High-m Modes Research Articles

Determination of MHD mode structures using soft x-ray diagnostics in TCV

A forward modeling technique is developed for determining the characteristic features of observed MHD modes from the line-of-sight data of the soft x-ray (SXR) diagnostics with 64 vertical lines-of-sight in theTokamak à Configuration Variable tokamak. Using diagnostics with excellent spatial resolution, this technique is shown to evaluate the poloidal mode numbers m, radial location and ballooning character of the MHD modes. In the first stage, the poloidal mode structures are modeled by the radially localized Gaussian-shaped emission regions rotating along the magnetic flux surfaces. In the second stage, the space structures of observed 95–100 kHz toroidal Alfvén eigenmodes (TAEs) are investigated numerically using HELENA, CSCAS and MISHKA codes. The calculated TAE eigenfunctions are then used in our forward modeling similarly to the approach of contrast imaging diagnostics (Edlund et al 2009 Phys. Rev. Lett. 102 165003) and SXR diagnostics (Piovesan et al 2008 Nucl. Fusion 48 065001). A ballooning structure of the observed n= 1 TAE mode can be analyzed more easily due to the low poloidal mode number m in contrast to the high-m modes analyzed by a similar SXR technique in the W7-X stellarator (Dreval et al 2021 Plasma Phys. Control. Fusion 63 065006).

Linear analysis of plasma pressure-driven mode in reversed shear cylindrical tokamak plasmas

The linear behavior of the dominant unstable mode (m = 2, n = 1) and its high order harmonics (m = 2n, n ≥ 2) are numerically investigated in a reversed magnetic shear cylindrical plasma with two q = 2 rational surfaces on the basis of the non-reduced magnetohydrodynamics (MHD) equations. The results show that with low beta (beta is defined as the ratio of plasma pressure to magnetic field pressure), the dominant mode is a classical double tearing mode (DTM). However, when the beta is sufficiently large, the mode is driven mainly by plasma pressure. In such a case, both the linear growth rate and mode structures are strongly affected by pressure, while almost independent of the resistivity. This means that the dominant mode undergoes a transition from DTM to pressure-driven mode with the increase of pressure, which is consistent with the experimental result in ASDEX Upgrade. The simulations also show that the distance between two rational surfaces has an important influence on the pressure needed in mode transition. The larger the distance between two rational surfaces, the larger the pressure for driving the mode transition is. Motivated by the phenomena that the high-m modes may dominate over low-m modes at small inter-resonance distance, the high-m modes with different pressures and q profiles are studied too.

Determination of poloidal mode numbers of MHD modes and their radial location using a soft x-ray camera array in the Wendelstein 7-X stellarator

A forward modeling technique is developed for determining the characteristic features of observed MHD modes from the line-of-sight data of the soft x-ray (SXR) tomography diagnostics in the Wendelstein 7-X (W7-X) stellarator. In particular, forward modeling is used to evaluate the poloidal mode numbers m, radial location, poloidal rotation direction and ballooning character of the MHD modes. The poloidal mode structures have been modeled by the radially localized Gaussian-shaped emission regions rotating along the magnetic surfaces. In the present study the cases of rigid-shape emission regions and flexible emission regions are modeled. Various mode phase velocity dependences on the magnetic surface position are simulated. The modeled phase dynamics of line-integrated oscillations and the distribution of oscillation amplitudes are compared with the experimental signals of the SXR cameras which observe the plasma at various viewing angles in the poloidal cross-section. Application of this technique enables describing of the 1–50 kHz modes. In particular, in the discharge W7X-PID 20180918.045 three identified branches with the poloidal mode numbers m= 8, m= 10 and m= 11 localized at ρ ≈ 0.3 are rotating in the clockwise poloidal direction. The present paper reports the first application of the forward modeling technique to the data from the SXR diagnostics in W7-X. The high m-modes are identified by forward modeling in W7-X.

Solar Rossby waves observed in GONG++ ring-diagram flow maps

Context. Solar Rossby waves have only recently been unambiguously identified in Helioseimsic and Magnetic Imager (HMI) and Michelson Doppler Imager maps of flows near the solar surface. So far this has not been done with the Global Oscillation Network Group (GONG) ground-based observations, which have different noise properties. Aims. We use 17 years of GONG++ data to identify and characterize solar Rossby waves using ring-diagram helioseismology. We compare directly with HMI ring-diagram analysis. Methods. Maps of the radial vorticity were obtained for flows within the top 2 Mm of the surface for 17 years of GONG++ data. The data were corrected for systematic effects including the annual periodicity related to the B0 angle. We then computed the Fourier components of the radial vorticity of the flows in the co-rotating frame. We performed the same analysis on the HMI data that overlap in time. Results. We find that the solar Rossby waves have measurable amplitudes in the GONG++ sectoral power spectra for azimuthal orders between m = 3 and m = 15. The measured mode characteristics (frequencies, lifetimes, and amplitudes) from GONG++ are consistent with the HMI measurements in the overlap period from 2010 to 2018 for m ≤ 9. For higher-m modes the amplitudes and frequencies agree within two sigmas. The signal-to-noise ratio of modes in GONG++ power spectra is comparable to those of HMI for 8 ≤ m ≤ 11, but is lower by a factor of two for other modes. Conclusions. The GONG++ data provide a long and uniform data set that can be used to study solar global-scale Rossby waves from 2001.

Long-lasting poloidal ULF waves observed by multiple satellites and high-latitude SuperDARN radars.

Poloidal ultra-low frequency (ULF) waves between 5-10 mHz were observed by multiple satellites and three high-latitude Super Dual Auroral Radar Network (SuperDARN) radars during the recovery phase of a moderate geomagnetic storm on Jan 24-27, 2016. The long-lasting ULF waves were observed in the magnetic field and energetic particle flux perturbations during three successive passes by two Geostationary Operational Environmental Satellites (GOES) through the dayside magnetosphere, during which plasmasphere expansion and refilling were observed by two Time History of Events and Macroscale Interactions during Substorms (THEMIS) probes. The radial magnetic field oscillation was in phase (~ 180° out of phase) with the northward (southward) moving proton flux oscillation at 95 keV, consistent with high-energy drift-bounce resonance signatures of protons with second harmonic poloidal standing Alfvén waves. The longitudinal extent of the waves approached 10 hours in local time on the dayside and gradually decreased with time. High-time resolution (~ 6 s) data from three high-latitude SuperDARN radars show that the wave intensification region was localized in latitude with a radial extent of ~ 135-225 km in the subauroral ionosphere. No signature of these waves were observed by ground-based magnetometers colocated with the GOES satellites suggesting that the poloidal waves were high-m mode and thus screened by the ionosphere. During this interval one of the THEMIS probes observed a bump-on-tail ion distribution at 1-3 keV which we suggest is the source of the long-lasting second harmonic poloidal ULF waves.

The q-profile effect on high-order harmonic q = 1 tearing mode generation during sawtooth crashes

The effect of q-profiles on the excitation of high-order harmonic q=1 tearing modes during sawtooth crashes is investigated by a collisionless fluid model with the electron inertia term in Ohm’s law. It is found that for a flat q-profile in the core region, the high-order harmonics, such as m/n=2/2 and/or m/n=3/3 modes, comparable to or stronger than the m/n=1/1 component, can be excited during tokamak sawteeth. The stronger the magnetic shear on the q=1 surface is, the more unstable the higher-m modes are. For smoothly monotonously increased q-profiles, a lower q value on the plasma edge tends to easily excite higher-m harmonics at the same level as the m = 1 mode simultaneously. The spatial characteristics of the eigenmodes in the cases with the typical q-profiles are also discussed. In addition, the basic feature of the magnetic island structures in the nonlinear evolution is numerically obtained, which is consistent qualitatively with the experimentally reconstructed phenomenon.

Validation of electron temperature gradient turbulence in the Columbia Linear Machine

The electron temperature gradient (ETG) mode, which is a universal mechanism for turbulent electron thermal transport in plasmas, is produced and verified in steady-state, collisionless hydrogen plasma of the Columbia Linear Machine. Electron temperature profiles with strong gradients are produced by DC acceleration in a remote biased mesh and subsequent thermalization. Finite amplitude ∼5%, steady-state oscillations at ∼0.3-0.5MHz (in the plasma frame), with azimuthal wave numbers m∼14-16 and parallel wave number k∥∼0.01cm-1 are measured. The massively parallel gyrokinetic toroidal code is used to study these modes. The results show that in the linear phase, the dispersion relation is consistent with kinetic theory. In the nonlinear stage, very strong nonlinear wave coupling gives rise to an inverse cascade of the energy from the fastest growing high-m modes to low-m nonlinear oscillations, which are consistent with the measured azimuthal mode spectrum. The radial structure of the fluctuation also agrees with the experiment. An inward radial shift of the peak of the potential fluctuation occurs during the nonlinear saturation and fluctuation fingers extend radially out to the edge plasma. Three-wave coupling mechanism is involved in the saturation of ETG modes. The simulations show a power law spectrum of the turbulence which suggests that the renormalization theory is appropriate to interpret the turbulent thermal flux.

Dynamics of resistive double tearing modes with broad linear spectra

The nonlinear evolution of resistive double tearing modes (DTMs) with safety factor values q=1 and q=3 is studied with a reduced cylindrical model of a tokamak plasma. We focus on cases where the resonant surfaces are a small distance apart. Recent numerical studies have shown that in such configurations high-m modes are strongly unstable and may peak around m=mpeak∼10. In this paper, it is first demonstrated that this result agrees with existing linear theory for DTMs. Based on this theory, a semiempirical formula for the dependence of mpeak on the system parameters is proposed. Second, with the use of nonlinear simulations, it is shown that the presence of fast growing high-m modes leads to a rapid turbulent collapse in an annular region, where small magnetic island structures form. Furthermore, consideration is given to the evolution of low-m modes, in particular the global m=1 internal kink, which can undergo nonlinear driving through coupling to fast growing linear high-m DTMs. Factors influencing the details of the dynamics are discussed. These results may be relevant to the understanding of the magnetohydrodynamic activity near the minimum of q and may thus be of interest for studies on stability and confinement of advanced tokamaks.

Fast growing double tearing modes in a tokamak plasma

Configurations with nearby multiple resonant surfaces have broad spectra of linearly unstable coupled tearing modes with dominant high poloidal mode numbers m. This was recently shown for the case of multiple q=1 resonances [Bierwage et al., Phys. Rev. Lett. 94 65001 (2005)]. In the present work, similar behavior is found for double tearing modes (DTM) on resonant surfaces with q⩾1. A detailed analysis of linear instability characteristics of DTMs with various mode numbers m is performed using numerical simulations. The mode structures and dispersion relations for linearly unstable modes are calculated. Comparisons between low- and higher-m modes are carried out, and the roles of the inter-resonance distance and of the magnetic Reynolds number SHp are investigated. High-m modes are found to be destabilized when the distance between the resonant surfaces is small. They dominate over low-m modes in a wide range of SHp, including regimes relevant for tokamak operation. These results may be readily applied to configurations with more than two resonant surfaces.

Excitation of the centrifugally driven interchange instability in a plasma confined by a magnetic dipole

The centrifugally driven electrostatic interchange instability is excited for the first time in a laboratory magnetoplasma. The plasma is confined by a dipole magnetic field, and the instability is excited when an equatorial mesh is biased to induce a radial current that creates rapid axisymmetric plasma rotation. The observed instabilities appear quasicoherent in the lab frame of reference; they have global radial mode structures and low azimuthal mode numbers, and they are modified by the presence of energetic, magnetically confined electrons. The mode structure is measured using a multiprobe correlation technique as well as a novel 96-point polar imaging diagnostic which measures particle flux along field lines that map to the pole. Interchange instabilities caused by hot electron pressure are simultaneously observed at the hot electron drift frequency. Adjusting the hot electron fraction α modifies the stability as well as the structures of the centrifugally driven modes. In the presence of larger fractions of energetic electrons, m=1 is observed to be the dominant mode. For faster rotating plasmas containing fewer energetic electrons, m=2 dominates. Results from a self-consistent nonlinear simulation reproduce the measured mode structures in both regimes. The low azimuthal mode numbers seen in the experiment and simulation can also be interpreted with a local, linear dispersion relation of the electrostatic interchange instability. Drift resonant hot electrons give the instability a real frequency, inducing stabilizing ion polarization currents that preferentially suppress high-m modes.

Tangential x-ray imaging system for analysis of the small-scale modes in the T-10 tokamak

A toroidally viewing x-ray imaging system is used in the T-10 tokamak for reconstruction of the small-scale plasma perturbations (high-m modes). The imaging system is based on the matrix array of the silicon surface-barrier diodes with a two-dimensional Soller collimator and an automatic positioning system. An additional hard x-ray detector and magnetic probe are placed inside the tokamak vessel for measurements of the nonthermal emission and fast magnetic field perturbations. Initial experiments in T-10 indicated improved sensitivity of the system for identification of the small-scale plasma perturbations prior the density limit disruptions.

Resistive scalings of modes at finite pressure

Stability analyses of internal resistive MHD modes resonating at surfaces with q = 1,2 or higher values in Tokamak are examined. The resistive scalings for modes with different poloidal mode numbers are studied. Special attention is paid to the transitions induced either by reducing the resistivity, or by the β value, or by shaping the current density profile. For the m/n = 2/1 modes a kink-tearing transition appears while for the higher-m modes mutual transitions take place between the kink modes and the interchange modes. Global and local shaping of the current and the pressure profiles can dramatically suppress the low-n modes in agreement with experimental results. For typical experimental values of resistivity, violation of the Suydam criterion does not have a noticeable effect on the growth rate. However, the ideal modes will dominate for high β.

The ideal MHD stability of line-tied coronal loops: A truncated Fourier series approach

The stability behaviour of a line-tied cylindrically symmetric coronal loop is investigated using a general method presented by Velli, Einaudi and Hood. The plasma perturbation in the linearised equation of motion is expanded in a truncated Fourier series in the poloidal and axial directions, and the resulting system of ordinary differential equations for the radial displacement is solved as an eigenvalue problem. The eigenvalue, be it the critical loop length or the growth rate, is found to converge rapidly with the order of the truncation (approximately as N−2, where N is the number of terms in the Fourier series).Results for the non-force-free uniform-twist field of Gold and Hoyle are presented and compared with a previous study based on the energy principle. The instability threshold for the m = 1 kink mode and the m = ∞ balloning mode are found to be of the same order of magnitude when substantial pressure gradients are present. Preliminary results for higher-m modes indicate that they yield instability thresholds comparable to the ballooning threshold for smaller gradients too.

Linear MHD stability of infinitely long coronal flux tubes

Twisted magnetic flux tubes are often used to model the filed in coronal loops, and much attention has been given to analysing their stability. Previous astrophysical studies have concentrated on establishing the existence of an instability or determining stability bounds, and little information seems available on the associated eigenvalues, which give crucial information on the energy released. This paper develops methods of determining eigenvalues for infinitely long flux tubes. The most striking feature of the results is that the eigenvalues are always small-of order 10−2 (in dimensionless units) even for the fastest helical kink modes (m=1). The more localized higher-m modes have even smaller eigenvalues. A family of flux tubes with field line twist proportional torν is investigated, and it appears that the most energetic instabilities occur in the Gold-Hoyle tube with uniform twist (ν=0). Implications of these results are discussed.

Turbulent stabilization of the tearing mode in tokamak plasmas

The effect of turbulent electron diffusion from stochastic electron orbits on the stability of low-beta fluctuations is considered through the use of the normal stochastic approximation. A set of coupled, self-adjoint equations is derived for the fluctuation potentials φ̃ and Ã∥. Solutions to this set of equations describe both unstable finite-β drift waves when analyzed for high-m modes and the tearing mode when analyzed for low-m modes. For the tearing mode, it is shown that stability is obtained for sufficiently large values of the diffusion coefficient. Provided De∼1/n, this implies that a density threshold must be surpassed before the tearing mode is observed. Physically, turbulent electron diffusion prohibits the formation of a perturbed current within a finite region about the rational surface. At higher densities, the inclusion of a finite electrostatic potential φ̃ gives an additional stabilizing term to the dispersion relation, which physically represents ion inertial effects. This ion inertial effect implies that, in the absence of diffusion, the tearing mode is stabilized for ion betas βi above some critical value.

Toroidal plasma produced by the stabilized double-theta pinch

The properties of the toroidal plasma produced by the stabilized double-theta pinch are investigated experimentally and theoretically. The toroidal plasma produced has a strongly elongated crosssection with the half-axis ratio b/a ∼ 14 and an aspect ratio R0/a ∼ 8 at t = 3 μs, when the values of β and βp are approximately 0.23 and 0.30, respectively. It is shown that in the vicinity of the median plane of the plasma the Suydam criterion S ≥ 0 for the high m-mode is satisfied throughout the cross-section of the plasma produced. With the one-dimensional approximation, an analytic solution of MHD equilibrium is presented which always satisfies the Suydam criterion.