Ideal MHD Research Articles

A High-Order Residual-Based Viscosity Finite Element Method for the Ideal MHD Equations

We present a high order, robust, and stable shock-capturing technique for finite element approximations of ideal MHD. The method uses continuous Lagrange polynomials in space and explicit Runge-Kutta schemes in time. The shock-capturing term is based on the residual of MHD which tracks the shock and discontinuity positions, and adds sufficient amount of viscosity to stabilize them. The method is tested up to third order polynomial spaces and an expected fourth-order convergence rate is obtained for smooth problems. Several discontinuous benchmarks such as Orszag-Tang, MHD rotor, Brio-Wu problems are solved in one, two, and three spacial dimensions. Sharp shocks and discontinuity resolutions are obtained.

Rigorous results on conserved and dissipated quantities in ideal MHD turbulence

We review recent mathematical results on the theory of ideal MHD turbulence. On the one hand, we explain a mathematical version of Taylor's conjecture on magnetic helicity conservation, both for simply and multiply connected domains. On the other hand, we describe how to prove the existence of weak solutions conserving magnetic helicity but dissipating cross helicity and energy in 3D Ideal MHD. Such solutions are bounded. In fact, we show that as soon as we are below the critical integrability for magnetic helicity conservation, there are weak solutions which do not preserve even magnetic helicity. These mathematical theorems rely on understanding the mathematical relaxation of MHD which is used as a model of the macroscopic behaviour of solutions of various nonlinear partial differential equations. Thus, on the one hand, we present results on the existence of weak solutions consistent with what is expected from experiments and numerical simulations, on the other hand, we show that below certain thresholds, there exist pathological solutions which should be excluded from physical grounds. It is still an outstanding open problem to find suitable admissibility conditions that are flexible enough to allow the existence of weak solutions but rigid enough to rule out physically unrealistic behaviour.

A subgrid turbulent mean field dynamo model for cosmological galaxy formation simulations

Abstract Magnetic fields have been included in cosmological simulations of galaxy formation only recently. In this paper, we develop a new subgrid model for the turbulent dynamo that takes place in the supersonic interstellar medium in star-forming galaxies. It is based on a mean-field approach that computes the turbulent kinetic energy at unresolved scales and modifies the induction equation to account for the corresponding α dynamo. Our subgrid model depends on one free parameter, the quenching parameter, that controls the saturation of the subgrid dynamo. Thanks to this mean-field approach, we can now model the fast amplification of the magnetic field inside turbulent star-forming galaxies without using prohibitively expensive high-resolution simulations. We show that the evolution of the magnetic field in our zoom-in Milky Way-like galaxy is consistent with a simple picture, in which the field is in equipartition with the turbulent kinetic energy inside the star-forming disc, with a field strength around 10 μG at low redshift, while at the same time strong galactic outflows fill the halo with a slightly weaker magnetic field, whose strength (10 nG) is consistent will the ideal MHD dilution factor. Our results are in good agreement with recent theoretical and numerical predictions. We also compare our simulation with Faraday depth observations at both low and high redshift, seeing overall good agreement with some caveats. Our model naturally predicts stronger magnetic fields at high redshift (around 100 μG in the galaxy and 1 μG in the halo), but also stronger depolarisation effects due to stronger turbulence at early time.

Formation and termination of runaway beams during vertical displacement events in tokamak disruptions

A simple 0D model which mimics the plasma surrounded by the conducting structures (Kiramov and Breizman 2017 Phys. Plasmas 24 100702) and including self-consistently the vertical plasma motion and the generation of runaway electrons during the disruption is used for an assessment of the effect of vertical displacement events on the runaway current formation and termination. The total plasma current and runaway current at the time the plasma hits the wall is estimated and the effect of injecting impurities into the plasma is evaluated. In the case of ITER, with a highly conducting wall, although the total plasma current when the plasma touches the wall is the same for any number of injected impurities, however the fraction of the plasma current carried by runaway electrons can significantly decrease for large enough amounts of impurities. The plasma velocity is larger and the time when the plasma hits the wall shorter for lower runaway currents, which are obtained when larger amounts of impurities are injected. When the plasma reaches the wall, the scraping-off of the runaway beam occurs and the current is terminated. During this phase, the plasma vertical displacement velocity and electric field can substantially increase leading to the deposition of a noticeable amount of energy on the runaway electrons (∼hundreds of MJ). It is found that an early second impurity injection reduces somewhat the amount of energy deposited by the runaways. Also larger temperatures of the companion plasma during the scraping-off might be efficient in reducing the power fluxes due to the runaways onto the PFCs. The plasma reaches the q a = 2 limit before the runaway electron current is terminated and by that time the amount of energy deposited on the runaway electrons can be substantially lower than that expected until the beam is fully terminated. Negligible additional conversion of magnetic into runaway kinetic energy is predicted during the runaway deconfinement following the large magnetic fluctuations after q a = 2 is crossed for characteristic deconfinement times lower than 0.1 ms which is a characteristic timescale for ideal MHD instabilities to develop.

Three dimensional ideal MHD equilibria with non-parallel flow and pressure anisotropy

We extend previous work of [9,10] to the case of ideal MHD equations with incompressible flow non-parallel to the magnetic field and pressure anisotropy. Classes of exact three-dimensional solutions are obtained with straight magnetic axes and closed nested pressure-surface intersections with the poloidal plane, for moderate values of the flow velocity parameters whereas these surfaces are destroyed for higher velocities.

Relaxed magnetohydrodynamics with ideal Ohm's law constraint

The gap between a recently developed dynamical version of relaxed magnetohydrodynamics (RxMHD) and ideal MHD (IMHD) is bridged by approximating the zero-resistivity ‘ideal’ Ohm's law (IOL) constraint using an augmented Lagrangian method borrowed from optimization theory. The augmentation combines a pointwise vector Lagrange multiplier method and global penalty function method and can be used either for iterative enforcement of the IOL to arbitrary accuracy, or for constructing a continuous sequence of magnetofluid dynamics models running between RxMHD (no IOL) and weak IMHD (IOL almost everywhere). This is illustrated by deriving dispersion relations for linear waves on an MHD equilibrium.

Optimizations of CFETR steady state H-mode scenario with localized reversed shear enhanced internal transport barrier

Both a fully noninductive steady state operation scenario and a hybrid scenario with fusion power ∼ 1 GW and fusion gain >10 are being considered to fulfill the mission of a Chinese fusion engineering testing reactor. Compared to the hybrid scenario, plasma current is generally lower in steady state operation, so that better confinement and stabilization of MHD instability introduced by higher normalized beta (possibly beyond the ideal MHD limit without a wall) are required to achieve the same fusion performance. Integrated modeling is used to find candidate scenarios to match both these requirements at the same time. By creating a localized strong reversed magnetic shear using radio frequency wave driven current, a strong off-axis internal transport barrier is formed, so that the target fusion power and fusion gain are achieved for Chinese fusion engineering testing reactor steady state operation. Further optimizing the location of the reversed magnetic shear by modifying radio frequency wave launch parameters can keep the normalized beta below the ideal MHD no-wall limit while the fusion power remains beyond 1 GW. Based on this finding, several combinations of heating and current drives are proposed with fusion gain close to 12.5.

Certain clarifications on the resistive wall mode theorem and extensions

In a previous paper by V. D. Pustovitov [Phys. Plasmas 24, 112513 (2017)], it is claimed that the proofs of the “Resistive-Wall-Mode theorem” by Pfirsch and Tasso [Nucl. Fusion 11, 259 (1971)] and extensions of that theorem for time dependent wall resistivity and equilibrium plasma flow are not detailed and that there are limitations restricting their applicability. In response, we provide here pertinent detailed derivations, showing that the proofs of the above-mentioned theorems are rigorous and complete, unlike the considerations of V. D. Pustovitov [Phys. Plasmas 24, 112513 (2017)], which ignore the self-adjointness of the operator ∇×∇× and the fact that the force operator in the linearized ideal MHD momentum equation remains self-adjoint in the presence of equilibrium flows.

Dyadic models for ideal MHD

We study two dyadic models for incompressible ideal magnetohydrodynamics, one with a uni-directional energy cascade and the other one with both forward and backward energy cascades. Global existence of weak solutions and local well-posedness are established for both models. In addition, solutions to the model with uni-directional energy cascade associated with positive initial data are shown to develop blow-up at a finite time. Moreover, a set of fixed points is found for each model. Linear instability about some particular fixed points is proved.

Partial Differential Equations with Quadratic Nonlinearities Viewed as Matrix-Valued Optimal Ballistic Transport Problems

We study a rather general class of optimal “ballistic” transport problems for matrix-valued measures. These problems naturally arise, in the spirit of Brenier (Commun Math Phys 364(2):579–605, 2018), from a certain dual formulation of nonlinear evolutionary equations with a particular quadratic structure reminiscent both of the incompressible Euler equation and of the quadratic Hamilton–Jacobi equation. The examples include the ideal incompressible MHD, the template matching equation, the multidimensional Camassa–Holm (also known as the $$H({{\,\mathrm{div}\,}})$$ geodesic equation), EPDiff, Euler- $$\alpha $$ , KdV and Zakharov–Kuznetsov equations, the equations of motion for the incompressible isotropic elastic fluid and for the damping-free Maxwell’s fluid. We prove the existence of the solutions to the optimal “ballistic” transport problems. For formally conservative problems, such as the above mentioned examples, a solution to the dual problem determines a “time-noisy” version of the solution to the original problem, and the latter one may be retrieved by time-averaging. This yields the existence of a new type of absolutely continuous in time generalized solutions to the initial-value problems for the above mentioned PDE. We also establish a sharp upper bound on the optimal value of the dual problem, and explore the weak–strong uniqueness issue.

Non-Linear Damping of Surface Alfvén Waves Due to Uniturbulence

This investigation is concerned with uniturbulence associated with surface Alfvén waves that exist in a Cartesian equilibrium model with a constant magnetic field and a piece-wise constant density. The surface where the equilibrium density changes in a discontinuous manner are the source of surface Alfvén waves. These surface Alfvén waves create uniturbulence because of the variation of the density across the background magnetic field. The damping of the surface Alfvén waves due to uniturbulence is determined using the Elsässer formulation. Analytical expressions for the wave energy density, the energy cascade, and the damping time are derived. The study of uniturbulence due to surface Alfvén waves is inspired by the observation that (the fundamental radial mode of) kink waves behave similarly to surface Alfvén waves. The results for this relatively simple case of surface Alfvén waves can help us understand the more complicated case of kink waves in cylinders. We perform a series of 3D ideal MHD simulations for a numerical demonstration of the non-linearly self-cascading model of unidirectional surface Alfvén waves using the code MPI-AMRVAC. We show that surface Alfvén waves damping time in the numerical simulations follows well our analytical prediction for that quantity. Analytical theory and the simulations show that the damping time is inversely proportional to the amplitude of the surface Alfvén waves and the density contrast. This unidirectional cascade may play a role in heating the coronal plasma.

LIE SYMMETRIES, GROUP INVARIANT SOLUTIONS AND CONSERVATION LAWS OF IDEAL MHD EQUATIONS

In this paper, the method of Lie symmetry is used to study the incompressible ideal MHD equations. We derive the infinitesimal generators of MHD by using Lie symmetry method. Then, we get the group invariant solutions to MHD based on the group transformation and known solutions. An one-dimensional optimal system which only depends on the commutator of Lie algebras is constructed. Through the similarity transformation of the optimal system, we obtain the 1 + 1-dimensional reduced partial differential equations, and further obtain the exact solutions of these equations, such as Bell soliton solution, junction soliton solution and seed solution. In addition, we prove the nonlinear self-adjoint and conservation laws of the MHD equations using the Ibragimov method. Finally, the classical symmetry and solutions to incompressible ideal MHD equations with initial conditions left invariant are discussed.

Implementation of the Hll-Grp Solver for Multidimensional Ideal Mhd Simulations Based on Finite Volume Method

Implementation of the Hll-Grp Solver for Multidimensional Ideal Mhd Simulations Based on Finite Volume Method

COCONUT MHD coronal model as a basis for EUHFORIA 2.0 space weather forecast

We have developed COCONUT -- a novel MHD solar corona model as a basis for EUHFORIA 2.0 space weather forecasting. The steady-state solution in a numerical domain extending from the solar surface up to 0.1 AU is computed. Here the pre-processed magnetic maps prescribed at the inner boundary determine the magnetic topology. These magnetic maps consist of radial components of Helioseismic and Magnetic Imager magnetograms projected on spherical harmonics with a selected maximum frequency. Within the numerical domain a set of ideal MHD equations with gravity is solved implicitly on an unstructured mesh. This choice of grid allows us to omit the issue of singularities presented at the poles in typically used structured grids. Through heavy parallelization, an optimized CFL profile and other numerical acceleration techniques, we achieved superior convergence times with high robustness and reliability even for numerically challenging cases of maximum solar activity. The MHD solution is generally obtained within a few hours of computation, which is crucial for space weather forecasting systems. Our model passed a series of tests and was further validated using detailed comparison with observations. It successfully recreated magnetic structures (streamers, pseudo streamers, coronal holes) observed during the solar eclipses in March 2015 and in July 2019, thus corresponding to maximum and minimum of solar activity. The COCONUT code will be next extended to include the heating and radiation terms for a more realistic physical description. In the future, a multispecies formulation will be implemented allowing us to consider also the impact of electrons as well as of neutrals on the solution. Figs 3, Refs 10.

Pellet triggering of edge localized modes in low collisionality pedestals at DIII-D

Edge localized modes (ELMs) are triggered using deuterium pellets injected into plasmas with ITER-relevant low collisionality pedestals, and the resulting peak ELM energy fluence is reduced by approximately 25%–50% relative to natural ELMs destabilized at similar pedestal pressures. Cryogenically frozen deuterium pellets are injected from the low-field side of the DIII-D tokamak at frequencies lower than the natural ELM frequency, and heat flux is measured by infrared cameras. Ideal MHD pedestal stability calculations show that without pellet injection, these low collisionality pedestals were limited by their current density (peeling-limited) rather than their pressure gradient (ballooning-limited). ELM triggering success correlates strongly with pellet mass, consistent with the theory that a large pressure perturbation is required to trigger an ELM in low collisionality discharges that are far from the ballooning stability boundary. For sufficiently large pellets, both instantaneous and time-integrated ELM energy deposition measured by infrared cameras is reduced with respect to naturally occurring ELMs at the inner strike point, which is the position where it is largest for natural ELMs. Energy fluence at the outer strike point is less effected. Cameras observing both heat flux and D-alpha emission often find significant toroidally asymmetric striations in the outboard far scrape-off layer resulting from ELMs that are triggered by pellets. Toroidal asymmetries at the inner strike point are similar between natural and pellet-triggered ELMs, suggesting that the reduction in peak heat flux and total fluence at that location is robust for the conditions reported here.

Heavy impurity transport in tokamaks subject to plasma rotation, NTV and the influence of saturated ideal MHD perturbations

Strongly peaked tungsten accumulation is a common feature of high performance plasma scenarios in JET with the ITER-like wall, particularly during MHD activity induced by continuous modes. This study investigates the effect of long living internal kink modes on heavy impurity transport in the presence of strong flows and NTV ambipolar electric field. A novel formulation which includes these effects is presented and applied in the VENUS-LEVIS code in order to follow tungsten ions in a saturated JET-like internal kinked toroidally rotating plasma configuration. The synergy between 3D magnetic fields, strong flows and NTV is seen to cause tungsten accumulation in contrast to what is observed in similar axisymmetric configurations. Rapid inward transport of impurities in JET plasmas following the triggering of continuous modes is explained by the work presented here, and we use the same theory to postulate why outward transport can occur in kinked ASDEX-U plasmas.

Overdense Threads in the Solar Corona Induced by Torsional Alfvén Waves

Abstract High-resolution and high-cadence observations have shown that Alfvén waves are ubiquitous in the solar atmosphere. Theoretical works suggest their ability to transfer large energy fluxes from the photosphere to the corona and solar wind. In this proof-of-concept Letter we show that torsional Alfvén waves can induce the formation of filamentary plasma structures in the solar corona. We perform high-resolution 3D ideal MHD simulations in an initially uniform coronal plasma permeated by a line-tied twisted magnetic field. We find that torsional Alfvén waves develop Kelvin–Helmholtz instabilities as a result of the phase mixing process. The Kelvin–Helmholtz instability drives plasma compression that breaks the uniformity of density, creating elongated overdense threads aligned with the direction of the magnetic field. With synthetic modeling of SDO/AIA imaging we show that the overdense filaments could be seen in observations as fine strands that illuminate the underlying magnetic structure.

2D continuous Chebyshev-Galerkin time-spectral method

A fully spectral multi-domain method has been developed and applied to three applications within ideal MHD, compressible Navier-Stokes, and a two-fluid plasma turbulence model named the Weiland model. The time-spectral method employed is the Generalized Weighted Residual Method (GWRM), where all domains such as space, time, and parameter space are spectrally decomposed with Chebyshev polynomials. The spectral decomposition of the temporal domain allows the GWRM to reach spectral accuracy in all dimensions. The GWRM linear/nonlinear algebraic equations are solved using an Anderson Acceleration (AA) method and a newly developed Quasi Semi-Implicit root solver (Q-SIR). Up to 85% improved convergence rate was obtained for Q-SIR as compared to AA and in certain cases only Q-SIR converged. In the most challenging simulations, featuring steep gradients, the GWRM converged for time intervals roughly two times larger than typical time steps for explicit time-marching schemes, being limited by the CFL condition. Time intervals up to 70 times larger than those of explicit time-marching schemes were used in smooth regions. Furthermore, the most computationally expensive algorithm, namely the product of two Chebyshev series, has been GPU accelerated with speedup gains of several thousands compared to a CPU.

A 3D magnetohydrodynamic simulation of the propagation of a plasma plume transverse to applied magnetic field

We have carried out a 3D ideal-MHD (magnetohydrodynamic) simulation to study the evolution of a laser-generated plasma plume in a moderate external magnetic field (0.13 T) oriented perpendicular to the flow direction of the plasma plume. The simulation shows that the plasma plume pushes the external magnetic field lines outward in the direction of the expansion. This leads to the compression and bending of the magnetic field lines. The force resulting from the change in shape and the density of the magnetic field lines opposes the expansion of the plume. An elliptic layer of shocked plasma is formed at the plasma/external field interface leaving a cavity in the plume core due to the outward expansion and the inertia of the plume. As the plasma pressure drops due to expansion, the imbalance between the magnetic energy and the internal energy results in the collapse of the cavity. These observations have striking similarities with the observations of the experiments (Behera et al 2017 Phys. Plasmas 24 033511) performed recently to study the plasma plume expansion in the presence of an external transverse magnetic field. This similarity indicates that the physical mechanisms dominantly governing the plasma plume expansion in the moderate magnetic field are aptly described in the ideal MHD regime. The studies thus show that the laser-generated plasma plume can be utilized to carry out interesting experiments on MHD phenomena in a simple laboratory setup.

A new locally divergence-free WLS-ENO scheme based on the positivity-preserving finite volume method for ideal MHD equations

In this paper, the WLS-ENO (Weighted-Least-Squares based Essentially Non-Oscillatory) reconstruction is modified to maintain the conservation of the cell average values. Furthermore, the divergence-free constraint is combined with the conservative WLS-ENO reconstruction, which can make the magnetic field locally divergence-free. The main merit of the proposed reconstruction scheme is that it can keep both the divergence-free constraint and ENO property for the magnetic field without using any limiter. We apply the scheme to the simulations of ideal MHD equations within the framework of a positivity-preserving finite volume method. The convergence, the magnetic field divergence error and the capability for low plasma-beta of the scheme are tested by some MHD benchmark problems.