Pressure-driven relaxation instability in a current-free high-shear helical system

The TJ toroidal tearing mode code is used to make realistic predictions of the electron cyclotron emission (ECE) signals generated by neoclassical tearing modes (NTMs) in an ITER-like tokamak plasma equilibrium. In the so-called ``outer region'', which comprises the bulk of the plasma, helical harmonics of the magnetic field with the same toroidal mode number as the NTM, but different poloidal mode numbers, are coupled together by the Shafranov shift and shaping of the equilibrium magnetic flux-surfaces. In the ``inner region'', which is localized in the vicinity of the NTM rational surface, helical harmonics whose poloidal and toroidal mode numbers are in the same ratio as those of the NTM are coupled together nonlinearly to produce a radially asymmetric magnetic island chain. The solutions in the inner and outer regions are asymptotically matched to one another. The asymptotic matching process determines the overall magnetic structure of the NTM, as well as the global perturbation to the electron temperature caused by the mode. A simulated ECE diagnostic is developed that accounts for the downshifting and broadening in frequency of the signal due to the relativistic mass increase of the emitting electrons. It
is estimated that ECE can reliably detect NTM island chains in ITER whose widths
exceed 1% of the plasma minor radius (in the absence of noise).

The effect of boundary conditions on plasma confinement in the reversed field pinch (RFP) was studied. Early theoretical studies of the magnetohydrodynamic stability of the reversed field pinch had shown that the pinch was unstable unless it was surrounded by a conducting shell which prevented the penetration of the magnetic field during the lifetime of the pinch. The presence of the sheer, however, complicated construction and prevented control of the plasma equilibrium by feedback systems. This research project was undertaken to determine the effect on confinement of varying the magnetic penetration time of the shell. A reversed field pinch, the Reversatron, was constructed so that shells of varying thickness could be installed. Shells were made with magnetic penetration times both longer and shorter than the duration of the plasma discharge. These shells allowed the performance to be determined as a function of the magnetic penetration time of the shell. This research showed that a shell with a short penetration time degraded the discharges. Diagnostic measurements showed reductions in plasma current, discharge duration, and the Doppler temperature of impurity ions and showed increases in the plasma resistivity and loop voltage. The greater power input and lower temperature implied a reduced energy confinement time. Magnetic diagnostics showed that two classes of MHD instabilities with poloidal mode number m = 1 were responsible for the degraded confinement. These were the dynamo modes resonant on the magentic axis (with toroidal mode numbers -9, -10, -11, -12) and the external kink modes (with toroidal mode numbers 3, 4, 5, and 6).

A two-dimensional array of saddle coils at Mc poloidal and Nc toroidal positions is used on the EXTRAP T2R reversed-field pinch (Brunsell P R et al 2001 Plasma Phys. Control. Fusion 43 1457) to study active control of resistive wall modes (RWMs). Spontaneous growth of several RWMs with poloidal mode number m = 1 and different toroidal mode number n is observed experimentally, in agreement with linear MHD modelling. The measured plasma response to a controlled coil field and the plasma response computed using the linear circular cylinder MHD model are in quantitive agreement. Feedback control introduces a linear coupling of modes with toroidal mode numbers n, n′ that fulfil the condition |n − n′| = Nc. Pairs of coupled unstable RWMs are present in feedback experiments with an array of Mc × Nc = 4 × 16 coils. Using intelligent shell feedback, the coupled modes are generally not controlled even though the field is suppressed at the active coils. A better suppression of coupled modes may be achieved in the case of rotating modes by using the mode control feedback scheme with individually set complex gains. In feedback with a larger array of Mc × Nc = 4 × 32 coils, the coupling effect largely disappears, and with this array, the main internal RWMs n = −11, −10, +5, +6 are all simultaneously suppressed throughout the discharge (7–8 wall times). With feedback there is a two-fold extension of the pulse length, compared to discharges without feedback.

Mode structures and excitation conditions for the low-frequency modes (LFMs) have been investigated in experimental advanced superconducting tokamak (EAST) plasmas with q min ⩽ 2. Two different stages/categories of the LFM instabilities are observed during the oscillation of annular/central collapse events: (I) the upward sweeping frequency of LFMs; (II) the upward frequency jumpsof LFMs. The annular/central events are triggered by the m/n = 2/1 double tearing modes with different q-profiles, while the LFMs are characterized by higher mode numbers m/n = 4/2, 6/3, …, where m and n are the poloidal and toroidal mode numbers, respectively. The maximum radial coverage of the LFMs is located in the annular region of 1.97 2.07 m with the normalized minor radius 0.2 0.4, while the higher-frequency (or upward sweeping frequency) branch is more localized to the radial position of 2 2.02 m (q min). The frequency characteristics of upward sweeps or upward jumps of the LFMs are mainly attributed to the change in the q-profile, e.g. the upward sweeping frequency in stage I is caused by q min decreasing. Accordingly, the linear wave properties of LFMs in EAST with weak/reversed magnetic shear are studied numerically and analytically based on a general fishbone-like dispersion relation. Without considering the contribution of energetic ions, it is shown that the LFM with Alfvénic polarization is an MHD-unstable kinetic ballooning mode with frequency of the order of the ion diamagnetic drift frequency. Several important factors for the excitation of LFM instability are analyzed: (1) the role of energetic ions is unimportant, and the LFMs can be excited under the two conditions of with/without energetic ions; (2) the higher τ = T e/T i with larger η i = L ni /L Ti are required, namely the normalized pressure gradient α ∝ (1 + τ)(1 + η i) should be large enough to overcome the stability effect of finite field line bending; (3) the weak/reversed shear q-profile with q min ⩽ 2 and suitable S ≡ (r/q)(q″)1/2 are required.

The excitation of an initially stable neoclassical tearing mode (NTM) is investigated in Experimental Advanced Superconducting Tokamak (EAST) low-β p H-mode plasmas (β p is defined as the ratio of the thermal pressure to the poloidal magnetic pressure). Using similar plasma parameters, n = 1 resonant magnetic perturbation (RMP) cannot always successfully excite the m/n = 2/1 NTM with the same RMP coil current setup (n and m are the toroidal and poloidal mode numbers, respectively). Data from a gas electron multiplier camera shows that NTM destabilization is related to RMP-induced crashes at the q = 1 resonant surface during the RMP ramp-up phase. The second RMP-induced crash amplitude decays exponentially as β p increases. There is a critical value above which the crash amplitude (or seed island width) is too small (below the critical island width) to trigger an NTM. Observation and analysis indicate that the m/n = 2/1 NTM is not forcibly driven by the n = 1 RMP (such as the m/n = 2/1 component), but is probably due to electron heat transport between the q = 1 and the q = 2 resonant surfaces. This paper describes experimental observations of NTM excitation which also have implications for further investigations of NTM locking and disruptions.

Studies of large-size (R = 1.5 m, a = 0.5 m), moderate current (I < 750 kA) reversed-field pinch (RFP) plasmas are carried out in the Madison Symmetric Torus in order to evaluate and improve RFP confinement, study general toroidal plasma MHD issues, determine the mechanism of the RFP dynamo, and measure fluctuation-induced transport and anomalous ion heating. MST confinement has been improved by reduction of magnetic field errors with correction coils in the primary circuit and reduction of impurities using boronization; high densities have been achieved with hydrogen pellet injection. MHD tearing modes with poloidal mode number m = 1 and toroidal mode numbers n = 5--7 are prevalent and nonlinearly couple to produce sudden relaxations akin to tokamak sawteeth. Edge fluctuation-induced transport has been measured with a variety of insertable probes. Ions exhibit anomalous heating, with increases of ion temperature occuring during strong MHD relaxation. The RFP dynamo has been studied with attention to various possible mechanisms, including motion-EMF drive, the Hall effect, and superthermal electrons. Initial profile control experiments have begun using insertable biased probes and plasma guns. The toroidal field capacity of MST will be upgraded during Summer, 1993 to allow low-current tokamak operation as well as improved RFP operation.

Two-dimensional (2D) plasma profiles, gradient-driven low frequency electrostatic waves and anomalous particle transport are studied experimentally in a purely toroidal plasma configuration. The plasma is produced by a negatively biased, emissive cathode in conjunction with a variably biased anode plate localized on the same field line. Nearly circularly symmetric potential profiles, positive or negative, can be obtained by varying the anode plate bias. Confinement loss in the form of plasma ejection in the major radius direction is only observed when the anode is biased close to ground potential. For the cases of circularly symmetric plasma flow electrostatic flute modes with poloidal mode number m = 1 and m = 2 are identified as unstable on the low-field side of the density maximum, but indications are also given that they coexist with spatially quasi-uniform oscillations and, for low frequencies, with an ion acoustic mode with toroidal mode number n = 1 propagating parallel to the magnetic field. The anomalous particle flux density is found to be non-uniformly distributed on a toroidal surface, and the main flux is passed through two lobes directed almost vertically upwards and downwards, while almost no flux takes place along the major radius direction.

The density limit for electron plasmas confined by toroidal magnetic surfaces is investigated. In a cylinder, the well-known limit is the Brillouin density, nB≡ϵ0B2∕2me. In an axisymmetric torus, the confining region shifts outward in major radius, and this shift is shown to equal half the plasma radius when n∕nB≈ι2a∕R0, where ι=1∕q is the rotational transform of the magnetic field and a∕R0 is the inverse aspect ratio of the torus. In a nonaxisymmetric torus, electron confinement is found to be lost due to stochasticity effects when n∕nB≈(ι2∕8M2)(a∕R0)2∕δMN. The asymmetry amplitudes δMN are the fractional variations in n∕B2 on a magnetic surface in the poloidal mode number M and the toroidal mode number N≈ιM.

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.

In the Large Helical Device (LHD), edge coherent MHD modes such as m/n = 2/3 or 1/2 (m, n: poloidal and toroidal mode numbers), of which the rational surface is located in the plasma edge region of the magnetic hill, are strongly enhanced after an L–H transition. In this case, an edge MHD mode localized in a region further in than the location of the enhanced MHD modes is often stabilized because of the decrease in the pressure gradient there. Enhancement of the edge MHD mode amplitude stops a substantial rise in the edge pressure gradient after the transition. In an L–H transition plasma, soft x-ray (SX) fluctuations, δIsx, related to edge MHD modes are clearly detected. The radial profiles of the amplitude and the phase differences among the SX-detector channels are sensitively dependent on the m-number. The relative amplitudes of the SX fluctuations, δIsx/Isx, which may reflect the eigenfunction of the MHD mode, rapidly increase towards the plasma edge in which the relevant rational surface exists.

Progress in the understanding of magnetohydrodynamic (MHD) stability is summarized on JT-60U tokamak discharges with improved confinement such as the (hot-ion) H-mode, high-βp mode, high-βp H-mode, and reversed shear discharges. Transport barriers, which are essential for the improved confinement, play key roles in the local and global MHD stability owing to the local large pressure gradient and the related bootstrap current. Disruptive β limits of these discharges are consistent with theoretical ideal kink-ballooning stability limits with low toroidal mode numbers n. Achievable β limit is improved by broadening of the pressure profile with high plasma internal inductance, plasma shaping, and wall stabilization. Edge localized modes (ELMs) and barrier localized modes (BLMs), which are associated with edge and internal transport barriers, respectively, are analyzed carefully. Resistive interchange modes with n ≤ 3 are excited in the negative shear region in reversed shear discharges with the internal transport barrier and lead to major collapse occasionally through nonlinear coupling with a tearing mode in the positive shear region. MHD characteristics of low m/n (m: poloidal mode number) tearing modes, which are attributed to the neoclassical tearing mode, are investigated. Stabilization of tearing modes and control of sawtooth activity are demonstrated using the fundamental O-mode electron cyclotron wave injection. Resistive wall modes associated with current-driven and pressure-driven low n external kink modes are identified.

When resistive interchange modes (RIMs) are unstable in aStellarator/Heliotron device, they may change the pressure profile even forβ values lower than the ideal β limit. The nonlinear RIM withn/m = 2/3 modifies the density profile in the vicinity of the resonantsurface with ι = 2/3 easily under the assumption of constanttemperature. Here m(n) is a poloidal (toroidal) mode number andι is a rotational transform. Also poloidal shear flow is generatedthere, which limits the region of density profile modificationsubstantially. The sign of the density gradient becomes positive at theresonant surface. A quasi-stationary profile is obtained forlow-collisional plasmas. When the collision frequency is high, localhumps are produced in the background density profile, and they have atendency to move to the resonant surface. It is also shown that thedensity profile including humps is correlated with the poloidal velocityshear profile. Here the typical shearing rate of the poloidal flow at thesaturated state exceeds the linear growth rate of the (m,n) = (3,2) mode.

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.

The ELM wetted area is a key factor in the peak power load during an ELM, as it sets the region over which the ELM energy is deposited. The deposited heat flux at the target is seen to have striations in the profiles that are generated by the arrival of filaments ejected from the confined plasma. The effect of the filaments arriving at the target on the ELM wetted area, and the relation to the midplane mode number is investigated in this paper using infrared (IR) thermography and high speed visible imaging (>10 kHz). Type I ELMs are analysed, as these have the largest heat fluxes and are observed to have toroidal mode numbers of between 5 and 15. The IR profiles during the ELMs show clear filamentary structures that evolve during the ELM cycle. An increasing number of striations at the target is seen to correspond to an increase in the wetted area. Analysis shows that the ratio of the ELM wetted area to the inter-ELM wetted area, a key parameter for ITER, for the type I ELMs is between 3 and 6 for lower single null plasmas and varies with the ELM midplane mode number, as determined by visible measurements. Monte-Carlo modelling of the ELMs is used to understand the variation seen in the wetted area and the effect of an increased mode number; the modelling replicates the trends seen in the experimental data and supports the observation of increased toroidal mode number generating larger target ELM wetted areas. ITER is thought to be peeling unstable which would imply a lower ELM mode number compared to MAST which is peeling–ballooning unstable. The results of this analysis suggest that the lower n peeling unstable ELMs expected for ITER will have smaller wetted areas than peeling–ballooning unstable ELMs. A smaller wetted area will increase the level of ELM control required, therefore a key prediction required for ITER is the expected ELM mode number.

Two novel nonlinear mode coupling processes for reversed shear Alfvén eigenmode (RSAE) nonlinear saturation are proposed and investigated. In the first process, the RSAE nonlinearly couples to a co-propagating toroidal Alfvén eigenmode (TAE) with the same toroidal and poloidal mode numbers, and generates a geodesic acoustic mode. In the second process, the RSAE couples to a counter-propagating TAE and generates an ion acoustic wave quasi-mode. The condition for the two processes to occur is favored during current ramp. Both the processes contribute to effectively saturate the Alfvénic instabilities, as well as nonlinearly transfer of energy from energetic fusion alpha particles to fuel ions in burning plasmas.