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# Finite volume WENO methods for hyperbolic conservation laws on Cartesian grids with adaptive mesh refinement

We present a WENO finite volume method for the approximation of hyperbolic conservation laws on adaptively refined Cartesian grids. On each single patch of the AMR grid, we use a modified dimension-by-dimension WENO method, which was recently developed by Buchmuller and Helzel (2014) [1]. This method retains the full spatial order of accuracy of the underlying one-dimensional WENO reconstruction for nonlinear multidimensional problems, and requires only one flux computation per interface. It is embedded into block-structured AMR through conservative interpolation functions and a numerical flux fix that transfers data between different levels of grid refinement. Numerical tests illustrate the accuracy of the new adaptive WENO finite volume method. Compared to the classical dimension-by-dimension approach, the new method is much more accurate while it is only slightly more expensive. Furthermore, we also show results of an accuracy study for an adaptive WENO method which uses multidimensional reconstruction of the conserved quantities and a high-order quadrature formula to compute the fluxes. While the accuracy of such a method is comparable with our new approach, it is about three times more expensive than the latter.

Applied Mathematics and Computation 272 (2016) 460

# Numerical magnetohydrodynamic simulations of expanding flux ropes: Influence of boundary driving

The expansion dynamics of a magnetized, current-carrying plasma arch is studied by means of time-dependent ideal MHD simulations. Initial conditions model the setup used in recent laboratory experiments that in turn simulate coronal loops [J. Tenfelde et al., Phys. Plasmas 19, 072513 (2012); E. V. Stenson and P. M. Bellan, Plasma Phys. Controlled Fusion 54, 124017 (2012)]. Boundary conditions of the electric field at the "lower" boundary, intersected by the arch, are chosen such that poloidal magnetic flux is injected into the domain, either localized at the arch footpoints themselves or halfway between them. These conditions are motivated by the tangential electric field expected to exist in the laboratory experiments due to the external circuit that drives the plasma current. The boundary driving is found to systematically enhance the expansion velocity of the plasma arch. While perturbations at the arch footpoints also deform its legs and create characteristic elongated segments, a perturbation between the footpoints tends to push the entire structure upwards, retaining an ellipsoidal shape.

Physics of Plasmas 20 (2013) 72104

# Numerical Simulation of Current Sheet Formation in a Quasi-Separatrix Layer using Adaptive Mesh Refinement

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

Physics of Plasmas 18 (2011) 32902

# Lyapunov exponents and information dimension of the mass distribution in turbulent compressible flows

Turbulent density fluctuations in isothermal highly compressible turbulent flows are highly clumped and can be quantified by the scaling properties of powers of the mass distribution. This Eulerian quantity can be related to Lagrangian properties of the system given by the Lyapunov exponents of tracer particles advected with the flow. Using highly resolved numerical simulations, we show that the Kaplan-Yorke conjecture holds within numerical uncertainties.

Physics Lett. A 374 (2010) 1039

# FlareLab: early results

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

Plasma Phys. Control. Fusion 52 (2010) 124030

# Numerical simulations of possible finite time singularities in the incompressible Euler equations: comparison of numerical methods

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

Physica D 237 (2008) 1932

# Massively Parallel Simulations of Solar Flares and Plasma Turbulence

Some of the outstanding problems in space- and astrophysical plasmasystems include solar flares and hydro- or magnetohydrodynamic turbulence (e.g. in the interstellar medium). Both fields demand for high resolution and thus numerical simulations need an efficient parallel implementation. We will describe the physics behind these problems and present the numerical frameworks for solving these problems on massive parallel computers.

Parallel Computing: Architectures, Algorithms and Applications 15 (2008) 467

# Density-PDFs and Lagrangian Statistics of highly compressible Turbulence

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

Physics Letters A 372 (2008) 3037

# Three-dimensional MHD simulation of expanding magnetic flux ropes

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

Phys. Plasmas 15 (2008) 42106

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

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

Comp. Phys. Comm. 178 (2008) 553

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

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

Comp. Phys. Comm. 177 (2007) 560

# Femtosecond laser microfabrication of subwavelength structures in photonics

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

Proc SPIE 6459 (2007) 64590

# Role of Plasma in Femtosecond Laser Pulse Propagation

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

AIP Conf. Proc. 876 (2006) 169

# Adaptive modeling of the femtosecond inscription in silica

We present an adaptive mesh approach to high performance comprehensive investigation of dynamics of light and plasma pattens during the process of direct laser inscription. The results reveal extreme variations of spatial and temporal scales and tremendous complexity of these patterns which was not feasible to study previously.

Proc. SPIE 6107 (2006) 6107

# Formation and disruption of Alfvénic filaments in Hall-magnetohydrodynamics

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

Phys. Plasmas 12 (2005) 52319

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

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

Physica Scripta 72 (2005) 450

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

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

Parallel Computing 31 (2005) 913

# Numerical modeling of quasiplanar giant water waves

In this work we present a further analytical development and a numerical implementation of the recently suggested theoretical model for highly nonlinear potential long-crested water waves, where weak three-dimensional effects are included as small corrections to exact two-dimensional equations written in the conformal variables {[}V. P. Ruban, Phys. Rev. E 71, 055303(R) (2005)]. Numerical experiments based on this theory describe the spontaneous formation of a single weakly three-dimensional large-amplitude wave (alternatively called freak, killer, rogue, or giant wave) on the deep water.

Phys. Rev. E 72 (2005) 66303

# Numerical studies on the dynamics of the ionospheric current system

We study the dynamics of the ionospheric current system by means of computational simulations. The initial configuration for the simulations represents numerical solutions of a stationary two-dimensional fluid model that focuses on the closure of one pair of field-aligned electric currents in the lower ionosphere. In this model the momentum transfer between the plasma and the neutral gas is included as well as the Hall term. The dynamical evolution of the current system caused by enhanced magnetospheric sheared plasma flows is investigated by means of numerical simulations carried out with a plasma-neutral gas fluid code. It is shown that in a self-consistent fluid model the intensification of electric field dominated auroral electrojets in the morning and evening sectors of the auroral oval can be understood as the ionospheric response to enhanced magnetospheric plasma convection.

Advances in Space Research 22 (1998) 1369

# Magnetic reconnection in a current sheet six-pack: A numerical experiment

Two-dimensional magneto-hydrodynamic (MHD) simulations are carried out to study magnetic reconnection in a configuration that is characterized by six current sheets extending radially from the z-axis. In the initial configuration, the current density within the current sheets sharply increases with increasing radial distance from the origin which makes it necessary to introduce a numerical wall-stabilization. With this, the interior region appears to be stable in the case of ideal Ohm's law, but finite resistivity leads to spontaneous magnetic reconnection, resulting in considerable plasma acceleration. The plasma flow is almost incompressible. The high symmetry of the initial configuration is approximately conserved even when a non-symmetric distribution of the resistivity is assumed.

Physica Scripta T74 (1998) 46

# Towards a self-consistent model for the convective auroral electrojet

It is shown that the intensification of electric field dominated auroral electrojets in the morning and evening sectors of the auroral oval on be understood in a self-consistent fluid model as the ionospheric response to enhanced magnetospheric plasma convection. Starting from a self-consistently calculated equilibrium characterized by current-closure in the partially ionized ionosphere we perform fluid simulations to study the dynamics of the ionospheric current system in response to enhanced magnetospheric convection.

Physica Scripta T74 (1998) 54

# Three dimensional simulation studies on bright points in the solar corona

The interaction phase of the Converging Flux Model (Priest ct al., 1994) for the formation of coronal X-ray bright points is investigated self-consistently by means of magnetohydrodynamical simulations. In these simulations the three-dimensional dynamical evolution of two mutually approaching magnetic structures of opposite polarity is studied. During the ideal phase of the approach a current sheet forms in the region above the polarity inversion line and the low-beta coronal plasma is compressed and heated locally. Stronger plasma heating on shorter time scales occurs due to magnetic reconnection as soon as a finite anomalous resistivity leads to a violation of the ideal Ohm's law. Both processes together might account for quasi-stationary soft X-ray emissivity of bright points, flaring of tiny filaments within X-ray bright points, and jet-like plasma hows in the vicinity of bright point features.

Advances in Space Research 19 (1997) 1861

# Three-dimensional numerical study of converging flux events

The first self-consistent three-dimensional magnetohydrodynamical simulations of converging magnetic flux events associated with the formation of coronal X-ray bright points are presented. The initial magnetic field results from two magnetic dipoles located below the photosphere at positions r(1) and r(2), respectively, and an additional horizontal magnetic field parallel to the line <(r(1)r(2))over bar>. Both dipole moments are vertical and have equal magnitude but opposite orientation. During the dynamical evolution, a prescribed photospheric convection pattern causes the magnetic dipole-like structures to approach one another. In the early phase of the evolution a current sheet forms in the central region above the polarity inversion line. When the current density exceeds a critical value, anomalous resistivity due to microturbulence is assumed to break the ideal Ohm's law. As a consequence, magnetic reconnection sets in and results in a jet-like plasma flow. The localized plasma heating associated with the reconnection process might account for the flaring of tiny filaments within bright point structures.

Astronomy and Astrophysics 323 (1997) 593

# On the self-consistent description of dynamic magnetosphere-ionosphere coupling phenomena with resolved ionosphere

A set of model equations is presented that allows for the self-consistent description of dynamic magnetosphere-ionosphere coupling phenomena with finite ionosphere. The model is very similar to magnetohydrodynamics (MHD), but the plasma-neutral gas interaction in the lower ionosphere is taken into account by a frictional force between,ions and neutrals and an ionization-recombination term in the plasma transport equation. Further, the Hall term and the electron pressure term are retained in Ohm's law. It is shown that for the collision-dominated E layer, the familiar Pedersen and Hall conductivities can be derived directly from the basic equations. The model is used to numerically simulate the dynamic formation of a magnetospheric-ionospheric current system as the response to prescribed localized magnetospheric convection. Apart from a pair of Birkeland current sheets that are closed by Pedersen currents, associated Hall currents are generated. In addition, density irregularities in the E layer form as a direct consequence of the current closure. They are, however, not related to electron precipitation. For small length scales (approximate to 10 km), these density perturbations result in considerably enhanced conductivities below the upward Birkeland currents which in turn lead to a spatial concentration of the latter compared to the downward currents. The timescale for the relaxation of the current system and the ionospheric convection toward a stationary state after the onset of magnetospheric convection is longer than the Alfven travel time and depends on the height-integrated Pedersen conductivity. This is in good agreement with earlier theoretical predictions [Southwood and Kivelson, 1991].

Journal of Geophysical Research - Space Physics 102 (1997) 85

# Particle simulations of collisionless reconnection in magnetotail configuration including electron dynamics

The role of electron dynamics in collisionless magnetotail reconnection is investigated by using two-dimensional full particle simulations with ion-to-electron mass ratios m(i)/m(e) set to 1 and 10. The results indicate that the electrons have a stabilizing influence on the ion tearing instability in the sense that growth rates are reduced in runs with the initial electron Larmor radius being sufficiently small in comparison with the ion Larmor radius. A comparison of our results with those obtained by Pritchett [1994] using a significantly different numerical model shows encouraging agreement. The attempt to achieve an approach closer to realistic magnetotail conditions is inhibited by present technical restrictions.

Journal of Geophysical Research - Space Physics 101 (1996) 27375

# Dynamics of sheared flow driven drift-Alfven waves in nonuniform plasmas

A two-fluid model is used to derive a set of nonlinear equations governing the dynamics of low-frequency drift-dispersive Alfven waves in a nonuniform plasma containing equilibrium density gradients and sheared equilibrium plasma flows. In the linear limit, we present a local dispersion relation, which is analyzed in some interesting limiting cases. It is found that sheared equilibrium flows can cause instability of the Alfven waves even in the absence of a density inhomogeneity. Furthermore, it is demonstrated that possible stationary solutions of the nonlinear equations governing the dynamics of dispersive Alfven waves with sheared equilibrium flows in cold plasmas can be represented in the form of coherent vortical structures. The results of these investigations should improve the understanding of current filamentation and vortex motions in space and laboratory plasmas.

Plasma Physics Reports 22 (1996) 818

# A simple model of core field generation during plasmoid evolution

Bipolar magnetic field signatures in the far magnetotail observed by the ISEE 3 spacecraft are commonly interpreted as signatures of a passing magnetic bubble, or plasmoid. A large number of such plasmoid-type variations in the north-south component of the magnetic field are accompanied by large core magnetic fields which are directed primarily in the cross-tail direction, indicating a flux rope like structure. Similar signatures are also found in a recent examination of GEOTAIL deep tail data. The fact that more of these flux ropelike plasmoids are encountered in the far tail than closer to the Earth raises the question whether they are the result of an evolution from no or low core fields to high core fields or whether plasmoids without core fields and flux ropes are entirely different entities. We present a model which explains the evolution of a looplike plasmoid in the near tail to a thinner flux rope in the far tail. The transition is accomplished by magnetic reconnection, which progressively connects the plasmoid magnetic field lines to the colder plasma in the low-latitude boundary layer and magnetosheath. The connection leads to a draining of hot plasma from plasmoid field lines and a subsequent collapse due to the plasma pressure reduction. The collapse causes a strong enhancement of any preexisting cross-tail magnetic field component, until a quasi-force-free state is reached. We also present MHD simulations to demonstrate the process. Last, we show that this mechanism call produce core field enhancements beyond the ambient lobe field strength.

Journal of Geophysical Research - Space Physics 101 (1996) 10797

# Role of magnetic reconnection in venus ionopause activity

The role of magnetic reconnection in the dynamics of the Venus dayside ionopause is investigated. The dispersion relation of the resistive tearing mode for the asymmetric ionopause equilibrium is derived and solved. It is shown that typical signatures of this mode correspond to helical magnetic flux tubes (flux ropes) as observed by Pioneer Venus Orbiter. Nonlinear numerical MHD-simulations were carried out in addition to the analytical investigations. The results of these simulations support our view that magnetic reconnection plays an essential role in the dynamic interaction of the solar wind with the Venus ionosphere.

Journal of Geophysical Research - Space Physics 100 (1995) 14833