
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 realworld 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 + 1dimensional 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 lognormally 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 timedependent parameter ξ(t) follows a lognormal 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 multiwaveletbased 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 FreidlinWentzell 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 boundarytype 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 shorttime large deviations of the onedimensional KardarParisiZhang equation with flat initial profile.
Journal of Statistical Physics 190 (2023) 50


F. AllmannRahn, 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 standalone use of models as well as a modelbased dynamic and adaptive domain decomposition. This requires ensuring excellent conservation properties, careful treatment of innerdomain model boundaries for model coupling, and robust time stepping algorithms, especially with the use of electron subcycling. This multiphysics approach is implemented in the muphyII code, tested on different scenarios of
space plasma reconnection and evaluated against space probe data and higherfidelity simulation results from literature. Adaptive model refinement is highlighted in particular, and a hybrid model with kinetic ions, pressuretensor fluid electrons, and Maxwell fields is appraised.
submitted (2023)


Timo Schorlepp, Tobias Grafke, Sandra May and Rainer Grauer
Spontaneous Symmetry Breaking for Extreme Vorticity and Strain in the 3D NavierStokes Equations
We investigate the spatiotemporal structure of the most likely
configurations realising extremely high vorticity or strain in the
stochastically forced 3D incompressible NavierStokes 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. Highvorticity
configurations are identified as pinched vortex filaments with
swirl, while highstrain configurations correspond to counterrotating
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 vortexsheet configuration.
Phil. Trans. R. Soc. A. 380 (2022) 20210051


Florian AllmannRahn, 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 phasespace grid are highly accurate but also numerically demanding. Coarse velocity space resolutions, which are unproblematic in particleincell (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 AllmannRahn, Rainer Grauer, Katharina Kormann
A Parallel LowRank Solver for the SixDimensional VlasovMaxwell 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, lowrank approximations can be employed. The so far available lowrank 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 lowrank solver for the full sixdimensional electromagnetic VlasovMaxwell equations with a compression of the particle distribution function in velocity space. Special attention is paid to mass conservation and Gauss's law. The lowrank 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 NavierStokes equations for arbitrary vessel geometries by combining a FourierSpectral method with a direct forcing Immersed Boundary method which allows to consider solidfluid interactions. The approach is applied to a paradigmatic setup motivated by the precession dynamo experiment currently under construction at Helmholtz Zentrum DresdenRossendorf (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. AllmannRahn, S. Lautenbach, R. Grauer, and R. D. Sydora
Fluid simulations of threedimensional reconnection that capture the lowerhybrid 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 threedimensional 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 lowerhybrid drift instability within a reconnection simulation. Characteristics of the instability are examined with fluid and kinetic continuum models, and its role in the threedimensional 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
Multilevel stochastic refinement for complex time series and fields: A DataDriven Approach
Spatiotemporally 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 multiscale 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'fandYaglom 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'fandYaglom 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 selfsimilar 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 selfsimilarity and may therefore require a more elaborate description. In the context of turbulence, the KolmogorovOboukhov 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 OrnsteinUhlenbeck 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 nonGaussian 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 KramersMoyal expansion of the velocity increment probability density function that is associated to a Markov process. We compare the different sets of KramersMoyal 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 KramersMoyal 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 selfsimilarity (purely nonlocal case) to intermittent behavior (intermediate case that agrees with Yakhot's mean field theory
[Phys. Rev. E 63 026307 (2001)]) to shocklike behavior (purely nonlinear Burgers case).
Atmosphere 11 (2020) 1003


J. Friedrich, S. Gallon, A. Pumir, R. Grauer
Multipoint 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 nonequidistant 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 nonselfsimilar 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 spacetime 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 spacetime histories of the dynamical degrees of freedom under the influence of random noise. First, we validate and benchmark the HMC algorithm by reproducing multiscale properties of the onedimensional Burgers equation driven by Gaussian and whiteintime noise. Second, we show how to implement an importance sampling protocol to significantly enhance, by orderofmagnitudes, 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 nonequilibrium system governed by a stochastic partial differential equation (SPDE) with additive forcing. To this end, we deploy the socalled instanton formalism that corresponds to a saddlepoint 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 onedimensional 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 instantonbased 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 Burgersshocks 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 multiincrement probability density functions derived directly from the Burgers equation.
in Complexity and Synergetics (Springer), S.C. Müller et al. (eds.) (2018) 39


F. AllmannRahn, 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
Multiscale velocity correlations in turbulence and Burgerlence: fusion rules, Markov processes in scale, and multifractal predictions
We compare different approaches towards an effective description of multiscale velocity field correlations in turbulence. Predictions made by the operator product expansion, the socalled 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 sixdimensional 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 10moment 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 10moment fluid model as well. It may turn out that in some spatial regions far away from the reconnection zone the temperature tensor in the 10moment description is nearly isotopic. In this case it is even possible to switch to a 5moment description. This change can be done separately for ions and electrons. To test this multiphysics approach, we apply this full physicsadaptive 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 vonKarman 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
SingleParticle Motion and Vortex Stretching in ThreeDimensional Turbulent Flows
Threedimensional 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 Reynoldsnumber magnetohydrodynamic turbulence
We investigate the scaling behavior of longitudinal and transverse structure functions in homogeneous and isotropic magnetohydrodynamic (MHD) turbulence by means of an exact hierarchy of structure function equations as well as by direct numerical simulations of two and threedimensional 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 twofluid codes on GPUs
We present a way to combine Vlasov and twofluid 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 5moment twofluid 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 speedup 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 VandenEijnden
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 MultiDimensional 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 multidimensional 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 nongradient, nonlinear 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 threedimensional (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 VandenEijnden
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 FreidlinWentzell action functional that arises in the context of large deviation theory for stochastic differential equations. The method is particularly wellsuited to calculate expectations dominated by noiseinduced excursions from deterministically stable fixpoints. Its simplicity and computational efficiency are illustrated here using several examples: a finitedimensional stochastic dynamical system (an OrnsteinUhlenbeck 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 TaylorGreen Dynamo
The evolution of a TaylorGreen 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, nonlinear and saturated regime. We find that the probability density functions (PDFs) of the magnetic field change from strongly nonGaussian PDFs in the kinematic regime to quasiGaussian 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 quasiisotropic (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 largeeddy turnover 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 impellerdriven 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 counterrotating 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 superfluid 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 pseudospectral 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 NavierStokes equation
We introduce a modification of the NavierStokes 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 NavierStokes turbulence. The numerically calculated scaling of structure functions is compared with a phenomenological model based on the SheL\'ev\^eque approach. Finally, for this equation we demonstrate global wellposedness 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 ChernykhStepanov 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 finitetime Euler singularities in threedimensional incompressible fluid flow
We address the question whether a singularity in a threedimensional incompressible inviscid fluid flow can occur in finite time. Analytical considerations and numerical simulations suggest highsymmetry flows being a promising candidate for a finitetime blowup. Utilizing Lagrangian and geometric nonblowup criteria, we present numerical evidence against the formation of a finitetime singularity for the highsymmetry 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 nonblowup 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 highresolution direct numerical simulations. The measurements are carried out using a novel scheme to integrate the twoway cou pling between the particle and the incompressible surrounding fluid flow maintained in a highReynoldsnumber turbulent regime. The main idea consists in combining a Fourier pseudospectral method for the fluid with an immersedboundary technique to impose the noslip 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 torquefree rotation rate of a spherical particle in various configurations of an upstreamlaminar 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 selfsimilarity 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 finitetime Euler singularities
We present a numerical method of analyzing possibly singular incompressible 3D Euler flows using massively parallel highresolution adaptively refined numerical simulations up to 8192^3 mesh points. Geometrical properties of Lagrangian vortex line segments are used in combination with analytical nonblowup criteria by Deng et al [Commun. PDE 31 (2006)] to reliably distinguish between singular and nearsingular flow evolution. We then apply the presented technique to a class of highsymmetry initial conditions and present numerical evidence against the formation of a finitetime 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 nonGaussian and governed by saddlepoints of the corresponding path integrals, socalled “instantons”.
Understanding intermittency in turbulent systems is still one of the open problems in classical physics. Since intermittency is governed by the nonGaussianity 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 ChernykhStepanov 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 Reynoldsnumber 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 orderdependent 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 NavierStokes 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 highReynolds
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 uǫl ) conditioned on a scaleaveraged 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 quasiGaussian
probability distribution functions (PDFs). Concerning Lagrangian statistics we found that conditioning on
a trajectory averaged energydissipation 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 K41prediction of the PDF of acceleration. Conditioned structure functions
show approximately K41scaling 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 QuasiSeparatrix Layer using Adaptive Mesh Refinement
The formation of a thin current sheet in a magnetic quasiseparatrix layer (QSL) is investi
gated by means of numerical simulation using a simplified ideal, lowβ, 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 quasistatic evolution. The sheet is locally onedimensional and the plasma flow in its
vicinity, when transformed into a comoving 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 KaplanYorke 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 plasmafilled 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 archshaped 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 Bfield and to form an expanding loop structure. The observed
dynamics of the magnetic flux tubes is analysed by means of threedimensional
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, pointlike particles by incompressible, resistive magnetohydrodynamic (MHD) turbulence is investigated by means of highresolution 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 Lorentzforce. 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 twodimensional 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 nonGaussian 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 EulerianLagrangian velocity increment in fully developped turbulence
We investigate the relationship between Eulerian and Lagrangian probability density functions obtained from numerical simulations of twodimensional as well as threedimensional 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 3D MHD studies of CME expansion
A recent thirdorder, essentially nonoscillatory central scheme to advance the equations of singlefluid magnetohydrodynamics (MHD) in time has been implemented into a new numerical code. This code operates on a 3D Cartesian, nonstaggered grid, and is able to handle shocklike 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 Parkerlike steady state for both hydrodynamic and MHD winds. The model is subsequently applied to expansion studies of CMElike 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 selfsimilar 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 solverfree numerical scheme for highMach 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 twophase 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 oftenunderestimated influence of varying lineofsight 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 KidaPelz 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 stateoftheart experiments and numerical simulations on turbulent velocity statistics along particle trajectories obtained in different flows with Reynolds numbers in the range Rlambda 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. ParisiFrisch 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 largescale MHD and multifluid models that explicitly incorporate effects of kinetic processes occuring on micro or mesoscales. 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 CWENObased 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
DensityPDFs and Lagrangian Statistics of highly compressible Turbulence
In isothermal, highly compressible turbulent flows, density fluctuations follow a lognormal distribution. We establish a connection between these density fluctuations and the probabilitydensityfunctions (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
Threedimensional MHD simulation of expanding magnetic flux ropes
Threedimensional, timedependent 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 HallMHD solver using whistler wave preconditioning
The dispersive character of the HallMHD 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 semiimplicit 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 Xline 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 threedimensional 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 singleparticle diffusion exhibits essentially the same behavior in NavierStokes and MHD turbulence, twoparticle relative dispersion in the MHD case differs significantly from the NavierStokes 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 NavierStokes 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 floatingpoint precision and interpolation scheme on the results of DNS of turbulence by pseudospectral codes
In this paper we investigate the impact of the floatingpoint precision and interpolation scheme on the results of direct numerical simulations (DNS) of turbulence by pseudospectral codes. Three different types of floatingpoint 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 tricubic scheme results in a slightly broader acceleration probability density function than a trilinear 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 plasmaassisted microfabrication of subwavelength structures by means of pointby point femtosecond laser inscription. It is shown that the spatiotemporal 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 selffocusing 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 NavierStokes and MHDTurbulence
We report on a comparison of highresolution 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 NavierStokes 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 NavierStokes 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 NavierStokes 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 NavierStokes 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 nonoscillatory central scheme
Here we present a new, semidiscrete, central scheme for the numerical solution of onedimensional systems of hyperbolic conservation laws. The method presented in this paper is an extension of the centrally weighted nonoscillatory 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 Machnumber 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 offdiagonal elements of the electron pressure tensor could be confirmed. The inductive electric field at the XLine is found to be dominated by the nongyrotropic 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 nongyrotropic 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 ultrashort 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 micromachining (ablation) and microfabrication etc. These applications require very high intensity of the laser field, typically 10^1310^15 TW/cm^2. Such high intensity leads to significant ionisation and creation of electronion or electronhole 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 laserplasma interaction during femtosecond laser pulse propagation are derived and discussed. It turns out that the strongly nonlinear effects such selffocusing followed by the pulse splitting are essential. These phenomena feature extremely complex dynamics of both the electromagnetic field and plasma density with different spatiotemporal 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
DarwinVlasov 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 fluxconservative 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 Hallmagnetohydrodynamics
Magnetohydrodynamics with Hall effect (HallMHD) 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 quasimonochromatic 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 finitedifference 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 HallMHD: A tendency towards finitetime singularity formation
Spontaneous development of shocklike singularities in axisymmetric solutions of the HallMHD equations is discussed. It is shown that the Hallterm in Ohm's law leads to a Burgerstype 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 HallMHD systems
The dependence of the Hallterm on the width of the magnetic islands of the tearingmode is examined. We applied the center manifold (CMF) theory to a Magnetohydrodynamic (MHD)system. The MHDsystem was chosen to be incompressible and includes in addition to viscosity the Hallterm in Ohm's law. For certain values of physical parameters the corresponding center manifold is twodimensional and therefore the original partial differential equations could be reduced to a twodimensional system of ordinary ones. This amplitude equations exhibit a pitchforkbifurcation which corresponds to the occurrence of the tearingmode. Eigenvalueproblems 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 MeshAdaptive Framework for Hyperbolic Conservation Laws
We report on the development of a computational framework for the parallel, meshadaptive solution of systems of hyperbolic conservation laws like the timedependent Euler equations in compressible gas dynamics orMagnetoHydrodynamics (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 finitedi erences and central schemes with Runge Kutta integrators. Parallel execution is achieved through a configurable hybrid of multithreading and MPIdistribution with dynamic load balancing. One two and threedimensional 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 KrylovSchwarz 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 (semiimplicit and implicit numerical methods), it becomes necessary to solve global equations.
This paper focuses on the application and performance of wellestablished preconditioned KrylovSchwarz solvers in an AMR context, using a KrylovSchwarz method to accelerate convergence while exploiting the hierarchical structure of AMR grids for multilevel 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 threedimensional Euler equations in the search for a finitetime 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
Threedimensional MHD highresolution computations with CWENO employing adaptive mesh refinement
Until recently, numerical simulations of discontinuities in highly superAlfvénic 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 NonOscillatory (CWENO) scheme. Combining this technique with that of adaptive mesh refinement (AMR), we have developed a thirdorder 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 Sedovtype explosion.
Comp. Phys. Comm. 158 (2004) 47


R. Grauer and F. Spanier
A note on the use of central schemes for incompressible NavierStokes 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 selfconsistent 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 timedependent. Significant progress w.r.t. their numerical realization was achieved recently with the Central Weighted Essentially NonOscillaroty 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, 914 September 2002 (ESA SP506, 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 selfconsistent 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 twofluid 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, 914 September 2002 (ESA SP506, December (2002) 21


L. Berge, K. Germaschewski, R. Grauer, and J. Juul Rasmussen
Hyperbolic Shock Waves of the Optical SelfFocusing with Normal GroupVelocity Dispersion
The theory of focusing light pulses in Kerr media with normal groupvelocity 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 selffocusing from
the domain of temporal dispersion. Justiﬁed by a selfsimilar approach, this property is conﬁrmed by
numerical simulations using an adaptivemesh 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 3D 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 selffocusing from the domain of temporal dispersion. Justified by a selfsimilar approach, this property is confirmed by numerical simulations using an adaptivemesh 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 twodimensional 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 socalled Burgers turbulence. The most striking feature is that the tails of the probability distribution of velocity increments could be calculated using information of the preshocks present in the flow. The situation is expected to be similar in the NavierStokes and MHD equations, where vortex tubes or current sheets may substitute the role of preshocks 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 twodimensional electron magnetohydrodynamic flows
Electron magnetohydrodynamic flows have recently attracted considerable interest not only in the field of collisionless reconnection but also as the first twodimensional 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 logPoisson 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 blowup 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 welldefined 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 nonadaptive 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 HasegawaMima 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 HasegawaMima 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 HasegawaMima equation which includes in addition a socalled 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. 191198). 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 randomforcedriven Burgers equation
The statistics of solutions to a family of onedimensional randomforcedriven advectiondiffusion equations is studied using high resolution numerical simulations. The equation differs from the usual Burgers equation by the nonlocal form of the nonlinear interaction term mimicking the nonlocality of the NavierStokes 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 nonadaptive 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 twodimensional hydrodynamic flows. Finally, we discuss the applicability of operatorproduct 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
Threedimensional 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 twodimensional magnetohydrodynamic flows
In twodimensional magnetohydrodynamic turbulence, the KraichnanIroshnikov 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 selfsimilarity 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 threedimensional NavierStokes 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 KahunenLoeve 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 lowdimensional behavior.
Phys. Lett. A 198 (1995) 383


R. Grauer and C. Marliani
Structure functions in magnetohydrodynamic turbulence
Higher order structure functions of twodimensional 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 highorder 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 sineGordon breathers
The dynamics of a breather in the damped and parametrically driven sineGordon equation is investigated both numerically and analytically. The KahunenLoeve 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 sineGordon equation showing perfect agreement. Information from the periodic spectral theory and linear stability analysis is used to identify the KahunenLoeve modes.
in Nonlinear coherent structures in physics and biology, Bayreuth, Germany, edited by K. H. Spatschek and F. G. Mertens, pages 381384, New York, 1994, Plenum. (1994)


B. Birnir and R. Grauer
An Explicit Description of the Global Attractor of the Damped and Driven SineGordon Equation
We prove that the size of the finitedimensional attractor of the damped and driven sineGordon equation stays small as the damping and driving amplitude become small. A decomposition of finitedimensional attractors in Banach space is found, into a part B that attracts all of phase space, except sets whose finitedimensional projections have Lebesgue measure zero, and a part C that only attracts sets whose finitedimensional projections have Lebesgue measure zero. We describe the components of the Battractor and C, which is called the "hyperbolic" structure, for the damped and driven sineGordon equation. B is lowdimensional 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 Battractor in this small parameter region.
Comm. Math. Phys. 162 (1994) 539


R. Grauer and Y. Kivshar
Chaotic and phaselocked breather dynamics in the damped and parametrically driven sineGordon equation
The dynamics of a breather in the damped and parametrically driven sineGordon equation is investigated both numerically and analytically. The KahunenLoeve 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 sineGordon 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 KahunenLoeve expansion and direct simulations are discussed. Finally, information from the periodic spectral theory and linear stability analysis is used to identify the KahunenLoeve 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 LowFrequency Excitations in Magnetized Plasmas
For the dissipative trappedion mode a simple onedimensional 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_{1x}As quantum wells in intense farinfrared 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 doublewell 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, doublewell and singlewell, are possible. Individual chaotic proton motion is triggered by the oscillations of the lattice. Depending on the system parameters, both the onewell and the crosswell attractors can be either periodic or chaotic. This has some interesting consequences for the interpretation and understanding of propagation of ionic defects in hydrogenbonded chains in the presence of external oscillating fields. A new frequencylocked propagating kink is found.
Phys. Rev. E 47 (1993) 236


R. Grauer and B. Birnir
The center manifold and bifurcations of the sineGordon equation
The generic bifurcations of breathers in the damped and driven sineGordon 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 sineGordon 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 threedimensional incompressible Euler equations. Using rotational symmetry, the Euler equations reduce to a twodimensional 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 multivortex mode in the square sineGordon 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 sineGordon system
This paper shows by use of simple physical arguments the existence of a particular multivortex dynamical configuration in a  perturbed or not  twodimensional sineGordon system. The stability of such modes is numerically checked. Their main topological invariant is the total length of the +/ 2Pi  wavefronts entering as elementary kinklike 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
Cellmapping description of coexisting phaselocked soliton states in a long Josephson junction
The coexistence of phaselocked soliton states in a long acbiased 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 phaselocked states consist of the well known zeroFieldStep (shuttling regime of solitons) and the socalled Ccycle 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 currentvoltage curve.
Phys. Rev. B 42 (1990) 9987


J. C. Fernandez, R. Grauer, K. Pinnow, and G. Reinisch
Phaselocked dynamical regimes to an external microwave field in a long, unbiased Josephson junction
We show the existence of asymmetrical (in the phase space) phaselocked limit cycles of (anti) kinks in inhomogeneously ac driven sineGordon 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 2d 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 phaselocked to an external microwave field in a long, unbiased Josephson junction
We demonstrate the existence of phaselocked limit cycles in inhomogenously driven sineGordon 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 sineGordon systems
A possible route to temporal chaos with coherent stable spatial structures is proposed for the driven damped sineGordon equation. For nearconservative perturbations, the dynamics of a breather is investigated numerically and semianalytically in the presence of an ac driver and a simple damping term. For moderate driver strength, a flat (spaceindependent) attractor exists whereas above a threshold a phaselocked breather coexists. The latter can undergo a perioddoubling 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 nearconservative perturbations and the low driver strengths allow to interpret the results within a simple model originating from the socalled nonlinear Schrödinger limit. In fact, within this limit (small amplitude breather), the chaotic (or not) transitions are dominated by interactions between breatherlike solutions and radiation (mostly k = 0 mode). Therefore, three collective coordinates, i.e. the amplitude of the phaselocked 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 fourdimensional. 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 spatiotemporal 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 fourdimensional nullspace corresponding to two different marginal tearing modes. With the center manifold theory the original partial differential equations could be reduced to a fourdimensional 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 spatiotemporal 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) 257282).
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 crosssection 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 crosssection and flat current profile
To calculate the equilibrium and stability of finite aspect ratio tokamaks, even for circular and elliptic plasma crosssections, 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 crosssections. In this paper, finite aspect ratio equilibrium solutions are derived for circular, prolate and oblate elliptical crosssections and for a flat current profile. In addition, the problem of axisymmetric stability is studied for prolate elliptical crosssections, 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 crosssection
in Proceedings of the 11th European Conference on Controlled Fusion and Plasma Physics, Aachen, Germany , European Physical Society 197 (1983)
