# List of Publications

H. Homann, O. Kamps, R. Friedrich and R. Grauer

#### Bridging from Eulerian to Lagrangian statistics in 3D hydro- and magnetohydrodynamic turbulent flows

Abstract:

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

Abstract:

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

submitted to Physics of Plasmas (2009)

O. Kamps, R. Friedrich, and R. Grauer

#### An exact relation between Eulerian and Lagrangian velocity increment statistics

Abstract:

We present a formal connection between Lagrangian and Eulerian velocity increment distributions which is applicable to a wide range of turbulent systems ranging from turbulence in incompressible fluids to magnetohydrodynamic turbulence. For the case of the inverse cascade regime of two-dimensional turbulence we numerically estimate the transition probabilities involved in this connection. In this context we are able to directly identify the processes leading to strongly non-Gaussian statistics for the Lagrangian velocity increments.

Phys. Rev. E 79 (2009) 066301.

R. Friedrich, R. Grauer, H. Homann, and O. Kamps

#### Statistics of a mixed Eulerian-Lagrangian velocity increment in fully developped turbulence

Abstract:

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

Physica Scripta 79 (2009) 55403.

J. Kleimann, A. Kopp, H. Fichtner, and R. Grauer

#### A novel code for numerical 3-D MHD studies of CME expansion

Abstract:

A recent third-order, essentially non-oscillatory central scheme to advance the equations of single-fluid magnetohydrodynamics (MHD) in time has been implemented into a new numerical code. This code operates on a 3-D Cartesian, non-staggered grid, and is able to handle shock-like gradients without producing spurious oscillations.

To demonstrate the suitability of our code for the simulation of coronal mass ejections (CMEs) and similar heliospheric transients, we present selected results from test cases and perform studies of the solar wind expansion during phases of minimum solar activity. We can demonstrate convergence of the system into a stable Parker-like steady state for both hydrodynamic and MHD winds. The model is subsequently applied to expansion studies of CME-like plasma bubbles, and their evolution is monitored until a stationary state similar to the initial one is achieved. In spite of the model's (current) simplicity, we can confirm the CME's nearly self-similar evolution close to the Sun, thus highlighting the importance of detailed modelling especially at small heliospheric radii.

Ann. Geophys. 27 (2009) 989-1004.

R. Kissmann, J. Kleimann, H. Fichtner and R. Grauer

#### Local turbulence simulations for the multiphase ISM

Abstract:

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

Mon. Not. R. Astron. Soc. 391 (2008) 1577-1588.

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

Abstract:

The numerical simulation of the 3D incompressible Euler equation is analyzed with respect to different integration methods. The numerical schemes we considered include spectral methods with different strategies for dealiasing and two variants of finite difference methods. Based on this comparison, a Kida-Pelz like initial condition is integrated using adaptive mesh refinement and estimates on the necessary numerical resolution are given. This estimate is based on analyzing the scaling behavior similar to the procedure in critical phenomena and present simulations are put into perspective.

Physica D 237 (2008) 1932.

A. Arneodo, R. Benzi, J. Berg, L. Biferale, E. Bodenschatz, A. Busse, E. Calzavarini, B. Castaing, M. Cencini, L. Chevillard, R.T. Fisher, R. Grauer, H. Homann, D. Lamb, A.S. Lanotte, E. Leveque, B. Luthi, J. Mann, N. Mordant, W.C. Mueller, S. Ott, N.T. Ouellette, J.F. Pinton, S.B. Pope, S.G. Roux, F. Toschi, H. Xu, P.K. Yeung

#### Universal intermittent properties of particle trajectories in highly turbulent flows

Abstract:

We present a collection of eight data sets from state-of-the-art experiments and numerical simulations on turbulent velocity statistics along particle trajectories obtained in different flows with Reynolds numbers in the range R-lambda is an element of [120740]. Lagrangian structure functions from all data sets are found to collapse onto each other on a wide range of time lags, pointing towards the existence of a universal behavior, within present statistical convergence, and calling for a unified theoretical description. Parisi-Frisch multifractal theory, suitably extended to the dissipative scales and to the Lagrangian domain, is found to capture the intermittency of velocity statistics over the whole three decades of temporal scales investigated here.

Phys. Rev. Lett. 100 (2008) 254504.

L. Arnold, C. Beetz, J. Dreher, H. Homann, C. Schwarz and R. Grauer

#### Massively Parallel Simulations of Solar Flares and Plasma Turbulence

Abstract:

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
editors C. Bischof, M. Bücker, P. Gibbon, G. Joubert, T. Lippert, B. Mohr, F. Peters
IOS Press, Advances in Parallel Computing, Volume 15 (2008)

H. Fichtner, A. Kopp, J. Kleimann, and R. Grauer

#### On MHD modelling of Coronal Mass Ejections

Abstract:

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

Astron. Soc. Pac. Conf. Ser. 385 (2008) 151.

C. Beetz, C. Schwarz, J. Dreher, R. Grauer

#### Density-PDFs and Lagrangian Statistics of highly compressible Turbulence

Abstract:

In isothermal, highly compressible turbulent flows, density fluctuations follow a log-normal distribution. We establish a connection between these density fluctuations and the probability-density-functions (PDF) of Lagrangian tracer particles advected with the flow. Our predicted particle statistics is tested against large scale numerical simulations, which were performed with $512^3$ collocation points and 2 million tracer particles integrated over several dynamical times.

Physics Letters A 372 (2008) 3037.

L. Arnold, J. Dreher, R. Grauer, H. Soltwisch, H. Stein

#### Three-dimensional MHD simulation of expanding magnetic flux ropes

Abstract:

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

Phys. Plasmas 15 (2008) 042106.

A. Busse, W.C Müller, H. Homann and R. Grauer

#### Statistics of passive tracers in three-dimensional magnetohydrodynamic turbulence

Abstract:

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

Phys. Plasmas 14 (2007) 122303.

L. Arnold, J. Dreher and R. Grauer

#### A semi implicit Hall-MHD solver using whistler wave preconditioning

Abstract:

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

Comp. Phys. Comm. 178 (2008) 553.

H. Homann, J. Dreher and R. Grauer

#### Impact of the floating-point precision and interpolation scheme on the results of DNS of turbulence by pseudo-spectral codes

Abstract:

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

Comp. Phys. Comm. 177 (2007) 560.

V. Mezentsev, J.S. Petrovic, M. Dubov, I. Bennion, J. Dreher, H. Schmitz, and R. Grauer

#### Femtosecond laser microfabrication of subwavelength structures in photonics

Abstract:

This paper describes experimental and numerical results of the plasma-assisted microfabrication of subwavelength structures by means of point-by point femtosecond laser inscription. It is shown that the spatio-temporal evolution of light and plasma patterns critically depend on input power. Subwavelength inscription corresponds to the supercritical propagation regimes when pulse power is several times self-focusing threshold. Experimental and numerical profiles show quantitative agreement.

Proc SPIE 6459 (2007) B4590, doi:10.1117/12.705800

H. Homann, R. Grauer, A. Busse and W.C. Müller

#### Lagrangian Statistics of Navier-Stokes- and MHD-Turbulence

Abstract:

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

J. Plasma Phys. 73 (2007) 821, doi: 10.1017/S0022377807006575

R. Kissmann and R. Grauer

#### A low dissipation essentially non-oscillatory central scheme

Abstract:

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

Comp. Phys. Comm. 176 (2007) 522.

H. Schmitz and R. Grauer

#### Vlasov simulations of collisionless magnetic reconnection without background density

Abstract:

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 Alfven speed remains finite far away from the current sheet.We present a fully kinetic Vlasov simulation of the perturbed Harris equilibrium using a Vlasov code. Parameters are chosen to match the Geospace Environment Modeling (GEM) Magnetic Reconnection Challenge but excluding the background density. This allows to compare with earlier simulations [Schmitz, Grauer, Phys. Plasmas 13 (2006) 092309] which include the background density. It is found that the absence of a background density causes the reconnection rate to be higher. On the other hand, the time until the onset of reconnection is hardly affected. Again the off diagonal elements of the pressure tensor are found to be important on the X-line but with modified importance for the individual terms.

Communications in Nonlinear Science and Numerical Simulation 13 (2008) 169, doi:10.1016/j.cnsns.2007.04.003

H. Schmitz and R. Grauer

#### Kinetic Vlasov Simulations of collisionless magnetic Reconnection

Abstract:

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

Physics of Plasmas 13 (2006) 092309.

V. Mezentsev, M. Dubov, J. Petrovic, I. Bennion, J. Dreher, and R. Grauer

#### Role of Plasma in Femtosecond Laser Pulse Propagation

Abstract:

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

AIP Conf. Proc. 876 (2006) 169, doi:10.1063/1.2406026

V. Mezentsev, J. Petrovic, J. Dreher, and R. Grauer

#### Adaptive modeling of the femtosecond inscription in silica

Abstract:

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

H. Schmitz and R. Grauer

#### Comparison of time splitting and backsubstitution methods for integrating Vlasov?s equation with magnetic fields

Abstract:

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.

H. Schmitz and R. Grauer

#### Darwin-Vlasov Simulations of magnetized Plasmas

Abstract:

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

J. Comp. Phys. 214 (2006) 738 - 756.

J. Dreher, D. Laveder, R. Grauer, T. Passot and P.L. Sulem

#### Formation and disruption of Alfvénic filaments in Hall-magnetohydrodynamics

Abstract:

Magnetohydrodynamics with Hall effect (Hall-MHD) allows one to take into account scales of the order of the ion inertial length and the dispersive character of media like the Earth magnetosheath. In these conditions, weakly nonlinear quasi-monochromatic Alfv\'en waves propagating along an ambient magnetic field can be subject to transverse instabilities leading to the formation of intense magnetic filaments.
This "filamentation" phenomenon, predicted by amplitude equations of nonlinear Schrödinger type and also observed in direct numerical simulations using spectral method is here reconsidered using a finite-difference adaptive mesh refinement code (AMR). This approach allows the simulation to be proceeded long enough to capture the destabilization of the filamentary structures and the formation of gradient singularities in the longitudinal direction, associated with the development of intense current sheets and with a strong acceleration of the plasma.

Phys. Plasmas 12 (2005) 052319.

J. Dreher, V. Ruban, and R. Grauer

#### Axisymmetric flows in Hall-MHD: A tendency towards finite-time singularity formation

Abstract:

Spontaneous development of shock-like singularities in axisymmetric solutions of the Hall-MHD equations is discussed. It is shown that the Hall-term in Ohm's law leads to a Burgers-type equation for the magnetic field evolution in weakly compressible regime. Numerical simulations are used to investigate the validity of this approximation for a particular class of initial conditions.

Physica Scripta 72 (2005) 450.

H. Homann and R. Grauer

#### Bifurcation analysis of magnetic reconnection in Hall-MHD systems

Abstract:

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

Physica D 208 (2005) 59-72.

J. Dreher and R. Grauer

#### Racoon: A Parallel Mesh-Adaptive Framework for Hyperbolic Conservation Laws

Abstract:

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

Parallel Computing 31 (2005) 913-932.

K. Germaschewski, A. Bhattacharjee, R. Grauer, D. Keyes, and B. Smith

#### Using Krylov-Schwarz methods in an adaptive mesh refinement enviroment

Abstract:

Much of the previous work in AMR methods has concentrated on solving hyperbolic equations with explicit timestepping. However, for many problems, either due to their physical nature (e.g. incompressible flows) or for performance reasons (semi-implicit and implicit numerical methods), it becomes necessary to solve global equations.
This paper focuses on the application and performance of well-established preconditioned Krylov-Schwarz solvers in an AMR context, using a Krylov-Schwarz method to accelerate convergence while exploiting the hierarchical structure of AMR grids for multi-level preconditioning using the fast adaptive composite (FAC) algorithm. We present an implementation which allows us to leverage the powerful supply of preconditioners and linear solvers from the PETSc library.
We apply this method to solve the three-dimensional Euler equations in the search for a finite-time singularity.

in Adaptive Mesh Refinement - Theory and Applications, Lecture Notes in Computational Sciences and Engineering (LNCSE) series, editors Tomasz Plewa, Timur Linde and V. Gregory Weirs, (Springer 2004)

J. Kleimann, A. Kopp, H. Fichtner, R. Grauer, and K. Germaschewski

#### Three-dimensional MHD high-resolution computations with CWENO employing adaptive mesh refinement

Abstract:

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

Comp. Phys. Comm. 158 (2004) 47.

R. Grauer and F. Spanier

#### A note on the use of central schemes for incompressible Navier-Stokes flows

J. Comp. Phys. 192 (2003) 727.

J. Kleimann, H, Fichtner, A. Kopp, K. Germaschewski, and R. Grauer

#### On the dynamics of the solar corona: the numerics behind a self-consistent 3D MHD Model

Abstract:

Space missions like SOHO have renewed the interest in the physics of the solar corona. This complex system is not yet fully understood due to lack of sufficiently detailed observations, and also because realistic models should cover processes occuring on various spatial scales, while being both multidimensional and time-dependent. Significant progress w.r.t. their numerical realization was achieved recently with the Central Weighted Essentially Non-Oscillaroty scheme. A 3rd order CWENO scheme efficiently capturing strong gradients forms the basis of our new code. After describing the algorithm and its implementaion, we present test results as well as comparisons with preexisting codes.

Proc. 10th. European Solar Physics Meeting, 'Solar Variability: From Core to Outer Frontiers', Prague, Czech Republic, 9-14 September 2002 (ESA SP-506, December 2002) 51-54.

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

Abstract:

A newly developed self-consistent 3D MHD code is applied to the problem of the dynamics of the solar corona. First, we present the basic system of equations for a two-fluid description of the solar wind plasma and point out possible numerical difficulties arising from an improper choice of variables. Second, we perform a study of the solar wind expansion during phases of minimum solar activity, serving mainly as a first `real world' test case. Third, we discuss first results of the application of the model to propagating disturbances, such as coronal mass ejections and/or shocks.

Proc. 10th. European Solar Physics Meeting, 'Solar Variability: From Core to Outer Frontiers', Prague, Czech Republic, 9-14 September 2002 (ESA SP-506, December 2002) 21-24.

L. Bergé, K. Germaschewski, R. Grauer, and J. Juul Rasmussen

#### Hyperbolic shock waves of the optical self-focusing with normal GVD

Abstract:

The theory of focusing light pulses in Kerr media with normal group velocity dispersion (GVD) in (2+1) and (3+1) dimensions is revisited. It is shown that pulse splitting introduced by GVD follows from shock fronts that develop along hyperbolas separating the region of transverse self-focusing from the domain of temporal dispersion. Justified by a self-similar approach, this property is confirmed by numerical simulations using an adaptive-mesh refinement code.

Phys. Rev. Lett. 89 (2002) 153902.

K. Germaschewski, R. Grauer, L. Bergé, V.K. Mezentsev, and J. Juul Rasmussen

#### Splittings, coalescence, bunch and snake patterns in the 3-D nonlinear Schrödinger equation with anisotropic dispersion

Abstract:

The self-focusing 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. Wave-packets having a power far above the self-focusing threshold undergo a transversal compression and are shown to split into two symmetric peaks. These peaks can sequentially decay into smaller-scale 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, bunch-type and snake-type 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) 175-198.

R. Grauer and C. Marliani

#### Current Sheet Formation in 3D Ideal Incompressible Magnetohydrodynamics

Abstract:

The evolution of current density and vorticity in the ideal, inviscid incompressible magnetohydrodynamic equations in three dimensions is studied numerically. Highly effective resolutions are obtained by adaptive structured mesh refinement techniques. We report on results for three different initial conditions showing similar behavior: in the early stage of the evolution a fast increase in vorticity and current density is observed. Thereafter, the evolution towards nearly two-dimensional current sheets results in a depletion of nonlinearity.

Phys. Rev. Lett. 84 (2000) 4850-4853.

R. Grauer

#### Modeling of strong MHD turbulence

Abstract:

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

in Plasma Turbulence and Energetic Particles in Astrophysics, Obserwatorium Astronomiczne, Uniwersytet Jagielloriski, Krakow, (1999) 197-204.

K. Germaschewski and R. Grauer

#### Longitudinal and transversal structure functions in two-dimensional electron magnetohydrodynamic flows

Abstract:

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

Physics of Plasmas 6 (1999) 3788-3793.

R. Grauer

#### Adaptive Mesh Refinement for Singular Structures in incompressible hydro- and magnetohydrodynamic flows

Abstract:

The question whether finite time singularities develop in incompressible hydro- and magnetohydrodynamic systems starting from smooth initial conditions is still an open problem. Here we present numerical simulations using the technique of adaptive mesh refinement which show evidence that in the 3D incompressible Euler equations a finite time blow-up in the vorticity occurs whereas in the 2D incompressible magnetohydrodynamic equations only exponential growth of vorticity and current density is observed.

in Hyperbolic Problems: Theory, Numerics, Applications Vol. I, ISNM, Birkhäuser (1999) 401-411.

M. Blüming, K. H. Spatschek, and R. Grauer

#### Center manifold approach to the reduced magnetohydrodynamic bifurcations with diffusive magnetic field lines

Abstract:

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) 1083-1092.

H. Friedel, R. Grauer, and K. H. Spatschek

#### Controlling chaotic states of a Pierce diode

Abstract:

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

Physics of Plasmas 9 (1998) 3187-3194.

R. Grauer, C. Marliani, and K. Germaschewski

#### Adaptive mesh refinement for singular solutions of the incompressible Euler equations

Abstract:

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

Phys. Rev. Lett. 84 (1998) 4850-4853.

R. Grauer

#### An Energy estimate for a perturbed Hasegawa-Mima equation

Abstract:

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

Nonlinearity 11 (1998) 659-666.

R. Grauer and C. Marliani

#### Geometry of singular structures in magnetohydrodynamic flows

Abstract:

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

Physics of Plasmas 5 (1998) 2544-2552.

H. Friedel, R. Grauer, and C. Marliani

#### Center manifold approach to controlling chaos

Abstract:

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) 45-52.

O. Zikanov, A. Thess, and R. Grauer

#### Statistics of turbulence in a generalized random-force-driven Burgers equation

Abstract:

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

Physics of Fluids 9 (1997) 1362-1367.

H. Friedel, R. Grauer, and C. Marliani

#### Adaptive mesh refinement for singular current sheets in incompressible magnetohydrodynamic flows

Abstract:

The formation of current sheets in ideal incompressible magnetohydrodynamic flows in two dimensions is studied numerically using the technique of adaptive mesh refinement. The growth of current density is in agreement with simple scaling assumptions. As expected, adaptive mesh refinement shows to be very efficient for studying singular structures compared to non-adaptive treatments.

J. Comp. Phys. 134 (1997) 190-198.

R. Grauer and C. Marliani

#### Analytical and numerical approaches to structure functions in magnetohydrodynamic turbulence

Abstract:

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 two-dimensional hydrodynamic flows. Finally, we discuss the applicability of operator-product expansions on a direct cascade in strongly turbulent systems.

Physica Scripta. T67 (1996) 38-42.

R. Grauer and T. C. Sideris

#### Finite time singularities in ideal fluids with swirl

Abstract:

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

Physica D 88 (1995) 116-132.

R. Grauer and C. Marliani

#### Numerical and analytical estimates for the structure functions in two-dimensional magnetohydrodynamic flows

Abstract:

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

Physics of Plasmas 2 (1995) 41-47.

R. Grauer and C. Marliani

#### Structure functions in magnetohydrodynamic turbulence

Abstract:

Higher order structure functions of two-dimensional magnetohydrodynamic flows are studied numerically using a second order upwind projection method. In addition, an analytical estimate for the first structure function based on a geometric technique is derived. Finally, a phenomenological model without adjustable parameters is presented.

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 high-order structure functions in magnetohydrodynamic turbulence

Abstract:

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) 335-338.

T. Eickermann, R. Grauer, and K. H. Spatschek

#### Identification of mass capturing structures in a perturbed nonlinear Schrödinger equation

Abstract:

The numerical solutions of a standard damped and driven nonlinear Schrödinger equation are compared with a systematic reduction obtained by the Kahunen-Loeve expansion. The role of the mass containing modes is clarified by analyzing the spectral data of the underlying periodic direct problem. A one to one correspondence to the marginally stable modes obtained from the linearized Bäcklund transformation is found which explains the observed low-dimensional behavior.

Phys. Lett. A 198 (1995) 383-388.

R. Grauer and Y. Kivshar

#### Dynamics of parametrically driven sine-Gordon breathers

Abstract:

The dynamics of a breather in the damped and parametrically driven sine-Gordon equation is investigated both numerically and analytically. The Kahunen-Loeve expansion is applied to extract the energetically dominant localized modes. These modes are used in a Galerkin approximation to the original partial differential equation. Solutions of the resulting amplitude equations are then compared to numerical simulations of the perturbed sine-Gordon equation showing perfect agreement. Information from the periodic spectral theory and linear stability analysis is used to identify the Kahunen-Loeve modes.

in Nonlinear coherent structures in physics and biology, Bayreuth, Germany, edited by K. H. Spatschek and F. G. Mertens, pages 381-384, New York, 1994, Plenum.

B. Birnir and R. Grauer

#### An Explicit Description of the Global Attractor of the Damped and Driven Sine-Gordon Equation

Abstract:

We prove that the size of the finite-dimensional attractor of the damped and driven sine-Gordon equation stays small as the damping and driving amplitude become small. A decomposition of finite-dimensional attractors in Banach space is found, into a part B that attracts all of phase space, except sets whose finite-dimensional projections have Lebesgue measure zero, and a part C that only attracts sets whose finite-dimensional projections have Lebesgue measure zero. We describe the components of the B-attractor and C, which is called the "hyperbolic" structure, for the damped and driven sine-Gordon equation. B is low-dimensional but the dimension of C, which is associated with transients, is much larger. We verify numerically that this is a complete description of the attractor for small enough damping and driving parameters and describe the bifurcations of the B-attractor in this small parameter region.

Comm. Math. Phys. 162 (1994) 539-590.

R. Grauer and Y. Kivshar

#### Chaotic and phase-locked breather dynamics in the damped and parametrically driven sine-Gordon equation

Abstract:

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

Phys. Rev. E 48 (1993) 4791-4800.

P. Beyer, R. Grauer, and K. H. Spatschek

#### Center Manifold Theory for Low-Frequency Excitations in Magnetized Plasmas

Abstract:

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

Phys. Rev. E 48 (1993) 4665-4673.

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, Vol. 312, pages 109-112, Plenum Press, 1993.

B. Birnir, B. Galdrikian, R. Grauer, and M. Sherwin

#### Nonperturbative resonances in periodically driven quantum wells

Abstract:

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

Phys. Rev. B 47 (1993) 6795-6798.

R. Grauer, K. H. Spatschek, and A. V. Zolotaryuk

#### Chaotic proton dynamics in the hydrogen bond

Abstract:

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

Phys. Rev. E 47 (1993) 236-247.

R. Grauer and B. Birnir

#### The center manifold and bifurcations of the sine-Gordon equation

Abstract:

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

Physica D 56 (1992) 165-184.

R. Grauer and T. C. Sideris

#### Numerical computation of 3D incompressible ideal fluids with swirl

Abstract:

We investigate numerically the question of blowup in finite time for the "swirling flow" of the three-dimensional incompressible Euler equations. Using rotational symmetry, the Euler equations reduce to a two-dimensional problem which is numerically solved by finite differences. The elliptic equation relating vorticity to velocity is solved with the multigrid method. Calculations were performed with 896 x 640 mesh points.

Phys. Rev. Lett. 25 (1991) 3511-3514.

E. Turlot, D. Esteve, C. Urbina, M. Devoret, R. Grauer, J. C. Fernandez, and G. Reinisch

#### The dynamical isoperimeter multi-vortex mode in the square sine-Gordon system

in Nonlinear World, edited by V. G. Baryakhtar, V. M. Chernosenko, N. S. Erokhin, A. G. Sitenko, and V. E. Zakharov, pages 363-383, 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 sine-Gordon system

Abstract:

This paper shows by use of simple physical arguments the existence of a particular multi-vortex dynamical configuration in a - perturbed or not - two-dimensional sine-Gordon system. The stability of such modes is numerically checked. Their main topological invariant is the total length of the +/- 2Pi - wavefronts entering as elementary kink-like patterns the constitution of the whole configuration. This is consistent with the conservation of energy in all situations which are considered in this paper.

Phys. Rev. B 42 (1990) 8418-8425.

J. C. Fernandez, R. Grauer, K. Pinnow, and G. Reinisch

#### Cell-mapping description of coexisting phase-locked soliton states in a long Josephson junction

Abstract:

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

Phys. Rev. B 42 (1990) 9987-9995.

J. C. Fernandez, R. Grauer, K. Pinnow, and G. Reinisch

#### Phase-locked dynamical regimes to an external microwave field in a long, unbiased Josephson junction

Abstract:

We show the existence of asymmetrical (in the phase space) phase-locked limit cycles of (anti) kinks in inhomogeneously ac driven sine-Gordon systems and point out their possible experimental verification by use of Josephson devices.

Phys. Lett. A 145 (1990) 333-339.

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 2-d isoperimeter solitonic mode propagating simultaneously along the two diagonals of a square Josephson junction of intermediate length.

in Nonlinear Coherent Structures,
edited by M. Barthes and J. Leon, Lecture Notes in Physics, Vol. 353, pages 181-192, Springer, 1989.

J. C. Fernandez, R. Grauer, and G. Reinisch

#### Dynamical regimes phase-locked to an external microwave field in a long, unbiased Josephson junction

Abstract:

We demonstrate the existence of phase-locked limit cycles in inhomogenously driven sine-Gordon systems and point out their possible experimental verification by use of Josephson devices.

in Nonlinear Coherent Structures, edited by M.Barthes and J. Leon, Lecture Notes in Physics, Vol. 353, pages 213-219, Springer, 1989.

M. Taki, K. H. Spatschek, J. C. Fernandez, R. Grauer, and G. Reinisch

#### Breather dynamics in the nonlinear Schrödinger regime of perturbed sine-Gordon systems

Abstract:

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

Physica D 40 (1989) 65-82.

R. Grauer

#### Codimension two interactions of tearing modes

Abstract:

The interaction of two different tearing modes is examined. Certain physical parameter values could be found such that the nullspace of the linearized problem is four-dimensional. The center manifold theory allows a reduction of the partial differential equations (PDE's) describing the tearing instabilities to a four dimensional system of ordinary ones (ODE's). Due to the symmetries of the problem many interesting spatial- and spatio-temporal solutions are possible in the neighbourhood of the codimension two point.

in Singular Behavior and Nonlinear Dynamics, Samos, Greece edited by St. Pnevmatikos, T. Bountis, and Sp. Pnevmatikos, pages 267-276, Singapore, 1989, World Scientific.

R. Grauer

#### Nonlinear interactions of tearing modes in the vicinity of a bifurcation point of codimension two

Abstract:

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

Physica D 35 (1989) 107-126.

E. Rebhan and R. Grauer

#### Tokamak profiles through constrained minimization of the entropy production

Abstract:

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

in Proceedings of the 14th European Conference on Controlled Fusion and Plasma Physics, Madrid, Spain, pages 1072-1075, European Physical Society, 1987.

R. Grauer and E. Rebhan

#### Analysis of ideal MHD equilibrium and axisymmetric stability for finite aspect ratio tokamaks with elliptic cross-section and flat current profile

Abstract:

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

J. Plasma Physics 22 (1984) 99-117.

R. Grauer and E. Rebhan

#### Semianalytic calculation of equilibrium and axisymmetric stability of finite aspect ratio tokamaks with elliptic cross-section

in Proceedings of the 11th European Conference on Controlled Fusion and Plasma Physics, Aachen, Germany , pages 197-200, European Physical Society, 1983.