Ideal Interchange Mode Research Articles

Benign Saturation of Ideal Ballooning Instability in a High-Performance Stellarator.

To examine the robustness of the designed 5% β limit for high-performance operation in the W7-X stellarator, we undertake nonlinear magnetohydrodynamic (MHD) simulations of pressure-driven instabilities using the m3d-c1 code. Consistent with linear analyses, ideal ballooning instabilities occur as β exceeds 5% in the standard configuration. Nonetheless, the modes saturate nonlinearly at relatively low levels without triggering large-scale crashes, even though confinement degradation worsens as β increases. In contrast, in an alternative configuration with nearly zero magnetic shear, ideal interchange modes induce a pressure crash at β=1%. These results suggest that the standard W7-X configuration might have a soft β limit that is fairly immune to major MHD events, which enhances the appeal of the stellarator approach to steady-state fusion reactors.

Gradient-driven turbulence in Texas Helimak

We investigate the turbulence level dependence on plasma profiles in experiments in Texas Helimak, a toroidal basic plasma device, with long stable electron cyclotron resonant heating (ECRH) discharges and great flexibility to alter the equilibrium magnetic field. A large set of Langmuir probes is used to obtain the turbulence level and also the plasma radial profiles for several magnetic field intensities with the same safety factor and field line pitch profiles. As a consequence of the ECRH heating, changing the toroidal magnetic field, the equilibrium density profiles are radially displaced. For all the analyzed discharges, with constant magnetic field curvature and shear profiles, we verify that the plasma turbulence has a critical dependence on the equilibrium density profile. Namely, radial regions with negative density radial gradient, i.e., in the opposite direction of the magnetic curvature, present high turbulence level. By properly comparing the turbulence radial profiles with the density peak position, we show that the negative density gradient is the main cause of high amplitude turbulence, in agreement with predictions for ideal interchange modes. Furthermore, intermittence analysis shows that the extreme events (bursts) contribution for the probability density functions (PDFs) is also related to the relative position with respect to the density peak, and that the turbulence level enhancement is likely due to the increase in burst occurrence.

Self-organized confinement in whole-device modeling of laboratory magnetospheres

Turbulent self-organization driven by global ideal interchange modes in a dipole-confined plasma is explored with self-consistent, whole-plasma simulations using a flux-tube averaged magnetohydrodynamic model in dipole magnetic geometry. We show the existence of robust particle pinch driven by ideal interchange-mode fluctuations, in which the particles are transported up the density gradient. It is found that the plasma profiles in a dipole field spontaneously relax to a marginally stable state as centrally peaked pressure and density are created by global interchange-mode transport.

Linear MHD analyses of locked-mode-like instabilities in LHD

To investigate the driving mechanism of the locked-mode-like instability observed in the large helical device, we reconstruct the magnetohydrodynamic (MHD) equilibria consistent with the measurement and identify a dominant MHD instability in the precursor phase based on linear MHD analyses. From the dependence of the linear growth rate on the magnetic Reynolds number, the radial mode structure of the electrostatic potential fluctuation and other indices, the ideal interchange mode is found to be dominant. Moreover, it is found that the Mercier parameter, D I, becomes much larger than 0.3 during the phase, while the precursor has constant frequency and fluctuation amplitude. Therefore, D I ≫ 0.3 is a good index of the on-set condition of the minor collapse itself. It is also found that the achievement of D I ≫ 0.3 is due to the movement of the resonant surface to the inner plasma region, which corresponds to the larger pressure gradient region.

Ion kinetic effects on linear pressure driven magnetohydrodynamic instabilities in helical plasmas

The linear MHD (magnetohydrodynamic) stability for high beta plasmas in the inward shifted Large Helical Device (LHD) configurations has been investigated for a wide range of magnetic Reynolds numbers $S$ using numerical simulations based on the kinetic MHD model with kinetic thermal ions where the beta is the ratio of the plasma pressure to the magnetic pressure. It is found that the dependence of the linear growth rate of the resistive ballooning modes on the $S$ number changes from $\unicode[STIX]{x1D6FE}\propto S^{-1/3}$ to $\unicode[STIX]{x1D6FE}\propto S^{-1}$ by the kinetic thermal ion effects so that the resistive ballooning modes are significantly suppressed as the $S$ number increases. For a high $S$ number comparable to experimental values, the most unstable modes are interchange modes. The kinetic thermal ion effects change the most unstable interchange mode from the ideal mode to the resistive mode. This transition of the interchange modes by kinetic thermal ion effects is consistent with the shift of the marginal stability boundary for the ideal interchange modes observed in the LHD experiments.

A new explanation of the sawtooth phenomena in tokamaks

The ubiquitous sawtooth phenomena in tokamaks are so named because the central temperature rises slowly and falls rapidly, similar to the blades of a saw. First discovered in 1974, it has so far eluded a theoretical explanation that is widely accepted and consistent with experimental observations. We propose here a new theory for the sawtooth phenomena in auxiliary heated tokamaks, which is motivated by our recent understanding of “magnetic flux pumping.” In this theory, the role of the (m,n)=(1,1) mode is to generate a dynamo voltage, which keeps the central safety factor, q0, just above 1.0 with low central magnetic shear. When central heating is present, the temperature on axis will increase until at some point, and the configuration abruptly becomes unstable to ideal MHD interchange modes with equal poloidal and toroidal mode numbers, m=n>1. It is these higher order modes and the localized magnetic stochasticity they produce that cause the sudden crash of the temperature profile, not magnetic reconnection. Long time 3D MHD simulations demonstrate these phenomena, which appear to be consistent with many experimental observations.

Drift-ideal magnetohydrodynamic simulations of m = 0 modes in Z-pinch plasmas

The effects of m = 0 modes on equilibrium Z-pinch plasmas are studied in this paper using a drift-ideal magnetohydrodynamic (MHD) model. The model equations are an extension of ideal MHD to include finite-ion-inertial-length/cyclotron-frequency (Ωi) effects in Ohm's law and in the electron and ion heat transport equations. The linear modes contained in this model include the ideal interchange (sausage) mode and in the magnetized limit, Ωiτi≫1 with τi the ion collision time, nonideal entropy modes. It is well known that these two modes are decoupled in the kρs≪1 limit, where k is the axial mode number and ρs=cs/Ωi is the gyro-Bohm scale with cs the sound speed [B. Kadomtsev, Sov. Phys. JETP-USSR 10, 780 (1960)]. For Bennett equilibrium profiles, it is shown that the regions of stability for both modes are completely governed by the adiabatic coefficient γ in these limits. Equilibria with Bennett profiles are stable to entropy modes for γ < 2 but unstable to ideal modes and vice versa for γ > 2. However, these modes are no longer decoupled when kρs≳1. The simulation results of the fully nonlinear set of equations in the magnetized limit show that seeded modes with kρs≳1 and γ = 5/3 display the characteristics of both ideal and entropy modes. The general heat flux for both ions and electrons as a function of the species magnetization is retained in the model. Both the linear and nonlinear behaviors of seeded modes for kρs≳1 display a strong dependence on the magnetization of the ions. The growth rate increases linearly with k at large kρs when the ions are magnetized but decreases with increasing k when Ωiτi≲1.

Effects of fast ions on interchange modes in the Large Helical Device plasmas

Effects of fast ions on the magnetohydrodynamic (MHD) instabilities in a Large Helical Device (LHD) plasma with the central beta value (=pressure normalized by the magnetic pressure) 4% have been investigated with hybrid simulations for energetic particles interacting with an MHD fluid. When fast ions are neglected, it is found that the dominant instability is an ideal interchange mode with the dominant harmonic m/n = 2/1, where m, n are respectively the poloidal and toroidal numbers. The spatial peak location of the m/n = 2/1 harmonic is close to the ι = 1/2 magnetic surface located at r/a = 0.29, where ι is the rotational transform and r/a is the normalized radius. The second unstable mode is a resistive interchange mode with m/n =3/2 that peaks at r/a = 0.65 nearby the ι = 2/3 surface, which grows more slowly than the m/n = 2/1 mode. The nonlinear coupling of the m/n = 3/2 and 2/1 mode results in the growth of the m/n = 5/3 mode and other modes leading to the global reduction and flattening of the pressure profile. When fast ions are considered with the central beta value 0.2% and the total pressure profile is kept the same, the ideal interchange mode with m/n = 2/1 located close to the plasma center is stabilized while the resistive interchange mode with m/n = 3/2 located far from the plasma center is less affected. The stabilization is attributed to the reduction of bulk pressure gradient, which is the dilution of the free energy source, because the energy transfer between the fast ions and the interchange modes is found to be negligible. For higher fast-ion pressure, Alfvén eigenmodes are destabilized by fast ions.

Effects of trapped energetic ions on the interchange mode

On the basis of the kinetic energy principle, effects on trapped energetic ions on the ideal interchange mode in helical systems are investigated. Approximating that the spatial profile of the ideal interchange mode is given, we introduce an extended dispersion relation of the ideal interchange mode. Numerical analyses of the dispersion relation show that the unstable ideal interchange mode is destabilized by the trapped energetic ions and has a finite rotation frequency due to the precession drift of the trapped energetic ions. We also derive a generalized eigenvalue equation of the ideal interchange mode in the presence of trapped energetic ions. By numerically solving the eigenvalue equation, it is found that a radial profile of the ideal interchange mode is spread by trapped energetic ions.

Characteristics of MHD instabilities limiting the beta value in LHD

Effects of low-n magnetohydrodynamic instabilities on plasma performance have been assessed in the regime where an achieved beta value is limited by instabilities. The unstable regime of an ideal interchange mode is characterized by enhanced magnetic hill and reduced magnetic shear. Experiments have clarified that (i) low-n modes are significantly destabilized in the ideal-unstable configurations and lead to degradation of central beta by at most 60%, and (ii) the degree of their damages strongly depends on the mode rotation velocity. The occurrence of the minor collapse is independent of an existence of an error field.

Interchange mode excited by trapped energetic ions

The kinetic energy principle describing the interaction between ideal magnetohydrodynamic (MHD) modes with trapped energetic ions is revised. A model is proposed on the basis of the reduced ideal MHD equations for background plasmas and the bounce-averaged drift-kinetic equation for trapped energetic ions. The model is applicable to large-aspect-ratio toroidal devices. Specifically, the effect of trapped energetic ions on the interchange mode in helical systems is analyzed. Results show that the interchange mode is excited by trapped energetic ions, even if the equilibrium states are stable to the ideal interchange mode. The energetic-ion-induced branch of the interchange mode might be associated with the fishbone mode in helical systems.

Novel Theory of Energetic-Ion-Induced Interchange Mode

On the basis of a kinetic energy principle, it is shown that the interchange mode in helical systems is excited by trapped energetic ions, where the ideal interchange mode is stable. The mode has a rotation frequency comparable to precession drift frequencies of trapped energetic ions. The theory explains how to apply the fishbone mode theory originally developed in tokamaks to helical systems.

Investigation of the long-lived saturated internal mode and its control on the HL-2A tokamak

HL-2A plasmas heated by neutral beam injection (NBI) regularly exhibit n = 1 long-lived saturated magnetohydrodynamic instabilities. A reduction in the electron density and plasma stored energy and an increase in fast ion losses are usually observed in the presence of such perturbations. The observed long-lived saturated internal mode (LLM) occurs when the safety factor profile has a weak shear in a broad range of the plasma centre with qmin around unity. It is found that the ideal interchange mode can become marginally stable due to the weak magnetic shear reaching a critical value. The LLM, due to its pressure-driven feature, is destabilized by the strong interaction with fast ions in the low-shear region during the NBI. Furthermore, for the first time it is clearly observed that the LLMs can be suppressed by electron cyclotron resonant heating (ECRH), or by supersonic molecular beam injection in HL-2A plasmas. Low-n sidebands observed during the LLM are also suppressed by increasing the ECRH power. The control of LLMs is due to the change in the magnetic shear or in the pressure profile induced by the local heating or fuelling.

Boundary induced amplification and nonlinear instability of interchange modes

It is shown that small distortions on the boundaries are amplified in the core of a magnetized plasma if the system is close to marginal stability for the ideal magnetohydrodynamic interchange mode. It is also shown that such marginal systems can be nonlinearly unstable. The combination of boundary amplification and nonlinearity is shown to result in a nonlinear instability. The induced instability is highly sensitive to the boundary in that, if the fractional deviation from marginality is a small parameter b, the system can go unstable from fractional boundary distortions of O(b3/2).

Mode locking phenomena observed near the stability boundary of the ideal interchange mode of LHD

This paper describes the first observation of mode locking phenomena in the Large Helical Device. In the experiments, a rotation of the m/n = 1/1 mode slowed down and stopped when the plasma approached the stability boundary of the ideal interchange mode. The minor collapse was caused by the growth of the mode just after mode locking. The mode before the locking has a radial structure with no inversion around the resonance, which implies that the mode has an interchange-type structure at least before the rotation stops. The location of the mode after the locking depends on that of the error or residual field in reduced error field operations.

Suprathermal ion transport in simple magnetized torus configurations

Inspired by suprathermal ion experiments in the basic plasma experiment TORPEX, the transport of suprathermal ions in ideal interchange mode turbulence is theoretically examined in the simple magnetized torus configuration. We follow ion tracer trajectories as specified by ideal interchange mode turbulence imported from a numerical simulation of drift-reduced Braginskii equations. Using the variance of displacements, σ2(t)∼tγ, we find that γ depends strongly on suprathermal ion injection energy and the relative magnitude of turbulent fluctuations. The value of γ also changes significantly as a function of time after injection, through three distinguishable phases: ballistic, interaction, and asymmetric. During the interaction phase, we find the remarkable presence of three regimes of dispersion: superdiffusive, diffusive, and subdiffusive, depending on the energy of the suprathermal ions and the amplitude of the turbulent fluctuations. We contrast these results with those from a “slab” magnetic geometry in which subdiffusion does not occur during the interaction phase. Initial results from TORPEX are consistent with data from a new synthetic diagnostic used to interpret our simulation results. The simplicity of the simple magnetized torus makes the present work of interest to analyses of more complicated contexts ranging from fusion devices to astrophysics and space plasma physics.

Nonlinear stability of the ideal magnetohydrodynamic interchange mode at marginal conditions in a transverse magnetic field

The stability of the ideal magnetohydrodynamic (MHD) interchange mode at marginal conditions is studied. A sufficiently strong constant magnetic field component transverse to the direction of mode symmetry provides the marginality conditions. A systematic perturbation analysis in the smallness parameter, |b2/Bc|1/2, is carried out, where Bc is the critical transverse magnetic field for the zero-frequency ideal mode and b2 is the deviation from Bc. The calculation is carried out to third order including nonlinear terms. It is shown that the system is nonlinearly unstable in the short wavelength limit, i.e., a large enough perturbation results in instability even if b2/Bc > 0 (linearly stable). The normalized amplitude for instability is shown to scale as |b2/Bc|1/2. A nonlinear, compressible, MHD simulation is done to check the analytic result. Good agreement is found, including the critical amplitude scaling.

Multi-scale MHD analysis incorporating pressure transport equation for beta-increasing LHD plasma

A multi-scale MHD numerical scheme is developed for analysis of nonlinear evolution of a beta-increasing plasma. The scheme is based on iterative calculations of nonlinear dynamics based on the reduced MHD (RMHD) equations and three-dimensional static equilibrium. The equation for average pressure in the RMHD equations plays the role of a transport equation that involves a heat source term and background pressure diffusion terms. The heat source term is controlled so that the beta value should be increased at a constant rate. The scheme is applied to a Large Helical Device (LHD) plasma up to average beta of 1.05%, which is unstable against linear ideal interchange modes while beta values much higher than the stability limit are obtained in the experiments. The result with the multi-scale scheme indicates that many local flat regions are generated in the background pressure profile in the nonlinear evolution of the interchange modes. This structure of the pressure profile suppresses disruptive phenomena because it reduces the driving force of the modes at higher beta value. Such self-organization in the pressure profile is considered to be the stabilizing mechanism in the plasma.

Blob-induced toroidal momentum transport in simple magnetized plasmas

The link between toroidal flows and density blobs is experimentally demonstrated in TORPEX simple magnetized plasmas: momentum is transferred from an ideal-interchange mode to density blobs. The phase shift between the toroidal flow and the density perturbations observed in the interchange mode where the blob is born is conserved along the blob radial trajectory. This leads to dipolar structures of the blob-induced flow or to monopolar perturbations, so large that the toroidal flow gets transiently reversed. The turbulent toroidal momentum flux is dominated either by the nonlinear flux or by the convective part but not by the Reynolds stress component.

Study of MHD Stability in LHD

This paper reviews progress in the study of pressure-driven interchange stability in the Large Helical Device (LHD) for 10 years. When the plasma approaches the boundary of ideal interchange mode, a strong magnetohydrodynamic (MHD) mode appears, leading to a distortion of pressure profile, although no major disruption is caused. The experiments for investigating magnetic shear effects in the magnetic hill configuration indicate that the reduction of magnetic shear leads to a minor collapse due to an excitation of low-order MHD mode. In the high-beta regime of more than 4%, MHD modes excited in the periphery with magnetic hill are observed to dominate, and it was found that the amplitude depends on the magnetic Reynolds number as well as the pressure gradient, which is qualitatively consistent with the prediction of resistive interchange mode. Also, experiments and theory for finding parameter dependence of the onset of the mode indicate that the onset is related to both the magnetic Reynolds number and the stability index of resistive interchange mode.