Multi-dimensional Magnetohydrodynamic Research Articles

A fast multi-dimensional magnetohydrodynamic formulation of the transition region adaptive conduction (TRAC) method

We have demonstrated that the transition region adaptive conduction (TRAC) method permits fast and accurate numerical solutions of the field-aligned hydrodynamic equations, successfully removing the influence of numerical resolution on the coronal density response to impulsive heating. This is achieved by adjusting the parallel thermal conductivity, radiative loss, and heating rates to broaden the transition region (TR), below a global cutoff temperature, so that the steep gradients are spatially resolved even when using coarse numerical grids. Implementing the original 1D formulation of TRAC in multi-dimensional magnetohydrodynamic (MHD) models would require tracing a large number of magnetic field lines at every time step in order to prescribe a global cutoff temperature to each field line. In this paper, we present a highly efficient formulation of the TRAC method for use in multi-dimensional MHD simulations, which does not rely on field line tracing. In the TR, adaptive local cutoff temperatures are used instead of global cutoff temperatures to broaden any unresolved parts of the atmosphere. These local cutoff temperatures are calculated using only local grid cell quantities, enabling the MHD extension of TRAC to efficiently account for the magnetic field evolution, without tracing field lines. Consistent with analytical predictions, we show that this approach successfully preserves the properties of the original TRAC method. In particular, the total radiative losses and heating remain conserved under the MHD formulation. Results from 2D MHD simulations of impulsive heating in unsheared and sheared arcades of coronal loops are also presented. These simulations benchmark the MHD TRAC method against a series of 1D models and demonstrate the versatility and robustness of the method in multi-dimensional magnetic fields. We show, for the first time, that pressure differences, generated during the evaporation phase of impulsive heating events, can produce current layers that are significantly narrower than the transverse energy deposition.

A low-dissipation HLLD approximate Riemann solver for a very wide range of Mach numbers

We propose a new Harten-Lax-van Leer discontinuities (HLLD) approximate Riemann solver to improve the stability of shocks and the accuracy of low-speed flows in multidimensional magnetohydrodynamic (MHD) simulations. Stringent benchmark tests verify that the new solver is more robust against a numerical shock instability and is more accurate for low-speed, nearly incompressible flows than the original solver, whereas additional computational costs are quite low. The novel ability of the new solver enables us to tackle MHD systems, including both high and low Mach number flows.

Transition region adaptive conduction (TRAC) in multidimensional magnetohydrodynamic simulations

Context. In solar physics, a severe numerical challenge for modern simulations is properly representing a transition region between the million-degree hot corona and a much cooler plasma of about 10 000 K (e.g., the upper chromosphere or a prominence). In previous 1D hydrodynamic simulations, the transition region adaptive conduction (TRAC) method has been proven to capture aspects better that are related to mass evaporation and energy exchange. Aims. We aim to extend this method to fully multidimensional magnetohydrodynamic (MHD) settings, as required for any realistic application in the solar atmosphere. Because modern MHD simulation tools efficiently exploit parallel supercomputers and can handle automated grid refinement, we design strategies for any-dimensional block grid-adaptive MHD simulations. Methods. We propose two different strategies and demonstrate their working with our open-source MPI-AMRVAC code. We benchmark both strategies on 2D prominence formation based on the evaporation–condensation scenario, where chromospheric plasma is evaporated through the transition region and then is collected and ultimately condenses in the corona. Results. A field-line-based TRACL method and a block-based TRACB method are introduced and compared in block grid-adaptive 2D MHD simulations. Both methods yield similar results and are shown to satisfactorily correct the underestimated chromospheric evaporation, which comes from a poor spatial resolution in the transition region. Conclusions. Because fully resolving the transition region in multidimensional MHD settings is virtually impossible, TRACB or TRACL methods will be needed in any 2D or 3D simulations involving transition region physics.

A Fully Self-consistent Model for Solar Flares

Abstract The “standard solar flare model” collects all physical ingredients identified by multiwavelength observations of our Sun: magnetic reconnection, fast particle acceleration, and the resulting emission at various wavelengths, especially in soft to hard X-ray channels. Its cartoon representation is found throughout textbooks on solar and plasma astrophysics and guides interpretations of unresolved energetic flaring events on other stars, accretion disks, and jets. To date, a fully self-consistent model that reproduces the standard scenario in all its facets is lacking, since this requires the combination of a large-scale, multidimensional magnetohydrodynamic (MHD) plasma description with a realistic fast electron treatment. Here we demonstrate such a novel combination, where MHD combines with an analytic fast electron model, adjusted to handle time-evolving, reconnecting magnetic fields and particle trapping. This allows us to study (1) the role of fast electron deposition in the triggering of chromospheric evaporation flows, (2) the physical mechanisms that generate various hard X-ray sources at chromospheric footpoints or looptops, and (3) the relationship between soft X-ray and hard X-ray fluxes throughout the entire flare loop evolution. For the first time, this self-consistent solar flare model demonstrates the observationally suggested relationship between flux swept out by the hard X-ray footpoint regions and the actual reconnection rate at the X-point, which is a major unknown in flaring scenarios. We also demonstrate that a looptop hard X-ray source can result from fast electron trapping.

MPI-AMRVAC 2.0 for Solar and Astrophysical Applications

Abstract We report on the development of MPI-AMRVAC version 2.0, which is an open-source framework for parallel, grid-adaptive simulations of hydrodynamic and magnetohydrodynamic (MHD) astrophysical applications. The framework now supports radial grid stretching in combination with adaptive mesh refinement (AMR). The advantages of this combined approach are demonstrated with one-dimensional, two-dimensional, and three-dimensional examples of spherically symmetric Bondi accretion, steady planar Bondi–Hoyle–Lyttleton flows, and wind accretion in supergiant X-ray binaries. Another improvement is support for the generic splitting of any background magnetic field. We present several tests relevant for solar physics applications to demonstrate the advantages of field splitting on accuracy and robustness in extremely low-plasma β environments: a static magnetic flux rope, a magnetic null-point, and magnetic reconnection in a current sheet with either uniform or anomalous resistivity. Our implementation for treating anisotropic thermal conduction in multi-dimensional MHD applications is also described, which generalizes the original slope-limited symmetric scheme from two to three dimensions. We perform ring diffusion tests that demonstrate its accuracy and robustness, and show that it prevents the unphysical thermal flux present in traditional schemes. The improved parallel scaling of the code is demonstrated with three-dimensional AMR simulations of solar coronal rain, which show satisfactory strong scaling up to 2000 cores. Other framework improvements are also reported: the modernization and reorganization into a library, the handling of automatic regression tests, the use of inline/online Doxygen documentation, and a new future-proof data format for input/output.

Efficient magnetohydrodynamic simulations on distributed multi-GPU systems using a novel GPU Direct–MPI hybrid approach

Modern graphics processing units (GPUs) have been widely utilized in magnetohydrodynamic (MHD) simulations in recent years. Due to the limited memory of a single GPU, distributed multi-GPU systems are needed to be explored for large-scale MHD simulations. However, the data transfer between GPUs bottlenecks the efficiency of the simulations on such systems. In this paper we propose a novel GPU Direct–MPI hybrid approach to address this problem for overall performance enhancement. Our approach consists of two strategies: (1) We exploit GPU Direct 2.0 to speedup the data transfers between multiple GPUs in a single node and reduce the total number of message passing interface (MPI) communications; (2) We design Compute Unified Device Architecture (CUDA) kernels instead of using memory copy to speedup the fragmented data exchange in the three-dimensional (3D) decomposition. 3D decomposition is usually not preferable for distributed multi-GPU systems due to its low efficiency of the fragmented data exchange. Our approach has made a breakthrough to make 3D decomposition available on distributed multi-GPU systems. As a result, it can reduce the memory usage and computation time of each partition of the computational domain. Experiment results show twice the FLOPS comparing to common 2D decomposition MPI-only implementation method. The proposed approach has been developed in an efficient implementation for MHD simulations on distributed multi-GPU systems, called MGPU–MHD code. The code realizes the GPU parallelization of a total variation diminishing (TVD) algorithm for solving the multidimensional ideal MHD equations, extending our work from single GPU computation (Wong et al., 2011) to multiple GPUs. Numerical tests and performance measurements are conducted on the TSUBAME 2.0 supercomputer at the Tokyo Institute of Technology. Our code achieves 2 TFLOPS in double precision for the problem with 12003 grid points using 216 GPUs.

CONSTRAINED-TRANSPORT MAGNETOHYDRODYNAMICS WITH ADAPTIVE MESH REFINEMENT IN CHARM

We present the implementation of a three-dimensional, second order accurate Godunov-type algorithm for magneto-hydrodynamic (MHD), in the adaptive-mesh-refinement (AMR) cosmological code {\tt CHARM}. The algorithm is based on the full 12-solve spatially unsplit Corner-Transport-Upwind (CTU) scheme. The fluid quantities are cell-centered and are updated using the Piecewise-Parabolic-Method (PPM), while the magnetic field variables are face-centered and are evolved through application of the Stokes theorem on cell edges via a Constrained-Transport (CT) method. The multidimensional MHD source terms required in the predictor step for high-order accuracy are applied in a simplified form which reduces their complexity in three dimensions without loss of accuracy or robustness. The algorithm is implemented on an AMR framework which requires specific synchronization steps across refinement levels. These include face-centered restriction and prolongation operations and a {\it reflux-curl} operation, which maintains a solenoidal magnetic field across refinement boundaries. The code is tested against a large suite of test problems, including convergence tests in smooth flows, shock-tube tests, classical two- and three-dimensional MHD tests, a three-dimensional shock-cloud interaction problem and the formation of a cluster of galaxies in a fully cosmological context. The magnetic field divergence is shown to remain negligible throughout.

Magnetically driven hyper-velocity launch capability at the Sandia Z accelerator

The intense magnetic field generated by the Z accelerator at Sandia National Laboratories is used as a pressure source for material science studies. A current of ∼20 MA can be delivered to the loads used in experiments on a time scale of ∼100–600 ns. Magnetic fields (pressures) exceeding 1200 T (600 GPa) have been produced in planar configurations. In one application we have developed, the magnetic pressure launches a flyer plate to ultra-high velocity in a plate impact experiment; equation of state data is obtained on the Hugoniot of a material that is shock compressed to multi-megabar pressure. This capability has been enhanced by the recent development of a planar stripline configuration that increases the magnetic pressure for a given current. Furthermore, the cross sectional area of a stripline flyer plate is larger than in previous coaxial loads; this improves the planarity of the flyer thereby reducing measurement uncertainty. Results of experiments and multi-dimensional magneto hydrodynamic (MHD) simulation are presented for ultra-high velocity aluminum and copper flyer plates. Aluminum flyer plates with dimensions ∼25 mm by ∼13 mm by ∼1 mm have been launched to velocities up to ∼45 km/s; for copper the peak velocity is ∼22 km/s. The significance of these results is that part of the flyer material remains solid at impact with the target; an accomplishment that is made possible by shaping the dynamic pressure (current) ramp so that the flyer compresses quasi-isentropically (i.e., shocklessly) during acceleration.

AN EXPLICIT SCHEME FOR INCORPORATING AMBIPOLAR DIFFUSION IN A MAGNETOHYDRODYNAMICS CODE

We describe a method for incorporating ambipolar diffusion in the strong coupling approximation into a multidimensional magnetohydrodynamic (MHD) code based on the total variation diminishing scheme. Contributions from ambipolar diffusion terms are included by explicit finite-difference operators in a fully unsplit way, maintaining second-order accuracy. The divergence-free condition of magnetic fields is exactly ensured at all times by a flux-interpolated constrained transport scheme. The super time-stepping method is used to accelerate the time step in high-resolution calculations and/or in strong ambipolar diffusion. We perform two test problems, the steady-state oblique C-type shocks and the decay of Alfven waves, confirming the accuracy and robustness of our numerical approach. Results from the simulations of the compressible MHD turbulence with ambipolar diffusion show the flexibility of our method as well as its ability to follow complex MHD flows in the presence of ambipolar diffusion. These simulations show that the dissipation rate of the MHD turbulence is strongly affected by the strength of ambipolar diffusion.

Magnetoacoustic Waves in the Solar Stratified Atmosphere

The propagation of magnetoacoustic waves in the solar atmosphere consistingof the photosphere, chromosphere and corona has been studied numericallyby time-dependent multi-dimensional magnetohydrodynamic (MHD)simulation. Pressure disturbances are introduced at the bottom of thechromosphere and at the bottom of the corona, respectively. Thecomputational results show that incurred fast and slow MHD wavespropagate away from the source of the disturbances. The fast MHD wavepropagates as an expansive wave in the radial direction, while the slowone steepens and it may evolve into a slow shock. We suggest that theextreme ultraviolet imaging telescope wave observed by the SOHO andMoreton wave are a fast MHD wave propagating in the corona and in thechromosphere, respectively.

Interplanetary studies: Propagation of disturbances between the Sun and the magnetosphere

This review is concerned with the interplanetary ‘transmission line’ between the Sun and the Earth's magnetosphere. It starts with comments about coronal mass ejections (CMEs) that are associated with various forms of solar activities. It then continues with some of the current views about their continuation through the heliosphere to Earth and elsewhere. The evolution of energy, mass, and momentum transfer is of prime interest since the temporal/spatial/magnitude behavior of the interplanetary electric field and transient solar wind dynamic pressure is relevant to the magnetospheric response (the presence or absence of geomagnetic storms and substorms) at Earth. Energetec particle flux predictions are discussed in the context of solar activity (flares, prominence eruptions) at various positions on the solar disk relative to Earth's central meridian. A number of multi-dimensional magnetohydrodynamic (MHD) models, applied to the solar, near-Sun, and interplanetary portions of the ‘transmission line’, are discussed. These model simulations, necessary to advancing our understanding beyond the phenomenological or morphological stages, are directed to deceptively simple questions such as the following: can one-to-one associations be made between specific forms of solar activity and magnetosphere response?