Resistive Magnetohydrodynamics Research Articles

Smoothed particle hydrodynamics method for pinch plasma simulation with non-ideal MHD model

When plasma is compressed by magnetic forces, a pinch phenomenon is observed. Pinch plasma has received significant attention as an efficient source of radiation and a way for high-density plasma physics analysis. In this study, a non-ideal magnetohydrodynamics (MHD) model is applied to a smoothed particle hydrodynamics (SPH) framework to analyze pinch plasmas whose local resistivity varies with temperature and pressure. The proposed SPH model incorporates several numerical treatments, such as a correction term to satisfy the ∇·B constraint and some artificial dissipation terms to govern the shock wave. Moreover, it includes the evaluation of a novel SPH discretization for non-ideal MHD terms, including current density calculations. Furthermore, the proposed model is validated with three benchmark cases: (1) Brio and Wu shock tube (ideal MHD), (2) resistive MHD shock simulation, and (3) magnetized Noh Z-pinch problem. The simulation results are compared with the results of some reference Eulerian MHD simulations and analytical solutions. The simulations agree well with the reference data, and the introduced numerical treatments are effective. Finally, X-pinch simulations are performed using the proposed model. The simulations well produce the micro Z-pinch and jet shapes, which are important X-pinch features. Overall, the proposed SPH model has extensive potential for studying the complex pinch plasma phenomena.

Self-consistent integrated modeling of combined hybrid discharge-laser produced plasma devices for extreme ultraviolet metrology

Discharge- and laser-produced plasma (DLPP) devices are being used as light sources for extreme ultraviolet (EUV) generation. A key challenge for both, DPP and LPP, is achieving sufficient brightness to support the throughput requirements of nanometrology tools. To simulate the environment of a hybrid DLPP device and optimize the EUV output, we have developed an integrated HEIGHTS-DLPP computer simulation package. The package integrates simulation of two evolving plasmas (DPP and LPP) and includes modeling of a set of integrated self-consistent processes: external power source and plasma energy balance, plasma resistive magnetohydrodynamics, plasma heat conduction, detailed radiation transport (RT), and laser absorption and refraction. We simulated and optimized DLPP devices using Xe gas as a target material. We synchronized the external circuit parameters, chamber gas parameters, and laser beam temporal and spatial profiles to achieve maximum EUV output. The full 3D Monte Carlo scheme was integrated for detailed RT and EUV output calculations in Xe using more than 3600 spectral groups. The modeling results are in good agreement with Julich Forschungszentrum experimental data. Theoretical models, developed and integrated into the HEIGHTS package, showed wide capabilities and flexibility. The models and package can be used for optimization of the experimental parameters and settings, investigation of DLPP devices with complex design, analyzing the impact of integrated spatial effects and working timeline arrangement on the final EUV output, and EUV source size, shape, and angular distribution.

Dark photon superradiance: Electrodynamics and multimessenger signals

We study the electrodynamics of a kinetically mixed dark photon cloud that forms through superradiance around a spinning black hole, and design strategies to search for the resulting multimessenger signals. A dark photon superradiance cloud sources a rotating dark electromagnetic field which, through kinetic mixing, induces a rotating visible electromagnetic field. Standard model charged particles entering this field initiate a transient phase of particle production that populates a plasma inside the cloud and leads to a system which shares qualitative features with a pulsar magnetosphere. We study the electrodynamics of the dark photon cloud with resistive magnetohydrodynamics methods applicable to highly magnetized plasma, adapting techniques from simulations of pulsar magnetospheres. We identify turbulent magnetic field reconnection as the main source of dissipation and electromagnetic emission, and compute the peak luminosity from clouds around solar-mass black holes to be as large as $10^{43}$ erg/s for open dark photon parameter space. The emission is expected to have a significant X-ray component and is potentially periodic, with period set by the dark photon mass. The luminosity is comparable to the brightest X-ray sources in the Universe, allowing for searches at distances of up to hundreds of Mpc with existing telescopes. We discuss observational strategies, including targeted electromagnetic follow-ups of solar-mass black hole mergers and targeted continuous gravitational wave searches of anomalous pulsars.

Mallat Scattering Transformation based surrogate for Magnetohydrodynamics

A Machine and Deep Learning (MLDL) methodology is developed and applied to give a high fidelity, fast surrogate for 2D resistive MagnetoHydroDynamic (MHD) simulations of Magnetic Liner Inertial Fusion (MagLIF) implosions. The resistive MHD code GORGON is used to generate an ensemble of implosions with different liner aspect ratios, initial gas preheat temperatures (that is, different adiabats), and different liner perturbations. The liner density and magnetic field as functions of x, y, and z were generated. The Mallat Scattering Transformation (MST) is taken of the logarithm of both fields and a Principal Components Analysis (PCA) is done on the logarithm of the MST of both fields. The fields are projected onto the PCA vectors and a small number of these PCA vector components are kept. Singular Value Decompositions of the cross correlation of the input parameters to the output logarithm of the MST of the fields, and of the cross correlation of the SVD vector components to the PCA vector components are done. This allows the identification of the PCA vectors vis-a-vis the input parameters. Finally, a Multi Layer Perceptron (MLP) neural network with ReLU activation and a simple three layer encoder/decoder architecture is trained on this dataset to predict the PCA vector components of the fields as a function of time. Details of the implosion, stagnation, and the disassembly are well captured. Examination of the PCA vectors and a permutation importance analysis of the MLP show definitive evidence of an inverse turbulent cascade into a dipole emergent behavior. The orientation of the dipole is set by the initial liner perturbation. The analysis is repeated with a version of the MST which includes phase, called Wavelet Phase Harmonics (WPH). While WPH do not give the physical insight of the MST, they can and are inverted to give field configurations as a function of time, including field-to-field correlations.

CMR Exploration. I. Filament Structure with Synthetic Observations

In this paper, we carry out a pilot parameter exploration for the collision-induced magnetic reconnection (CMR) mechanism that forms filamentary molecular clouds. Following Kong et al., we utilize Athena++ to model CMR in the context of resistive magnetohydrodynamics (MHD), considering the effect from seven physical conditions, including the ohmic resistivity (η), the magnetic field (B), the cloud density (ρ), the cloud radius R, the isothermal temperature T, the collision velocity v x , and the shear velocity v z . Compared to their fiducial model, we consider a higher and a lower value for each one of the seven parameters. We quantify the exploration results with five metrics, including the density probability distribution function (ρ-PDF), the filament morphology (250 μm dust emission), the B–ρ relation, the dominant fiber width, and the ringiness that describes the significance of the ringlike substructures. The exploration forms straight and curved CMR filaments with rich substructures that are highly variable in space and time. The variation translates to fluctuation in all five metrics, reflecting the chaotic nature of magnetic reconnection in CMR. A temporary B ∝ ρ relation is noticeable during the first 0.6 Myr. Overall, the exploration provides useful initial insights into the CMR mechanism.

Local well-posedness of the free-boundary problem in compressible resistive magnetohydrodynamics

We prove the local well-posedness in Sobolev spaces of the free-boundary problem for compressible inviscid resistive isentropic MHD system under the Rayleigh-Taylor physical sign condition, which describes the motion of a free-boundary compressible plasma in an electro-magnetic field with magnetic diffusion. We use Lagrangian coordinates and apply the tangential smoothing method introduced by Coutand-Shkoller to construct the approximation solutions. One of the key observations is that the Christodoulou-Lindblad type elliptic estimate together with magnetic diffusion not only gives the common control of magnetic field and fluid pressure directly, but also controls the Lorentz force which is a higher order term in the energy functional.

Relativistic resistive magneto-hydrodynamics code for high-energy heavy-ion collisions

We construct a relativistic resistive magneto-hydrodynamic (RRMHD) numerical simulation code for high-energy heavy-ion collisions as a first designed code in the Milne coordinates. We split the system of differential equations into two parts, a non-stiff and a stiff part. For the non-stiff part, we evaluate the numerical flux using HLL approximated Riemann solver and execute the time integration by the second-order of Runge–Kutta algorithm. For the stiff part, which appears in Ampere’s law, we integrate the equations using semi-analytic solutions of the electric field. We employ the generalized Lagrange multiplier method to ensure the divergence-free constraint for the magnetic field and Gauss’s law. We confirm that our code reproduces well the results of standard RRMHD tests in the Cartesian coordinates. In the Milne coordinates, the code with high conductivity is validated against relativistic ideal MHD tests. We also verify the semi-analytic solutions of the accelerating longitudinal expansion of relativistic resistive magneto-hydrodynamics in high-energy heavy-ion collisions in comparison with our numerical result. Our numerical code reproduces these solutions.

Charge-dependent anisotropic flow in high-energy heavy-ion collisions from a relativistic resistive magneto-hydrodynamic expansion

We have investigated the charge-dependent anisotropic flow in high-energy heavy-ion collisions, using relativistic resistive magneto-hydrodynamics (RRMHD). We consider the optical Glauber model as an initial model of the quark-gluon plasma (QGP) and the solution of the Maxwell equations with source term of the charged particles in two colliding nuclei as initial electromagnetic fields. The RRMHD simulation is performed with these initial conditions in Au-Au and Cu-Au collisions at $\sqrt{s_{\mathrm{NN}}} = 200$ GeV. We have calculated the charge-odd contribution to the directed flow $\Delta v_1$ and elliptic flow $\Delta v_2$ in both collisions based on electric charge distributions as a consequence of RRMHD. Our results show that the $\Delta v_1$ and $\Delta v_2$ are approximately proportional to the electrical conductivity ($\sigma$) of the medium. In the $\sigma=0.023~\mathrm{fm}^{-1}$ case, our result of $\Delta v_1$ is consistent with STAR data in Au-Au collisions. Furthermore, in Cu-Au collisions, $\Delta v_1$ has a non-zero value at $\eta = 0$. We conclude that the charge-dependent anisotropic flow is a good probe to extract the electrical conductivity of the QGP medium in high-energy heavy-ion experiments.

Charge diffusion in relativistic resistive second-order dissipative magnetohydrodynamics

We study charge diffusion in relativistic resistive second-order dissipative magnetohydrodynamics. In this theory, charge diffusion is not simply given by the standard Navier-Stokes form of Ohm's law, but by an evolution equation which ensures causality and stability. This, in turn, leads to transient effects in the charge diffusion current, the nature of which depends on the particular values of the electrical conductivity and the charge-diffusion relaxation time. The ensuing equations of motion are of so-called stiff character, which requires special care when solving them numerically. To this end, we specifically develop an implicit-explicit Runge-Kutta method for solving relativistic resistive second-order dissipative magnetohydrodynamics and subject it to various tests. We then study the system's evolution in a simplified 1+1-dimensional scenario for a heavy-ion collision, where matter and electromagnetic fields are assumed to be transversely homogeneous, and investigate the cases of an initially non-expanding fluid and a fluid initially expanding according to a Bjorken scaling flow. In the latter case, the scale invariance is broken by the ensuing self-consistent dynamics of matter and electromagnetic fields. However, the breaking becomes quantitatively important only if the electromagnetic fields are sufficiently strong. The breaking of scale invariance is larger for smaller values of the conductivity. Aspects of entropy production from charge diffusion currents and stability are also discussed.

The virtual element method for the 3D resistive magnetohydrodynamic model

We present a four-field virtual element discretization for the time-dependent resistive magnetohydrodynamics equations in three space dimensions, focusing on the semi-discrete formulation. The proposed method employs general polyhedral meshes and guarantees velocity and magnetic fields that are divergence free up to machine precision. We provide a full convergence analysis under suitable regularity assumptions, which is validated by some numerical tests.

Monolithic multigrid for implicit Runge–Kutta discretizations of incompressible fluid flow

Most research on preconditioners for time-dependent PDEs has focused on implicit multi-step or diagonally-implicit multi-stage temporal discretizations. In this paper, we consider monolithic multigrid preconditioners for fully-implicit multi-stage Runge–Kutta (RK) time integration methods. These temporal discretizations have very attractive accuracy and stability properties, but they couple the spatial degrees of freedom across multiple time levels, requiring the solution of very large linear systems. We extend the classical Vanka relaxation scheme to implicit RK discretizations of saddle point problems. We present numerical results for the incompressible Stokes, Navier–Stokes, and resistive magnetohydrodynamics equations, in two and three dimensions, confirming that these relaxation schemes lead to robust and scalable monolithic multigrid methods for a challenging range of incompressible fluid-flow models.

Directed flow in relativistic resistive magneto-hydrodynamic expansion for symmetric and asymmetric collision systems

We construct a dynamical model for high-energy heavy-ion collisions based on the relativistic resistive magneto-hydrodynamic framework. Using our newly developed ($3+1$)-dimensional relativistic resistive magneto-hydrodynamics code, we investigate magneto-hydrodynamic expansion in symmetric and asymmetric collision systems as the first application to high-energy heavy-ion collisions. As a realistic initial condition for electromagnetic fields, we consider the solutions of the Maxwell equations with the source term of point charged particles moving in the direction of the beam axis, including finite constant electrical conductivity of the medium. We evaluate the directed flow in the symmetric and asymmetric collisions at BNL Relativistic Heavy Ion Collider energy. We find a significant effect of finite electrical conductivity on the directed flow in the asymmetric collision system. We confirm that a certain amount of energy transfer by dissipation associated with Ohmic conduction occurs in the asymmetric collision system because of the asymmetry of the electric field produced by two different colliding nuclei. Because this energy transfer makes the pressure gradient of the medium flatter, the growth of directed flow decreases.

Modeling disks and magnetic outflows around a forming massive star

Context. Similar to their lower mass siblings, massive protostars can be expected to: (a) be surrounded by circumstellar disks, and (b) launch magnetically driven jets and outflows. The disk formation and global evolution is thereby controlled by advection of angular momentum from large scales, the efficiency of magnetic braking and the resistivity of the medium, and the internal thermal and magnetic pressures of the disk. Aims. We determine the dominant physical mechanisms that shape the appearance of these circumstellar disks, their sizes, and aspect ratios. Methods. We performed a series of 30 simulations of a massive star forming from the gravitational collapse of a molecular cloud threaded by an initially uniform magnetic field, starting from different values for the mass of the cloud, its initial density and rotation profiles, its rotational energy content, the magnetic field strength, and the resistivity of the material. The gas and dust was modeled with the methods of resistive magnetohydrodynamics, also considering radiation transport of thermal emission and self-gravity. We checked for the impact of spatial resolution in a dedicated convergence study. Results. After the initial infall phase dominated by the gravitational collapse, an accretion disk was formed, shortly followed by the launching of magnetically driven outflows. Two layers can be distinguished in the accretion disk: a thin layer, vertically supported by thermal pressure, and a thick layer, vertically supported by magnetic pressure. Both regions exhibit Keplerian-like rotation and grow outward over time. We observed the effects of magnetic braking in the inner ~50 au of the disk at late times in our fiducial case. The parameter study reveals that the size of the disk is mostly determined by the density and rotation profiles of the initial mass reservoir and not by the magnetic field strength. We find that the disk size and protostellar mass gain scale with the initial mass of the cloud. Magnetic pressure can slightly increase the size of the accretion disk, while magnetic braking is more relevant in the innermost parts of the disk as opposed to the outer disk. From the parameter study, we infer that multiple initial conditions for the onset of gravitational collapse are able to produce a given disk size and protostellar mass.

A multilevel block preconditioner for the HDG trace system applied to incompressible resistive MHD

We present a scalable block preconditioning strategy for the trace system arising from the high-order hybridized discontinuous Galerkin (HDG) discretization of incompressible resistive magnetohydrodynamics (MHD). We construct the block preconditioner with a least squares commutator (BFBT) approximation for the inverse of the Schur complement that segregates out the pressure unknowns of the trace system. The remaining velocity, magnetic field, and Lagrange multiplier unknowns form a coupled nodal unknown block for which a system algebraic multigrid (AMG) is used to approximate the inverse. The complexity of the MHD equations together with the algebraic nature of the statically condensed HDG trace system makes the choice of smoother in the system AMG part critical for the convergence and performance of the block preconditioner. Our numerical experiments show GMRES preconditioned by ILU(0) of overlap zero as a smoother inside system AMG performs best in terms of robustness, time per nonlinear iteration and memory requirements. With several transient test cases in 2D and 3D including the island coalescence problem at high Lundquist number we demonstrate the robustness and parallel scalability of the block preconditioner. Additionally for the upper block a preliminary study of an alternate nodal block system solver based on a multilevel approximate nested dissection is presented. On a 2D island coalescence problem the multilevel approximate nested dissection preconditioner shows better scalability with respect to mesh refinement than the system AMG, but is relatively less robust with respect to Lundquist number scaling. In the Appendix B, we rigorously show: (1) the uniqueness of the solution of the nonlinear MHD system for small time, (2) the convergence of the Picard iterations for each backward Euler step, and (3) the convergence of the entire time stepping procedure.

Dust motion and possibility of dust growth in a growing circumstellar disk

ABSTRACT We calculate the evolution of a star-forming cloud core using a three-dimensional resistive magnetohydrodynamics simulation, treating dust grains as Lagrangian particles, to investigate the dust motion in the early star formation stage. We prepare six different-sized set of dust particles in the range ad = 0.01–$1000\, \mu$m, where ad is the dust grain size. In a gravitationally collapsing cloud, a circumstellar disk forms around a protostar and drives a protostellar outflow. Almost all the small dust grains (ad ≲ 10–$100\, \mu$m) initially distributed in the region θ0 ≲ 45° are ejected from the center by the outflow, where θ0 is the initial zenith angle relative to the rotation axis, whereas only a small number of the large dust grains ($a_{\rm d} \gtrsim 100\, \mu$m) distributed in the region are ejected. All other grains fall onto either the protostar or disk without being ejected by the outflow. Regardless of the dust grain size, the behavior of the dust motion is divided into two trends after dust particles settle into the circumstellar disk. The dust grains reaching the inner disk region from the upper envelope preferentially fall onto the protostar, while those reaching the outer disk region or disk outer edge from the envelope can survive without an inward radial drift. These surviving grains can induce dust growth. Thus, we expect that the outer disk regions could be a favored place of planet formation.

A study of global magnetic helicity in self-consistent spherical dynamos

ABSTRACT Magnetic helicity is a fundamental constraint in both ideal and resistive magnetohydrodynamics. Measurements of magnetic helicity density on the Sun and other stars are used to interpret the internal behaviour of the dynamo generating the global magnetic field. In this note, we study the behaviour of the global relative magnetic helicity in three self-consistent spherical dynamo solutions of increasing complexity. Magnetic helicity describes the global linkage of the poloidal and toroidal magnetic fields (weighted by magnetic flux), and our results indicate that there are preferred states of this linkage. This leads us to propose that global magnetic reversals are, perhaps, a means of preserving this linkage, since, when only one of the poloidal or toroidal fields reverses, the preferred state of linkage is lost. It is shown that magnetic helicity indicates the onset of reversals and that this signature may be observed at the outer surface.

Two-dimensional MHD equilibria of diamagnetic bubble in gas-dynamic trap

This article presents a magnetohydrodynamic (MHD) two-dimensional numerical model of diamagnetic bubble equilibria in an axisymmetric open trap. The theoretical model consists of the Grad–Shafranov equilibrium equation and the transport equation obtained within the resistive single-fluid MHDs with isotropic pressure. Found are the numerical solutions corresponding to the diamagnetic confinement mode. In particular, the equilibria of the diamagnetic bubble in the gas-dynamic multimirror trap are calculated. We investigate the effect of magnetic field corrugation on the equilibrium; the corrugation of the vacuum field is shown to lead to a rather moderate corrugation of the bubble boundary if the period of corrugation is sufficiently small. A valuable numerical result is the distribution of the diamagnetic field, which would be useful for optimizing the position of the wall-stabilization plates.

An Extension of Gmunu: General-relativistic Resistive Magnetohydrodynamics Based on Staggered-meshed Constrained Transport with Elliptic Cleaning

We present the implementation of general-relativistic resistive magnetohydrodynamics solvers and three divergence-free handling approaches adopted in the General-relativistic multigrid numerical (Gmunu) code. In particular, implicit–explicit Runge–Kutta schemes are used to deal with the stiff terms in the evolution equations for small resistivity. The three divergence-free handling methods are (i) hyperbolic divergence cleaning (also known as the generalized Lagrange multiplier), (ii) staggered-meshed constrained transport schemes, and (iii) elliptic cleaning through a multigrid solver, which is applicable in both cell-centered and face-centered (stagger grid) magnetic fields. The implementation has been tested with a number of numerical benchmarks from special-relativistic to general-relativistic cases. We demonstrate that our code can robustly recover from the ideal magnetohydrodynamics limit to a highly resistive limit. We also illustrate the applications in modeling magnetized neutron stars, and compare how different divergence-free handling methods affect the evolution of the stars. Furthermore, we show that the preservation of the divergence-free condition of the magnetic field when using staggered-meshed constrained transport schemes can be significantly improved by applying elliptic cleaning.

Gravitoturbulent dynamo in global simulations of gaseous disks

Context. The turbulence driven by gravitational instabilities (GIs) can amplify magnetic fields in massive gaseous disks. This GI dynamo may appear in young circumstellar disks, whose weak ionization challenges other amplification routes, as well as in active galactic nuclei. Although regarded as a large-scale dynamo, only local simulations have so far described its kinematic regime. Aims. We study the GI dynamo in global magnetohydrodynamic (MHD) models of accretion disks, focusing on its kinematic phase. Methods. We perform resistive MHD simulations with the PLUTO code for different radiative cooling times and electrical resistivities. A weak magnetic field seeds the dynamo, and we adopt mean-field and heuristic models to capture its essence. Results. We recover the same induction process leading to magnetic field amplification as previously identified in local simulations. The dynamo is, however, global in nature, connecting distant annuli of the disk via a large-scale dynamo mode of a fixed growth rate. This large-scale amplification can be described by a mean-field model that does not rely on conventional α-Ω effects. When varying the disk parameters we find an optimal resistivity that facilitates magnetic amplification, whose magnetic Reynolds number, ℛm ≲ 10, is substantially smaller than in local simulations. Unlike local simulations, we find an optimal cooling rate and the existence of global oscillating dynamo modes. The nonlinear saturation of the dynamo puts the disk in a strongly magnetized turbulent state on the margins of the effective range of GI. In our simulations, the accretion power eventually exceeds the threshold required by local thermal balance against cooling, leaving the long-term nonlinear outcome of the GI dynamo uncertain.

A conservative finite element solver for the induction equation of resistive MHD: Vector potential method and constraint preconditioning

A new conservative finite element solver for the three-dimensional steady magnetohydrodynamic (MHD) kinematics equations is presented. The solver utilizes magnetic vector potential and current density as solution variables, which are discretized by H(curl)-conforming edge-element and H(div)-conforming face element respectively. As a result, the divergence-free constraints of discrete current density and magnetic induction are both satisfied. Moreover the solutions also preserve the total magnetic helicity. The generated linear algebraic equation is a typical dual saddle-point problem that is ill-conditioned and indefinite. To efficiently solve it, we develop a block preconditioner based on constraint preconditioning framework and devise a preconditioned FGMRES solver. Numerical experiments verify the conservative properties, the convergence rate of the discrete solutions and the robustness of the preconditioner. Simulation of three-dimensional driven cavity flow problem using full MHD solver is also conducted.