Ideal MHD Research Articles

Finite-beta turbulence in Wendelstein 7-X enhanced by sub-threshold kinetic ballooning modes

Abstract Magnetic fluctuations affecting turbulence and transport, which are manifest at finite normalized plasma pressure beta, pose a significant challenge to magnetic confinement fusion devices aiming to achieve high performance. Such regimes are not yet comprehensively understood in stellarator geometry. This work presents simulations of electromagnetic instabilities and high-beta turbulence in the Wendelstein 7-X (W7-X) stellarator, showing how ion-temperature-gradient-driven (ITG) turbulence is enhanced by unconventional kinetic ballooning modes well below the ideal MHD threshold. These sub-threshold KBMs (stKBMs) become strongly excited in the turbulent state and enable higher fluxes via zonal-flow erosion. The threshold of stKBM impact on turbulent fluxes is heavily dependent on the pressure gradient, evidenced here by the enhanced destabilization and fluxes resulting from the inclusion of an electron temperature gradient. Understanding and controlling these stKBMs will be paramount for W7-X and potentially other stellarators to achieve optimal performance.

Global existence of bounded smooth solutions for the compressible ideal MHD system with planar symmetry

Global existence of bounded smooth solutions for the compressible ideal MHD system with planar symmetry

Global well-posedness of the free-surface incompressible ideal MHD equations with velocity damping

Global well-posedness of the free-surface incompressible ideal MHD equations with velocity damping

A multi-dimensional, robust, and cell-centered finite-volume scheme for the ideal MHD equations

We present a new multi-dimensional, robust, and cell-centered finite-volume scheme for the ideal MHD equations. This scheme relies on relaxation and splitting techniques and can be easily used at high order. A fully conservative version is not entropy satisfying but is observed experimentally to be more robust than standard constrained transport schemes at low plasma beta. At very low plasma beta and high Alfvén number, we have designed an entropy-satisfying version that is not conservative for the magnetic field but preserves admissible states and we switch locally a-priori between the two versions depending on the regime of plasma beta and Alfvén number. This strategy is robust in a wide range of standard MHD test cases, all performed at second order with a classic MUSCL-Hancock scheme.

Toroidal Alfvén eigenmodes excited by energetic electrons in EAST low-density Ohmic plasmas

Abstract Operation in the quiescent regime with abundant trapped energetic electrons (EEs) has been achieved during the current flattop in EAST low-density Ohmic plasmas. This was facilitated by increasing the electron density to a specified level and subsequently reducing it slowly, resulting in the accumulation of a sufficient number of trapped EEs within the energy range of 150–250 keV. During the phase of decreasing electron density, toroidal Alfvén eigenmodes (TAEs) were observed to be excited by these EEs, with frequencies falling within the range of about 100–300 kHz. The experimental parameters were carefully set to satisfy the resonance conditions for TAE excitation by EEs, aligning well with predictions from ideal MHD theory. Statistical analysis indicated different density dependencies between the frequencies of TAEs and the Alfvén frequencies, due to their different radial excitation positions. The radial positions of the TAEs were found to be influenced by the energy distribution and the evolution of trapped EEs, which in turn were affected by the decay rate of electron density and loop voltage. Measurements of Hard X-rays (HXR) confirmed an energy distribution characterized by a “bump-on-tail” shape, with the TAEs observed near the energy bump. Theoretical considerations also demonstrate the possibility that the e-TAE can be driven unstable under this experiment condition even if the mode does not rotate in the electron-diamagnetic drift direction.

The Centrifugal Acceleration and the Y-point of the Pulsar Magnetosphere

We investigate the centrifugal acceleration in an axisymmetric pulsar magnetosphere under the ideal MHD approximation. We solve the field-aligned equations of motion for flows inside the current sheet with finite thickness. We find that flows coming into the vicinity of a Y-point become super fast. The centrifugal acceleration takes place efficiently, and most of the Poynting energy is converted into kinetic energy. However, the super-fast flow does not provide enough centrifugal drift current to open the magnetic field. Opening of the magnetic field is possible by the plasmas that are accelerated in the azimuthal direction with a large Lorentz factor in the closed-field region. We find that this acceleration takes place if the field strength increases toward the Y-point from inside. The accelerated plasma is transferred from the closed-field region to the open-field region by magnetic reconnection with plasmoid emission. We also estimate the Lorentz factor to be reached in the centrifugal wind.

Stabilization of magneto-Rayleigh-Taylor instability with non-uniform density and magnetic field profiles in Cartesian Geometry

Managing the Rayleigh–Taylor instability growth rate is one of the main challenges in inertial fusion. Among the proposed methods to tackle this issue, the application of a pre-embedded axial magnetic field and plasma density gradient of layers can be mentioned. Recent experimental evidence in Z-pinch devices shows that it is possible to suppress the growth rate of the Rayleigh-Taylor instability by applying a power law density gradient profile with exponent 2. In this work, in the framework of Cartesian coordinates, for a confined plasma between two fixed planes, the dispersion relation of a magnetized plasma has been obtained using linearization of the ideal MHD equations. Here, plasma with a power-law density profile of ∝r3 and magnetic field with the exponential profile is used. It is shown that for the relative strength of the magnetic field with a value of, ωf*=0.4 and more, the growth rate of instability appears only in a limited range of the range of reduced wave numbers, k*. By increasing this ratio, stability conditions are much improved. On the other hand, recent experimental studies show that self-generated plasma rotation is produced in an imploding Z-pinch device with an initial axial magnetic field. In the second part, the effect of plasma rotation on plasma stability conditions is investigated. Again, a new dispersion relation of the rotating magnetized plasma was derived. It is shown that in the non-magnetized case, a large relative angular velocity, Ω, is necessary to suppress the instability. However, for relative magnetic field strength, ωf*=0.4 and more, stable conditions can be achieved with a relatively lower angular velocity of about 5.

Calculation of toroidal Alfvén eigenmode mode structure in general axisymmetric toroidal geometry

A workflow is developed based on the ideal MHD model to investigate the linear physics of various Alfvén eigenmodes in general axisymmetric toroidal geometry by solving the coupled shear Alfvén wave (SAW) and ion sound wave (ISW) equations in ballooning space. The model equations are solved by the FALCON code in the singular layer, and the corresponding solutions are then taken as the boundary conditions for calculating parallel mode structures in the whole ballooning space. As an application of the code, the frequencies and mode structures of toroidal Alfvén eigenmode (TAE) are calculated in the reference equilibria of the Divertor Tokamak Test facility with positive and negative triangularities, respectively. As typical result for reactor relevant plasma conditions, which are strongly triangular in the outer core region where magnetic shear is of order unity, we show that the triangularity effect on TAE is generally small. Furthermore, by properly handling the boundary conditions, we demonstrate finite TAE damping due to coupling with the local acoustic continuum and find that the damping rate is small for typical plasma parameters.

Variable-spectrum mode control of high poloidal beta discharges

DIII-D experiments demonstrate that high pressure, broad current profile equilibria can be accessed in the high poloidal beta regime by optimizing the MHD mode control poloidal spectrum. A novel, variable spectrum (VS) magnetic feedback scheme implemented using the DIII-D internal non-axisymmetric coils (I-coils) facilitated access to reduced internal inductance ℓi operation above the no-wall beta limit compared with both no feedback and fixed spectrum feedback. In addition, the VS feedback helped avoid beta collapses caused by marginally unstable resistive wall mode activity. The lower and upper I-coil rows were configured in two independent feedback loops, allowing the feedback field’s poloidal spectrum to vary and track changes in the plasma mode structure as the edge safety factor q 95 varied from 11 to 6 during the discharges. The q 95 dependence of the measured phase difference between the lower and upper I-coil rows during VS feedback is qualitatively compatible with ideal MHD simulations of the least-stable plasma kink mode and with plasma response simulations that included kinetic modifications to ideal MHD. The VS feedback approach is a straightforward way to improve resilience to variations in mode structure that occur as plasma parameters change. The demonstrated expansion of the operating space to lower ℓi is expected to improve the coupling of the plasma kink mode to external fields and beneficial wall eddy currents, and is compatible with high bootstrap fraction operation.

FLIPEC, an ideal MHD free-boundary axisymmetric equilibrium solver in the presence of macroscopic flows

The most relevant features of FLIPEC (Free fLow Iterative Plasma Equilibrium Code) are presented. This new code iteratively calculates free-boundary, axisymmetric ideal MHD equilibria with arbitrary poloidal and toroidal plasma flows. FLIPEC is a mature code that has emerged from a complete overhaul of a previous version (F-Torija Daza 2022 et al Nucl. Fusion 62 126044). It uses a (inverse) curvilinear coordinate representation for the Grad–Shafranov–Bernoulli equation system, which allows FLIPEC to extend its free-boundary capabilities to arbitrary plasma shapes and removes many limitations with regards to the distance between plasma and external coils. Run-time stabilization of vertical modes has also been implemented by means of artificial feedback coils. Finally, active targeting schemes have also been included. These capabilities are illustrated on two very different cases: the ITER tokamak baseline configuration and a NSTX spherical tokamak equilibrium.

Modelling and experiment to stabilize disruptive tearing modes in the ITER baseline scenario in DIII-D

The achievement of high gain, stationary conditions in a tokamak scenario aimed at producing fusion energy in the ITER Project is crucial to the demonstration that this form of energy can be used in future reactors to provide cheap and clean energy globally. Disruptions are a challenge for the fusion energy field, in particular for the ‘ITER Baseline Scenario’ (IBS), as reproduced in the DIII-D tokamak. This work shows that a solution has been found for the m= 2/n = 1 tearing modes that have consistently caused disruptions in the IBS: stable operation down to zero input torque was achieved by modifying the current density profile at the beginning of the pressure flattop and the ELM character later in the discharges, guided by previous results showing that the most likely cause of these instabilities is the current density profile. The coupling between sawteeth, n>2 modes and the 2/1 TMs is shown to not be statistically significant, nor the leading origin for the evolution towards instability. Ideal and resistive MHD modeling provide positive verification that a steeper ‘well’ in the region of the q = 2 rational surface leads to worse ideal stability, higher tearing index Δ’ and lower threshold Δ’c for resistive instabilities, consistent with the experimental results. This provides confidence that the methods used in this work can be extrapolated to other devices and applied to avoid disruptions in ITER and pulsed fusion devices worldwide.

High-order genuinely multidimensional finite volume methods via kernel-based WENO

In this paper a family of fully multidimensional kernel-based reconstruction schemes for use in finite volume methods (FVMs) will be presented. These methods are intended for use in shock dominated problems, and stability is achieved through a suitable adaptation of the Adaptive Order Weighted Essentially Non-Oscillatory (WENO-AO) method to the proposed kernel-based reconstruction schemes. There are a number of key difficulties in the design of high-order finite volume schemes which will be discussed and addressed. High (4th and 6th) order convergence will be demonstrated on smooth exact solutions of the ideal MHD equations. The very same scheme will then be applied to extremely stringent astrophysical benchmark problems.

Gas-kinetic scheme for partially ionized plasma in hydrodynamic regime

Most plasmas are only partially ionized. To better understand the dynamics of these plasmas, the behaviors of a mixture of neutral species and plasma in ideal magnetohydrodynamic states are investigated. The current approach is about the construction of coupled kinetic models for the neutral gas, electron, and proton, and the development of the corresponding gas-kinetic scheme (GKS) for the solution in the continuum flow regime. The scheme is validated in the 1D Riemann problem for an enlarged system with the interaction from the Euler waves of the neutral gas and magnetohydrodynamic ones of the plasma. Additionally, the Orszag–Tang vortex problem across different ionized states is tested to examine the influence of neutrals on the MHD wave evolution. These tests demonstrate that the proposed scheme can capture the fundamental features of ideal partially ionized plasma, and a transition in the wave structure from the ideal MHD solution of the fully ionized plasma to the Euler solution of the neutral gas is obtained.

A New S-M Limiter Entropy Stable Scheme Based on Moving Mesh Method for Ideal MHD and SWMHD Equations

A New S-M Limiter Entropy Stable Scheme Based on Moving Mesh Method for Ideal MHD and SWMHD Equations

Refined conserved quantities criteria for the ideal MHD equations in a bounded domain

In this paper, we present two sufficient conditions keeping the total energy and cross-helicity conservation in spaces B _ 3 , V M O α ( Ω ) \underline {B}^{\alpha }_{3,VMO}(\Omega ) and B 3 , ∞ β ( Ω ) B^{\beta }_{3,\infty }(\Omega ) for the ideal magnetohydrodynamic equations in a bounded domain. This generalizes the sufficient criteria for conserved quantities recently obtained by Bardos, Gwiazda, Świerczewska Gwiazda, Titi and Wiedemann [Proc. A. 475 (2019), 18 pp.] and by Wang and Zuo [J. Differential Equations 268 (2020), pp. 4079–4101].

PROTO-SPHERA: a magnetic confinement experiment which emulates the jet + torus astrophysical plasmas

The PROTO-SPHERA experiment, built at the CR-ENEA laboratory in Frascati, was in part inspired by the jet + torus astrophysical plasmas, a rather common morphology in Astrophysics. This paper illustrates how the said plasma morphology can be reproduced in a laboratory with the setup of the PROTO-SPHERA experiment. The experiment as such displayed the appearance and sustainment of a plasma torus around an internal magnetized plasma centerpost (jet) by self-organisation; an entirely unexplored phenomenon to date. The remarkable ideal MHD stability of the PROTO-SPHERA plasma is extremely significant, as it is obtained in a simply connected geometry, inside a perfectly insulating vacuum vessel, and without the need of a nearby stabilizing conducting shell. The concluding sections of this paper deal with application of force-free fields to the Pulsar Wind Nebulae morphology and present an extension of the well-known split-dipole model. Such an extension provides a natural description of the presence of tori around the Pulsar plasma jets. In addition, similarities and differences between the laboratory and the astrophysical jet + torus plasmas are detailed.

A New Discretely Divergence-Free Positivity-Preserving High-Order Finite Volume Method for Ideal MHD Equations

A New Discretely Divergence-Free Positivity-Preserving High-Order Finite Volume Method for Ideal MHD Equations

Scenario optimization for the tokamak ramp-down phase in RAPTOR: Part B. safe termination of DEMO plasmas

An optimized plasma current ramp-down strategy is critical for safe and fast termination of plasma discharges in a tokamak demonstration fusion reactor (DEMO), both in planned and emergency scenarios, avoiding plasma disruptions and excessive heat loads to the first wall. Plasma stability limits and machine-specific technical requirements constrain the stable envelope through which the plasma must be navigated. Large amounts of auxiliary heating are required throughout the ramp-down phase, to avoid a radiative collapse in the presence of intrinsic tungsten and seeded xenon impurities, as quantitatively estimated in this work. As the plasma current is reduced, the current density becomes increasingly peaked, reflected by a growing value of the internal inductance ℓi3 , resulting in reduced controllability of the vertical position of the plasma. The feasibility of different plasma current ramp-down rates is tested by applying an automated optimization framework embedding the RAPTOR core transport solver. Optimal time traces for plasma current Ip(t) and plasma elongation κ(t) are proposed, to satisfy an Ip -dependent upper limit on the plasma internal inductance, as obtained from vertical stability studies using the CREATE-NL code, as well as a constraint on the time evolution of q 95, to avoid an ideal MHD mode. A negative current density near the plasma edge is observed in our simulations, even for the most conservative Ip ramp-down rate, indicating significant transient dynamics due to a large resistive time.

Optimal decay rate and convergence of the incompressible viscous resistive MHD system

In this work, we investigate the optimal Lp(p≥2) time decay rate of global solutions to the incompressible viscous resistive MHD system with degeneration, and study its convergence to the incompressible viscous ideal MHD system as the resistivity ε goes to zero. The analysis is based on the high-low frequency decomposition and the uniform energy estimate of the system with respect to ε.

Bounded solutions in incompressible hydrodynamics

In this article, we study bounded solutions of Euler-type equations on Rd which have no integrability at |x|→+∞. As has been previously noted, such solutions fail to achieve uniqueness in an initial value problem, even under strong smoothness conditions. This contrasts with well-posedness results that have been obtained by using the Leray projection operator in these equations. This apparent paradox is solved by noting that using the Leray projector requires an extra condition the solutions must fulfill at |x|→+∞. Our goal is to find one such condition which is sharp. We then apply the methods we develop to prove a full uniqueness result for Besov-Lipschitz solutions, as to the theory of Serfati solutions. In the last Section, we see how these techniques also apply to the Elsässer variables used in ideal MHD.