence. Enhanced transport of charged particles is investigated as well. This method presents significant progress towards physically faithful synthetic turbulence. submitted (2024)

Jeremiah Lübke, Jan Friedrich, Rainer Grauer Stochastic interpolation of sparsely sampled time series by a superstatistical random process and its synthesis in Fourier and wavelet space We present a novel method for stochastic interpolation of sparsely sampled time signals based on a superstatistical random process generated from a multivariate Gaussian scale mixture. In com- parison to other stochastic interpolation methods such as Gaussian process regression, our method possesses strong multifractal properties and is thus applicable to a broad range of real-world time series, e.g. from solar wind or atmospheric turbulence. Furthermore, we provide a sampling algo- rithm in terms of a mixing procedure that consists of generating a 1 + 1-dimensional field u(t, ξ), where each Gaussian component uξ(t) is synthesized with identical underlying noise but differ- ent covariance function Cξ(t,s) parameterized by a log-normally distributed parameter ξ. Due to the Gaussianity of each component uξ(t), we can exploit standard sampling alogrithms such as Fourier or wavelet methods and, most importantly, methods to constrain the process on the sparse measurement points. The scale mixture u(t) is then initialized by assigning each point in time t a ξ(t) and therefore a specific value from u(t, ξ), where the time-dependent parameter ξ(t) follows a log-normal process with a large correlation time scale compared to the correlation time of u(t, ξ). We juxtapose Fourier and wavelet methods and show that a multiwavelet-based hierarchical ap- proximation of the interpolating paths, which produce a sparse covariance structure, provide an adequate method to locally interpolate large and sparse datasets. Journal of Physics: Complexity 4 (2023) 15005

T. Schorlepp, T. Grafke, R. Grauer Symmetries and zero modes in sample path large deviations Sharp large deviation estimates for stochastic differential equations with small noise, based on minimizing the Freidlin-Wentzell action functional under appropriate boundary conditions, can be obtained by integrating certain matrix Riccati differential equations along the large deviation minimizers or instantons, either forward or backward in time. Previous works in this direction often rely on the existence of isolated minimizers with positive definite second variation. By adopting techniques from field theory and explicitly evaluating the large deviation prefactors as functional determinant ratios using Forman’s theorem, we extend the approach to general systems where degenerate submanifolds of minimizers exist. The key technique for this is a boundary-type regularization of the second variation operator. This extension is particularly relevant if the system possesses continuous symmetries that are broken by the instantons. We find that removing the vanishing eigenvalues associated with the zero modes is possible within the Riccati formulation and amounts to modifying the initial or final conditions and evaluation of the Riccati matrices. We apply our results in multiple examples including a dynamical phase transition for the average surface height in short-time large deviations of the one-dimensional Kardar-Parisi-Zhang equation with flat initial profile. Journal of Statistical Physics 190 (2023) 50

Timo Schorlepp, Tobias Grafke, Sandra May and Rainer Grauer Spontaneous Symmetry Breaking for Extreme Vorticity and Strain in the 3D Navier-Stokes Equations We investigate the spatio-temporal structure of the most likely configurations realising extremely high vorticity or strain in the stochastically forced 3D incompressible Navier-Stokes equations. Most likely configurations are computed by numerically finding the highest probability velocity field realising an extreme constraint as solution of a large optimisation problem. High-vorticity configurations are identified as pinched vortex filaments with swirl, while high-strain configurations correspond to counter-rotating vortex rings. We additionally observe that the most likely configurations for vorticity and strain spontaneously break their rotational symmetry for extremely high observable values. Instanton calculus and large deviation theory allow us to show that these maximum likelihood realisations determine the tail probabilities of the observed quantities. In particular, we are able to demonstrate that artificially enforcing rotational symmetry for large strain configurations leads to a severe underestimate of their probability, as it is dominated in likelihood by an exponentially more likely symmetry broken vortex-sheet configuration. Phil. Trans. R. Soc. A. 380 (2022) 20210051

Florian Allmann-Rahn, Simon Lautenbach, Rainer Grauer An Energy Conserving Vlasov Solver That Tolerates Coarse Velocity Space Resolutions: Simulation of MMS Reconnection Events Vlasov solvers that operate on a phase-space grid are highly accurate but also numerically demanding. Coarse velocity space resolutions, which are unproblematic in particle-in-cell (PIC) simulations, lead to strong numerical heating or oscillations in standard continuum Vlasov methods. We present a new dual Vlasov solver which is based on an established positivity preserving advection scheme for the update of the distribution function and an energy conserving partial differential equation solver for the kinetic update of mean velocity and temperature. The solvers work together via moment fitting during which the maximum entropy part of the distribution function is replaced by the solution from the partial differential equation solver. This numerical scheme makes continuum Vlasov methods competitive with PIC methods concerning computational cost and enables us to model large scale reconnection in Earth's magnetosphere with a fully kinetic continuum method. The simulation results agree well with measurements by the MMS spacecraft. Journal of Geophysical Research - Space Physics 127 (2022) 29976

Florian Allmann-Rahn, Rainer Grauer, Katharina Kormann A Parallel Low-Rank Solver for the Six-Dimensional Vlasov-Maxwell Equations Continuum Vlasov simulations can be utilized for highly accurate modelling of fully kinetic plasmas. Great progress has been made recently regarding the applicability of the method in realistic plasma configurations. However, a reduction of the high computational cost that is inherent to fully kinetic simulations would be desirable, especially at high velocity space resolutions. For this purpose, low-rank approximations can be employed. The so far available low-rank solvers are restricted to either electrostatic systems or low dimensionality and can therefore not be applied to most space, astrophysical and fusion plasmas. In this paper we present a new parallel low-rank solver for the full six-dimensional electromagnetic Vlasov-Maxwell equations with a compression of the particle distribution function in velocity space. Special attention is paid to mass conservation and Gauss's law. The low-rank Vlasov solver is applied to standard benchmark problems of plasma turbulence and magnetic reconnection and compared to the full grid method. It yields accurate results at significantly reduced computational cost. J. Comp. Phys. 469 (2022) 111562

Mike Wilbert, André Giesecke, and Rainer Grauer Numerical Investigation of the Flow inside a Precession driven cylindrical Cavity with additional Baffles using an Immersed Boundary Method In this paper we present a numerical approach to solve the Navier-Stokes equations for arbitrary vessel geometries by combining a Fourier-Spectral method with a direct forcing Immersed Boundary method which allows to consider solid-fluid interactions. The approach is applied to a paradigmatic setup motivated by the precession dynamo experiment currently under construction at Helmholtz- Zentrum Dresden-Rossendorf (HZDR) are presented. The experiment consists of a fluid filled cylinder rotating about 2 axes which induces a precession driven flow inside the cavity. The cylinder is also equipped with baffles at the end caps with adjustable penetration depth to impact the flow. The numerical details as well as simulation results for the spin up and precession driven flow in a circular cylinder with additional baffles are presented. Phys. Fluids 34 (2022) 96607

F. Allmann-Rahn, S. Lautenbach, R. Grauer, and R. D. Sydora Fluid simulations of three-dimensional reconnection that capture the lower-hybrid drift instability Fluid models that approximate kinetic effects have received attention recently in the modelling of large scale plasmas such as planetary magnetospheres. Where fully kinetic computations are not an option, fluid or hybrid models can be an excellent replacement. In three-dimensional reconnection, however, both reconnection itself and current sheet instabilities need to be represented appropriately, which has been an issue before. We show that a heat flux closure based on pressure gradients enables a ten moment fluid model to capture the lower-hybrid drift instability within a reconnection simulation. Characteristics of the instability are examined with fluid and kinetic continuum models, and its role in the three-dimensional reconnection simulation is analysed. It is found that the initial perturbation level has significant impact on the resulting turbulence. Journal of Plasma Physics 87 (2021) 905870115

M. Sinhuber, J. Friedrich, R. Grauer, M. Wilczek Multi-level stochastic refinement for complex time series and fields: A Data-Driven Approach Spatio-temporally extended nonlinear systems often exhibit a remarkable complexity in space and time. In many cases, extensive records of such data sets are difficult to obtain, yet needed for a range of applications. Here, we present a method to generate synthetic time series or fields that reproduce statistical multi-scale features of complex systems. The method is based on a hierarchical refinement employing transition probability density functions (PDFs) from one scale to another. We address the case in which such PDFs can be obtained from experimental measurements or simulations and then used to generate arbitrarily large synthetic data sets. The validity of our approach is demonstrated at the example of an experimental dataset of high Reynolds number turbulence. New Journal of Physics 23 (2021) 63063

Timo Schorlepp, Tobias Grafke, Rainer Grauer Gel'fand-Yaglom type equations for calculating fluctuations around Instantons in stochastic systems In recent years, instanton calculus has successfully been employed to estimate tail probabilities of rare events in various stochastic dynamical systems. Without further corrections, however, these estimates can only capture the exponential scaling. In this paper, we derive a general, closed form expression for the leading prefactor contribution of the fluctuations around the instanton trajectory for the computation of probability density functions of general observables. The key technique is applying the Gel'fand-Yaglom recursive evaluation method to the suitably discretized Gaussian path integral of the fluctuations, in order to obtain matrix evolution equations that yield the fluctuation determinant. We demonstrate agreement between these predictions and direct sampling for examples motivated from turbulence theory. Journal of Physics A: Mathematical and Theoretical 54 (2021) 235003

J. Friedrich, J. Peinke, A. Pumir, R. Grauer Explicit construction of joint multipoint statistics in complex systems Complex systems often involve random fluctuations for which self-similar properties in space and time play an important role. Fractional Brownian motions, characterized by a single scaling exponent, the Hurst exponent H, provide a convenient tool to construct synthetic signals that capture the statistical properties of many processes in the physical sciences and beyond. However, in certain strongly interacting systems, e.g., turbulent flows, stock market indices, or cardiac interbeats, multiscale interactions lead to significant deviations from self-similarity and may therefore require a more elaborate description. In the context of turbulence, the Kolmogorov-Oboukhov model (K62) describes anomalous scaling, albeit explicit constructions of a turbulent signal by this model are not available yet. Here, we derive an explicit formula for the joint multipoint probability density function of a multifractal field. To this end, we consider a scale mixture of fractional Ornstein-Uhlenbeck processes and introduce a fluctuating length scale in the corresponding covariance function. In deriving the complete statistical properties of the field, we are able to systematically model synthetic multifractal phenomena. We conclude by giving a brief outlook on potential applications which range from specific tailoring or stochastic interpolation of wind fields to the modeling of financial data or non-Gaussian features in geophysical or geospatial settings. Journal of Physics: Complexity 2 (2021) 45006

J. Friedrich and R. Grauer Generalized description of intermittency in turbulence via stochastic methods We present a generalized picture of intermittency in turbulence that is based on the theory of stochastic processes. To this end, we rely on the experimentally and numerically verified finding by R.~Friedrich and J. Peinke [Phys. Rev. Lett. 78, 863 (1997)] that allows for an interpretation of the turbulent energy cascade as a Markov process of velocity increments in scale. It is explicitly shown that all known phenomenological models of turbulence can be reproduced by the Kramers-Moyal expansion of the velocity increment probability density function that is associated to a Markov process. We compare the different sets of Kramers-Moyal coefficients of each phenomenology and deduce that an accurate description of intermittency should take into account an infinite number of coefficients. This is demonstrated in more detail for the case of Burgers turbulence that exhibits pronounced intermittency effects. Moreover, the influence of nonlocality on Kramers-Moyal coefficients is investigated by direct numerical simulations of a generalized Burgers equation. Depending on the balance between nonlinearity and nonlocality, we encounter different intermittency behavior that ranges from self-similarity (purely nonlocal case) to intermittent behavior (intermediate case that agrees with Yakhot's mean field theory [Phys. Rev. E 63 026307 (2001)]) to shock-like behavior (purely nonlinear Burgers case). Atmosphere 11 (2020) 1003

J. Friedrich, S. Gallon, A. Pumir, R. Grauer Multi-point fractional Brownian bridges and their applications We propose and test a method to interpolate sparsely sampled signals by a stochastic process with a broad range of spatial and/or temporal scales. To this end, we extend the notion of a fractional Brownian bridge, defined as fractional Brownian motion with a given scaling (Hurst) exponent H and with prescribed start and end points, to a bridge process with an arbitrary number of prescribed intermediate and non-equidistant points. We demonstrate the validity of our method on a signal from fluid turbulence in a high Reynolds number flow. Furthermore, we discuss possible extensions of the present work to include the non-self-similar character of the signal. The derived method could be instrumental within a variety of fields such as astrophysics, particle tracking, specific tailoring of surrogate data, and spatial planning. Phys. Rev. Lett. 125 (2020) 170602

G. Margazoglou, L. Biferale, R. Grauer, K. Jansen, D. Mesterházy, T. Rosenow, R. Tripiccione A Hybrid Monte Carlo algorithm for sampling rare events in space-time histories of stochastic fields We introduce a variant of the Hybrid Monte Carlo (HMC) algorithm to address large deviation statistics in stochastic hydrodynamics. Based on the path integral approach to stochastic (partial) differential equations, our HMC algorithm samples space-time histories of the dynamical degrees of freedom under the influence of random noise. First, we validate and benchmark the HMC algorithm by reproducing multi-scale properties of the one-dimensional Burgers equation driven by Gaussian and white-in-time noise. Second, we show how to implement an importance sampling protocol to significantly enhance, by order-of-magnitudes, the probability to sample extreme and rare events, making it possible for the first time to estimate moments of field variables of extremely high order (up to 30 and more). By employing reweighting techniques, we map the biased configurations back to the original probability measure in order to probe their statistical importance. Finally, we show that by biasing the system towards very intense negative gradients, the HMC algorithm is able to explore the statistical fluctuations around instanton configurations. Our results will also be interesting and relevant in lattice gauge theory since they provide a new insight on reweighting techniques. Phys. Rev. E 99 (2019) 53303

Lasse Ebener, Georgios Margazoglou, Jan Friedrich, Luca Biferale, Rainer Grauer Instanton based importance sampling for rare events in stochastic PDEs We present a new method for sampling rare and large fluctuations in a non-equilibrium system governed by a stochastic partial differential equation (SPDE) with additive forcing. To this end, we deploy the so-called instanton formalism that corresponds to a saddle-point approximation of the action in the path integral formulation of the underlying SPDE. The crucial step in our approach is the formulation of an alternative SPDE that incorporates knowledge of the instanton solution such that we are able to constrain the dynamical evolutions around extreme flow configurations only. Finally, a reweighting procedure based on the Girsanov theorem is applied to recover the full distribution function of the original system. The entire procedure is demonstrated on the example of the one-dimensional Burgers equation. Furthermore, we compare our method to conventional direct numerical simulations as well as to Hybrid Monte Carlo methods. It will be shown that the instanton-based sampling method outperforms both approaches and allows for an accurate quantification of the whole probability density function of velocity gradients from the core to the very far tails. Chaos 29 (2019) 63102

J. Friedrich and R. Grauer Markov Property of Velocity Increments in Burgers Turbulence We investigate the intermittency properties of a turbulent flow without pressure described by the Burgers equation. To this end, we make use of a phe- nomenogical description devised by R. Friedrich and J. Peinke [Phys. Rev. Lett. 78, 863 (1997)] that interprets the turbulent energy cascade as a Markov process in scale. The impact of Burgers-shocks on the Markov property of the velocity incre- ments is discussed and compared to numerical simulations. Furthermore, we give a brief outlook on the use of the Markov property as a possible closure of a hierarchy of multi-increment probability density functions derived directly from the Burgers equation. in Complexity and Synergetics (Springer), S.C. Müller et al. (eds.) (2018) 39

F. Allmann-Rahn, T. Trost and R. Grauer Temperature gradient driven heat flux closure in fluid simulations of collisionless reconnection Recent efforts to include kinetic effects in fluid simulations of plasmas have been very promising. Concerning collisionless magnetic reconnection, it has been found before that damping of the pressure tensor to isotropy leads to good agreement with kinetic runs in certain scenarios. An accurate representation of kinetic effects in reconnection was achieved in a study by Wang et al. (Phys. Plasmas, volume 22, 2015, 012108) with a closure derived from earlier work by Hammett and Perkins (PRL, volume 64, 1990, 3019). Here, their approach is analyzed on the basis of heat flux data from a Vlasov simulation. As a result, we propose a new local closure in which heat flux is driven by temperature gradients. That way, a more realistic approximation of Landau damping in the collisionless regime is achieved. Previous issues are addressed and the agreement with kinetic simulations in different reconnection setups is improved significantly. To the authors’ knowledge, the new fluid model is the first to perform well in simulations of the coalescence of large magnetic islands. J. Plasma Phys. 84 (2018) 905840307

Jan Friedrich, Georgios Margazoglou, Luca Biferale, Rainer Grauer Multi-scale velocity correlations in turbulence and Burgerlence: fusion rules, Markov processes in scale, and multifractal predictions We compare different approaches towards an effective description of multi-scale velocity field correlations in turbulence. Predictions made by the operator product expansion, the so-called fusion rules, are placed in juxtaposition to an approach that interprets the turbulent energy cascade in terms of a Markov process of velocity increments in scale. We explicitly show that the fusion rules are a direct consequence of the Markov property provided that the structure functions exhibit scaling in the inertial range. Furthermore, the limit case of joint velocity gradient and velocity increment statistics is discussed and put into the context of the notion of dissipative anomaly. We generalize a prediction made by the multifractal (MF) approach derived in [Phys. Rev. Lett. 80, 3244 (1998)] to correlations among inertial range velocity increment and velocity gradients of any order. We show that for the case of squared velocity gradients such a relation can be derived from "first principles" in the case of Burgers equation. Our results are benchmarked by intensive direct numerical simulations of Burgers turbulence. Phys. Rev. E 98 (2018) 23104

Simon Lautenbach, Rainer Grauer Multiphysics simulations of collisionless plasmas Collisionless plasmas, mostly present in astrophysical and space environments, often require a kinetic treatment as given by the Vlasov equation. Unfortunately, the six-dimensional Vlasov equation can only be solved on very small parts of the considered spatial domain. However, in some cases, e.g. magnetic reconnection, it is sufficient to solve the Vlasov equation in a localized domain and solve the remaining domain by appropriate fluid models. In this paper, we describe a hierarchical treatment of collisionless plasmas in the following way. On the finest level of description, the Vlasov equation is solved both for ions and electrons. The next courser description treats electrons with a 10-moment fluid model incorporating a simplified treatment of Landau damping. At the boundary between the electron kinetic and fluid region, the central question is how the fluid moments influence the electron distribution function. On the next coarser level of description the ions are treated by an 10-moment fluid model as well. It may turn out that in some spatial regions far away from the reconnection zone the temperature tensor in the 10-moment description is nearly isotopic. In this case it is even possible to switch to a 5-moment description. This change can be done separately for ions and electrons. To test this multiphysics approach, we apply this full physics-adaptive simulations to the Geospace Environmental Modeling (GEM) challenge of magnetic reconnection. Frontiers in Physics 6 (2018) 113

S. Kreuzahler, Y. Ponty, N. Plihon, H. Homann, R. Grauer Dynamo enhancement and mode selection triggered by high magnetic permeability We present results from consistent dynamo simulations, where the electrically conducting and incompressible flow inside a cylinder vessel is forced by moving impellers numerically implemented by a penalization method. The numerical scheme models jumps of magnetic permeability for the solid impellers, resembling various configurations tested experimentally in the von-Karman Sodium experiment. The most striking experimental observations are reproduced in our set of simulations. In particular, we report on the existence of a time averaged axisymmetric dynamo mode, self- consistently generated when the magnetic permeability of the impellers exceeds a threshold. We describe a possible scenario involving both the turbulent flow in the vicinity of the impellers and the high magnetic permeability of the impellers. Phys. Rev. Lett. 119 (2017) 234501

Alain Pumir, Haitao Xu, Eberhard Bodenschatz and Rainer Grauer Single-Particle Motion and Vortex Stretching in Three-Dimensional Turbulent Flows Three-dimensional turbulent flows are characterized by a flux of energy from large to small scales, which breaks the time reversal symmetry. The motion of tracer particles, which tend to lose energy faster than they gain it, is also irreversible. Here, we connect the time irreversibility in the motion of single tracers with vortex stretching and thus with the generation of the smallest scales. Phys. Rev. Lett. 116 (2016) 124502

J. Friedrich, H. Homann, T. Schäfer, R. Grauer Longitudinal and transverse structure functions in high Reynolds-number magneto-hydrodynamic turbulence We investigate the scaling behavior of longitudinal and transverse structure functions in homogeneous and isotropic magneto-hydrodynamic (MHD) turbulence by means of an exact hierarchy of structure function equations as well as by direct numerical simulations of two- and three-dimensional MHD turbulence. In particular, rescaling relations between longitudinal and transverse structure functions are derived and utilized in order to compare different scaling behavior in the inertial range. It is found that there are no substantial differences between longitudinal and transverse structure functions in MHD turbulence. This finding stands in contrast to the case of hydrodynamic turbulence which shows persistent differences even at high Reynolds numbers. We propose a physical picture that is based on an effective reduction of pressure contributions due to local regions of same magnitude and alignment of velocity and magnetic field fluctuations. Finally, our findings underline the importance of the pressure term for the actually observed scaling differences in hydrodynamic turbulence. New J. Phys. 18 (2016) 125008

M. Rieke, T. Trost and R. Grauer Coupled Vlasov and two-fluid codes on GPUs We present a way to combine Vlasov and two-fluid codes for the simulation of a collisionless plasma in large domains while keeping full information on the velocity distribution in localised areas of interest. This is made possible by solving the full Vlasov equation in one region while the remaining area is treated by a 5-moment two-fluid code. In such a treatment, the main challenge of coupling kinetic and fluid descriptions is the interchange of physically correct boundary conditions between the different plasma models. In contrast to other treatments, we do not rely on any specific form of the distribution function, e.g. a Maxwellian type. Instead, we combine an extrapolation of the distribution function and a correction of the moments based on the fluid data. Thus, throughout the simulation both codes provide the necessary boundary conditions for each other. A speed-up factor of around 10 is achieved by using GPUs for the computationally expensive solution of the Vlasov equation. Additional major savings are obtained due to the coupling where the amount of savings roughly corresponds to the fraction of the domain where the kinetic equations are solved. The coupled codes were then tested on the propagation of whistler waves and on the GEM reconnection challenge. JCP 283 (2015) 436

Tobias Grafke, Rainer Grauer, Tobias Schäfer, Eric Vanden-Eijnden Relevance of instantons in Burgers turbulence Instanton calculations are performed in the context of stationary Burgers turbulence to estimate the tails of the probability density function (PDF) of velocity gradients. These results are then compared to those obtained from massive direct numerical simulations (DNS) of the randomly forced Burgers equation. The instanton predictions are shown to agree with the DNS in a wide range of regimes, including those that are far from the limiting cases previously considered in the literature. These results settle the controversy of the relevance of the instanton approach for the prediction of the velocity gradient PDF tail exponents. They also demonstrate the usefulness of the instanton formalism in Burgers turbulence, and suggest that this approach may be applicable in other contexts, such as 2D and 3D turbulence in compressible and incompressible flows. EPL 109 (2015) 34003

Tobias Grafke, Rainer Grauer, Stephan Schindel Efficient Computation of Instantons for Multi-Dimensional Turbulent Flows with Large Scale Forcing Extreme events play a crucial role in fluid turbulence. Inspired by methods from field theory, these extreme events, their evolution and probability can be computed with help of the instanton formalism as minimizers of a suitable action functional. Due to the high number of degrees of freedom in multi-dimensional fluid flows, traditional global minimization techniques quickly become prohibitive because of their memory requirements. We outline a novel method for finding the minimizing trajectory in a wide class of problems that typically occurs in the turbulence setup, where the underlying dynamical system is a non-gradient, non-linear partial differential equation. We demonstrate the efficiency of the algorithm in terms of performance and memory by computing high resolution instanton field configurations corresponding to viscous shocks for 1D and 2D compressible turbulent flows. Communications in Computational Physics 18 (2015) 577

Tobias Grafke, Rainer Grauer, Tobias Schäfer The instanton method and its numerical implementation in fluid mechanics A precise characterization of structures occurring in turbulent fluid flows at high Reynolds numbers is one of the last open problems of classical physics. In this review we discuss recent developments related to the application of instanton methods to turbulence. Instantons are saddle point configurations of the underlying path integrals. They are equivalent to minimizers of the related Freidlin–Wentzell action and known to be able to characterize rare events in such systems. While there is an impressive body of work concerning their analytical description, this review focuses on the question on how to compute these minimizers numerically. In a short introduction we present the relevant mathematical and physical background before we discuss the stochastic Burgers equation in detail. We present algorithms to compute instantons numerically by an efficient solution of the corresponding Euler–Lagrange equations. A second focus is the discussion of a recently developed numerical filtering technique that allows to extract instantons from direct numerical simulations. In the following we present modifications of the algorithms to make them efficient when applied to two- or three-dimensional (2D or 3D) fluid dynamical problems. We illustrate these ideas using the 2D Burgers equation and the 3D Navier–Stokes equations. Journal of Physics A: Mathematical and Theoretical (Topical Review) 48 (2015) 333001

Tobias Grafke, Rainer Grauer, Tobias Schäfer, Eric Vanden-Eijnden Arclength parametrized Hamilton's equations for the calculation of instantons A method is presented to compute minimizers (instantons) of action functionals using arclength parametrization of Hamilton's equations. This method can be interpreted as a local variant of the geometric minimum action method (gMAM) introduced to compute minimizers of the Freidlin-Wentzell action functional that arises in the context of large deviation theory for stochastic differential equations. The method is particularly well-suited to calculate expectations dominated by noise-induced excursions from deterministically stable fixpoints. Its simplicity and computational efficiency are illustrated here using several examples: a finite-dimensional stochastic dynamical system (an Ornstein-Uhlenbeck model) and two models based on stochastic partial differential equations: the $\phi^4$-model and the stochastically driven Burgers equation. SIAM: Multiscale Modeling and Simulation 12 (2014) 566

Holger Homann, Yannick Ponty, Giorgio Krstulovic, Rainer Grauer Structures and Lagrangian statistics of the Taylor-Green Dynamo The evolution of a Taylor-Green forced magnetohydrodynamic (MHD) system showing dynamo activity is analyzed via direct numerical simulations. The statistical properties of the velocity and magnetic field in Eulerian coordinates and along trajectories of fluid elements (Lagrangian coordinates) are studied during the kinematic, non-linear and saturated regime. We find that the probability density functions (PDFs) of the magnetic field change from strongly non-Gaussian PDFs in the kinematic regime to quasi-Gaussian PDFs in the saturated one. Their corresponding flatness give a precise handle on the definition of the limiting points of the different regimes. Fluid element (tracer) trajectories change from chaotic quasi-isotropic (kinematic phase) to strongly magnetic field aligned (saturated phase). This is connected to a dramatic increase of the correlation time of velocity and magnetic field fluctuations experienced by tracers largely exceeding one turbulent large-eddy turn-over time. A remarkable consequence is an intermittent scaling regime of the Lagrangian magnetic field structure functions at unusually long time scales. New J. Phys. 16 (2014) 75014

Sophia Kreuzahler, Daniel Schulz, Holger Homann, Yannick Ponty, Rainer Grauer Numerical study of impeller-driven von Karman flows via a volume penalization method Studying strongly turbulent flows is still a major challenge in fluid dynamics. It is highly desirable to have comparable experiments to obtain a better understanding of the mechanisms generating turbulence. The von Kármán flow apparatus is one of those experiments that has been used in various turbulence studies by different experimental groups over the last two decades. The von Kármán flow apparatus produces a highly turbulent flow inside a cylinder vessel driven by two counter-rotating impellers. The studies cover a broad range of physical systems including incompressible flows, especially water and air, magnetohydrodynamic systems using liquid metal for understanding the important topic of the dynamo instability, particle tracking to study Lagrangian type turbulence and recently quantum turbulence in super-fluid helium. Therefore, accompanying numerical studies of the von Kármán flow that compare quantitatively data with those from experiments are of high importance for understanding the mechanism producing the characteristic flow patterns. We present a direct numerical simulation (DNS) version the von Kármán flow, forced by two rotating impellers. The cylinder geometry and the rotating objects are modelled via a penalization method and implemented in a massive parallel pseudo-spectral Navier\u2013Stokes solver. From the wide range of different impellers used in von Kármán water and sodium experiments we choose a special configuration (TM28), in order to compare our simulations with the according set of well documented water experiments. Though this configuration is different from the one in the final VKS experiment (TM73), using our method it is quite easy to change the impeller shape to the one actually used in VKS. The decomposition into poloidal and toroidal components and the mean velocity field from our simulations are in good agreement with experimental results. In addition, we analysed the flow structure close to the impeller blades, a region hardly accessible to experiments. Depending on the blade geometry different vortex topologies are found. The very promising results imply that our numerical modelling could also be applied to other physical systems and configurations driven by the von Kármán flow. New J.Phys. 16 (2014) 103001

Tobias Grafke, Rainer Grauer, Thomas C. Sideris Turbulence properties and global regularity of a modified Navier-Stokes equation We introduce a modification of the Navier-Stokes equation that has the remarkable property of possessing an infinite number of conserved quantities in the inviscid limit. This new equation is studied numerically and turbulence properties are analyzed concerning energy spectra and scaling of structure functions. The dissipative structures arising in this new equation are curled vortex sheets contrary to vortex tubes arising in Navier-Stokes turbulence. The numerically calculated scaling of structure functions is compared with a phenomenological model based on the She-L\'ev\^eque approach. Finally, for this equation we demonstrate global well-posedness for sufficiently smooth initial conditions in the periodic case and in $\mathbb R^3$. The key feature is the availability of an additional estimate which shows that the $L^4$-norm of the velocity field remains finite. Physica D 254 (2013) 18

Tobias Grafke, Rainer Grauer, Tobias Schäfer Instanton filtering for the stochastic Burgers equation We address the question whether one can identify instantons in direct numerical simulations of the stochastically driven Burgers equation. For this purpose, we first solve the instanton equations using the Chernykh-Stepanov method [Phys. Rev. E 64, 026306 (2001)]. These results are then compared to direct numerical simulations by introducing a filtering technique to extract prescribed rare events from massive data sets of realizations. Using this approach we can extract the entire time history of the instanton evolution which allows us to identify the different phases predicted by the direct method of Chernykh and Stepanov with remarkable agreement. Journal of Physics A: Mathematical and Theoretical (FAST TRACK COMMUNICATION) 46 (2013) 62002

Tobias Grafke and Rainer Grauer Lagrangian approach for finite-time Euler singularities in three-dimensional incompressible fluid flow We address the question whether a singularity in a three-dimensional incompressible inviscid fluid flow can occur in finite time. Analytical considerations and numerical simulations suggest high-symmetry flows being a promising candidate for a finite-time blowup. Utilizing Lagrangian and geometric non-blowup criteria, we present numerical evidence against the formation of a finite-time singularity for the high-symmetry vortex dodecapole initial condition. We use data obtained from high resolution adaptively refined numerical simulations and inject Lagrangian tracer particles to monitor geometric properties of vortex line segments. We then verify the assumptions made by analytical non-blowup criteria introduced by Deng et. al [Commun. PDE 31 (2006)] connecting vortex line geometry (curvature, spreading) to velocity increase to rule out singular behavior. Applied Mathematics Letters 26 (2013) 500

Holger Homann, Jeremie Bec and Rainer Grauer Effect of turbulent fluctuations on the drag and lift forces on a towed sphere and its boundary layer The impact of turbulent fluctuations on the forces exerted by a fluid on a towed spher- ical particle is investigated by means of high-resolution direct numerical simulations. The measurements are carried out using a novel scheme to integrate the two-way cou- pling between the particle and the incompressible surrounding fluid flow maintained in a high-Reynolds-number turbulent regime. The main idea consists in combining a Fourier pseudo-spectral method for the fluid with an immersed-boundary technique to impose the no-slip boundary condition on the surface of the particle. This scheme is shown to converge as the power 3/2 of the spatial resolution. This behaviour is explained by the L2 convergence of the Fourier representation of a velocity field displaying discontinu- ities of its derivative. Benchmarking of the code is performed by measuring the drag and lift coefficients and the torque-free rotation rate of a spherical particle in various configurations of an upstream-laminar carrier flow. Such studies show a good agreement with experimental and numerical measurements from other groups. A study of the tur- bulent wake downstream the sphere is also reported. The mean velocity deficit is shown to behave as the inverse of the distance from the particle, as predicted from classical similarity analysis. This law is reinterpreted in terms of the principle of “permanence of large eddies” that relates infrared asymptotic self-similarity to the law of decay of energy in homogeneous turbulence. The developed method is then used to attack the problem of an upstream flow that is in a developed turbulent regime. It is shown that the average drag force increases as a function of the turbulent intensity and the particle Reynolds number. This increase is sig- nificantly larger than predicted by standard drag correlations based on laminar upstream flows. It is found that the relevant parameter is the ratio of the viscous boundary layer thickness to the dissipation scale of the ambient turbulent flow. The drag enhancement can be motivated by the modification of the mean velocity and pressure profile around the sphere by small scale turbulent fluctuations. It is demonstrated that the variance of the drag force fluctuations can be modelled by means of standard drag correlations. Temporal correlations of the drag and lift forces are also presented. J. Fluid Mech. 721 (2013) 155179

T. Grafke and R. Grauer Lagrangian and geometric analysis of finite-time Euler singularities We present a numerical method of analyzing possibly singular incompressible 3D Euler flows using massively parallel high-resolution adaptively refined numerical simulations up to 8192^3 mesh points. Geometrical properties of Lagrangian vortex line segments are used in combination with analytical non-blowup criteria by Deng et al [Commun. PDE 31 (2006)] to reliably distinguish between singular and near-singular flow evolution. We then apply the presented technique to a class of high-symmetry initial conditions and present numerical evidence against the formation of a finite-time singularity in this case. Procedia IUTAM 9 (2013) 32

T. Grafke, R. Grauer and T. Schäfer Instanton filtering for the stochastic Burgers equation Extreme events in stochastic nonlinear systems play an essential role in nature. Characterizing their likelihood is a fundamental albeit challenging problem since the tails of the underlying probability distributions are usually non-Gaussian and governed by saddlepoints of the corresponding path integrals, so-called “instantons”. Understanding intermittency in turbulent systems is still one of the open problems in classical physics. Since intermittency is governed by the non-Gaussianity of rare fluctuations, instantons might offer a way to better understand the behavior of turbulent systems. In the present work we concentrate on rare fluctuations in Burgers turbulence and we address the question whether one can identify instantons in direct numerical simulations of the stochastically driven Burgers equation. This is of special importance since this demonstrates that instantons indeed form the skeleton of rare turbulent fluctuations. For this purpose, we first solve the instanton equations using the Chernykh-Stepanov method [Phys. Rev. E 64, 026306 (2001)]. These results are then compared to direct numerical simulations by introducing a filtering technique to extract prescribed rare events from massive data sets of realizations. Using this approach we can extract the entire time history of the instanton evolution, which allows us to identify the different phases predicted by the direct method of Chernykh and Stepanov with remarkable agreement. European Physics News (Highlight) 43 (2013) 12

R. Grauer, H. Homann, and J.F. Pinton Longitudinal and Transverse structure functions in high Reynolds-number turbulence Using exact relations between velocity structure functions [1–3] and neglecting pressure contributions in a first approximation, we obtain a closed system and derive simple order-dependent rescaling relationships between longitudinal and transverse structure functions. By means of numerical data with turbulent Reynolds numbers ranging from λ = 320 to λ = 730, we establish a clear correspondence between their respective scaling range, while confirming that their scaling exponents do differ. This difference does not seem to depend on Reynolds number. Making use of the Mellin transform, we further map longitudinal to (rescaled) transverse probability density functions. New Journal of Physics 14 (2012) 63016

Th. Hater, H. Homann, and R. Grauer A Lagrangian model for the evolution of turbulent magnetic and passive scalar field In this paper we present an extension of the Recent Fluid Deformation (RFD) closure introduced by Chevillard and Meneveau [1] which was developed for modeling the time evolution of Lagrangian fluctuations in incompressible Navier-Stokes turbulence. We apply the RFD closure to study the evolution of magnetic and passive scalar fluctuations. This comparison is especially interesting since the stretching term for the magnetic field and for the gradient of the passive scalar are similar but differ by a sign such that the effect of stretching and compression by the turbulent velocity field is reversed. Probability density functions (PDFs) of magnetic fluctuations and fluctuations of the gradient of the passive scalar obtained from the RFD closure are compared against PDFs obtained from direct numerical simulations. Phys. Rev. E 83 (2011) 17302

H. Homann, D. Schulz, and R. Grauer Conditional Eulerian and Lagrangian velocity increment statistics of fully developed turbulent ﬂow Conditional statistics of homogeneous isotropic turbulent ﬂow is investigated by means of high-Reynolds number direct numerical simulations performed with 20483 collocation points. Eulerian as well as Lagrangian velocity increment statistics under several conditions are analyzed and compared. In agreement with experimental data longitudinal probability density functions P (δl u\|ǫl ) conditioned on a scale-averaged energy dissipation rate are close to Gaussian distributions over all scales within the inertial range of scales. Also trans- verse increments conditioned on either the dissipation rate or the square of the vorticity have quasi-Gaussian probability distribution functions (PDFs). Concerning Lagrangian statistics we found that conditioning on a trajectory averaged energy-dissipation rate ǫτ signiﬁcantly reduces the scale dependence of the increment PDFs P (δτ ui \|ǫτ ). By means of dimensional arguments we propose a novel condition for Lagrangian incre- ments which is shown to reduce even more the ﬂatness of the corresponding PDFs and thus intermittency in the inertial range of scales. The conditioned PDF corresponding to the smallest increment considered is reasonably well described by the K41-prediction of the PDF of acceleration. Conditioned structure functions show approximately K41-scaling with a larger scaling range than the unconditioned ones. Physics of Fluids 23 (2011) 55102

F. Effenberger, K. Thust, L. Arnold, R. Grauer, and J. Dreher Numerical Simulation of Current Sheet Formation in a Quasi-Separatrix Layer using Adaptive Mesh Refinement The formation of a thin current sheet in a magnetic quasi-separatrix layer (QSL) is investi- gated by means of numerical simulation using a simplified ideal, low-β, MHD model. The initial configuration and driving boundary conditions are relevant to phenomena observed in the solar corona and were studied earlier by Aulanier et al., A&A 444, 961 (2005). In extension to that work, we use the technique of adaptive mesh refinement (AMR) to significantly enhance the local spatial resolution of the current sheet during its formation, which enables us to follow the evolution into a later stage. Our simulations are in good agreement with the results of Aulanier et al. up to the calculated time in that work. In a later phase, we observe a basically unarrested collapse of the sheet to length scales that are more than one order of magnitude smaller than those reported earlier. The current density attains correspondingly larger maximum values within the sheet. During this thinning process, which is finally limited by lack of resolution even in the AMR studies, the current sheet moves upward, following a global expansion of the magnetic structure during the quasi-static evolution. The sheet is locally one-dimensional and the plasma flow in its vicinity, when transformed into a co-moving frame, qualitatively resembles a stagnation point flow. In conclusion, our simulations support the idea that extremely high current densities are generated in the vicinities of QSLs as a response to external perturbations, with no sign of saturation. Physics of Plasmas 18 (2011) 32902

Ch. Schwarz, Ch. Beetz, J. Dreher, and R. Grauer Lyapunov exponents and information dimension of the mass distribution in turbulent compressible flows Turbulent density fluctuations in isothermal highly compressible turbulent flows are highly clumped and can be quantified by the scaling properties of powers of the mass distribution. This Eulerian quantity can be related to Lagrangian properties of the system given by the Lyapunov exponents of tracer particles advected with the flow. Using highly resolved numerical simulations, we show that the Kaplan-Yorke conjecture holds within numerical uncertainties. Physics Lett. A 374 (2010) 1039

H. Soltwisch, P. Kempkes, F. Mackel, H. Stein, J. Tenfelde, L. Arnold, J. Dreher and R. Grauer FlareLab: early results The FlareLab experiment at Bochum University has been constructed to generate and investigate plasma-filled magnetic flux tubes similar to archshaped solar prominences, which often result in coronal mass ejections (CMEs). In its first version, the device has been used to reproduce and extend previous studies of Bellan et al (1998 Phys. Plasmas 5 1991). Here the plasma source consists of two electrodes, which can be connected to a 1.0 kJ capacitor bank, and of a horseshoe magnet, which provides an arch-shaped guiding field. The discharge is ignited in a cloud of hydrogen gas that has been puffed into the space above the electrodes. In the first few microseconds the plasma current rises at a rate of several kAμs−1, causing the plasma column to pinch along the guiding B-field and to form an expanding loop structure. The observed dynamics of the magnetic flux tubes is analysed by means of three-dimensional MHD simulations in order to determine the influence of parameters like the initial magnetic field geometry on magnetic stability. At present, FlareLab is redesigned to mimic a model that was proposed by Titov and D´emoulin (1999 Astron. Astrophys. 351 707) to investigate twisted magnetic configurations in solar flares. Plasma Phys. Control. Fusion 52 (2010) 124030

H. Homann, O. Kamps, R. Friedrich and R. Grauer Bridging from Eulerian to Lagrangian statistics in 3D hydro- and magnetohydrodynamic turbulent flows We present measurements of conditional PDFs which allow to systematically bridge from Eulerian to Lagrangian statistics in fully developed 3D turbulence. The transition is investigated both for hydro- as well as magnetohydrodynamic flows and comparisons are drawn. Significant differences in the transition PDFs are observed for these flows and traced back to the differing coherent structures. In particular we address the problem of an increasing degree of intermittency going from Eulerian to Lagrangian coordinates by means of the conditional PDFs involved in this transformation. First simple models of these PDFs are investigated in order to distinguish different contributions to the degree of Lagrangian intermittency. New Journal of Physics 11 (2009) 73020

H. Homann, J. Bec, H. Fichtner, and R. Grauer Clustering of passive impurities in MHD turbulence The transport of heavy, neutral or charged, point-like particles by incompressible, resistive magnetohydrodynamic (MHD) turbulence is investigated by means of high-resolution numerical simulations. The spatial distribution of such impurities is observed to display strong deviations from homogeneity, both at dissipative and inertial range scales. Neutral particles tend to cluster in the vicinity of coherent vortex sheets due to their viscous drag with the flow, leading to the simultaneous presence of very concentrated and almost empty regions. The signature of clustering is different for charged particles. These exhibit in addition to the drag the Lorentz-force. The regions of spatial inhomogeneities change due to attractive and repulsive vortex sheets. While small charges increase clustering, larger charges have a reverse effect. Physics of Plasmas 16 (2009) 82308

O. Kamps, R. Friedrich, and R. Grauer An exact relation between Eulerian and Lagrangian velocity increment statistics We present a formal connection between Lagrangian and Eulerian velocity increment distributions which is applicable to a wide range of turbulent systems ranging from turbulence in incompressible fluids to magnetohydrodynamic turbulence. For the case of the inverse cascade regime of two-dimensional turbulence we numerically estimate the transition probabilities involved in this connection. In this context we are able to directly identify the processes leading to strongly non-Gaussian statistics for the Lagrangian velocity increments. Phys. Rev. E 79 (2009) 66301

R. Friedrich, R. Grauer, H. Homann, and O. Kamps Statistics of a mixed Eulerian-Lagrangian velocity increment in fully developped turbulence We investigate the relationship between Eulerian and Lagrangian probability density functions obtained from numerical simulations of two-dimensional as well as three-dimensional turbulence. We show that in contrast to the structure functions of the Lagrangian velocity increment $\delta_\tau {\bf v}({\bf y})= {\bf u}({\bf x}({\bf y},\tau),\tau)- {\bf u}({\bf y},0)$, where ${\bf u}({\bf x},t)$ denotes the Eulerian velocity and ${\bf x}({\bf y},t)$ the particle path initially starting at ${\bf x}({\bf y},0)={\bf y}$, the structure functions of the velocity increment $\delta_\tau {\bf w}({\bf y}) ={\bf u}({\bf x}({\bf y},\tau),\tau)-{\bf u}({\bf y},\tau)$ exhibits a wide range of scaling behavior. Similar scaling indices are detected for the structure functions for particles diffusing in frozen turbulent fields. Furthermore, we discuss a connection to the scaling of Eulerian transversal structure functions. Physica Scripta 79 (2009) 55403

J. Kleimann, A. Kopp, H. Fichtner, and R. Grauer A novel code for numerical 3-D MHD studies of CME expansion A recent third-order, essentially non-oscillatory central scheme to advance the equations of single-fluid magnetohydrodynamics (MHD) in time has been implemented into a new numerical code. This code operates on a 3-D Cartesian, non-staggered grid, and is able to handle shock-like gradients without producing spurious oscillations. To demonstrate the suitability of our code for the simulation of coronal mass ejections (CMEs) and similar heliospheric transients, we present selected results from test cases and perform studies of the solar wind expansion during phases of minimum solar activity. We can demonstrate convergence of the system into a stable Parker-like steady state for both hydrodynamic and MHD winds. The model is subsequently applied to expansion studies of CME-like plasma bubbles, and their evolution is monitored until a stationary state similar to the initial one is achieved. In spite of the model's (current) simplicity, we can confirm the CME's nearly self-similar evolution close to the Sun, thus highlighting the importance of detailed modelling especially at small heliospheric radii. Ann. Geophys. 27 (2009) 989

R. Kissmann, J. Kleimann, H. Fichtner and R. Grauer Local turbulence simulations for the multiphase ISM In this paper, we show results of numerical simulations for the turbulence in the interstellar medium (ISM). These results were obtained using a Riemann solver-free numerical scheme for high-Mach number hyperbolic equations. Here, we especially concentrate on the physical properties of the ISM. That is, we do not present turbulence simulations trimmed to be applicable to the ISM. The simulations are rather based on physical estimates for the relevant parameters of the interstellar gas. Applying our code to simulate the turbulent plasma motion within a typical interstellar molecular cloud, we investigate the influence of different equations of state (isothermal and adiabatic) on the statistical properties of the resulting turbulent structures. We find slightly different density power spectra and dispersion maps, while both cases yield qualitatively similar dissipative structures, and exhibit a departure from the classical Kolmogorov case towards a scaling described by the She¿Leveque model. Solving the full energy equation with realistic heating/cooling terms appropriate for the diffuse interstellar gas (DIG), we are able to reproduce a realistic two-phase distribution of cold and warm plasma. When extracting maps of polarized intensity from our simulation data, we find encouraging similarity to actual observations. Finally, we compare the actual magnetic field strength of our simulations to its value inferred from the rotation measure. We find these to be systematically different by a factor of about 1.15, thus highlighting the often-underestimated influence of varying line-of-sight particle densities on the magnetic field strength derived from observed rotation measures. Mon. Not. R. Astron. Soc. 391 (2008) 1577

T. Grafke, H. Homann, J. Dreher, and R. Grauer Numerical simulations of possible finite time singularities in the incompressible Euler equations: comparison of numerical methods The numerical simulation of the 3D incompressible Euler equation is analyzed with respect to different integration methods. The numerical schemes we considered include spectral methods with different strategies for dealiasing and two variants of finite difference methods. Based on this comparison, a Kida-Pelz like initial condition is integrated using adaptive mesh refinement and estimates on the necessary numerical resolution are given. This estimate is based on analyzing the scaling behavior similar to the procedure in critical phenomena and present simulations are put into perspective. Physica D 237 (2008) 1932

A. Arneodo, R. Benzi, J. Berg, L. Biferale, E. Bodenschatz, A. Busse, E. Calzavarini, B. Castaing, M. Cencini, L. Chevillard, R.T. Fisher, R. Grauer, H. Homann, D. Lamb, A.S. Lanotte, E. Leveque, B. Luthi, J. Mann, N. Mordant, W.C. Mueller, S. Ott, N.T. Ouellette, J.F. Pinton, S.B. Pope, S.G. Roux, F. Toschi, H. Xu, P.K. Yeung Universal intermittent properties of particle trajectories in highly turbulent flows We present a collection of eight data sets from state-of-the-art experiments and numerical simulations on turbulent velocity statistics along particle trajectories obtained in different flows with Reynolds numbers in the range R-lambda is an element of [120740]. Lagrangian structure functions from all data sets are found to collapse onto each other on a wide range of time lags, pointing towards the existence of a universal behavior, within present statistical convergence, and calling for a unified theoretical description. Parisi-Frisch multifractal theory, suitably extended to the dissipative scales and to the Lagrangian domain, is found to capture the intermittency of velocity statistics over the whole three decades of temporal scales investigated here. Phys. Rev. Lett. 100 (2008) 254504

L. Arnold, C. Beetz, J. Dreher, H. Homann, C. Schwarz and R. Grauer Massively Parallel Simulations of Solar Flares and Plasma Turbulence Some of the outstanding problems in space- and astrophysical plasmasystems include solar flares and hydro- or magnetohydrodynamic turbulence (e.g. in the interstellar medium). Both fields demand for high resolution and thus numerical simulations need an efficient parallel implementation. We will describe the physics behind these problems and present the numerical frameworks for solving these problems on massive parallel computers. Parallel Computing: Architectures, Algorithms and Applications 15 (2008) 467

H. Fichtner, A. Kopp, J. Kleimann, and R. Grauer On MHD modelling of Coronal Mass Ejections We give at first a brief overview of the motivation for magnetohydrodynamic simulations of coronal mass ejections that can be classified as principal, local, and global, and discuss some of the present problems with the modelling. Besides the initiation, acceleration, evolution, and interactions of CMEs with each other and with planetary magnetospheres, we identify the need for large-scale MHD and multi-fluid models that explicitly incorporate effects of kinetic processes occuring on micro- or meso-scales. Concentrating on the accel- eration and heating of the solar wind and CMEs by plasma waves, we describe an alternative route to this goal. Subsequently, we present CWENO-based local CME simulations and relate them to observations with the ACE spacecraft near 1 AU. Astron. Soc. Pac. Conf. Ser. 385 (2008) 151

C. Beetz, C. Schwarz, J. Dreher, R. Grauer Density-PDFs and Lagrangian Statistics of highly compressible Turbulence In isothermal, highly compressible turbulent flows, density fluctuations follow a log-normal distribution. We establish a connection between these density fluctuations and the probability-density-functions (PDF) of Lagrangian tracer particles advected with the flow. Our predicted particle statistics is tested against large scale numerical simulations, which were performed with $512^3$ collocation points and 2 million tracer particles integrated over several dynamical times. Physics Letters A 372 (2008) 3037

L. Arnold, J. Dreher, R. Grauer, H. Soltwisch, H. Stein Three-dimensional MHD simulation of expanding magnetic flux ropes Three-dimensional, time-dependent numerical simulations of the dynamics of magnetic flux ropes are presented. The simulations are targeted towards an experiment previously conducted at CalTech (Bellan, P. M. and J. F. Hansen, Phys. Plasmas, {\bf 5}, 1991 (1998)) which aimed at simulating Solar prominence eruptions in the laboratory. The plasma dynamics is described by ideal MHD using different models for the evolution of the mass density. Key features of the reported experimental observations like pinching of the current loop, its expansion and distortion into helical shape are reproduced in the numerical simulations. Details of the final structure depend on the choice of a specific model for the mass density. Phys. Plasmas 15 (2008) 42106

L. Arnold, J. Dreher and R. Grauer A semi implicit Hall-MHD solver using whistler wave preconditioning The dispersive character of the Hall-MHD solutions, in particular the whistler waves, is a strong restriction to numerical treatments of this system. Numerical stability demands a time step dependence of the form $\Delta t\propto (\Delta x)^2$ for explicit calculations. A new semi--implicit scheme for integrating the induction equation is proposed and applied to a reconnection problem. It it based on a fix point iteration with a physically motivated preconditioning. Due to its convergence properties, short wavelengths converge faster than long ones, thus it can be used as a smoother in a nonlinear multigrid method. Comp. Phys. Comm. 178 (2008) 553

H. Schmitz and R. Grauer Vlasov simulations of collisionless magnetic reconnection without background density A standard starting point for the simulation of collisionless reconnection is the Harris equilibrium which is made up of a current sheet that separates two regions of opposing magnetic field. Magnetohydrodynamic simulations of collisionless reconnection usually include a homogeneous background density for reasons of numerical stability. While, in some cases, this is a realistic assumption, the background density may introduce new effects both due to the more involved structure of the distribution function or due to the fact that the Alfv`en speed remains finite far away from the current sheet.We present a fully kinetic Vlasov simulation of the perturbed Harris equilibrium using a Vlasov code. Parameters are chosen to match the Geospace Environment Modeling (GEM) Magnetic Reconnection Challenge but excluding the background density. This allows to compare with earlier simulations [Schmitz, Grauer, Phys. Plasmas 13 (2006) 092309] which include the background density. It is found that the absence of a background density causes the reconnection rate to be higher. On the other hand, the time until the onset of reconnection is hardly affected. Again the off diagonal elements of the pressure tensor are found to be important on the X-line but with modified importance for the individual terms. Communications in Nonlinear Science and Numerical Simulation 13 (2008) 169

A. Busse, W.C Müller, H. Homann and R. Grauer Statistics of passive tracers in three-dimensional magnetohydrodynamic turbulence Magnetohydrodynamic (MHD) turbulence is studied from the Lagrangian viewpoint by following fluid particle tracers in high resolution direct numerical simulations. Results regarding turbulent diffusion and dispersion as well as Lagrangian structure functions are presented. Whereas turbulent single-particle diffusion exhibits essentially the same behavior in Navier-Stokes and MHD turbulence, two-particle relative dispersion in the MHD case differs significantly from the Navier-Stokes behavior. This observation is linked to the local anisotropy of MHD turbulence which is clearly reflected by quantities measured in a Lagrangian frame of reference. In the MHD case the Lagrangian structure functions display a lower level of intermittency as compared to the Navier-Stokes case contrasting Eulerian results. This is not only true for short time increments [Homann, \emph{et al.}, to be published in J.\ Plasma Phys. (2007)] but also holds for increments up to the order of the integral time scale. The apparent discrepancy can be explained by the difference in the characteristic shapes of fluid particle trajectories in the vicinity of most singular dissipative structures. Phys. Plasmas 14 (2007) 122303

H. Homann, J. Dreher and R. Grauer Impact of the floating-point precision and interpolation scheme on the results of DNS of turbulence by pseudo-spectral codes In this paper we investigate the impact of the floating-point precision and interpolation scheme on the results of direct numerical simulations (DNS) of turbulence by pseudo-spectral codes. Three different types of floating-point precision configurations show no differences in the statistical results. This implies that single precision computations allow for increased Reynolds numbers due to the reduced amount of memory needed. The interpolation scheme for obtaining velocity values at particle positions has a noticeable impact on the Lagrangian acceleration statistics. A tri-cubic scheme results in a slightly broader acceleration probability density function than a tri-linear scheme. Furthermore the scaling behavior obtained by the cubic interpolation scheme exhibits a tendency towards a slightly increased degree of intermittency compared to the linear one. Comp. Phys. Comm. 177 (2007) 560

V. Mezentsev, J.S. Petrovic, M. Dubov, I. Bennion, J. Dreher, H. Schmitz, and R. Grauer Femtosecond laser microfabrication of subwavelength structures in photonics This paper describes experimental and numerical results of the plasma-assisted microfabrication of subwavelength structures by means of point-by point femtosecond laser inscription. It is shown that the spatio-temporal evolution of light and plasma patterns critically depend on input power. Subwavelength inscription corresponds to the supercritical propagation regimes when pulse power is several times self-focusing threshold. Experimental and numerical profiles show quantitative agreement. Proc SPIE 6459 (2007) 64590

H. Homann, R. Grauer, A. Busse and W.C. Müller Lagrangian Statistics of Navier-Stokes- and MHD-Turbulence We report on a comparison of high-resolution numerical simulations of Lagrangian particles advected by incompressible turbulent hydro- and magnetohydrodynamic (MHD) flows. Numerical simulations were performed with up to $1024^3$ collocation points and 10 million particles in the Navier-Stokes case and $512^3$ collocation points and 1 million particles in the MHD case. In the hydrodynamics case our findings compare with recent experiments from Mordant et al. [1] and Xu et al. [2]. They differ from the simulations of Biferale et al. [3] due to differences of the ranges choosen for evaluating the structure functions. In Navier-Stokes turbulence intermittency is stronger than predicted by a multifractal approach of [3] whereas in MHD turbulence the predictions from the multifractal approach are more intermittent than observed in our simulations. In addition, our simulations reveal that Lagrangian Navier-Stokes turbulence is more intermittent than MHD turbulence, whereas the situation is reversed in the Eulerian case. Those findings can not consistently be described by the multifractal modeling. The crucial point is that the geometry of the dissipative structures have different implications for Lagrangian and Eulerian intermittency. Application of the multifractal approach for the modeling of the acceleration PDFs works well for the Navier-Stokes case but in the MHD case just the tails are well described. J. Plasma Phys. 73 (2007) 821

R. Kissmann and R. Grauer A low dissipation essentially non-oscillatory central scheme Here we present a new, semidiscrete, central scheme for the numerical solution of one-dimensional systems of hyperbolic conservation laws. The method presented in this paper is an extension of the centrally weighted non-oscillatory schemes (CWENO) presented in [7], [5] and [6]. The method suggested in this manuscript is derived independently of the order of the scheme. The gain in this new method is a decreased dissipation especially for high Mach-number flows, which are frequently encountered, e. g., in astrophysical contexts or turbulent systems. Comp. Phys. Comm. 176 (2007) 522

H. Schmitz and R. Grauer Kinetic Vlasov Simulations of collisionless magnetic Reconnection A fully kinetic Vlasov simulation of the Geospace Environment Modeling (GEM) Magnetic Reconnection Challenge is presented. Good agreement is found with previous kinetic simulations using particle in cell (PIC) codes, confirming both the PIC and the Vlasov code. In the latter the complete distribution functions $f_k$ ($k=i,e$) are discretised on a numerical grid in phase space. In contrast to PIC simulations, the Vlasov code does not suffer from numerical noise and allows a more detailed investigation of the distribution functions. The role of the different contributions of Ohm's law are compared by calculating each of the terms from the moments of the $f_k$. The important role of the off--diagonal elements of the electron pressure tensor could be confirmed. The inductive electric field at the X--Line is found to be dominated by the non--gyrotropic electron pressure, while the bulk electron inertia is of minor importance. Detailed analysis of the electron distribution function within the diffusion region reveals the kinetic origin of the non--gyrotropic terms. Physics of Plasmas 13 (2006) 92309

V. Mezentsev, M. Dubov, J. Petrovic, I. Bennion, J. Dreher, and R. Grauer Role of Plasma in Femtosecond Laser Pulse Propagation This paper describes physics of nonlinear ultra-short laser pulse propagation affected by plasma created by the pulse itself. Major applications are also discussed. Nonlinear propagation of the femtosecond laser pulses in gaseous and solid transparent dielectric media is a fundamental physical phenomenon in a wide range of important applications such as laser lidars, laser micro-machining (ablation) and microfabrication etc. These applications require very high intensity of the laser field, typically 10^13-10^15 TW/cm^2. Such high intensity leads to significant ionisation and creation of electron-ion or electron-hole plasma. The presence of plasma results into significant multiphoton and plasma absorption and plasma defocusing. Consequently, the propagation effects appear extremely complex and result from competitive counteraction of the above listed effects and Kerr effect, diffraction and dispersion. The theoretical models used for consistent description of laser-plasma interaction during femtosecond laser pulse propagation are derived and discussed. It turns out that the strongly nonlinear effects such self-focusing followed by the pulse splitting are essential. These phenomena feature extremely complex dynamics of both the electromagnetic field and plasma density with different spatio-temporal structures evolving at the same time. Some numerical approaches capable to handle all these complications are also discussed. AIP Conf. Proc. 876 (2006) 169

V. Mezentsev, J. Petrovic, J. Dreher, and R. Grauer Adaptive modeling of the femtosecond inscription in silica We present an adaptive mesh approach to high performance comprehensive investigation of dynamics of light and plasma pattens during the process of direct laser inscription. The results reveal extreme variations of spatial and temporal scales and tremendous complexity of these patterns which was not feasible to study previously. Proc. SPIE 6107 (2006) 6107

H. Schmitz and R. Grauer Comparison of time splitting and backsubstitution methods for integrating Vlasov's equation with magnetic fields The standard approach for integrating the multidimensional Vlasov equation using grid based, conservative schemes is based on a time splitting approach. Here, we show that although the truncation error is of second order, time splitting introduces systematic heating of the plasma. We introduce a backsubstitution method, which not only avoids this deficiency but also is computationally less expensive. The general approach is demonstrated in conjunction with Boris? scheme for evaluating the characteristics. Comp. Phys. Comm. 175 (2006) 86

H. Schmitz and R. Grauer Darwin-Vlasov Simulations of magnetized Plasmas We present a new Vlasov code for collisionless plasmas in the nonrelativistic regime. A Darwin approximation is used for suppressing electromagnetic vacuum modes. The spatial integration is based on an extension of the flux-conservative scheme introduced by Filbet et al. \cite{FIL01}. Performance and accuracy is demonstrated by comparing it to a standard finite differences scheme for two test cases, including a Harris sheet magnetic reconnection scenario. J. Comp. Phys. 214 (2006) 738

J. Dreher, D. Laveder, R. Grauer, T. Passot and P.L. Sulem Formation and disruption of Alfvénic filaments in Hall-magnetohydrodynamics Magnetohydrodynamics with Hall effect (Hall-MHD) allows one to take into account scales of the order of the ion inertial length and the dispersive character of media like the Earth magnetosheath. In these conditions, weakly nonlinear quasi-monochromatic Alfv\'en waves propagating along an ambient magnetic field can be subject to transverse instabilities leading to the formation of intense magnetic filaments. This "filamentation" phenomenon, predicted by amplitude equations of nonlinear Schrödinger type and also observed in direct numerical simulations using spectral method is here reconsidered using a finite-difference adaptive mesh refinement code (AMR). This approach allows the simulation to be proceeded long enough to capture the destabilization of the filamentary structures and the formation of gradient singularities in the longitudinal direction, associated with the development of intense current sheets and with a strong acceleration of the plasma. Phys. Plasmas 12 (2005) 52319

J. Dreher, V. Ruban, and R. Grauer Axisymmetric flows in Hall-MHD: A tendency towards finite-time singularity formation Spontaneous development of shock-like singularities in axisymmetric solutions of the Hall-MHD equations is discussed. It is shown that the Hall-term in Ohm's law leads to a Burgers-type equation for the magnetic field evolution in weakly compressible regime. Numerical simulations are used to investigate the validity of this approximation for a particular class of initial conditions. Physica Scripta 72 (2005) 450

H. Homann and R. Grauer Bifurcation analysis of magnetic reconnection in Hall-MHD systems The dependence of the Hall-term on the width of the magnetic islands of the tearing-mode is examined. We applied the center manifold (CMF) theory to a Magnetohydrodynamic (MHD)-system. The MHD-system was chosen to be incompressible and includes in addition to viscosity the Hall-term in Ohm's law. For certain values of physical parameters the corresponding center manifold is two-dimensional and therefore the original partial differential equations could be reduced to a two-dimensional system of ordinary ones. This amplitude equations exhibit a pitchfork-bifurcation which corresponds to the occurrence of the tearing-mode. Eigenvalue-problems and linear equations due to the center manifold reduction were solved numerically with the Arpack++-library. Physica D 208 (2005) 59

J. Dreher and R. Grauer Racoon: A Parallel Mesh-Adaptive Framework for Hyperbolic Conservation Laws We report on the development of a computational framework for the parallel, mesh-adaptive solution of systems of hyperbolic conservation laws like the timedependent Euler equations in compressible gas dynamics orMagneto-Hydrodynamics (MHD) and similar models in plasma physics. Local mesh refinement is realized by the recursive bisection of grid blocks along each spatial dimension, implemented numerical schemes include standard finite-di erences and central schemes with Runge- Kutta integrators. Parallel execution is achieved through a configurable hybrid of multi-threading and MPI-distribution with dynamic load balancing. One- two- and three-dimensional test computations for the Euler equations have been carried out and show good parallel scaling behavior. The Racoon framework is currently used to study the formation of singularities in plasmas and fluids. Parallel Computing 31 (2005) 913

K. Germaschewski, A. Bhattacharjee, R. Grauer, D. Keyes, and B. Smith Using Krylov-Schwarz methods in an adaptive mesh refinement enviroment Much of the previous work in AMR methods has concentrated on solving hyperbolic equations with explicit timestepping. However, for many problems, either due to their physical nature (e.g. incompressible flows) or for performance reasons (semi-implicit and implicit numerical methods), it becomes necessary to solve global equations. This paper focuses on the application and performance of well-established preconditioned Krylov-Schwarz solvers in an AMR context, using a Krylov-Schwarz method to accelerate convergence while exploiting the hierarchical structure of AMR grids for multi-level preconditioning using the fast adaptive composite (FAC) algorithm. We present an implementation which allows us to leverage the powerful supply of preconditioners and linear solvers from the PETSc library. We apply this method to solve the three-dimensional Euler equations in the search for a finite-time singularity. in Adaptive Mesh Refinement - Theory and Applications, Lecture Notes in Computational Sciences and Engineering (LNCSE) series, editors Tomasz Plewa, Timur Linde (2004) 115

J. Kleimann, A. Kopp, H. Fichtner, R. Grauer, and K. Germaschewski Three-dimensional MHD high-resolution computations with CWENO employing adaptive mesh refinement Until recently, numerical simulations of discontinuities in highly super-Alfvénic plasmas have been severely limited by comparatively crude resolution and accuracy. Significant progress in the numerical simulation of such plasmas was achieved with the recently implemented Central Weighted Essentially Non-Oscillatory (CWENO) scheme. Combining this technique with that of adaptive mesh refinement (AMR), we have developed a third-order numerical scheme, which is able to efficiently capture strong gradients on spatial scales being small compared to the overall scale of the plasma system considered. Here, we first describe important algorithmic aspects of the scheme as well as the physics included in it. Second, we present the results of various performance tests. And, third, we illustrate its application to `real world problems' using the example of the dynamics of a Sedov-type explosion. Comp. Phys. Comm. 158 (2004) 47

R. Grauer and F. Spanier A note on the use of central schemes for incompressible Navier-Stokes flows J. Comp. Phys. 192 (2003) 727

J. Kleimann, H, Fichtner, A. Kopp, K. Germaschewski, and R. Grauer On the dynamics of the solar corona: the numerics behind a self-consistent 3D MHD Model Space missions like SOHO have renewed the interest in the physics of the solar corona. This complex system is not yet fully understood due to lack of sufficiently detailed observations, and also because realistic models should cover processes occuring on various spatial scales, while being both multidimensional and time-dependent. Significant progress w.r.t. their numerical realization was achieved recently with the Central Weighted Essentially Non-Oscillaroty scheme. A 3rd order CWENO scheme efficiently capturing strong gradients forms the basis of our new code. After describing the algorithm and its implementaion, we present test results as well as comparisons with preexisting codes. Proc. 10th. European Solar Physics Meeting, 'Solar Variability: From Core to Outer Frontiers', Prague, Czech Republic, 9-14 September 2002 (ESA SP-506, December (2002) 51

J. Kleimann, H, Fichtner, A. Kopp, K. Germaschewski, and R. Grauer On the dynamics of the solar corona: first results obtained with a new 3D MHD Model A newly developed self-consistent 3D MHD code is applied to the problem of the dynamics of the solar corona. First, we present the basic system of equations for a two-fluid description of the solar wind plasma and point out possible numerical difficulties arising from an improper choice of variables. Second, we perform a study of the solar wind expansion during phases of minimum solar activity, serving mainly as a first `real world' test case. Third, we discuss first results of the application of the model to propagating disturbances, such as coronal mass ejections and/or shocks. Proc. 10th. European Solar Physics Meeting, 'Solar Variability: From Core to Outer Frontiers', Prague, Czech Republic, 9-14 September 2002 (ESA SP-506, December (2002) 21

L. Berge, K. Germaschewski, R. Grauer, and J. Juul Rasmussen Hyperbolic Shock Waves of the Optical Self-Focusing with Normal Group-Velocity Dispersion The theory of focusing light pulses in Kerr media with normal group-velocity dispersion in (2+1) and (3+1) dimensions is revisited. It is shown that pulse splitting introduced by this dispersion follows from shock fronts that develop along hyperbolas separating the region of transverse self-focusing from the domain of temporal dispersion. Justiﬁed by a self-similar approach, this property is conﬁrmed by numerical simulations using an adaptive-mesh reﬁnement code. Phys. Rev. Lett. 89 (2002) 153902

K. Germaschewski, R. Grauer, L. Bergé, V.K. Mezentsev, and J. Juul Rasmussen Splittings, coalescence, bunch and snake patterns in the 3-D nonlinear Schrödinger equation with anisotropic dispersion The theory of focusing light pulses in Kerr media with normal group velocity dispersion (GVD) in (2+1) and (3+1) dimensions is revisited. It is shown that pulse splitting introduced by GVD follows from shock fronts that develop along hyperbolas separating the region of transverse self-focusing from the domain of temporal dispersion. Justified by a self-similar approach, this property is confirmed by numerical simulations using an adaptive-mesh refinement code. Physica D 151 (2001) 175

R. Grauer and C. Marliani Current Sheet Formation in 3D Ideal Incompressible Magnetohydrodynamics The evolution of current density and vorticity in the ideal, inviscid incompressible magnetohydrodynamic equations in three dimensions is studied numerically. Highly effective resolutions are obtained by adaptive structured mesh refinement techniques. We report on results for three different initial conditions showing similar behavior: in the early stage of the evolution a fast increase in vorticity and current density is observed. Thereafter, the evolution towards nearly two-dimensional current sheets results in a depletion of nonlinearity. Phys. Rev. Lett. 84 (2000) 4850

R. Grauer Modeling of strong MHD turbulence Intermittency in fully turbulent hydro- and magnetohydrodynamic flows is still a fascinating but unsolved problem. Recently, remarkable progress has been achieved in a model system of turbulence, the so-called Burgers turbulence. The most striking feature is that the tails of the probability distribution of velocity increments could be calculated using information of the pre-shocks present in the flow. The situation is expected to be similar in the Navier-Stokes and MHD equations, where vortex tubes or current sheets may substitute the role of pre-shocks of the Burgers equation. Numerical simulations using structured adaptive mesh refinement are presented to study the evolution of the singular structures in ideal equations without dissipation. In Plasma Turbulence and Energetic Particles in Astrophysics, Obserwatorium Astronomiczne, Uniwersytet Jagielloriski, Krakow (1999) 197

K. Germaschewski and R. Grauer Longitudinal and transversal structure functions in two-dimensional electron magnetohydrodynamic flows Electron magnetohydrodynamic flows have recently attracted considerable interest not only in the field of collisionless reconnection but also as the first two-dimensional turbulent system showing Kolmogorov like scaling for the energy spectrum. Here, longitudinal and transversal structure functions are calculated for varying Reynolds numbers. The simulations show that the differences between longitudinal and transversal structure functions are finite size effects for this type of flow and vanish in the limit of high Reynolds numbers. In addition, the scaling of structure functions for velocity and current density could be described by assuming a log-Poisson statistics with two atoms, as recently proposed for passive scalar advection. Physics of Plasmas 6 (1999) 3788

R. Grauer Adaptive Mesh Refinement for Singular Structures in incompressible hydro- and magnetohydrodynamic flows The question whether finite time singularities develop in incompressible hydro- and magnetohydrodynamic systems starting from smooth initial conditions is still an open problem. Here we present numerical simulations using the technique of adaptive mesh refinement which show evidence that in the 3D incompressible Euler equations a finite time blow-up in the vorticity occurs whereas in the 2D incompressible magnetohydrodynamic equations only exponential growth of vorticity and current density is observed. in Hyperbolic Problems: Theory, Numerics, Applications Vol. I, ISNM, Birkhäuser (1999) 401

M. Blüming, K. H. Spatschek, and R. Grauer Center manifold approach to the reduced magnetohydrodynamic bifurcations with diffusive magnetic field lines Bifurcations in plasmas are investigated on the basis of a reduced dissipative magnetohydrodynamic (MHD) model. In contrast to previous investigations, the diffusivity of magnetic field lines is taken into account. Making use of the center manifold theory for the first bifurcations, and Galerkin approximations for higher bifurcations, it is shown that the diffusion of magnetic field lines affects the transitions in the transients. When the mode which resembles the so called high confinement mode becomes unstable via a Hopf bifurcation, the changes in the oscillation frequencies are calculated. It is demonstrated that over a wide range of parameter values the so called electrostatic approximation is quite good. The strength of the generated magnetic field fluctuations is calculated, and the influence of the latter on a possible magnetic braiding is estimated. Physics of Plasmas 6 (1999) 1083

H. Friedel, R. Grauer, and K. H. Spatschek Controlling chaotic states of a Pierce diode A recently developed nonlinear approach to control chaos is applied to the Pierce diode. In the latter, both (kinetic) virtual cathode oscillations and (hydrodynamic) plasma oscillations appear. Via the period doubling route, the plasma oscillations can become chaotic. They are, however, usually superimposed by virtual cathode oscillations. Here it is shown that in the hydrodynamic as well as in the kinetic regime unstable periodic orbits can be stabilized. The results can be applied to bring the Pierce diode into a well-defined state of microwave oscillations. Physics of Plasmas 9 (1998) 3187

R. Grauer, C. Marliani, and K. Germaschewski Adaptive mesh refinement for singular solutions of the incompressible Euler equations The occurrence of a finite time singularity in the incompressible Euler equations in three dimensions is studied numerically using the technique of adaptive mesh refinement. As opposed to earlier treatments, a prescribed accuracy is guaranteed over the entire integration domain. A singularity in the vorticity could be traced down to 5 levels of refinement which corresponds to a resolution of $2048^3$ mesh points in a non-adaptive treatment. The growth of vorticity fits a power law behavior proportional to $1/(T^* - t)$ where $T^*$ denotes the time when the singularity occurs. Phys. Rev. Lett. 84 (1998) 4850

R. Grauer An Energy estimate for a perturbed Hasegawa-Mima equation It is commonly believed that drift waves and drift wave turbulence play a major role in understanding the anomalous transport at the plasma edge of a tokamak fusion reactor. A one field equation describing the electrostatic potential fluctuations in this regime is the so called Hasegawa-Mima equation. If this equation is driven by some instability and damped by some hyperviscous term, the energy grows exponentially in time which is not consistent with the approximations made in the derivation of the equation. Numerical simulations of a perturbed Hasegawa-Mima equation which includes in addition a so-called E x B nonlinearity showed that the energy saturates at a finite level. In this paper this numerical observation is proven analytically. Nonlinearity 11 (1998) 659

R. Grauer and C. Marliani Geometry of singular structures in magnetohydrodynamic flows The flattening of current sheets is investigated by means of numerical simulations of the ideal incompressible magnetohydrodynamic equations in two dimensions. The use of adaptive mesh refinement techniques allows to resolve the more and more singular structures and to follow the exponential growth of current density. The numerical results are in good agreement with a scaling ansatz proposed by Sulem et al. (J. Plasma Phys., 33, 1985, pp. 191-198). The geometry of the current sheets is characterized by the alignment properties of the deformation matrices. Physics of Plasmas 5 (1998) 2544

H. Friedel, R. Grauer, and C. Marliani Center manifold approach to controlling chaos Existing methods of controlling chaos can be generalized using ideas of center manifold theory. This approach extends the existing linear theory into the nonlinear regime, thus enlarging the range in phase space where control is possible. At the same time, sensitivity of the stabilized system against noise is reduced. In addition, this procedure leads to nonlinear time delay feedback rules in a constructive way. Physics Letters A 236 (1997) 45

O. Zikanov, A. Thess, and R. Grauer Statistics of turbulence in a generalized random-force-driven Burgers equation The statistics of solutions to a family of one-dimensional random-force-driven advection-diffusion equations is studied using high resolution numerical simulations. The equation differs from the usual Burgers equation by the non-local form of the nonlinear interaction term mimicking the non-locality of the Navier-Stokes equation. It is shown that under an appropiate choice of random forcing the statistical properties of the solution (energy spectrum and scaling exponents of structure functions)coincide with those of Kolmogorov turbulence. Also, a generalization is proposed which allows intermittency effects to be modeled. Physics of Fluids 9 (1997) 190

H. Friedel, R. Grauer, and C. Marliani Adaptive mesh refinement for singular current sheets in incompressible magnetohydrodynamic flows The formation of current sheets in ideal incompressible magnetohydrodynamic flows in two dimensions is studied numerically using the technique of adaptive mesh refinement. The growth of current density is in agreement with simple scaling assumptions. As expected, adaptive mesh refinement shows to be very efficient for studying singular structures compared to non-adaptive treatments. J. Comp. Phys. 134 (1997) 190

R. Grauer and C. Marliani Analytical and numerical approaches to structure functions in magnetohydrodynamic turbulence In magnetohydrodynamic turbulence, the classical theory by Kraichnan and Iroshnikov based on dimensional analysis gives a linear dependence of the exponents ζp=p/4 of the structure functions for the Elsässer variables z±=u±B. This linear behavior contradicts observations of MHD turbulence in the solar wind, where anomalous scaling was found similar as in hydrodynamic turbulence. Since the experimentally observed scaling can not yet be derived by analytical theories, one is dependent also on numerical simulations. As an alternative to direct numerical simulations we present a stochastic approach that recently was introduced for two-dimensional hydrodynamic flows. Finally, we discuss the applicability of operator-product expansions on a direct cascade in strongly turbulent systems. Physica Scripta T67 (1996) 38

R. Grauer and T. C. Sideris Finite time singularities in ideal fluids with swirl Three-dimensional ideal, incompressible fluids with swirl are studied numerically using two different methods: standard finite differences and a projection method based on upwind differencing. Both methods give quantitatively similar results, leading to the conclusion that singularities form in finite time in a manner consistent with known theoretical criteria. The effect of singularities in incompressible flows on nearby compressible flows is discussed. Physica D 88 (1995) 116

R. Grauer and C. Marliani Numerical and analytical estimates for the structure functions in two-dimensional magnetohydrodynamic flows In two-dimensional magnetohydrodynamic turbulence, the Kraichnan-Iroshnikov dimensional analysis suggests a linear scaling law for the exponents zeta_p = p/4 of the structure functions for the Elsässer variables z± =u±B. Numerical simulations are presented and higher order structure functions are calculated using the extended self-similarity hypotheses of Benzi et al. [Phys. Rev. E 48, (1993)]. In addition, an estimate for the first structure function zeta_1 \ge 1/4 is derived using a geometric technique introduced by Constantin and Procaccia [Phys. Rev. E 47, (1993)] in the the context of the transport of a passive scalar in three-dimensional Navier-Stokes turbulence. Physics of Plasmas 2 (1995) 41

T. Eickermann, R. Grauer, and K. H. Spatschek Identification of mass capturing structures in a perturbed nonlinear Schrödinger equation The numerical solutions of a standard damped and driven nonlinear Schrödinger equation are compared with a systematic reduction obtained by the Kahunen-Loeve expansion. The role of the mass containing modes is clarified by analyzing the spectral data of the underlying periodic direct problem. A one to one correspondence to the marginally stable modes obtained from the linearized Bäcklund transformation is found which explains the observed low-dimensional behavior. Phys. Lett. A 198 (1995) 383

R. Grauer and C. Marliani Structure functions in magnetohydrodynamic turbulence Higher order structure functions of two-dimensional magnetohydrodynamic flows are studied numerically using a second order upwind projection method. In addition, an analytical estimate for the first structure function based on a geometric technique is derived. Finally, a phenomenological model without adjustable parameters is presented. Structure and Dynamics of Nonlinear Waves in Fluids, edited by K. Kirchgässner and A. Mielke, London, World Scientific. 7 (1994) 239

R. Grauer, J. Krug, and C. Marliani Scaling of high-order structure functions in magnetohydrodynamic turbulence A phenomenological model for the description of intermittency corrections in magnetohydrodynamic flows is presented. The strength of the model lies in its lack of adjustable parameters. A comparison to measurements in the solar wind is presented. Phys. Lett. A 195 (1994) 335

R. Grauer and Y. Kivshar Dynamics of parametrically driven sine-Gordon breathers The dynamics of a breather in the damped and parametrically driven sine-Gordon equation is investigated both numerically and analytically. The Kahunen-Loeve expansion is applied to extract the energetically dominant localized modes. These modes are used in a Galerkin approximation to the original partial differential equation. Solutions of the resulting amplitude equations are then compared to numerical simulations of the perturbed sine-Gordon equation showing perfect agreement. Information from the periodic spectral theory and linear stability analysis is used to identify the Kahunen-Loeve modes. in Nonlinear coherent structures in physics and biology, Bayreuth, Germany, edited by K. H. Spatschek and F. G. Mertens, pages 381-384, New York, 1994, Plenum. (1994)

B. Birnir and R. Grauer An Explicit Description of the Global Attractor of the Damped and Driven Sine-Gordon Equation We prove that the size of the finite-dimensional attractor of the damped and driven sine-Gordon equation stays small as the damping and driving amplitude become small. A decomposition of finite-dimensional attractors in Banach space is found, into a part B that attracts all of phase space, except sets whose finite-dimensional projections have Lebesgue measure zero, and a part C that only attracts sets whose finite-dimensional projections have Lebesgue measure zero. We describe the components of the B-attractor and C, which is called the "hyperbolic" structure, for the damped and driven sine-Gordon equation. B is low-dimensional but the dimension of C, which is associated with transients, is much larger. We verify numerically that this is a complete description of the attractor for small enough damping and driving parameters and describe the bifurcations of the B-attractor in this small parameter region. Comm. Math. Phys. 162 (1994) 539

R. Grauer and Y. Kivshar Chaotic and phase-locked breather dynamics in the damped and parametrically driven sine-Gordon equation The dynamics of a breather in the damped and parametrically driven sine-Gordon equation is investigated both numerically and analytically. The Kahunen-Loeve expansion is applied to extract the energetically dominant localized modes. These modes are used in a Galerkin approximation to the original partial differential equation. Solutions of the resulting amplitude equations are then compared to numerical simulations of the perturbed sine-Gordon equation showing perfect agreement. In addition, two collective coordinate models (bases on a direct approach and on the inverse scattering transform) are constructed and their limitations in comparison with the Kahunen-Loeve expansion and direct simulations are discussed. Finally, information from the periodic spectral theory and linear stability analysis is used to identify the Kahunen-Loeve modes and to show why this approach gives rather good results. Phys. Rev. E 48 (1993) 4791

P. Beyer, R. Grauer, and K. H. Spatschek Center Manifold Theory for Low-Frequency Excitations in Magnetized Plasmas For the dissipative trapped-ion mode a simple one-dimensional nonlinear model equation, including effects of instability, dissipation, and dispersion, is investigated. The center manifold theory is applied to the situation of more than one marginally stable mode, and the dynamics in the neighborhood of the onset of instability is elucidated. Depending on the (three) relevant parameters, stable solitary waves, mixed modes, heteroclinic orbits etc can exist, and a scenario for the nonlinear dynamical behavior is developed. The bifurcation diagrams are drawn with quantitative predictions in parameter space. An important conclusion is that the used codimension two analysis can predict successive bifurcations which cannot be captured by simple analysis of one unstable mode. The analytical calculations are checked by numerical simulations. Phys. Rev. E 48 (1993) 4665

T. Eickermann, R. Grauer, and K. H. Spatschek New Aspects of chaotic dynamics in nonlinear Schrödinger systems in Future Directions of Nonlinear Dynamics in Physical and Biological Systems, edited by P. L. Christiansen, J. C. Eilbeck, and R. D. Parmentier, Nato Advanced Study Institute Series B 312 (1993) 109

B. Birnir, B. Galdrikian, R. Grauer, and M. Sherwin Nonperturbative resonances in periodically driven quantum wells Energy absorption characterestics are computed for a classical and a quantum model of an infinite square well, as a function of driving amplitude and frequency, Nonperturbative resonances are observed corresponding to the replacement of states localized in phase space by more extended states. Their presence is predicted by avoided crossings in the quasienergy spectrum of the Floquet operator. The conditions under which these resonances occur can be realized in experiments on GaAs/Al_x Ga_{1-x}As quantum wells in intense far-infrared radiation. Phys. Rev. B 47 (1993) 6795

R. Grauer, K. H. Spatschek, and A. V. Zolotaryuk Chaotic proton dynamics in the hydrogen bond The motion of a proton in a double-well potential created by the potentials of two heavy ions in a molecular chain is considered. The topology of the potential changes depending on the distance between the molecules. Both forms, double-well and single-well, are possible. Individual chaotic proton motion is triggered by the oscillations of the lattice. Depending on the system parameters, both the one-well and the cross-well attractors can be either periodic or chaotic. This has some interesting consequences for the interpretation and understanding of propagation of ionic defects in hydrogen-bonded chains in the presence of external oscillating fields. A new frequency-locked propagating kink is found. Phys. Rev. E 47 (1993) 236

R. Grauer and B. Birnir The center manifold and bifurcations of the sine-Gordon equation The generic bifurcations of breathers in the damped and driven sine-Gordon equation are investigated both numerically and analytically. The linear stability analysis and information from periodic spectral theory suggest that three modes are relevant for the system. They correspond to frequency and (temporal) phase changes and to the flat pendulum. Using these modes (nonautonomous) amplitude equations are derived and compared with numerical simulations of the perturbed sine-Gordon equation. Physica D 56 (1992) 165

R. Grauer and T. C. Sideris Numerical computation of 3D incompressible ideal fluids with swirl We investigate numerically the question of blowup in finite time for the "swirling flow" of the three-dimensional incompressible Euler equations. Using rotational symmetry, the Euler equations reduce to a two-dimensional problem which is numerically solved by finite differences. The elliptic equation relating vorticity to velocity is solved with the multigrid method. Calculations were performed with 896 x 640 mesh points. Phys. Rev. Lett. 25 (1991) 3511

E. Turlot, D. Esteve, C. Urbina, M. Devoret, R. Grauer, J. C. Fernandez, and G. Reinisch The dynamical isoperimeter multi-vortex mode in the square sine-Gordon system in Nonlinear World, edited by V. G. Baryakhtar, V. M. Chernosenko, N. S. Erokhin, A. G. Sitenko, and V. E. Zakharov, Singapore, World Scientific 363 (1990)

E. Turlot, D. Esteve, C. Urbina, M. Devoret, R. Grauer, J. C. Fernandez, and G. Reinisch Dynamical isoperimeter pattern in the square sine-Gordon system This paper shows by use of simple physical arguments the existence of a particular multi-vortex dynamical configuration in a - perturbed or not - two-dimensional sine-Gordon system. The stability of such modes is numerically checked. Their main topological invariant is the total length of the +/- 2Pi - wavefronts entering as elementary kink-like patterns the constitution of the whole configuration. This is consistent with the conservation of energy in all situations which are considered in this paper. Phys. Rev. B 42 (1990) 8418

J. C. Fernandez, R. Grauer, K. Pinnow, and G. Reinisch Cell-mapping description of coexisting phase-locked soliton states in a long Josephson junction The coexistence of phase-locked soliton states in a long ac-biased Josephson junction is pointed out on the basis of numerical calculations. We use a combined interpolation and cell mapping technique to calculate periodic orbits along with their stability and basins of attraction. The dominant coexistent phase-locked states consist of the well known zero-Field-Step (shuttling regime of solitons) and the so-called C-cycle dynamics. In the latter the soliton is bouncing only at one end of the junction, therefore producing no average voltage. The probability of reaching the basins of attraction of these different motions explain the hysteresis and the complicated fine structure in the current-voltage curve. Phys. Rev. B 42 (1990) 9987

J. C. Fernandez, R. Grauer, K. Pinnow, and G. Reinisch Phase-locked dynamical regimes to an external microwave field in a long, unbiased Josephson junction We show the existence of asymmetrical (in the phase space) phase-locked limit cycles of (anti) kinks in inhomogeneously ac driven sine-Gordon systems and point out their possible experimental verification by use of Josephson devices. Phys. Lett. A 145 (1990) 333

E. Turlot, D. Esteve, C. Urbina, M. Devoret, R. Grauer, J. C. Fernandez, and G. Reinisch Can a nontrivial solitonic mode be observed in a square Josephson junction? We show by use energy considerations and numerical simulations the existence of a stable 2-d isoperimeter solitonic mode propagating simultaneously along the two diagonals of a square Josephson junction of intermediate length. in Nonlinear Coherent Structures, edited by M. Barthes and J. Leon, Lecture Notes in Physics, Springer 353 (1989) 181

J. C. Fernandez, R. Grauer, and G. Reinisch Dynamical regimes phase-locked to an external microwave field in a long, unbiased Josephson junction We demonstrate the existence of phase-locked limit cycles in inhomogenously driven sine-Gordon systems and point out their possible experimental verification by use of Josephson devices. in Nonlinear Coherent Structures, edited by M.Barthes and J. Leon, Lecture Notes in Physics, Springer 353 (1989) 213

M. Taki, K. H. Spatschek, J. C. Fernandez, R. Grauer, and G. Reinisch Breather dynamics in the nonlinear Schrödinger regime of perturbed sine-Gordon systems A possible route to temporal chaos with coherent stable spatial structures is proposed for the driven damped sine-Gordon equation. For near-conservative perturbations, the dynamics of a breather is investigated numerically and semi-analytically in the presence of an ac driver and a simple damping term. For moderate driver strength, a flat (space-independent) attractor exists whereas above a threshold a phase-locked breather co-exists. The latter can undergo a period-doubling route to temporal chaos as is shown here for a certain parameter regime. Relations to other works which operate in a different parameter regime are discussed. The near-conservative perturbations and the low driver strengths allow to interpret the results within a simple model originating from the so-called nonlinear Schrödinger limit. In fact, within this limit (small amplitude breather), the chaotic (or not) transitions are dominated by interactions between breather-like solutions and radiation (mostly k = 0 mode). Therefore, three collective coordinates, i.e. the amplitude of the phase-locked breather, its phase, as well as the complex amplitude of the k = 0 mode, are sufficient to construct a system of four ordinary differential equations of first order which reveal the basic features of partial differential equations in a satisfactory manner. Physica D 40 (1989) 65

R. Grauer Codimension two interactions of tearing modes The interaction of two different tearing modes is examined. Certain physical parameter values could be found such that the nullspace of the linearized problem is four-dimensional. The center manifold theory allows a reduction of the partial differential equations (PDE's) describing the tearing instabilities to a four dimensional system of ordinary ones (ODE's). Due to the symmetries of the problem many interesting spatial- and spatio-temporal solutions are possible in the neighbourhood of the codimension two point. in Singular Behavior and Nonlinear Dynamics, Samos, Greece edited by St. Pnevmatikos, T. Bountis, and Sp. Pnevmatikos, Singapore, World Scientific (1989) 267

R. Grauer Nonlinear interactions of tearing modes in the vicinity of a bifurcation point of codimension two The interaction of two different tearing modes in slab geometry is examined. For certain physical parameter values the linearised problem has a four-dimensional nullspace corresponding to two different marginal tearing modes. With the center manifold theory the original partial differential equations could be reduced to a four-dimensional system of ordinary ones. These amplitude equations are equivariant under O(2)-actions due to the symmetries of the physical problem. Because of these symmetries there exists many interesting spatial- and spatio-temporal solutions in the neighbourhood of the critical parameters such as standing, travelling and modulated waves and a structural stable heteroclinic orbit (see Ambruster, Guckenheimer, Holmes, Physica D 29 (1988) 257-282). Physica D 35 (1989) 107

E. Rebhan and R. Grauer Tokamak profiles through constrained minimization of the entropy production In the present paper, considering a cylindrical plasma with circular cross-section we minimize the entropy production caused by Ohmic heating and classical heat conduction under the constraint that momentum and energy balance be conserved. Although this is an heuristic approach there is some hope that it may nevertheless be reasonable. Firstly, the only effect of this constrained minimization consists effectively in removing an indeterminancy concerning the sources which are necessary for stationary mass flow. Secondly, the profiles which we obtain look quite reasonable. Finally, a somewhat related approach which was employed by Steenbeck (Phys. Z. 33 (1932) 809, Phys. Z. 38 (1937) 1099) for explaining arc discarges and which was later attributed to minimum entropy production by Peters (Z. Physik 144 (1956) 612) turned out quite successful. in Proceedings of the 14th European Conference on Controlled Fusion and Plasma Physics, Madrid, Spain, European Physical Society (1987) 1072

R. Grauer and E. Rebhan Analysis of ideal MHD equilibrium and axisymmetric stability for finite aspect ratio tokamaks with elliptic cross-section and flat current profile To calculate the equilibrium and stability of finite aspect ratio tokamaks, even for circular and elliptic plasma cross-sections, intricate numerical methods must usually be employed, which require a fast and large computer. So far, these calculations have been carried out analytically only for infinite aspect ratio. In fact certain analytical equilibrium solutions at finite aspect ratio are known, but these do not include such important cases as elliptical cross-sections. In this paper, finite aspect ratio equilibrium solutions are derived for circular, prolate and oblate elliptical cross-sections and for a flat current profile. In addition, the problem of axisymmetric stability is studied for prolate elliptical cross-sections, the approach being almost entirely analytical. J. Plasma Physics 22 (1984) 99

R. Grauer and E. Rebhan Semianalytic calculation of equilibrium and axisymmetric stability of finite aspect ratio tokamaks with elliptic cross-section in Proceedings of the 11th European Conference on Controlled Fusion and Plasma Physics, Aachen, Germany , European Physical Society 197 (1983)

Prof. Dr. Rainer Grauer

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Publikationen

F. Allmann-Rahn, S. Lautenbach, M. Deisenhofer, R. Grauer

The muphyII Code: Multiphysics Plasma Simulation on Large HPC Systems

Jeremiah Lübke, Frederic Effenberger, Mike Wilbert, Horst Fichtner, Rainer Grauer

Towards Synthetic Magnetic Turbulence with Coherent Structures

Jeremiah Lübke, Jan Friedrich, Rainer Grauer

Stochastic interpolation of sparsely sampled time series by a superstatistical random process and its synthesis in Fourier and wavelet space

T. Schorlepp, T. Grafke, R. Grauer

Symmetries and zero modes in sample path large deviations

Timo Schorlepp, Tobias Grafke, Sandra May and Rainer Grauer

Spontaneous Symmetry Breaking for Extreme Vorticity and Strain in the 3D Navier-Stokes Equations

Florian Allmann-Rahn, Simon Lautenbach, Rainer Grauer

An Energy Conserving Vlasov Solver That Tolerates Coarse Velocity Space Resolutions: Simulation of MMS Reconnection Events

Florian Allmann-Rahn, Rainer Grauer, Katharina Kormann

A Parallel Low-Rank Solver for the Six-Dimensional Vlasov-Maxwell Equations

Mike Wilbert, André Giesecke, and Rainer Grauer

Numerical Investigation of the Flow inside a Precession driven cylindrical Cavity with additional Baffles using an Immersed Boundary Method

F. Allmann-Rahn, S. Lautenbach, R. Grauer, and R. D. Sydora

Fluid simulations of three-dimensional reconnection that capture the lower-hybrid drift instability

M. Sinhuber, J. Friedrich, R. Grauer, M. Wilczek

Multi-level stochastic refinement for complex time series and fields: A Data-Driven Approach

Timo Schorlepp, Tobias Grafke, Rainer Grauer

Gel'fand-Yaglom type equations for calculating fluctuations around Instantons in stochastic systems

J. Friedrich, J. Peinke, A. Pumir, R. Grauer

Explicit construction of joint multipoint statistics in complex systems

J. Friedrich and R. Grauer

Generalized description of intermittency in turbulence via stochastic methods

J. Friedrich, S. Gallon, A. Pumir, R. Grauer

Multi-point fractional Brownian bridges and their applications

G. Margazoglou, L. Biferale, R. Grauer, K. Jansen, D. Mesterházy, T. Rosenow, R. Tripiccione

A Hybrid Monte Carlo algorithm for sampling rare events in space-time histories of stochastic fields

Lasse Ebener, Georgios Margazoglou, Jan Friedrich, Luca Biferale, Rainer Grauer

Instanton based importance sampling for rare events in stochastic PDEs

J. Friedrich and R. Grauer

Markov Property of Velocity Increments in Burgers Turbulence

F. Allmann-Rahn, T. Trost and R. Grauer

Temperature gradient driven heat flux closure in fluid simulations of collisionless reconnection

Jan Friedrich, Georgios Margazoglou, Luca Biferale, Rainer Grauer

Multi-scale velocity correlations in turbulence and Burgerlence: fusion rules, Markov processes in scale, and multifractal predictions

Simon Lautenbach, Rainer Grauer

Multiphysics simulations of collisionless plasmas

S. Kreuzahler, Y. Ponty, N. Plihon, H. Homann, R. Grauer

Dynamo enhancement and mode selection triggered by high magnetic permeability

Alain Pumir, Haitao Xu, Eberhard Bodenschatz and Rainer Grauer

Single-Particle Motion and Vortex Stretching in Three-Dimensional Turbulent Flows

J. Friedrich, H. Homann, T. Schäfer, R. Grauer

Longitudinal and transverse structure functions in high Reynolds-number magneto-hydrodynamic turbulence

M. Rieke, T. Trost and R. Grauer

Coupled Vlasov and two-fluid codes on GPUs

Tobias Grafke, Rainer Grauer, Tobias Schäfer, Eric Vanden-Eijnden

Relevance of instantons in Burgers turbulence

Tobias Grafke, Rainer Grauer, Stephan Schindel

Efficient Computation of Instantons for Multi-Dimensional Turbulent Flows with Large Scale Forcing

Tobias Grafke, Rainer Grauer, Tobias Schäfer

The instanton method and its numerical implementation in fluid mechanics

Tobias Grafke, Rainer Grauer, Tobias Schäfer, Eric Vanden-Eijnden

Arclength parametrized Hamilton's equations for the calculation of instantons

Holger Homann, Yannick Ponty, Giorgio Krstulovic, Rainer Grauer

Structures and Lagrangian statistics of the Taylor-Green Dynamo

Sophia Kreuzahler, Daniel Schulz, Holger Homann, Yannick Ponty, Rainer Grauer

Numerical study of impeller-driven von Karman flows via a volume penalization method

Tobias Grafke, Rainer Grauer, Thomas C. Sideris

Turbulence properties and global regularity of a modified Navier-Stokes equation

Tobias Grafke, Rainer Grauer, Tobias Schäfer

Instanton filtering for the stochastic Burgers equation

Tobias Grafke and Rainer Grauer

Lagrangian approach for finite-time Euler singularities in three-dimensional incompressible fluid flow

Holger Homann, Jeremie Bec and Rainer Grauer

Effect of turbulent fluctuations on the drag and lift forces on a towed sphere and its boundary layer

T. Grafke and R. Grauer

Lagrangian and geometric analysis of finite-time Euler singularities

T. Grafke, R. Grauer and T. Schäfer

Instanton filtering for the stochastic Burgers equation

R. Grauer, H. Homann, and J.F. Pinton

Longitudinal and Transverse structure functions in high Reynolds-number turbulence

Th. Hater, H. Homann, and R. Grauer

A Lagrangian model for the evolution of turbulent magnetic and passive scalar field

H. Homann, D. Schulz, and R. Grauer