Ideal Magnetohydrodynamic Model Research Articles

Upward and Downward Catastrophes of Coronal Magnetic Flux Ropes in Quadrupolar Magnetic Fields

Abstract Coronal magnetic flux ropes are closely related to large-scale solar activities. Using a 2.5-dimensional time-dependent ideal magnetohydrodynamic model in Cartesian coordinates, we carry out numerical simulations to investigate the evolution of a magnetic system consisting of a flux rope embedded in a fully closed quadrupolar magnetic field with different photospheric flux distributions. It is found that when the photospheric flux is not concentrated too much toward the polarity inversion line and the constraint exerted by the background field is not too weak, the equilibrium states of the system are divided into two branches: the rope sticks to the photosphere for the lower branch and levitates in the corona for the upper branch. These two branches are connected by an upward catastrophe (from the lower branch to the upper) and a downward catastrophe (from the upper branch to the lower). Our simulations reveal that there exist both upward and downward catastrophes in quadrupolar fields, which may be either force-free or non-force-free. The existence and the properties of these two catastrophes are influenced by the photospheric flux distribution, and a downward catastrophe is always paired with an upward catastrophe. Comparing the decay indices in catastrophic and noncatastrophic cases, we infer that torus unstable may be a necessary but not sufficient condition for a catastrophic system.

Turbulent Magnetic Relaxation in Pulsar Wind Nebulae

Abstract We present a model for magnetic energy dissipation in a pulsar wind nebula. A better understanding of this process is required to assess the likelihood that certain astrophysical transients may be powered by the spin-down of a “millisecond magnetar.” Examples include superluminous supernovae, gamma-ray bursts, and anticipated electromagnetic counterparts to gravitational wave detections of binary neutron star coalescence. Our model leverages recent progress in the theory of turbulent magnetic relaxation to specify a dissipative closure of the stationary magnetohydrodynamic (MHD) wind equations, yielding predictions of the magnetic energy dissipation rate throughout the nebula. Synchrotron losses are self-consistently treated. To demonstrate the model’s efficacy, we show that it can reproduce many features of the Crab Nebula, including its expansion speed, radiative efficiency, peak photon energy, and mean magnetic field strength. Unlike ideal MHD models of the Crab (which lead to the so-called σ-problem), our model accounts for the transition from ultra to weakly magnetized plasma flow and for the associated heating of relativistic electrons. We discuss how the predicted heating rates may be utilized to improve upon models of particle transport and acceleration in pulsar wind nebulae. We also discuss implications for the Crab Nebula’s γ-ray flares, and point out potential modifications to models of astrophysical transients invoking the spin-down of a millisecond magnetar.

Modeling of 3D magnetic equilibrium effects on edge turbulence stability during RMP ELM suppression in tokamaks

Recent experimental observations have found turbulent fluctuation structures that are non-axisymmetric in a tokamak with applied 3D fields. In this paper, two fluid resistive effects are shown to produce changes relevant to turbulent transport in the modeled 3D magnetohydrodynamic (MHD) equilibrium of tokamak pedestals with these 3D fields applied. Ideal MHD models are insufficient to reproduce the relevant effects. By calculating the ideal 3D equilibrium using the VMEC code, the geometric shaping parameters that determine linear turbulence stability, including the normal curvature and local magnetic shear, are shown to be only weakly modified by applied 3D fields in the DIII-D tokamak. These ideal MHD effects are therefore not sufficient to explain the observed changes to fluctuations and transport. Using the M3D-C1 code to model the 3D equilibrium, density is shown to be redistributed on flux surfaces in the pedestal when resistive two fluid effects are included, while islands are screened by rotation in this region. The redistribution of density results in density and pressure gradient scale lengths that vary within pedestal flux surfaces between different helically localized flux tubes. This would produce different drive terms for trapped electron mode and kinetic ballooning mode turbulence, the latter of which is expected to be the limiting factor for pedestal pressure gradients in DIII-D.

Stabilizing effect of resistivity towards ELM-free H-mode discharge in lithium-conditioned NSTX

Linear stability analysis of the national spherical torus experiment (NSTX) Li-conditioned ELM-free H-mode equilibria is carried out in the context of the extended magneto-hydrodynamic (MHD) model in NIMROD. The purpose is to investigate the physical cause behind edge localized mode (ELM) suppression in experiment after the Li-coating of the divertor and the first wall of the NSTX tokamak. Besides ideal MHD modeling, including finite-Larmor radius effect and two-fluid Hall and electron diamagnetic drift contributions, a non-ideal resistivity model is employed, taking into account the increase of Zeff after Li-conditioning in ELM-free H-mode. Unlike an earlier conclusion from an eigenvalue code analysis of these equilibria, NIMROD results find that after reduced recycling from divertor plates, profile modification is necessary but insufficient to explain the mechanism behind complete ELMs suppression in ideal two-fluid MHD. After considering the higher plasma resistivity due to higher Zeff, the complete stabilization could be explained. A thorough analysis of both pre-lithium ELMy and with-lithium ELM-free cases using ideal and non-ideal MHD models is presented, after accurately including a vacuum-like cold halo region in NIMROD to investigate ELMs.

MHD simulation of the distribution of the gasdynamic parameters and magnetic field behind the Earth’s bow shock under sharp variations in the solar wind dynamic pressure

The distributions of the gasdynamic parameters (density, pressure, and velocity) and the magnetic field behind the Earth’s bow shock (on the outer boundary of the magnetosheath) generated under sharp variations in the solar wind dynamic pressure are found in the three-dimensional non-planepolarized formulation with allowance for the interplanetary magnetic field within the framework of the ideal magnetohydrodynamic model using the solution to the MHD Riemann problem of breakdown of an arbitrary discontinuity. Such a discontinuity which depends on the inclination of an element of the bow shock surface arises when a contact discontinuity traveling together with the solar wind and on which the solar wind density and, consequently, the dynamic pressure, increases or decreases suddenly impinges on the Earth’s bow shock and propagates along its surface initiating the development of to six waves or discontinuities (shocks). The general interaction pattern is constructed for the entire bow shock surface as a mosaic of exact solutions to the MHD Riemann problem obtained on computer using an original software (MHD Riemann solver) so that the flow pattern is a function of the angular surface coordinates (latitude and longitude). The calculations are carried out for various jumps in density on the contact discontinuity and characteristics parameters of the solar wind and interplanetary magnetic field at the Earth’s orbit. It is found that there exist horseshoe zones on the bow shock in which the increase in the density and the magnetic field strength in the fast shock waves or their reduced decrease in the fast rarefaction waves penetrating into the magnetosheath and arising as a result of sharp variation in the solar wind dynamic pressure is superposed on significant drop in the density and growth in the magnetic field strength in slow rarefaction waves. The distributions of the hydrodynamic parameters and the magnetic field can be used to interpret measurements carried out on spacecraft in the solar wind at the libration point and orbiters in the neighborhood of the Earth’s magnetosphere.

Influence of Photospheric Magnetic Conditions on the Catastrophic Behaviors of Flux Ropes in Solar Active Regions

Since only the magnetic conditions at the photosphere can be routinely observed in current observations, it is of great significance to find out the influences of photospheric magnetic conditions on solar eruptive activities. Previous studies about catastrophe indicated that the magnetic system consisting of a flux rope in a partially open bipolar field is subject to catastrophe, but not if the bipolar field is completely closed under the same specified photospheric conditions. In order to investigate the influence of the photospheric magnetic conditions on the catastrophic behavior of this system, we expand upon the 2.5 dimensional ideal magnetohydrodynamic (MHD) model in Cartesian coordinates to simulate the evolution of the equilibrium states of the system under different photospheric flux distributions. Our simulation results reveal that a catastrophe occurs only when the photospheric flux is not concentrated too much toward the polarity inversion line and the source regions of the bipolar field are not too weak; otherwise no catastrophe occurs. As a result, under certain photospheric conditions, a catastrophe could take place in a completely closed configuration whereas it ceases to exist in a partially open configuration. This indicates that whether the background field is completely closed or partially open is not the only necessary condition for the existence of catastrophe, and that the photospheric conditions also play a crucial role in the catastrophic behavior of the flux rope system.

DOWNWARD CATASTROPHE OF SOLAR MAGNETIC FLUX ROPES

ABSTRACT 2.5-dimensional time-dependent ideal magnetohydrodynamic (MHD) models in Cartesian coordinates were used in previous studies to seek MHD equilibria involving a magnetic flux rope embedded in a bipolar, partially open background field. As demonstrated by these studies, the equilibrium solutions of the system are separated into two branches: the flux rope sticks to the photosphere for solutions at the lower branch but is suspended in the corona for those at the upper branch. Moreover, a solution originally at the lower branch jumps to the upper, as the related control parameter increases and reaches a critical value, and the associated jump is here referred to as an upward catastrophe. The present paper advances these studies in three aspects. First, the magnetic field is changed to be force-free; the system still experiences an upward catastrophe with an increase in each control parameter. Second, under the force-free approximation, there also exists a downward catastrophe, characterized by the jump of a solution from the upper branch to the lower. Both catastrophes are irreversible processes connecting the two branches of equilibrium solutions so as to form a cycle. Finally, the magnetic energy in the numerical domain is calculated. It is found that there exists a magnetic energy release for both catastrophes. The Ampère's force, which vanishes everywhere for force-free fields, appears only during the catastrophes and does positive work, which serves as a major mechanism for the energy release. The implications of the downward catastrophe and its relevance to solar activities are briefly discussed.

Non-axisymmetric ideal equilibrium and stability of ITER plasmas with rotating RMPs

The magnetic perturbations produced by the resonant magnetic perturbation (RMP) coils will be rotated in ITER so that the spiral patterns due to strike point splitting which are locked to the RMP also rotate. This is to ensure even power deposition on the divertor plates. VMEC equilibria are calculated for different phases of the RMP rotation. It is demonstrated that the off harmonics rotate in the opposite direction to the main harmonic. This is an important topic for future research to control and optimize ITER appropriately. High confinement mode (H-mode) is favourable for the economics of a potential fusion power plant and its use is planned in ITER. However, the high pressure gradient at the edge of the plasma can trigger periodic eruptions called edge localized modes (ELMs). ELMs have the potential to shorten the life of the divertor in ITER (Loarte et al 2003 Plasma Phys. Control. Fusion 45 1549) and so methods for mitigating or suppressing ELMs in ITER will be important. Non-axisymmetric RMP coils will be installed in ITER for ELM control. Sampling theory is used to show that there will be significant a harmonic sideband. There are nine coils toroidally in ITER so . This results in a significant n = 6 component to the applied field and a significant n = 5 component to the applied field. Although the vacuum field has similar amplitudes of these harmonics the plasma response to the various harmonics dictates the final equilibrium. Magnetic perturbations with toroidal mode number n = 3 and n = 4 are applied to a 15 MA, burning ITER plasma. We use a three-dimensional ideal magnetohydrodynamic model (VMEC) to calculate ITER equilibria with applied RMPs and to determine growth rates of infinite n ballooning modes (COBRA). The case shows little change in ballooning mode growth rate as the RMP is rotated, however there is a change with rotation for the case.

Identification of multi-modal plasma responses to applied magnetic perturbations using the plasma reluctance

Using the plasma reluctance, the Ideal Perturbed Equilibrium Code is able to efficiently identify the structure of multi-modal magnetic plasma response measurements and the corresponding impact on plasma performance in the DIII-D tokamak. Recent experiments demonstrated that multiple kink modes of comparable amplitudes can be driven by applied nonaxisymmetric fields with toroidal mode number n = 2. This multi-modal response is in good agreement with ideal magnetohydrodynamic models, but detailed decompositions presented here show that the mode structures are not fully described by either the least stable modes or the resonant plasma response. This work identifies the measured response fields as the first eigenmodes of the plasma reluctance, enabling clear diagnosis of the plasma modes and their impact on performance from external sensors. The reluctance shows, for example, how very stable modes compose a significant portion of the multi-modal plasma response field and that these stable modes drive significant resonant current. This work is an overview of the first experimental applications using the reluctance to interpret the measured response and relate it to multifaceted physics, aimed towards providing the foundation of understanding needed to optimize nonaxisymmetric fields for independent control of stability and transport.

Impact of the interplanetary magnetic field on thewave flow pattern in the neighborhood of the Earth’s bow shock under sharp variations in the solar wind dynamic pressure

The impact of the interplanetary magnetic field on transformation and disintegration of the Earth’s bow shock into a system of magnetohydrodynamic (MHD) shock waves, rotational discontinuities and rarefaction waves under the action of abrupt variations in the solar wind dynamic pressure is simulated in the three-dimensional non-plane-polarized formulation within the framework of the ideal magnetohydrodynamic model using the solution of the MHD Riemann problem of breakdown of an arbitrary discontinuity. This discontinuity arises when a contact discontinuity, on which the solar wind density increases or decreases suddenly and which travels together with the solar wind, impinges on the Earth’s bow shock and propagates along its surface. The interaction pattern is constructed in the quasisteady- state formulation as a mosaic of exact solutions obtained on computer using an original MHD Riemann solver. The wave flow patterns are found for all elements of the surface of the bow shock as functions of their latitude and longitude for various jumps in the density on the contact discontinuity and characteristics parameters of the solar wind and interplanetary magnetic field at the Earth’s orbit. It is found that when the solar wind dynamic pressure increases, a fast MHD shock wave, that first penetrates into the magnetosheath, is always formed. When the solar wind dynamic pressure decreases, the influence of the interplanetary magnetic field can lead to the development of the leading fast MHD shock wave in certain zones on the surface of the Earth’s bow shock. The solution obtained can be used to interpret measurements on spacecraft in the solar wind at the libration point and in the neighborhood of the Earth’s magnetosphere.

Nonlinear Alfvén waves in extended magnetohydrodynamics

Large-amplitude Alfvén waves are observed in various systems in space and laboratories, demonstrating an interesting property that the wave shapes are stable even in the nonlinear regime. The ideal magnetohydrodynamics (MHD) model predicts that an Alfvén wave keeps an arbitrary shape constant when it propagates on a homogeneous ambient magnetic field. However, such arbitrariness is an artifact of the idealized model that omits the dispersive effects. Only special wave forms, consisting of two component sinusoidal functions, can maintain the shape; we derive fully nonlinear Alfvén waves by an extended MHD model that includes both the Hall and electron inertia effects. Interestingly, these “small-scale effects” change the picture completely; the large-scale component of the wave cannot be independent of the small scale component, and the coexistence of them forbids the large scale component to have a free wave form. This is a manifestation of the nonlinearity-dispersion interplay, which is somewhat different from that of solitons.

Thermal loads of plasma focus chambers

The passage of current through the electrodes of a plasma focus chamber is analyzed taking into account the skin effect; the thermal load on the elliptical shaped electrodes is studied in the interaction with the current-plasma sheath, calculated within the two-dimensional ideal magnetohydrodynamic model. The heating of the anode of the plasma focus chamber by the discharge current is assessed under the limit modes of operation. Signs of the thermal action of plasma particles on the electrode surface are demonstrated.

Three-dimensional drift kinetic response of high-β plasmas in the DIII-D tokamak.

A quantitative interpretation of the experimentally measured high-pressure plasma response to externally applied three-dimensional (3D) magnetic field perturbations, across the no-wall Troyon β limit, is achieved. The self-consistent inclusion of the drift kinetic effects in magnetohydrodynamic (MHD) modeling [Y. Q. Liu etal., Phys. Plasmas 15, 112503 (2008)] successfully resolves an outstanding issue of the ideal MHD model, which significantly overpredicts the plasma-induced field amplification near the no-wall limit, as compared to experiments. The model leads to quantitative agreement not only for the measured field amplitude and toroidal phase but also for the measured internal 3D displacement of the plasma. The results can be important to the prediction of the reliable plasma behavior in advanced fusion devices, such as ITER [K. Ikeda, Nucl. Fusion 47, S1 (2007)].

Collision of an interplanetary rotational discontinuity with the Earth’s bow shock. Hydrodynamic parameters and magnetic field

Hydrodynamic parameters and magnetic fields generated in each of the waves in the neighborhood of the Earth’s bow shock are found within the framework of the ideal magnetohydrodynamic (MHD) model in the three-dimensional non-plane-polarized formulation when the front of an interplanetary rotational discontinuity, which travels from the Sun and rotates the interplanetary magnetic field by a given angle, impinges on the Earth’s bow shock and propagates along its surface. The interaction pattern is constructed in the quasi-steady-state formulation as a mosaic of exact solutions, obtained by means of a computer using an original MHD 3-D Riemann solver, to the Riemann problem of breakdown of a discontinuity between the states downstream of the rotational discontinuity and the bow shock on the traveling line of intersection of their fronts. The calculations are carried out for typical parameters of the quiescent solar wind and the interplanetary magnetic field at the Earth’s orbit. The solutions obtained can be used to interpret measurements carried out by spacecraft in the solar wind and in neighborhood of the Earth’s magnetosphere.

Temporal evolution of surface ripples on a finite plasma slab subject to the magneto-Rayleigh-Taylor instability

Using the ideal magnetohydrodynamic model, we calculate the temporal evolution of initial ripples on the boundaries of a planar plasma slab that is subjected to the magneto-Rayleigh-Taylor instability. The plasma slab consists of three regions. We assume that in each region the plasma density is constant with an arbitrary value and the magnetic field is also constant with an arbitrary magnitude and an arbitrary direction parallel to the interfaces. Thus, the instability may be driven by a combination of magnetic pressure and kinetic pressure. The general dispersion relation is derived, together with the feedthrough factor between the two interfaces. The temporal evolution is constructed from the superposition of the eigenmodes. Previously established results are recovered in the various limits. Numerical examples are given on the temporal evolution of ripples on the interfaces of the finite plasma slab.

Magnetic field reversals and long-time memory in conducting flows.

Employing a simple ideal magnetohydrodynamic model in spherical geometry, we show that the presence of either rotation or finite magnetic helicity is sufficient to induce dynamical reversals of the magnetic dipole moment. The statistical character of the model is similar to that of terrestrial magnetic field reversals, with the similarity being stronger when rotation is present. The connection between long-time correlations, 1/f noise, and statistics of reversals is supported, consistent with earlier suggestions.

FORMATION OF MAGNETIZED PRESTELLAR CORES WITH AMBIPOLAR DIFFUSION AND TURBULENCE

We investigate the roles of magnetic fields and ambipolar diffusion during prestellar core formation in turbulent giant molecular clouds (GMCs), using three-dimensional numerical simulations. Our simulations focus on the shocked layer produced by a converging flow within a GMC, and survey varying ionization and angle between the upstream flow and magnetic field. We also include ideal magnetohydrodynamic (MHD) and hydrodynamic models. From our simulations, we identify hundreds of self-gravitating cores that form within 1 Myr, with masses M ~ 0.04 - 2.5 solar-mass and sizes L ~ 0.015 - 0.07 pc, consistent with observations of the peak of the core mass function (CMF). Median values are M = 0.47 solar-mass and L = 0.03 pc. Core masses and sizes do not depend on either the ionization or upstream magnetic field direction. In contrast, the mass-to-magnetic flux ratio does increase with lower ionization, from twice to four times the critical value. The higher mass-to-flux ratio for low ionization is the result of enhanced transient ambipolar diffusion when the shocked layer first forms. However, ambipolar diffusion is not necessary to form low-mass supercritical cores. For ideal MHD, we find similar masses to other cases. These masses are 1 - 2 orders of magnitude lower than the value that defines a magnetically supercritical sphere under post-shock ambient conditions. This discrepancy is the result of anisotropic contraction along field lines, which is clearly evident in both ideal MHD and diffusive simulations. We interpret our numerical findings using a simple scaling argument which suggests that gravitationally critical core masses will depend on the sound speed and mean turbulent pressure in a cloud, regardless of magnetic effects.

Collision of an interplanetary shock wave with the Earth’s bow shock. Hydrodynamic parameters and magnetic field

Hydrodynamic parameters and magnetic field generated in each of the waves in neighborhood of the Earth’s bow shock when an interplanetary shock wave impinges on it and propagates along its surface are found in the three-dimensional non-plane-polarized formulation within the framework of the ideal magnetohydrodynamic model. The interaction pattern is constructed in the quasi-steady-state formulation as a mosaic of exact solutions, obtained by means of a computer, to the Riemann problem of breakdown of a discontinuity between the states downstream of the impinging wave and the bow shock on the traveling line of intersection of their fronts. The calculations are carried out for typical parameters of the quiescent solar wind and the interplanetary magnetic field in the Earth’s orbit when the plane front of a shock wave moves along the Sun-Earth radius with various given velocities. The solutions obtained can be used to interpret measurements carried out by spacecraft in the solar wind and in neighborhood of the Earth’s magnetosphere.

Global sound modes in mirror traps with anisotropic pressure

Global oscillations of inhomogeneous plasma with frequencies close to the bounce frequency of ions in mirror traps have been studied. It has been shown that, in some cases, the sound can be reflected from the axial plasma inhomogeneity. The ideal magnetohydrodynamic (MHD) model with Chew-Goldberger-Low approximation has been utilized to determine conditions of existence of the standing waves in the mirror-confined plasma. Linearized wave equation for the longitudinal plasma oscillations in thin anisotropic inhomogeneous plasma with finite β has been derived. The wave equation has been treated numerically. The oscillations are studied for the case of the trap with partially filled loss-cone and the trap with sloshing ions. It has been shown that in cells of the multiple-mirror trap standing waves can exist. The frequency of the wave is of the order of the mean bounce-frequency of ions. In the trap with sloshing ions, the mode supported by the pressure of fast ions could exist. The results of oscillations observation in the experiment on the Gas Dynamic Trap have been presented.

Effects of Hall current and electron temperature anisotropy on proton fire-hose instabilities

The standard magnetohydrodynamic (MHD) theory predicts that the Alfvén wave may become fire-hose unstable for β∥−β⊥>2. In this study, we examine the proton fire-hose instability (FHI) based on the gyrotropic two-fluid model, which incorporates the ion inertial effects arising from the Hall current and electron temperature anisotropy but neglects the electron inertia in the generalized Ohm's law. The linear dispersion relation is derived and analyzed which in the long wavelength approximation, λik→0 or αe=μ0(p∥,e−p⊥,e)/B2=1, recovers the ideal MHD model with separate temperature for ions and electrons. Here, λi is the ion inertial length and k is the wave number. For parallel propagation, both ion cyclotron and whistler waves become propagating and growing for β∥−β⊥>2+λi2k2(αe−1)2/2. For oblique propagation, the necessary condition for FHI remains to be β∥−β⊥>2 and there exist one or two unstable fire-hose modes, which can be propagating and growing or purely growing. For large λik values, there exists no nearly parallel FHI leaving only oblique FHI and the effect of αe>1 may greatly enhance the growth rate of parallel and oblique FHI. The magnetic field polarization of FHI may be reversed due to the sign change associated with (αe−1) and the purely growing FHI may possess linear polarization while the propagating and growing FHI may possess right-handed or left-handed polarization.