Magnetic Reconnection Problem Research Articles

Extending the Spectral Difference Method with Divergence Cleaning (SDDC) to the Hall MHD Equations

The Hall Magnetohydrodynamic (MHD) equations are an extension of the standard MHD equations that include the “Hall” term from the general Ohm’s law. The Hall term decouples ion and electron motion physically on the ion inertial length scales. Implementing the Hall MHD equations in a numerical solver allows more physical simulations for plasma dynamics on length scales less than the ion inertial scale length but greater than the electron inertial length. The present effort is an important step towards producing physically correct results to important problems, such as the Geospace Environmental Modeling (GEM) Magnetic Reconnection problem. The solver that is being modified is currently capable of solving the resistive MHD equations on unstructured grids using the spectral difference scheme which is an arbitrarily high-order method that is relatively simple to parallelize. The GEM Magnetic Reconnection problem is used to evaluate whether the Hall MHD equations have been correctly implemented in the solver using the spectral difference method with divergence cleaning (SDDC) algorithm by comparing against the reconnection rates reported in the literature.

Identification of high order closure terms from fully kinetic simulations using machine learning

Simulations of large-scale plasma systems are typically based on a fluid approximation approach. These models construct a moment-based system of equations that approximate the particle-based physics as a fluid, but as a result, they lack the small-scale physical processes available to fully kinetic models. Traditionally, empirical closure relations are used to close the moment-based system of equations, which typically approximate the pressure tensor or heat flux. The more accurate the closure relation, the stronger the simulation approaches kinetic-based results. In this paper, new closure terms are constructed using machine learning techniques. Two different machine learning models, a multi-layer perceptron and a gradient boosting regressor, synthesize a local closure relation for the pressure tensor and heat flux vector from fully kinetic simulations of a 2D magnetic reconnection problem. The models are compared to an existing closure relation for the pressure tensor, and the applicability of the models is discussed. The initial results show that the models can capture the diagonal components of the pressure tensor accurately and show promising results for the heat flux, opening the way for new experiments in multi-scale modeling. We find that the sampling of the points used to train both models plays a capital role in their accuracy.

Implicit hybridized discontinuous Galerkin methods for compressible magnetohydrodynamics

We present hybridized discontinuous Galerkin (HDG) methods for ideal and resistive compressible magnetohydrodynamics (MHD). The HDG methods are fully implicit, high-order accurate and endowed with a unique feature which distinguishes themselves from other discontinuous Galerkin (DG) methods. In particular, they reduce the globally coupled unknowns to the approximate trace of the solution on element boundaries, thereby resulting in considerably smaller global degrees of freedom than other DG methods. Furthermore, we develop a shock capturing method to deal with shocks by appropriately adding artificial bulk viscosity, molecular viscosity, thermal conductivity, and electric resistivity to the physical viscosities in the MHD equations. We show the optimal convergence of the HDG methods for ideal MHD problems and validate our resistive implementation for a magnetic reconnection problem. For smooth problems, we observe that employing a generalized Lagrange multiplier (GLM) formulation can reduce the errors in the divergence of the magnetic field by two orders of magnitude. We demonstrate the robustness of our shock capturing method on a number of test cases and compare our results, both qualitatively and quantitatively, with other MHD solvers. For shock problems, we observe that an effective treatment of both the shock wave and the divergence-free constraint is crucial to ensuring numerical stability.

Analysis of fast turbulent reconnection with self-consistent determination of turbulence timescale

We present results of a Reynolds-averaged turbulence model simulation on the problem of magnetic reconnection. In the model, in addition to the mean density, momentum, magnetic field, and energy equations, the evolution equations of the turbulent cross-helicity W, turbulent energy K, and its dissipation rate ε are simultaneously solved to calculate the rate of magnetic reconnection for a Harris-type current sheet. In contrast to previous studies based on algebraic modeling, the turbulence time scale is self-determined by the nonlinear evolutions of K and ε, their ratio being a time scale. We compare the reconnection rate produced by our mean-field model to the resistive nonturbulent MHD rate. To test whether different regimes of reconnection are produced, we vary the initial strength of turbulent energy and study the effect on the amount of magnetic flux reconnected in time. We obtain that the time scale of turbulence self-adjusts to always produce fast reconnection.

High-order two-fluid plasma solver for direct numerical simulations of plasma flows with full transport phenomena

The two-fluid plasma equations for a single ion species, with full transport terms, including temperature and magnetic field dependent ion and electron viscous stresses and heat fluxes, frictional drag force, and ohmic heating terms, have been implemented in the CFDNS code and solved by using sixth-order non-dissipative compact finite differences for plasma flows in several different regimes. In order to be able to fully resolve all the dynamically relevant time and length scales, while maintaining computational feasibility, the assumptions of infinite speed of light and negligible electron inertia have been made. Non-dimensional analysis of the two-fluid plasma equations shows that, by varying the characteristic/background number density, length scale, temperature, and magnetic strength, the corresponding Hall, resistive, and ideal magnetohydrodynamic equations can be recovered as limiting cases. The accuracy and robustness of this two-fluid plasma solver in handling plasma flows in different regimes have been validated against four canonical problems: Alfven and whistler dispersion relations, electromagnetic plasma shock, and magnetic reconnection. For all test cases, by using physical dissipation and diffusion, with negligible numerical dissipation/diffusion, fully converged Direct Numerical Simulation (DNS)-like solutions are obtained when the ion Reynolds number based on the grid size is smaller than a threshold value which is about 2.3 in this study. For the magnetic reconnection problem, the results show that the magnetic flux saturation time and value converge when the ion and magnetic Reynolds numbers are large enough. Thus, the DNS-like results become relevant to practical problems with much larger Reynolds numbers.

A reduced Landau-gyrofluid model for magnetic reconnection driven by electron inertia

An electromagnetic reduced gyrofluid model for collisionless plasmas, accounting for electron inertia, finite ion Larmor radius effects and Landau-fluid closures for the electron fluid is derived by means of an asymptotic expansion from a parent gyrofluid model. In the absence of terms accounting for Landau damping, the model is shown to possess a non-canonical Hamiltonian structure. The corresponding Casimir invariants are derived and use is made thereof, in order to obtain a set of normal field variables, in terms of which the Poisson bracket and the model equations take a remarkably simple form. The inclusion of perpendicular temperature fluctuations generalizes previous Hamiltonian reduced fluid models and, in particular, the presence of ion perpendicular gyrofluid temperature fluctuations reflects into the presence of two new Lagrangian invariants governing the ion dynamics. The model is applied, in the cold-ion limit, to investigate numerically a magnetic reconnection problem. The Landau damping terms are shown to reduce, by decreasing the electron temperature fluctuations, the linear reconnection rate and to delay the nonlinear island growth. The saturated island width, on the other hand, is independent of Landau damping. The fraction of magnetic energy converted into perpendicular kinetic energy also appears to be unaffected by the Landau damping terms, which, on the other hand, dissipate parallel kinetic energy as well as free energy due to density and electron temperature fluctuations.

Two-way coupled MHD-PIC simulations of magnetic reconnection in magnetic island coalescence

We present simulations of magnetic reconnection with a newly developed coupled MHD-PIC code. In this work a global magnetohydrodynamic (MHD) simulation receives kinetic feedback within an embedded region that is modeled by a kinetic particle-in-cell (PIC) code. The PIC code receives initial and boundary conditions from the MHD simulation, while the MHD solution is updated with the PIC state. We briefly describe this coupling mechanism. This method is suitable for simulating magnetic reconnection problems, as we show with the example of reconnection in the coalescence of magnetic islands. We compare the MHD, Hall-MHD, fully PIC and coupled MHD-PIC simulations of the magnetic island coalescence. We find that the kinetic simulations are very different from the MHD and Hall-MHD results, while the coupled MHD-PIC simulations can remedy this discrepancy while saving computing time. The diffusion region is well resolved in the kinetic simulations, which is also captured by the coupled MHD-PIC model. The coupled simulation also reproduces the kinetic Hall magnetic fields correctly. We calculate the reconnection rates and find differences between the MHD and kinetic results. We find that the coupled MHD-PIC code can reasonably reproduce the kinetic reconnection rate when a larger PIC feedback region is used, while still saving significant computing time.

Magnetic reconnection in terms of catastrophe theory

Magnetic field line reconnection (magnetic reconnection) is a phenomenon that occurs in space and laboratory plasma. Magnetic reconnection allows both the change the magnetic topology and the conversion of the magnetic energy into energy of fast particles. The critical point (critical line or plane in higher dimensional cases) of the magnetic field play an important role in process of magnetic reconnection, as in its neighborhood occurs a change of its topology of a magnetic field and redistribution of magnetic field energy. A lot of literature is devoted to the analytical and numerical investigation of the reconnection process. The main result of these investigations as the result of magnetic reconnection the current sheet is formed and the magnetic topology is changed. While the studies of magnetic reconnection in 2D and 3D configurations have a led to several important results, many questions remain open, including the behavior of a magnetic field in the neighborhood of a critical point of high order. The magnetic reconnection problem is closely related to the problem of the structural stability of vector fields. Since the magnetic field topology changes during both spontaneous and induced magnetic reconnection, it is natural to expect that the magnetic field should evolve from a structurally unstable into a structurally stable configuration. Note that, in this case, the phenomenon under analysis is more complicated since, during magnetic reconnection in a highly conducting plasma, we deal with the non-linear interaction between two vector fields: the magnetic field and the field of the plasma velocities. The aim of our article is to consider the process of magnetic reconnection and transformation of the magnetic topology from the viewpoint of catastrophe theory. Bifurcations in similar configurations (2D magnetic configuration with null high order point) with varying parameters were thoroughly discussed in a monograph by Poston and Stewart.

A Unified Gas Kinetic Scheme for Continuum and Rarefied Flows V: Multiscale and Multi-Component Plasma Transport

AbstractAs a continuation of developing multiscale method for the transport phenomena, a unified gas kinetic scheme (UGKS) for multi-scale and multi-component plasma simulation is constructed. The current scheme is a direct modeling method, where the time evolution solutions from the Vlasov-BGK equations of electron and ion and the Maxwell equations are used to construct a scale-dependent plasma simulation model. The modeling scale used in the UGKS is the mesh size scale, which can be comparable to or much larger than the local mean free path. As a result, with the variation of modeling scales in space and time through the so-called cell's Knudsen number and normalized Larmor radius, the discretized governing equations can recover a wide range of plasma evolution from the Vlasov equation in the kinetic scale to different-type of magnetohydrodynamic (MHD) equations in the hydrodynamic scale. The UGKS provides a general evolution model, which goes to the Vlasov equation in the kinetic scale and many types of MHD equations in the hydrodynamic scale, such as the two fluids model, the Hall, the resistive, and the ideal MHD equations. All current existing governing equations become the subsets of the UGKS, and the UGKS bridges these distinguishable governing equations seamlessly. The construction of UGKS is based on the implementation of physical conservation laws and the un-splitting treatment of particle collision, acceleration, and transport in the construction of a scale-dependent numerical flux across a cell interface. At the same time, the discretized plasma evolution equations are coupled with the Maxwell equations for electro-magnetic fields, which also cover a scale-dependent transition between the Ampére's law and the Ohm's law for the calculation of electric field. The time step of UGKS is not limited by the relaxation time, the cyclotron period, and the speed of light in the ideal-MHD regime. Our scheme is able to give a physically accurate solution for plasma simulation with a wide range of Knudsen number and normalized Larmor radius. It can be used to study the phenomena from the Vlasov limit to the scale of plasma skin depth for the capturing of two-fluid effect, and the phenomena in the plasma transition regime with a modest Knudsen number and Larmor radius. The UGKS is validated by numerical test cases, such as the Landau damping and two stream instability in the kinetic regime, and the Brio-Wu shock tube problem, and the Orszag-Tang MHD turbulence problem in the hydrodynamic regime. The scheme is also used to study the geospace environment modeling (GEM), such as the challenging magnetic reconnection problem in the transition regime. At the same time, the magnetic reconnection mechanism of the Sweet-Parker model and the Hall effect model can be connected smoothly through the variation of Larmor radius in the UGKS simulations. Overall, the UGKS is a physically reliable multi-scale plasma simulation method, and it provides a powerful and unified approach for the study of plasma physics.

Three-dimensional magnetic reconnection and its application to solar flares

Solar flares are powerful radiations occurring in the Sun’s atmosphere. They are powered by magnetic reconnection, a phenomenon that can convert magnetic energy into other forms of energy such as heat and kinetic energy, and which is believed to be ubiquitous in the universe. With the ever increasing spatial and temporal resolutions of solar observations, as well as numerical simulations benefiting from increasing computer power, we can now probe into the nature and the characteristics of magnetic reconnection in three dimensions to better understand the phenomenon’s consequences during eruptive flares in our star’s atmosphere. We review in the following the efforts made on different fronts to approach the problem of magnetic reconnection. In particular, we will see how understanding the magnetic topology in three dimensions helps in locating the most probable regions for reconnection to occur, how the current layer evolves in three dimensions and how reconnection leads to the formation of flux ropes, plasmoids and flaring loops.

Extended theory of the Taylor problem in the plasmoid-unstable regime

A fundamental problem of forced magnetic reconnection has been solved taking into account the plasmoid instability of thin reconnecting current sheets. In this problem, the reconnection is driven by a small amplitude boundary perturbation in a tearing-stable slab plasma equilibrium. It is shown that the evolution of the magnetic reconnection process depends on the external source perturbation and the microscopic plasma parameters. Small perturbations lead to a slow nonlinear Rutherford evolution, whereas larger perturbations can lead to either a stable Sweet-Parker-like phase or a plasmoid phase. An expression for the threshold perturbation amplitude required to trigger the plasmoid phase is derived, as well as an analytical expression for the reconnection rate in the plasmoid-dominated regime. Visco-resistive magnetohydrodynamic simulations complement the analytical calculations. The plasmoid formation plays a crucial role in allowing fast reconnection in a magnetohydrodynamical plasma, and the presented results suggest that it may occur and have profound consequences even if the plasma is tearing-stable.

A finite volume formulation of the multi-moment advection scheme for Vlasov simulations of magnetized plasma

We present a finite volume version of the multi-moment advection scheme (Minoshima et al., 2011, 2013). The scheme advances zeroth to second order piecewise moments at cell centers through their numerical fluxes obtained from one-dimensional high order interpolation. The modification simplifies the scheme without losing its high performance. We apply the scheme to two-dimensional electromagnetic Vlasov simulations of linear wave propagation and nonlinear magnetic reconnection problems. Our Vlasov simulation resolves microscopic structure of the non-Maxwellian plasma velocity distribution around the magnetic reconnection site as well as macroscopic structure of fluid quantities within the energy error of 2%, and is in good agreement with previous studies.

The Fluid-Kinetic Particle-in-Cell method for plasma simulations

A method that solves concurrently the multi-fluid and Maxwell's equations has been developed for plasma simulations. By calculating the stress tensor in the multi-fluid momentum equation by means of computational particles moving in a self-consistent electromagnetic field, the kinetic effects are retained while solving the multi-fluid equations. The Maxwell's and multi-fluid equations are discretized implicitly in time enabling kinetic simulations over time scales typical of the fluid simulations. The Fluid-Kinetic Particle-in-Cell method has been implemented in a three-dimensional electromagnetic code, and tested against the two-stream instability, the Weibel instability, the ion cyclotron resonance and magnetic reconnection problems. The method is a promising approach for coupling fluid and kinetic methods in a unified framework.

Application of PDSLin to the magnetic reconnection problem

Magnetic reconnection is a fundamental process in a magnetized plasma at both low and high magnetic Lundquist numbers (the ratio of the resistive diffusion time to the Alfven wave transit time), which occurs in a wide variety of laboratory and space plasmas, e.g. magnetic fusion experiments, the solar corona and the Earth's magnetotail. An implicit time advance for the two-fluid magnetic reconnection problem is known to be difficult because of the large condition number of the associated matrix. This is especially troublesome when the collisionless ion skin depth is large so that the Whistler waves, which cause the fast reconnection, dominate the physics (Yuan et al 2012 J. Comput. Phys. 231 5822–53). For small system sizes, a direct solver such as SuperLU can be employed to obtain an accurate solution as long as the condition number is bounded by the reciprocal of the floating-point machine precision. However, SuperLU scales effectively only to hundreds of processors or less. For larger system sizes, it has been shown that physics-based (Chacon and Knoll 2003 J. Comput. Phys. 188 573–92) or other preconditioners can be applied to provide adequate solver performance. In recent years, we have been developing a new algebraic hybrid linear solver, PDSLin (Parallel Domain decomposition Schur complement-based Linear solver) (Yamazaki and Li 2010 Proc. VECPAR pp 421–34 and Yamazaki et al 2011 Technical Report). In this work, we compare numerical results from a direct solver and the proposed hybrid solver for the magnetic reconnection problem and demonstrate that the new hybrid solver is scalable to thousands of processors while maintaining the same robustness as a direct solver.

Numerical simulation of four-field extended magnetohydrodynamics in dynamically adaptive curvilinear coordinates via Newton–Krylov–Schwarz

Numerical simulations of the four-field extended magnetohydrodynamics (MHD) equations with hyper-resistivity terms present a difficult challenge because of demanding spatial resolution requirements. A time-dependent sequence of r-refinement adaptive grids obtained from solving a single Monge–Ampère (MA) equation addresses the high-resolution requirements near the x-point for numerical simulation of the magnetic reconnection problem. The MHD equations are transformed from Cartesian coordinates to solution-defined curvilinear coordinates. After the application of an implicit scheme to the time-dependent problem, the parallel Newton–Krylov–Schwarz (NKS) algorithm is used to solve the system at each time step. Convergence and accuracy studies show that the curvilinear solution requires less computational effort than a pure Cartesian treatment. This is due both to the more optimal placement of the grid points and to the improved convergence of the implicit solver, nonlinearly and linearly. The latter effect, which is significant (more than an order of magnitude in number of inner linear iterations for equivalent accuracy), does not yet seem to be widely appreciated.

A Discontinuous Galerkin Method for Ideal Two-Fluid Plasma Equations

AbstractA discontinuous Galerkin method for the ideal 5 moment two-fluid plasma system is presented. The method uses a second or third order discontinuous Galerkin spatial discretization and a third order TVD Runge-Kutta time stepping scheme. The method is benchmarked against an analytic solution of a dispersive electron acoustic square pulse as well as the two-fluid electromagnetic shock [1] and existing numerical solutions to the GEM challenge magnetic reconnection problem [2]. The algorithm can be generalized to arbitrary geometries and three dimensions. An approach to maintaining small gauge errors based on error propagation is suggested.

The SHASTA Code Modified by Self-adaptive Mesh and Numerical Experiment of Magnetic Reconnections

SHASTA (Sharp and Smooth Transport Algorithm) is a code with single mesh to solve the 2-dimensional magnetohydrodynamic (MHD) equations. When SHASTA is used to the numerical simulation of magnetic reconnection problem, it is modified to be the code which adopts the method of the selfadaptive mesh. The modified code can carry out refined calculations in diffusion regions. In the process of the self-adaptive calculations with SHASTA, a “plugand-play” strategy is adopted and the original algorithm to solve 2-dimensional MHD partial differential equations is treated as an independent cell. In addition, the hierarchical data structure is used in this modification and parameters in each refined level are described by a 2-dimensional variable array. The regions where the distributions of magnetic field and pressure exhibit steep variations are marked as the refined regions. Then, the distributions of physical quantities and the boundary conditions in the grid points of refined levels are deduced via interpolation method. Finally, the refined calculated results of refined regions are assigned to the previous level of mesh and the existing results are updated. The numerical experiment of magnetic reconnections which adopts refined calculations indicates that compared with the code with single mesh, the resolution of details is improved and the corresponding increment of computing time is related to the selection of parameters in the simulation. The calculation accuracy and effect on instability, which are caused by a part of the self-adaptive code, depend on the boundary settings, push strategy over each single step as well as the interpolation algorithm.

Collisionless Magnetic Reconnection via Alfvén Eigenmodes

We propose an analytic approach to the problem of collisionless magnetic reconnection formulated as a process of Alfvén eigenmodes' generation and dissipation. Alfvén eigenmodes are confined by the current sheet in the same way that quantum mechanical waves are confined by the tanh;{2} potential. The dynamical time scale of reconnection is the system scale divided by the eigenvalue propagation velocity of the n = 1 mode. The prediction of the n = 1 mode shows good agreement with the in situ measurement of the reconnection-associated Hall fields.

The role of the Weibel instability at the reconnection jet front in relativistic pair plasma reconnection

The role of the Weibel instability is investigated for the first time in the context of the large-scale magnetic reconnection problem. A late-time evolution of magnetic reconnection in relativistic pair plasmas is demonstrated by particle-in-cell simulations. In the outflow regions, powerful reconnection jets pile up the magnetic fields and then a contact discontinuity appears there. Further downstream, it is found that the two-dimensional extension of the relativistic Weibel instability generates electromagnetic fields, which are comparable to the antiparallel or piled-up fields. In a microscopic viewpoint, the instability allows the plasma’s multiple interactions with the discontinuity. In a macroscopic viewpoint, the instability leads to rapid expansion of the current sheet and then the reconnection jet front further propagates into the downstream. Possible application to the three-dimensional case is briefly discussed.

Adiabatic plasma equilibrium and application to a reconnection problem

The evolution of many plasma systems is adiabatic, i.e., plasma entropy is conserved in each magnetic flux tube. An apparently surprising result found recently from simulations of a forced magnetic reconnection problem (the “Newton Challenge” [J. Birn et al., Phys. Plasmas 13, 092117 (2006)]) is that even in the presence of a dissipative process such as reconnection, the entropy within a flux tube can still be approximately conserved, due to the strong localization of the dissipation. To address plasma equilibrium with such adiabatic constraints, a novel code has been developed that computes equilibria with entropy profile as input, using the alternating dimension method [H. Grad et al., Proc. Natl. Acad. Sci. USA 72, 3789 (1975)]. The code alternates between solving the two–dimensional (2D) Grad-Shafranov equation to obtain the field configuration (flux function A) from the pressure profile P(A) and a 1D ordinary differential equation that uses the entropy conservation to derive the pressure function P(A) from a flux surface average. As a particular application, the code is used to compute equilibria relevant to the Newton Challenge, with a grid reflecting the reconnected state topology (with an X and an O point). The equilibria found agree very well with late stages of magnetohydrodynamic simulations of the Newton Challenge. This agreement not only validates the new code, but also proves that the final state approached by the reconnection simulations is indeed an equilibrium quasiadiabatically connected with the initial state. The results also show a significant release of magnetic energy through reconnection. Finally, other potential applications of the new code, especially to adiabatic evolution of space plasmas, are discussed.