Adaptive Mesh Refinement for Singular Current Sheets in Incompressible Magnetohydrodynamic Flows

The formation of current sheets in ideal incompressible magnetohydrodynamic flows in two dimensions is studied numerically using the technique of adaptive mesh refinement. The growth of current density is in agreement with simple scaling assumptions. As expected, adaptive mesh refinement shows to be very efficient for studying singular structures compared to non-adaptive treatments.Comment: 8 pages RevTeX, 13 Postscript figure

Multiphysics simulations of collisionless plasmas

Collisionless plasmas, mostly present in astrophysical and space environments, often require a kinetic treatment as given by the Vlasov equation. Unfortunately, the six-dimensional Vlasov equation can only be solved on very small parts of the considered spatial domain. However, in some cases, e.g. magnetic reconnection, it is sufficient to solve the Vlasov equation in a localized domain and solve the remaining domain by appropriate fluid models. In this paper, we describe a hierarchical treatment of collisionless plasmas in the following way. On the finest level of description, the Vlasov equation is solved both for ions and electrons. The next courser description treats electrons with a 10-moment fluid model incorporating a simplified treatment of Landau damping. At the boundary between the electron kinetic and fluid region, the central question is how the fluid moments influence the electron distribution function. On the next coarser level of description the ions are treated by an 10-moment fluid model as well. It may turn out that in some spatial regions far away from the reconnection zone the temperature tensor in the 10-moment description is nearly isotopic. In this case it is even possible to switch to a 5-moment description. This change can be done separately for ions and electrons. To test this multiphysics approach, we apply this full physics-adaptive simulations to the Geospace Environmental Modeling (GEM) challenge of magnetic reconnection.Comment: 13 pages, 5 figure

Lagrangian and geometric analysis of finite-time Euler singularities

We present a numerical method of analyzing possibly singular incompressible 3D Euler flows using massively parallel high-resolution adaptively refined numerical simulations up to 8192^3 mesh points. Geometrical properties of Lagrangian vortex line segments are used in combination with analytical non-blowup criteria by Deng et al [Commun. PDE 31 (2006)] to reliably distinguish between singular and near-singular flow evolution. We then apply the presented technique to a class of high-symmetry initial conditions and present numerical evidence against the formation of a finite-time singularity in this case.Comment: arXiv admin note: text overlap with arXiv:1210.253

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 time-dependent Euler equations in compressible gas dynamics or Magneto-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-differences as well as shock-capturing central schemes, both in connection with Runge-Kutta type integrators. Parallel execution is achieved through a configurable hybrid of POSIX-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.Comment: late submissio

The instanton method and its numerical implementation in fluid mechanics

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

Coupled Vlasov and two-fluid codes on GPUs

We present a way to combine Vlasov and two-fluid codes for the simulation of a collisionless plasma in large domains while keeping full information of the velocity distribution in localized areas of interest. This is made possible by solving the full Vlasov equation in one region while the remaining area is treated by a 5-moment two-fluid code. In such a treatment, the main challenge of coupling kinetic and fluid descriptions is the interchange of physically correct boundary conditions between the different plasma models. In contrast to other treatments, we do not rely on any specific form of the distribution function, e.g. a Maxwellian type. Instead, we combine an extrapolation of the distribution function and a correction of the moments based on the fluid data. Thus, throughout the simulation both codes provide the necessary boundary conditions for each other. A speed-up factor of around 20 is achieved by using GPUs for the computationally expensive solution of the Vlasov equation and an overall factor of at least 60 using the coupling strategy combined with the GPU computation. The coupled codes were then tested on the GEM reconnection challenge

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

In this paper we present an extension of the \emph{Recent Fluid Deformation (RFD)} closure introduced by Chevillard and Meneveau (2006) which was developed for modeling the time evolution of Lagrangian fluctuations in incompressible Navier-Stokes turbulence. We apply the RFD closure to study the evolution of magnetic and passive scalar fluctuations. This comparison is especially interesting since the stretching term for the magnetic field and for the gradient of the passive scalar are similar but differ by a sign such that the effect of stretching and compression by the turbulent velocity field is reversed. Probability density functions (PDFs) of magnetic fluctuations and fluctuations of the gradient of the passive scalar obtained from the RFD closure are compared against PDFs obtained from direct numerical simulations