MHD Limit Research Articles

The impact of energetic particles and rotation on tokamak plasmas

We discuss two contributions that elucidate the impact of energetic particles and rotation on tokamak plasmas: FLOW-M (M. J. Hole and G. Dennis, Plasma Phys. Control. Fusion 51, 035014, 2009), a generalisation of the ideal MHD flow code FLOW to multiple quasi-neutral fluids, and recent work on steady poloidal and toroidal bulk flows in tokamak plasmas [K. G. McClements and M.J. Hole, Phys. Plasmas 17, 082509 (2010)]. Hole and Dennis have generalized ideal MHD to consider multiple quasi-neutral fluids, each in thermal equilibrium and each thermally insulated from each other such that no population mixing occurs. Kinetically, such a model may be able to approximate the ion or electron distribution function in regions of velocity phase space with a large number of particles, at the expense of more weakly populated phase space, which may have uncharacteristically high temperature and hence pressure. As magnetic equilibrium effects increase with the increase in pressure, this work constitutes an upper limit to the effect of energetic particles. McClements and Hole have examined the effects of poloidal and toroidal flows on tokamak plasma equilibria in the MHD limit. Transonic poloidal flows, of the order of the sound speed multiplied by the ratio of poloidal magnetic field to total field Bθ/B, can cause the (normally elliptic) Grad-Shafranov (G-S) equation to become hyperbolic in part of the solution domain. The discontinuity in variables produced by this transition indicates a breakdown in the validity of the MHD model in tokamak plasmas. It is pointed out that the range of poloidal flows for which the G-S equation is hyperbolic increases with plasma beta and Bθ/B, thereby complicating the problem of determining spherical tokamak plasma equilibria with transonic poloidal flows. When the assumption of isentropic flux surfaces is replaced with the more tokamak-relevant one of isothermal flux surfaces, a simple expression can be obtained for the variation of density on a flux surface when poloidal and toroidal flows are simultaneously present. Combined with Thomson scattering measurements of density and temperature, this expression could be used to infer information on poloidal and toroidal flows on the high field side of a tokamak plasma, where direct measurements of flows are not generally possible.

DISK FORMATION ENABLED BY ENHANCED RESISTIVITY

Disk formation in magnetized cloud cores is hindered by magnetic braking. Previous work has shown that for realistic levels of core magnetization, the magnetic field suppresses the formation of rotationally supported disks during the protostellar mass accretion phase of low-mass star formation both in the ideal MHD limit and in the presence of ambipolar diffusion for typical rates of cosmic ray ionization. Additional effects, such as ohmic dissipation, the Hall effect, and protostellar outflow, are needed to weaken the magnetic braking and enable the formation of persistent, rotationally supported, protostellar disks. In this paper, we first demonstrate that the classic microscopic resistivity is not large enough to enable disk formation by itself. We then experiment with a set of enhanced values for the resistivity in the range $\eta=10^{17}$--$10^{22}$ cm^2/s. We find that a value of order $10^{19}$ cm^2/s is needed to enable the formation of a 100 AU-scale Keplerian disk; the value depends somewhat on the degree of core magnetization. The required resistivity is a few orders of magnitude larger than the classic microscopic values. Whether it can be achieved naturally during protostellar collapse remains to be determined.

Protostellar collapse: radiative and magnetic feedbacks on small-scale fragmentation

It is established that both radiative transfer and magnetic field have a strong impact on the collapse and the fragmentation of prestellar dense cores, but no consistent calculation exists yet at such scales. We present original AMR calculations including magnetic field (in the ideal MHD limit) and radiative transfer, within the Flux Limited Diffusion approximation, of the collapse of a 1 solar mass dense core. We compare the results with calculations performed with a barotropic EOS. We show that radiative transfer has an important impact on the collapse and the fragmentation, through the cooling or heating of the gas, and is complementary of the magnetic field. A larger field yields a stronger magnetic braking, increasing the accretion rate and thus the effect of the radiative feedback. Even for a strongly magnetized core, where the dynamics of the collapse is dominated by the magnetic field, radiative transfer is crucial to determine the temperature and optical depth distributions, two potentially accessible observational diagnostics. A barotropic EOS cannot account for realistic fragmentation. The diffusivity of the numerical scheme, however, is found to strongly affect the output of the collapse, leading eventually to spurious fragmentation. Both radiative transfer and magnetic field must be included in numerical calculations of star formation to obtain realistic collapse configurations and observable signatures. Nevertheless, the numerical resolution and the robustness of the solver are of prime importance to obtain reliable results. When using an accurate solver, the fragmentation is found to always remain inhibited by the magnetic field, at least in the ideal MHD limit, even when radiative transfer is included.

Overview of JT-60U results towards the establishment of advanced tokamak operation

Recent JT-60U experimental results towards the establishment of advanced tokamak (AT) operation are reviewed. We focused on the further expansion of the operational regime of AT plasmas towards higher βN regime with wall stabilization. After the installation of ferritic steel tiles in 2005, the high power heating in a large plasma cross-section in which the wall stabilization is expected has been possible. In 2007, the modification of power supply of NBIs improved the flexibility of the heating profile in long-pulse plasmas. The investigation of key physics issues for the establishment of steady-state AT operation is also in progress using new diagnostics and improved heating systems. In weak magnetic shear plasma, high βN ∼ 3 exceeding the ideal MHD limit without a conducting wall () is sustained for ∼5 s (∼3τR) with RWM stabilization by a toroidal rotation at the q = 2 surface. External current drivers of negative-ion based NB and lower-hybrid waves together with a large bootstrap current fraction (fBS) of 0.5 can sustain the whole plasma current of 0.8 MA for 2 s (1.5τR). In reversed magnetic shear plasma, high βN ∼ 2.7 (βp ∼ 2.3) exceeding with qmin ∼ 2.4 (q95 ∼ 5.3), HH98(y,2) ∼ 1.7 and fBS ∼ 0.9 is obtained with wall stabilization. These plasma parameters almost satisfy the requirement of ITER steady-state scenario. In long-pulse plasmas with positive magnetic shear, a high βNHH98(y,2) of 2.6 with βN ∼ 2.6 and HH98(y,2) ∼ 1 is sustained for 25 s, significantly longer than the current diffusion time (∼14τR) without neoclassical tearing modes (NTMs). A high G-factor, (a major of fusion gain), of 0.54 and a large fBS > 0.43 are suitable for ITER hybrid operation scenario. Based on the plasma for ITER hybrid operation scenario, the high βN of 2.1 with good thermal plasma confinement of HH98(y,2) > 0.85 is sustained for longer than 12 s at and frad > 0.79. Physics studies for the development of AT plasmas, physics studies of H-mode, pedestal and ELM characteristics and physics studies on impurity transport, SOL/divertor plasmas and plasma–wall interactions are also in progress. The active NTM stabilization system using modulated ECCD, which is synchronized to rotating island, has been developed and the efficiency of modulated ECCD in m/n = 2/1 NTM stabilization has been demonstrated. The intrinsic toroidal rotation driven by the ion pressure gradient and by the ECH is confirmed. The dedicated H-mode and pedestal experiments indicate two scalings, H-factor evaluated for the core plasma as and pedestal width scaling of . New fast diagnostics with high spatial and temporal resolutions reveals the different structures of pedestal pressure between co- and counter-rotating plasma, resulting in different ELM sizes determined by the radial penetration depth of the ELM crash. The tungsten accumulation becomes more significant with increasing toroidal rotation in the counter-direction.

Very high order PNPM schemes on unstructured meshes for the resistive relativistic MHD equations

In this paper we propose the first better than second order accurate method in space and time for the numerical solution of the resistive relativistic magnetohydrodynamics (RRMHD) equations on unstructured meshes in multiple space dimensions. The nonlinear system under consideration is purely hyperbolic and contains a source term, the one for the evolution of the electric field, that becomes stiff for low values of the resistivity. For the spatial discretization we propose to use high order P N P M schemes as introduced in Dumbser et al. [M. Dumbser, D. Balsara, E.F. Toro, C.D. Munz, A unified framework for the construction of one-step finite volume and discontinuous Galerkin schemes, Journal of Computational Physics 227 (2008) 8209–8253] for hyperbolic conservation laws and a high order accurate unsplit time-discretization is achieved using the element-local space–time discontinuous Galerkin approach proposed in Dumbser et al. [M. Dumbser, C. Enaux, E.F. Toro, Finite volume schemes of very high order of accuracy for stiff hyperbolic balance laws, Journal of Computational Physics 227 (2008) 3971–4001] for one-dimensional balance laws with stiff source terms. The divergence-free character of the magnetic field is accounted for through the divergence cleaning procedure of Dedner et al. [A. Dedner, F. Kemm, D. Kröner, C.-D. Munz, T. Schnitzer, M. Wesenberg, Hyperbolic divergence cleaning for the MHD equations, Journal of Computational Physics 175 (2002) 645–673]. To validate our high order method we first solve some numerical test cases for which exact analytical reference solutions are known and we also show numerical convergence studies in the stiff limit of the RRMHD equations using P N P M schemes from third to fifth order of accuracy in space and time. We also present some applications with shock waves such as a classical shock tube problem with different values for the conductivity as well as a relativistic MHD rotor problem and the relativistic equivalent of the Orszag–Tang vortex problem. We have verified that the proposed method can handle equally well the resistive regime and the stiff limit of ideal relativistic MHD. For these reasons it provides a powerful tool for relativistic astrophysical simulations involving the appearance of magnetic reconnection.

MAGNETIC BRAKING AND PROTOSTELLAR DISK FORMATION: AMBIPOLAR DIFFUSION

It is established that the formation of rotationally supported disks during the main accretion phase of star formation is suppressed by a moderately strong magnetic field in the ideal MHD limit. Nonideal MHD effects are expected to weaken the magnetic braking, perhaps allowing the disk to reappear. We concentrate on one such effect, ambipolar diffusion, which enables the field lines to slip relative to the bulk neutral matter. We find that the slippage does not sufficiently weaken the braking to allow rotationally supported disks to form for realistic levels of cloud magnetization and cosmic ray ionization rate; in some cases, the magnetic braking is even enhanced. Only in dense cores with both exceptionally weak fields and unreasonably low ionization rate do such disks start to form in our simulations. We conclude that additional processes, such as Ohmic dissipation or Hall effect, are needed to enable disk formation. Alternatively, the disk may form at late times when the massive envelope that anchors the magnetic brake is dissipated, perhaps by a protostellar wind.

Nonlinear magnetohydrodynamics in axisymmetric heterogeneous domains using a Fourier/finite element technique and an interior penalty method

The Maxwell equations in the MHD limit in heterogeneous axisymmetric domains composed of conducting and non-conducting regions are solved by using a mixed Fourier/Lagrange finite element technique. Finite elements are used in the meridian plane and Fourier modes are used in the azimuthal direction. Parallelization is made with respect to the Fourier modes. Continuity conditions across interfaces are enforced using an interior penalty technique. The performance of the method is illustrated on kinematic and full dynamo configurations.

Global plasma oscillations in electron internal transport barriers in TCV

In the Tokamak à Configuration Variable (TCV) (Hofmann F et al1994 Plasma Phys. Control. Fusion 36 B277), global plasma oscillations have been discovered in fully non-inductively driven plasmas featuring electron internal transport barriers (ITB) with strong ECRH/ECCD. These oscillations are linked to the destabilization and stabilization of MHD modes near the foot of the ITB and can lead to large oscillations of the total plasma current and line-averaged density, among others. They are intrinsically related to the fact that ITBs have large pressure gradients in a region of low magnetic shear. Therefore, the ideal MHD limit is relatively low and infernal modes can be unstable. Depending on the proximity to the ideal limit, small crashes or resistive modes can appear which affect the time evolution of the discharge. Being near marginal stability, the modes can self-stabilize due to the modification of the pressure gradient and local q-profile. The plasma recovers good confinement, reverses shear and the ITB builds up, until a new MHD mode is destabilized. TCV results show that this cycling behaviour can be controlled by modifying the current density or the pressure profiles, either with Ohmic current density perturbation or by modifying the ECH/ECCD power. It is demonstrated that many observations such as q ⩾ 2 sawteeth, beta collapses, minor disruptions and oscillation regimes in ITBs can be assigned to the same physics origin: the proximity to the infernal mode stability limit.

Time-dependent Force-free Pulsar Magnetospheres: Axisymmetric and Oblique Rotators

Magnetospheres of many astrophysical objects can be accurately described by the low-inertia (or "force-free") limit of MHD. We present a new numerical method for solution of equations of force-free relativistic MHD based on the finite-difference time-domain (FDTD) approach with a prescription for handling spontaneous formation of current sheets. We use this method to study the time-dependent evolution of pulsar magnetospheres in both aligned and oblique magnetic geometries. For the aligned rotator we confirm the general properties of the time-independent solution of Contopoulos et al. (1999). For the oblique rotator we present the 3D structure of the magnetosphere and compute, for the first time, the spindown power of pulsars as a function of inclination of the magnetic axis. We find the pulsar spindown luminosity to be L = (mu^2 Omega^4/c^3) (1+ sin^2(alpha)) for a star with the dipole moment "mu", rotation frequency "Omega", and magnetic inclination angle "alpha". We also discuss the effects of current sheet resistivity and reconnection on the structure and evolution of the magnetosphere.

A mechanism for parallel electric field generation in the MHD limit: possible implications for the coronal heating problem in the two stage mechanism

We solve numerically ideal, 2.5D, MHD equations in Cartesian coordinates, with a plasma beta of 0.0001 starting from the equilibrium that mimics a footpoint of a large curvature radius solar coronal loop or a polar region plume. On top of such an equilibrium, a purely Alfv\'enic, linearly polarised, plane wave is launched. In the context of the coronal heating problem a new two stage mechanism of plasma heating is presented by putting emphasis, first, on the generation of parallel electric fields within an ideal MHD description directly, rather than focusing on the enhanced dissipation mechanisms of the Alfv\'en waves and, second, dissipation of these parallel electric fields via {\it kinetic} effects. It is shown that a single Alfv\'en wave harmonic with frequency $\nu = 7$ Hz and longitudinal wavelength $\lambda_A = 0.63$ Mm, for a putative Alfv\'en speed of 4328 km s$^{-1}$, the generated parallel electric field could account for 10% of the necessary coronal heating requirement. We conjecture that wide spectrum (10$^{-4}-10^3$ Hz) Alfv\'en waves, based on the observationally constrained spectrum, could provide the necessary coronal heating requirement. The exact amount of energy that could be deposited by such waves through our mechanism of parallel electric field generation can only be calculated once a more complete parametric study is done. Thus, the "theoretical spectrum" of the energy stored in parallel electric fields versus frequency needs to be obtained.

Wave turbulence in incompressible Hall magnetohydrodynamics

We investigate the steepening of the magnetic fluctuation power law spectra observed in the inner solar wind for frequencies higher than 0.5 Hz. This high frequency part of the spectrum may be attributed to dispersive nonlinear processes. In that context, the long-time behavior of weakly interacting waves is examined in the framework of 3D incompressible Hall MHD turbulence. The Hall term added to the standard MHD equations makes the Alfv\'en waves dispersive and circularly polarized. We introduce the generalized Els\"asser variables and, using a complex helicity decomposition, we derive for three-wave interaction processes the general wave kinetic equations; they describe the nonlinear dynamics of Alfv\'en, whistler and ion cyclotron wave turbulence in the presence of a strong uniform magnetic field $B_0 \ep$. Hall MHD turbulence is characterized by anisotropies of different strength. We show that electron and standard MHD turbulence can be seen as two frequency limits of the present theory but the standard MHD limit is singular; additionally, we analyze in detail the ion MHD turbulence limit. Exact power law solutions of the master wave kinetic equations are given in the small and large scale limits for which we have, respectively, the total energy spectra $E(\kpn,\kpa) \sim \kpn^{-5/2} |\kpa|^{-1/2}$ and $E(\kpn,\kpa) \sim \kpn^{-2}$. An anisotropic phenomenology is developed to describe continuously the different scaling laws of the energy spectrum; one predicts $E(\kpn,\kpa) \sim \kpn^{-2} |\kpa|^{-1/2} (1+\kpn^2d_i^2)^{-1/4}$. Nonlocal interactions between Alfv\'en, whistler and ion cyclotron waves are investigated; a non trivial dynamics exists only when a discrepancy from the equipartition between the large scale kinetic and magnetic energies happens.

The role of ion cyclotron motion at Ganymede: Magnetic field morphology and magnetospheric dynamics

Ion cyclotron motion can play a role in shaping magnetospheres and governing magnetospheric dynamics, particularly in weakly magnetized systems such as the moons of outer planets. However, MHD explicitly neglects such effects. We demonstrate the importance of ion cyclotron motion in the near space environment of Ganymede using 3‐dimensional multi‐fluid simulations to account for Galileo magnetometer data from several flybys through various regions of Ganymede's magnetosphere. These simulations track several ion species and incorporate ion cyclotron effects through a comprehensive treatment of the plasma dynamics and the generalized Ohm's law equation. In the ideal MHD limit of the multi‐fluid formulation the size of Ganymede's magnetosphere is underestimated, while in the full multi‐fluid treatment the magnetosphere is shown to be in good agreement with magnetometer measurements from Galileo. The importance of treating the ion cyclotron motion is most noticeable near the magnetopause where magnetic field strengths approach zero.

-ray burst internal shocks with magnetization

(Please note that the abstract has been significantly shorted) We investigate Gamma-ray Burst (GRB) internal shocks with moderate magnetization, with the magnetization parameter $\sigma$ ranging from 0.001 to 10. Possible magnetic dissipation in the stripped magnetized shells is also taken into account through introducing a parameter $k$ ($0<k\leq 1$), which is the ratio of the electric field strength of the downstream and the upstream. By solving the general MHD jump conditions, we show that the dynamic evolution of the shock with magnetic dissipation is different from the familiar one obtained in the ideal MHD limit. In view of the possible high degree of linear polarization of GRB 021206 and the identification of a possible highly magnetized flow in GRB 990123 and GRB 021211, we suggest that a mildly magnetized internal shock model ($0.01<\sigma<1$) with moderate magnetic dissipation is a good candidate to explain the GRB prompt emission data.

MHD activity in Tore Supra gigajoule scenario

The Tore Supra gigajoule (GJ) scenario was developed during the 2002 experimental campaign, with the goal of realizing long pulses in a stationary state, at a loop voltage therefore close to zero. This challenge has led to very successful results, with a record injected energy of 0.75 GJ and perfect control of the plasma density over 4 min 25 s. The performance is however limited by the lower hybrid system reliability and MHD limits. This paper focuses on the three kinds of MHD activity that were observed in the development of this scenario. The first one, probably a manifestation of the feedback control on the plasma position, is benign and, once triggered, stays at a low level. The second one is the triggering of a fast growing instability, which leads to a strong degradation of the confinement in the core region and to a redistribution of the plasma current. After such events, a regime with a persistent MHD activity sets in and represents the third kind of MHD activity in the GJ experiments. This paper reports on the analysis of this MHD activity. Perspectives for the future development of the GJ scenario in Tore Supra are drawn.

Electron Inertial Effects on Rapid Energy Redistribution at Magnetic X‐Points

The evolution of nonpotential perturbations to a current-free magnetic X-point configuration is studied, taking into account electron inertial effects as well as resistivity. Electron inertia is shown to have a negligible effect on the evolution of the system whenever the collisionless skin depth is less than the resistive scale length. Nonpotential magnetic field energy in this resistive MHD limit initially reaches equipartition with flow energy, in accordance with ideal MHD, and is then dissipated extremely rapidly on an Alfvenic timescale that is essentially independent of Lundquist number. In agreement with resistive MHD results obtained by previous authors, the magnetic field energy and kinetic energy are then observed to decay on a longer timescale and exhibit oscillatory behavior, reflecting the existence of discrete normal modes with finite real frequency. When the collisionless skin depth exceeds the resistive scale length, the system again evolves initially according to ideal MHD. At the end of this ideal phase, the field energy decays typically on an Alfvenic timescale, while the kinetic energy (which is equally partitioned between ions and electrons in this case) is dissipated on the electron collision timescale. The oscillatory decay in the energy observed in the resistive case is absent, but short-wavelength structures appear in the field and velocity profiles, suggesting the possibility of particle acceleration in oppositely directed current channels. The model provides a possible framework for interpreting observations of energy release and particle acceleration on timescales down to less than a second in the impulsive phase of solar flares.

Ideal kink instability of a magnetic loop equilibrium

The force-free coronal loop model by [CITE] is found to be unstable with respect to the ideal kink mode, which suggests this instability as a mechanism for the initiation of flares. The long-wavelength () mode grows for average twists (at a loop aspect ratio of ≈5). The threshold of instability increases with increasing major loop radius, primarily because the aspect ratio then also increases. Numerically obtained equilibria at subcritical twist are very close to the approximate analytical equilibrium; they do not show indications of sigmoidal shape. The growth of kink perturbations is eventually slowed down by the surrounding potential field, which varies only slowly with radius in the model. With this field a global eruption is not obtained in the ideal MHD limit. Kink perturbations with a rising loop apex lead to the formation of a vertical current sheet below the apex, which does not occur in the cylindrical approximation.

MHD limits in the T-15M tokamak

The problem of plasma MHD stability in the T-15M tokamak is considered in realistic geometry. Stability of the external kink modes with the toroidal wavenumbers n=1, 2, and 3 is numerically investigated in equilibrium configurations similar to the systematically analyzed steady-state configurations with reversed shear in ITER. The stability limits in T-15M are found with and without account taken of the stabilizing influence of the first wall. The results of the calculations are used to compare T-15M with ITER. The stabilizing effect resulting from reducing the distance between the first wall and the plasma in T-15M is evaluated.

Resonant absorption of ULF waves near the ion cyclotron frequency: A simulation study

We present a numerical investigation of Alfven resonances when frequencies are comparable to the ion cyclotron frequency(ωci) and MHD approximations become no longer valid. We have developed a three‐dimensional multi‐fluid wave model, which can fully include the effects of multi‐ions and electron. We use simple box model of an inhomogeneous plasma to examine how wave coupling properties change when the characteristic MHD resonance modes become comparable to ωci. The wave spectra and energy relations are presented for both MHD waves and ion cyclotron waves. Our results show that previous resonant absorption in pure MHD waves is well confirmed in the multi‐fluid model. By adopting a set of new eigenmodes, we show that the resonant absorption persists even for the higher frequency. It is suggested that, near ωci, the left‐handed circularly polarized mode is likely to behave as a shear mode in the MHD limit.

Whistler‐mediated magnetic reconnection in large systems: Magnetic flux pileup and the formation of thin current sheets

We compute numerical solutions of the resistive Hall MHD equations corresponding to pairwise magnetic island coalescence. The simulation results can be organized according to the relative sizes of three length scales: the electron dissipation length, ℓe; the ion inertial length, di; and the island wavelength, λ. We identify three qualitatively distinct regimes of magnetic island coalescence: (1) the resistive MHD limit, di ≲ ℓe ≪ λ; (2) the “whistler‐mediated” limit, ℓe ≪ di ≪ λ; and (3) the “whistler‐driven” limit, ℓe ≪ λ ≲ di. In the resistive MHD limit, magnetic flux piles up outside thin current sheets between the islands. The upstream Alfvén speed increases with increasing Lundquist number, and the reconnection rate is insensitive to the Lundquist number. In the whistler‐driven limit, the electron and ion bulk flows decouple on the island wavelength scale. Magnetic flux pileup does not occur, and the coalescence proceeds on a whistler timescale that is much shorter than the Alfvén time. In the whistler‐mediated limit, electron and ion bulk flows decouple in spatially localized “ion inertial sheets” around the island separatrices. Flux pileup is reduced, and the upstream Alfvén speed approaches a nearly constant value as the Lundquist number is increased. The maximum reconnection rate in the whistler‐mediated limit is comparable to that observed in the resitive MHD limit over the Lundquist number range 500 &lt; Sλ &lt; 10000.

MHD stability of lower hybrid enhanced performance discharges on the Tore Supra tokamak

Lower hybrid enhanced performance (LHEP) discharges have been successfully operated on Tore Supra, with a significant improvement in electron core-confinement and remarkable steady-state features. However, recent attempts to operate such discharges with better confinement performance were hindered by the systematic onset of large MHD activity, preventing any good performance recovery. The associated MHD limit is found to result from reaching the tearing mode limit as the current profile is driven from centrally peaked towards off-axis peaked by the lower hybrid (LH) wave. The specific feature of the considered discharges is that they achieve a weak magnetic shear over a broader core region than past LHEP discharges, then leading to systematic fast MHD onset (m/n = 2/1 tearing mode) when qmin approaches 2. The present paper reports on experimental observations and modelling which support this finding and analyses the perspectives of improving the LHEP scenario by combining the current profile control through the LH wave with a significant increase in pressure through ion cyclotron wave coupling.