Ideal Magnetohydrodynamic Equations Research Articles

Toroidal Alfvén Eigenmodes and Fast Particles in Tokamaks

A general introduction is given to ideal magnetohydrodynamic (MHD) waves, using linear perturbations of ideal MHD equations. Subsequently the shear Alfvén waves are shown to form a discrete spectrum of Global Alfvén Eigenmodes in current-carrying cylindrical plasmas and Toroidal Alfvén Eigenmodes in toroidal plasmas. A comparison between theory and experiment is presented for the observed discrete spectra of Alfvén waves driven by energetic ions in Joint European Torus. The mechanism of excitation of Alfvén instabilities is described by considering particle-to-wave power transfer, and two main mechanisms of TAE-induced re-distribution and losses of energetic ions are discussed.

Multi-level Monte Carlo finite volume methods for nonlinear systems of conservation laws in multi-dimensions

We extend the multi-level Monte Carlo (MLMC) in order to quantify uncertainty in the solutions of multi-dimensional hyperbolic systems of conservation laws with uncertain initial data. The algorithm is presented and several issues arising in the massively parallel numerical implementation are addressed. In particular, we present a novel load balancing procedure that ensures scalability of the MLMC algorithm on massively parallel hardware. A new code is described and applied to simulate uncertain solutions of the Euler equations and ideal magnetohydrodynamics (MHD) equations. Numerical experiments showing the robustness, efficiency and scalability of the proposed algorithm are presented.

Magnetoacoustic waves in diagnostics of the flare current sheets

Aims. To obtain diagnostics tools for solar flare current sheets, we numerically studied impulsively generated magnetoacoustic waves in the Harris current sheet. Methods. We used two-dimensional (2-D) magnetohydrodynamic (MHD) and wave dispersion models. Tests of these models were performed for waves in the density slab, where analytical solutions are known. In the MHD model, we solved the full set of ideal MHD equations by means of the modified two-step Lax-Wendroff algorithm. The initial perturbation was chosen to generate preferentially the fast sausage magnetoacoustic waves. To determine the dispersion characteristics of MHD waves in the Harris current sheet, we numerically solved the equation of plasma motions by means of the Runge-Kutta fourth order method together with the bisection iteration one. To establish some diagnostics of these waves and their corresponding flare current sheets, we used the wavelet analysis method. Results. We find that the results of tests of our 2-D MHD and wave dispersion models for the density slab are in good agreement with analytical results. We analyze the magnetoacoustic waves in the current sheet and compare them with those in the density slab. In both cases, for similar geometrical and plasma parameters, we find that wave trains were generated and propagated in a similar way. Their signals registered at selected locations of the Harris current sheet and density slab are also similar. Nevertheless, a dependence of the period of the magnetoacoustic waves on the width of the Harris current sheet differs from that for the density slab. The form of the wave front inside the current sheet similarly differs from that in the density slab. We find that the wavelet spectra of the signals of incoming magnetoacoustic waves at selected locations in the current sheet have the form of wavelet tadpoles. We distinguish that the form of these wavelet tadpoles becomes longer and the heads of the wavelet tadpoles are detected later in time as the distance of the detection point from that of the initial wave perturbation increases. We also find that the wavelet tadpole period depends on the plasma beta parameter. The results are discussed from the point of view of their use as diagnostics of the flare current sheets or flare loops.

Arbitrary order exactly divergence-free central discontinuous Galerkin methods for ideal MHD equations

Ideal magnetohydrodynamic (MHD) equations consist of a set of nonlinear hyperbolic conservation laws, with a divergence-free constraint on the magnetic field. Neglecting this constraint in the design of computational methods may lead to numerical instability or nonphysical features in solutions. In our recent work [F. Li, L. Xu, S. Yakovlev, Central discontinuous Galerkin methods for ideal MHD equations with the exactly divergence-free magnetic field, Journal of Computational Physics 230 (2011) 4828–4847], second and third order exactly divergence-free central discontinuous Galerkin methods were proposed for ideal MHD equations. In this paper, we further develop such methods with higher order accuracy. The novelty here is that the well-established H(div)-conforming finite element spaces are used in the constrained transport type framework, and the magnetic induction equations are extensively explored in order to extract sufficient information to uniquely reconstruct an exactly divergence-free magnetic field. The overall algorithm is local, and it can be of arbitrary order of accuracy. Numerical examples are presented to demonstrate the performance of the proposed methods especially when they are fourth order accurate.

OBSERVATIONS OF POST-FLARE PLASMA DYNAMICS DURING AN M1.0 FLARE IN AR11093 BY THESOLAR DYNAMICS OBSERVATORY/ATMOSPHERIC IMAGING ASSEMBLY

We observe the motion of cool and hot plasma in a multi-stranded post flare loop system that evolved in the decay phase of a two ribbon M1.0 class flare in AR 11093 on 7 August 2010 using SDO/AIA 304 \AA\ and 171 \AA\ filters. The moving intensity feature and its reflected counterpart are observed in the loop system at multi-temperature. The observed hot counterpart of the plasma that probably envelopes the cool confined plasma, moves comparatively faster ($\sim$34 km s$^{-1}$) to the later (29 km s$^{-1}$) in form of the spreaded intensity feature. The propagating plasma and intensity reflect from the region of another footpoint of the loop. The subsonic speed of the moving plasma and associated intensity feature may be most likely evolved in the post flare loop system through impulsive flare heating processes. Complementing our observations of moving multi-temperature intensity features in the post flare loop system and its reflection, we also attempt to solve two-dimensional ideal magnetohydrodynamic equations numerically using the VAL-IIIC atmosphere as an initial condition to simulate the observed plasma dynamics. We consider a localized thermal pulse impulsively generated near one footpoint of the loop system during the flare processes, which is launched along the magnetic field lines at the solar chromosphere. The pulse steepens into a slow shock at higher altitudes while moving along this loop system, which triggers plasma perturbations that closely exhibit the observed plasma dynamics.

Thermal instability in molecular clouds, including dust particles, Hall effect and ambipolar diffusion

In ideal magnetohydrodynamics (MHD), ions and electrons are both tied to the magnetic field. The Hall effect arises in a plasma when electrons are able to drift with the magnetic field but ions cannot. In this case, the generalized Ohm law is modified, and it affects the ideal MHD equations. In this paper, we study thermal instability in molecular clouds, when it is affected by ambipolar and Hall diffusion and charged dust particles. For this purpose, we include their effects in the equations of weakly ionized molecular clouds. By using a linear perturbation method, we investigate the thermal instability of the system. In this way, we obtain a non-dimensional seventh-degree characteristic equation, which, in the absence of a magnetic field, leads to the prior characteristic equations. From a numerical manipulation of the characteristic equation, we conclude that the thermal instability allows compression along the magnetic field when the effects of both ambipolar and Hall diffusion are considered. Moreover, when the effect of Hall diffusion is increased, there is a compression at θ=π/4, where θ is angle between the magnetic field and the wave vector of perturbation. We also derive some new critical wavelengths that separate the spatial ranges of stability and instability.

Slow magnetacoustic waves in magnetic arcades

Aims. We investigate two-dimensional effects on the evolution of impulsively-generated slow magnetoacoustic waves in magnetic arcades of the solar corona. Methods. We used a two-dimensional box model of a coronal arcade, neglecting the effects of gravity and magnetic curvature. Ideal finite-β magnetohydrodynamic equations were employed. The plasma was taken to be uniform, penetrated by a straight and uniform magnetic field. We applied line-tying boundary conditions at the magnetic footpoints. Results. Running and standing slow magnetoacoustic waves develop across the magnetic field due to the reflection from the footpoints in the arcade. The perpendicular group speed is lower than both sound and Alfven speeds. The speed grows with the increase in plasma-β, which is consistent with analytical theory. Slow magnetoacoustic pulse perturbs the magnetic field and current density; this effect is stronger for higher amplitudes. Standing waves spread across the field, forming a characteristic phase-mixing pattern of antiparallel flows. The two-dimensional effects are more pronounced for higher β. Our results confirm the plausibility of the interpretation of the observed evolution of two-ribbon flares in terms of slow magnetoacoustic waves in magnetic arcades.

Numerical simulations of solar macrospicules

Context. We consider a localized pulse in the component of velocity, parallel to the ambient magnetic field lines, that is initially launched in the solar chromosphere. Aims. We aim to generalize the recent numerical model of spicule formation (Murawski & Zaqarashvili 2010) by implementing a VAL-C model of solar temperature. Methods. With the use of the code FLASH we solve two-dimensional ideal magnetohydrodynamic equations numerically to simulate the solar macrospicules. Results. Our numerical results reveal that the pulse located below the transition region triggers plasma perturbations, which exhibit many features of macrospicules. We also present an observational (SDO/AIA 304 A) case study of the macrospicule that approximately mimics the numerical simulations. Conclusions. In the frame of the model we devised, the solar macrospicules can be triggered by velocity pulses launched from the chromosphere.

Convergence of the compressible magnetohydrodynamic equations to incompressible magnetohydrodynamic equations

In this paper, we study the incompressible limit of the three-dimensional compressible magnetohydrodynamic equations, which models the dynamics of compressible quasi-neutrally ionized fluids under the influence of electromagnetic fields. Based on the convergence-stability principle, we show that, when the Mach number, the shear viscosity coefficient, and the magnetic diffusion coefficient are sufficiently small, the initial-value problem of the model has a unique smooth solution in the time interval where the ideal incompressible magnetohydrodynamic equations have a smooth solution. When the latter has a global smooth solution, the maximal existence time for the former tends to infinity as the Mach number, the shear viscosity coefficient, and the magnetic diffusion coefficient go to zero. Moreover, we obtain the convergence of smooth solutions for the model forwards those for the ideal incompressible magnetohydrodynamic equations with a sharp convergence rate.

Observations of a pulse-driven cool polar jet by SDO/AIA

Context. We observe a solar jet at north polar coronal hole (NPCH) using SDO AIA 304 {\deg}A image data on 3 August 2010. The jet rises obliquely above the solar limb and then retraces its propagation path to fall back. Aims. We numerically model this observed solar jet by implementing a realistic (VAL-C) model of solar temperature. Methods. We solve two-dimensional ideal magnetohydrodynamic equations numerically to simulate the observed solar jet. We consider a localized velocity pulse that is essentially parallel to the background magnetic field lines and initially launched at the top of the solar photosphere. The pulse steepens into a shock at higher altitudes, which triggers plasma perturbations that exhibit the observed features of the jet. The typical direction of the pulse also clearly exhibits the leading front of the moving jet. Results. Our numerical simulations reveal that a large amplitude initial velocity pulse launched at the top of the solar photosphere produces in general the observed properties of the jet, e.g., upward and backward average velocities, height, width, life-time, and ballistic nature. Conclusions. The close matching between the jet observations and numerical simulations provides first strong evidence for the formation of this jet by a single velocity pulse. The strong velocity pulse is most likely generated by the low- atmospheric reconnection in the polar region which results in triggering of the jet. The downflowing material of the jet most likely vanishes in the next upcoming velocity pulses from lower solar atmosphere, and therefore distinctly launched a single jet upward in the solar atmosphere is observed.

High-order dimensionally split Lagrange-remap schemes for ideal magnetohydrodynamics

We first propose a new class of high-order finite volume schemes for solving the 1-D ideal magnetohydrodynamics equations that is particularly well-suited for modern computer architectures. Applicable to arbitrary equations of state, these schemes, based on a Lagrange-remap approach, are high-order accurate in both space and time in the non-linear regime. A multidimensional extension on 2-D Cartesian grids using a high-order dimensional splitting technique is then proposed. Numerical results up to fourth-order on smooth and non-smooth test problems are also provided.

Tokamak equilibria with non field-aligned axisymmetric divergence-free rotational flows

Rotational ideal divergence-free magnetohydrodynamic (MHD) equations are expressed in terms of transformed variables w→*=(μρ)1/2v→ and μp* = (μp + w*2/2), where v→, p, and ρ are plasma velocity, pressure, and mass density, respectively. With divergence-free flows, ∇·v→=0, the plasma density ρ does not appear in the MHD equations written in terms of w→* and μp*. The non field-aligned rotational Grad-Shafranov equation is represented in spherical coordinates. Tokamak-like axisymmetric equilibria with v→ ⊥∇ρ are obtained analytically by solving for torus solutions under only three source functions.

Magnetoacoustic shock formation near a magnetic null point

We investigate the interaction of nonlinear fast magnetoacoustic waves with a magnetic null point in connection with the triggering of solar flares. Methods. We model the propagation of fast, initially axisymmetric waves towards a two-dimensional isothermal magnetic null point in terms of ideal magnetohydrodynamic equations. The numerical simulations are carried out with the Lagrangian remap code Lare2D. Results. Dynamics of initially axisymmetric fast pulses of small amplitude is found to be consistent with a linear analytical solution proposed earlier. The increase in the amplitude leads to the nonlinear acceleration of the compression pulse and deceleration of the rarefaction pulse and hence the distortion of the wave front. The pulse experiences nonlinear steepening in the radial direction either on the leading or the back slopes for the compression and rarefaction pulses, respectively. This effect is most pronounced in the directions perpendicular to the field. Hence, the nonlinear evolution of the fast pulse depends on the polar angle. The nonlinear steepening generates the sharp spikes of the electric current density. As in the uniform medium, the position of the shock formation also depends on the initial width of the pulse. Only sufficiently smooth and low-amplitude initial pulses can reach the vicinity of the null point, create there current density spikes, and initiate magnetic reconnection by seeding anomalous electrical resistivity. Steeper and higher amplitude initial pulses overturn at larger distance from the null point, and cannot trigger reconnection.

Efficient magnetohydrodynamic simulations on graphics processing units with CUDA

Magnetohydrodynamic (MHD) simulations based on the ideal MHD equations have become a powerful tool for modeling phenomena in a wide range of applications including laboratory, astrophysical, and space plasmas. In general, high-resolution methods for solving the ideal MHD equations are computationally expensive and Beowulf clusters or even supercomputers are often used to run the codes that implemented these methods. With the advent of the Compute Unified Device Architecture (CUDA), modern graphics processing units (GPUs) provide an alternative approach to parallel computing for scientific simulations. In this paper we present, to the best of the authorʼs knowledge, the first implementation of MHD simulations entirely on GPUs with CUDA, named GPU-MHD, to accelerate the simulation process. GPU-MHD supports both single and double precision computations. A series of numerical tests have been performed to validate the correctness of our code. Accuracy evaluation by comparing single and double precision computation results is also given. Performance measurements of both single and double precision are conducted on both the NVIDIA GeForce GTX 295 (GT200 architecture) and GTX 480 (Fermi architecture) graphics cards. These measurements show that our GPU-based implementation achieves between one and two orders of magnitude of improvement depending on the graphics card used, the problem size, and the precision when comparing to the original serial CPU MHD implementation. In addition, we extend GPU-MHD to support the visualization of the simulation results and thus the whole MHD simulation and visualization process can be performed entirely on GPUs.

Astrophysical weighted particle magnetohydrodynamics

This paper presents applications of weighted meshless scheme for conservation laws to the Euler equations and the equations of ideal magnetohydrodynamics. The divergence constraint of the latter is maintained to the truncation error by a new meshless divergence cleaning procedure. The physics of the interaction between the particles is described by an one-dimensional Riemann problem in a moving frame. As a result, necessary diffusion which is required to treat dissipative processes is added automatically. As a result, our scheme has no free parameters that controls the physics of inter-particle interaction, with the exception of the number of the interacting neighbours which control the resolution and accuracy. The resulting equations have the form similar to SPH equations, and therefore existing SPH codes can be used to implement the weighed particle scheme. The scheme is validated in several hydrodynamic and MHD test cases. In particular, we demonstrate for the first time the ability of a meshless MHD scheme to model magneto-rotational instability in accretion disks.

A finite volume implicit time integration method for solving the equations of ideal magnetohydrodynamics for the hyperbolic divergence cleaning approach

A finite volume numerical technique is proposed to solve the compressible ideal MHD equations for steady and unsteady problems based on a quasi-Newton implicit time integration strategy. The solenoidal constraint is handled by a hyperbolic divergence cleaning approach allowing its satisfaction up to machine accuracy. The conservation of the magnetic flux is computed in a consistent way using the numerical flux of the finite volume discretization. For the unsteady problem, the time accuracy is obtained by a Newton subiteration at each physical timestep thereby converging the solenoidal constraint to steady state. We perform extensive numerical experiments to validate and demonstrate the capabilities of the proposed numerical technique.

Analytic dispersion relation for resistive wall modes in rotating plasmas by generalized matching theory

Analytic dispersion relation is derived for resistive wall modes (RWMs) in rotating plasmas, which gives the growth rate and the real frequency. The given parameters are wall information (radius, thickness and volume resistivity) and equilibrium quantities at a plasma surface and at a singular point of the generalized Newcomb equation, which is an inertia-less linearized ideal magnetohydrodynamic equation with equilibrium rotation. Derivation of the dispersion relation is based on the generalized matching theory proposed by the present authors, which exploits the inner ‘region’ with finite width. It is found that the RWM stability is strongly affected by rotation shear, not at the rational surface, but at the singular point of the generalized Newcomb equation.

Central discontinuous Galerkin methods for ideal MHD equations with the exactly divergence-free magnetic field

In this paper, central discontinuous Galerkin methods are developed for solving ideal magnetohydrodynamic (MHD) equations. The methods are based on the original central discontinuous Galerkin methods designed for hyperbolic conservation laws on overlapping meshes, and use different discretization for magnetic induction equations. The resulting schemes carry many features of standard central discontinuous Galerkin methods such as high order accuracy and being free of exact or approximate Riemann solvers. And more importantly, the numerical magnetic field is exactly divergence-free. Such property, desired in reliable simulations of MHD equations, is achieved by first approximating the normal component of the magnetic field through discretizing induction equations on the mesh skeleton, namely, the element interfaces. And then it is followed by an element-by-element divergence-free reconstruction with the matching accuracy. Numerical examples are presented to demonstrate the high order accuracy and the robustness of the schemes.

The Brunt–Väisälä frequency of rotating tokamak plasmas

The continuous spectrum of analytical toroidally rotating magnetically confined plasma equilibria is investigated analytically and numerically. In the presence of purely toroidal flow, the ideal magnetohydrodynamic equations leave the freedom to specify which thermodynamic quantity is constant on the magnetic surfaces. Introducing a general parametrization of this quantity, analytical equilibrium solutions are derived that still posses this freedom. These equilibria and their spectral properties are shown to be ideally suited for testing numerical equilibrium and stability codes including toroidal rotation. Analytical expressions are derived for the low-frequency continuous Alfvén spectrum. These expressions still allow one to choose which quantity is constant on the magnetic surfaces of the equilibrium, thereby generalizing previous results. The centrifugal convective effect is shown to modify the lowest Alfvén continuum branch to a buoyancy frequency, or Brunt–Väisälä frequency. A comparison with numerical results for the case that the specific entropy, the temperature, or the density is constant on the magnetic surfaces yields excellent agreement, showing the usefulness of the derived expressions for the validation of numerical codes.

Multicomponent theory of buoyancy instabilities in magnetized plasmas: the case of magnetic field parallel to gravity

We investigate electromagnetic buoyancy instabilities of the electron-ion plasma with the heat flux based on not the magnetohydrodynamic (MHD) equations, but using the multicomponent plasma approach when the momentum equations are solved for each species. We consider a geometry in which the background magnetic field, gravity, and stratification are directed along one axis. The nonzero background electron thermal flux is taken into account. Collisions between electrons and ions are included in the momentum equations. No simplifications usual for the one-fluid MHD-approach in studying these instabilities are used. We derive a simple dispersion relation, which shows that the thermal flux perturbation generally stabilizes an instability for the geometry under consideration. This result contradicts to conclusion obtained in the MHD-approach. We show that the reason of this contradiction is the simplified assumptions used in the MHD analysis of buoyancy instabilities and the role of the longitudinal electric field perturbation which is not captured by the ideal MHD equations. Our dispersion relation also shows that the medium with the electron thermal flux can be unstable, if the temperature gradients of ions and electrons have the opposite signs. The results obtained can be applied to the weakly collisional magnetized plasma objects in laboratory and astrophysics.