List of Publications 
H. Homann, O. Kamps, R. Friedrich and R. Grauer
Bridging from Eulerian to Lagrangian statistics in 3D hydro and magnetohydrodynamic turbulent flowsAbstract: 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 turbulenceAbstract: 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.
submitted to Physics of Plasmas (2009) 

O. Kamps, R. Friedrich, and R. Grauer
An exact relation between Eulerian and Lagrangian velocity increment statisticsAbstract: 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) 066301. 

R. Friedrich, R. Grauer, H. Homann, and O. Kamps
Statistics of a mixed EulerianLagrangian velocity increment in fully developped turbulenceAbstract: 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. Download (pdf) 

J. Kleimann, A. Kopp, H. Fichtner, and R. Grauer
A novel code for numerical 3D MHD studies of CME expansionAbstract:
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.
Ann. Geophys. 27 (2009) 9891004. Download (pdf) 

R. Kissmann, J. Kleimann, H. Fichtner and R. Grauer
Local turbulence simulations for the multiphase ISMAbstract: 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) 15771588. 

T. Grafke, H. Homann, J. Dreher, and R. Grauer
Numerical simulations of possible finite time singularities in the incompressible Euler equations: comparison of numerical methodsAbstract: 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 flowsAbstract: 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 TurbulenceAbstract: 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 

H. Fichtner, A. Kopp, J. Kleimann, and R. Grauer
On MHD modelling of Coronal Mass EjectionsAbstract: 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 TurbulenceAbstract: 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 ropesAbstract: 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) 042106. 

A. Busse, W.C Müller, H. Homann and R. Grauer
Statistics of passive tracers in threedimensional magnetohydrodynamic turbulenceAbstract: 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. 

L. Arnold, J. Dreher and R. Grauer
A semi implicit HallMHD solver using whistler wave preconditioningAbstract: 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. Homann, J. Dreher and R. Grauer
Impact of the floatingpoint precision and interpolation scheme on the results of DNS of turbulence by pseudospectral codesAbstract: 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. Download (pdf) 

V. Mezentsev, J.S. Petrovic, M. Dubov, I. Bennion,
J. Dreher, H. Schmitz, and R. Grauer
Femtosecond laser microfabrication of subwavelength structures in photonicsAbstract: 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) B4590, doi:10.1117/12.705800 

H. Homann, R. Grauer, A. Busse and W.C. Müller
Lagrangian Statistics of NavierStokes and MHDTurbulenceAbstract: 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, doi: 10.1017/S0022377807006575 Download (pdf) 

R. Kissmann and R. Grauer
A low dissipation essentially nonoscillatory central schemeAbstract: 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. Download (pdf) 

H. Schmitz and R. Grauer
Vlasov simulations of collisionless magnetic reconnection without background densityAbstract: 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, doi:10.1016/j.cnsns.2007.04.003 Download (pdf) 

H. Schmitz and R. Grauer
Kinetic Vlasov Simulations of collisionless magnetic ReconnectionAbstract: 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) 092309. Download (pdf) 

V. Mezentsev, M. Dubov, J. Petrovic, I. Bennion, J. Dreher, and R. Grauer
Role of Plasma in Femtosecond Laser Pulse PropagationAbstract: 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, doi:10.1063/1.2406026 Download (pdf) 

V. Mezentsev, J. Petrovic, J. Dreher, and R. Grauer
Adaptive modeling of the femtosecond inscription in silicaAbstract: 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) 61070R, doi:10.1117/12.647303 Download (pdf) 

H. Schmitz and R. Grauer
Comparison of time splitting and backsubstitution methods for integrating Vlasov?s equation with magnetic fieldsAbstract: 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  92. Download (pdf) 

H. Schmitz and R. Grauer
DarwinVlasov Simulations of magnetized PlasmasAbstract: 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  756. Download (pdf) 

J. Dreher, D. Laveder, R. Grauer, T. Passot and P.L. Sulem
Formation and disruption of Alfvénic filaments in HallmagnetohydrodynamicsAbstract:
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.
Phys. Plasmas 12 (2005) 052319. Download (pdf) 

J. Dreher, V. Ruban, and R. Grauer
Axisymmetric flows in HallMHD: A tendency towards finitetime singularity formationAbstract: 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. Download (pdf) 

H. Homann and R. Grauer
Bifurcation analysis of magnetic reconnection in HallMHD systemsAbstract: 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) 5972. Download (pdf) 

J. Dreher and R. Grauer
Racoon: A Parallel MeshAdaptive Framework for Hyperbolic Conservation LawsAbstract: 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) 913932. Download (pdf) 

K. Germaschewski, A. Bhattacharjee, R. Grauer, D. Keyes, and B. Smith
Using KrylovSchwarz methods in an adaptive mesh refinement enviromentAbstract:
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.
in Adaptive Mesh Refinement  Theory and Applications, Lecture Notes in Computational Sciences and Engineering (LNCSE) series, editors Tomasz Plewa, Timur Linde and V. Gregory Weirs, (Springer 2004) Download (ps.gz) 

J. Kleimann, A. Kopp, H. Fichtner, R. Grauer, and K. Germaschewski
Threedimensional MHD highresolution computations with CWENO employing adaptive mesh refinementAbstract: 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. Download (ps.gz) 

R. Grauer and F. Spanier
A note on the use of central schemes for incompressible NavierStokes flowsJ. Comp. Phys. 192 (2003) 727. Download (pdf) 

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 ModelAbstract: 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) 5154. Download (ps.gz) 

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 ModelAbstract: 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) 2124. Download (ps.gz) 

L. Bergé, K. Germaschewski, R. Grauer, and J. Juul Rasmussen
Hyperbolic shock waves of the optical selffocusing with normal GVDAbstract: 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. 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 dispersionAbstract: The selffocusing and splitting mechanisms of waves governed by the cubic nonlinear Schrödinger equation with anisotropic dispersion are investigated numerically by means of an adaptive mesh refinement code. Wavepackets having a power far above the selffocusing threshold undergo a transversal compression and are shown to split into two symmetric peaks. These peaks can sequentially decay into smallerscale structures developing near the front edge, as long as their individual power remains above threshold, until the final dispersion of the wave. Their phase a nd amplitude dynamics are detailed and compared with those characterizing collapsing objects with no anisotropic dispersion. Their ability to mutually coalesce is also analyzed and modelled from the interaction of Gaussian components. Next, bunchtype and snaketype instabilities, which result from periodic modulations driven by even and odd localized modes, are studied. The influence of the initial wave amplitude, the amplitude and wavenumber of the perturbations on the interplay of snake and bunch patterns are finally discussed. Physica D 151 (2001) 175198. 

R. Grauer and C. Marliani
Current Sheet Formation in 3D Ideal Incompressible MagnetohydrodynamicsAbstract: 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) 48504853. Download (ps.gz) 

R. Grauer
Modeling of strong MHD turbulenceAbstract: 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) 197204. 

K. Germaschewski and R. Grauer
Longitudinal and transversal structure functions in twodimensional electron magnetohydrodynamic flowsAbstract: 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) 37883793. Download (ps.gz) 

R. Grauer
Adaptive Mesh Refinement for Singular Structures in incompressible hydro and magnetohydrodynamic flowsAbstract: 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) 401411. 

M. Blüming, K. H. Spatschek, and R. Grauer
Center manifold approach to the reduced magnetohydrodynamic bifurcations with diffusive magnetic field linesAbstract: 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) 10831092. 

H. Friedel, R. Grauer, and K. H. Spatschek
Controlling chaotic states of a Pierce diodeAbstract: 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) 31873194. Download (ps.gz) 

R. Grauer, C. Marliani, and K. Germaschewski
Adaptive mesh refinement for singular solutions of the incompressible Euler equationsAbstract: 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) 48504853. Download (ps.gz) 

R. Grauer
An Energy estimate for a perturbed HasegawaMima equationAbstract: 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) 659666. Download (ps.gz) 

R. Grauer and C. Marliani
Geometry of singular structures in magnetohydrodynamic flowsAbstract: 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) 25442552. 

H. Friedel, R. Grauer, and C. Marliani
Center manifold approach to controlling chaosAbstract: 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) 4552. Download (ps.gz) 

O. Zikanov, A. Thess, and R. Grauer
Statistics of turbulence in a generalized randomforcedriven Burgers equationAbstract: 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) 13621367. 

H. Friedel, R. Grauer, and C. Marliani
Adaptive mesh refinement for singular current sheets in incompressible magnetohydrodynamic flowsAbstract: 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) 190198. Download (ps.gz) 

R. Grauer and C. Marliani
Analytical and numerical approaches to structure functions in magnetohydrodynamic turbulenceAbstract: In magnetohydrodynamic turbulence, the classical theory by Kraichnan and Iroshnikov based on dimensional analysis gives a linear dependence of the exponents $\ζ$_{p}=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) 3842. Download (ps.gz) 

R. Grauer and T. C. Sideris
Finite time singularities in ideal fluids with swirlAbstract: 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) 116132. 

R. Grauer and C. Marliani
Numerical and analytical estimates for the structure functions in twodimensional magnetohydrodynamic flowsAbstract: 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) 4147. 

R. Grauer and C. Marliani
Structure functions in magnetohydrodynamic turbulenceAbstract: 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. in Structure and Dynamics of Nonlinear Waves in Fluids, edited by K. Kirchgässner and A. Mielke, London, 1994, World Scientific. 

R. Grauer, J. Krug, and C. Marliani
Scaling of highorder structure functions in magnetohydrodynamic turbulenceAbstract: 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) 335338. 

T. Eickermann, R. Grauer, and K. H. Spatschek
Identification of mass capturing structures in a perturbed nonlinear Schrödinger equationAbstract: 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) 383388. 

R. Grauer and Y. Kivshar
Dynamics of parametrically driven sineGordon breathersAbstract: 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. 

B. Birnir and R. Grauer
An Explicit Description of the Global Attractor of the Damped and Driven SineGordon EquationAbstract: 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) 539590. 

R. Grauer and Y. Kivshar
Chaotic and phaselocked breather dynamics in the damped and parametrically driven sineGordon equationAbstract: 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) 47914800. 

P. Beyer, R. Grauer, and K. H. Spatschek
Center Manifold Theory for LowFrequency Excitations in Magnetized PlasmasAbstract: 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) 46654673. 

T. Eickermann, R. Grauer, and K. H. Spatschek
New Aspects of chaotic dynamics in nonlinear Schrödinger systemsin 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, Vol. 312, pages 109112, Plenum Press, 1993. 

B. Birnir, B. Galdrikian, R. Grauer, and M. Sherwin
Nonperturbative resonances in periodically driven quantum wellsAbstract: 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) 67956798. 

R. Grauer, K. H. Spatschek, and A. V. Zolotaryuk
Chaotic proton dynamics in the hydrogen bondAbstract: 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) 236247. 

R. Grauer and B. Birnir
The center manifold and bifurcations of the sineGordon equationAbstract: 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) 165184. 

R. Grauer and T. C. Sideris
Numerical computation of 3D incompressible ideal fluids with swirlAbstract: 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) 35113514. 

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 systemin Nonlinear World, edited by V. G. Baryakhtar, V. M. Chernosenko, N. S. Erokhin, A. G. Sitenko, and V. E. Zakharov, pages 363383, Singapore, 1990, World Scientific. 

E. Turlot, D. Esteve, C. Urbina, M. Devoret,
R. Grauer, J. C. Fernandez, and G. Reinisch
Dynamical isoperimeter pattern in the square sineGordon systemAbstract: 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) 84188425. 

J. C. Fernandez, R. Grauer, K. Pinnow, and G. Reinisch
Cellmapping description of coexisting phaselocked soliton states in a long Josephson junctionAbstract: 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) 99879995. 

J. C. Fernandez, R. Grauer, K. Pinnow, and G. Reinisch
Phaselocked dynamical regimes to an external microwave field in a long, unbiased Josephson junctionAbstract: 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) 333339. 

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?Abstract: 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,


J. C. Fernandez, R. Grauer, and G. Reinisch
Dynamical regimes phaselocked to an external microwave field in a long, unbiased Josephson junctionAbstract: 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, Vol. 353, pages 213219, Springer, 1989. 

M. Taki, K. H. Spatschek, J. C. Fernandez,
R. Grauer, and G. Reinisch
Breather dynamics in the nonlinear Schrödinger regime of perturbed sineGordon systemsAbstract: 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) 6582. 

R. Grauer
Codimension two interactions of tearing modesAbstract: 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, pages 267276, Singapore, 1989, World Scientific. 

R. Grauer
Nonlinear interactions of tearing modes in the vicinity of a bifurcation point of codimension twoAbstract: 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) 107126. 

E. Rebhan and R. Grauer
Tokamak profiles through constrained minimization of the entropy productionAbstract: 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, pages 10721075, European Physical Society, 1987. 

R. Grauer and E. Rebhan
Analysis of ideal MHD equilibrium and axisymmetric stability for finite aspect ratio tokamaks with elliptic crosssection and flat current profileAbstract: 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) 99117. 

R. Grauer and E. Rebhan
Semianalytic calculation of equilibrium and axisymmetric stability of finite aspect ratio tokamaks with elliptic crosssectionin Proceedings of the 11th European Conference on Controlled Fusion and Plasma Physics, Aachen, Germany , pages 197200, European Physical Society, 1983. 
