Session VI2: MHD

Active Resistive Wall Mode Stabilization in Low Rotation, High Beta NSTX Plasmas

An active feedback system to stabilize the resistive wall mode (RWM) in the National Spherical Torus Experiment (NSTX) is used to maintain plasma stability for greater than 90 RWM growth times. These experiments are the first to demonstrate RWM active stabilization in high beta, low aspect ratio tokamak plasmas with toroidal plasma rotation significantly below the critical rotation profile for passive stability and in the range predicted for ITER. Actively stabilized, low rotation plasmas reached normalized beta of 5.6, and the ratio of normalized beta to the toroidal mode number, n = 1 and 2 ideal no-wall stability limits reached 1.2 and 1.15 respectively, determined by DCON stability analysis of the time-evolving reconstructed experimental equilibria. The significant, controlled reduction of the plasma rotation to less than one percent of the Alfven speed was produced by non-resonant magnetic braking by an applied n = 3 field. The observed plasma rotation damping is in quantitative agreement with neoclassical toroidal viscosity theory including trapped particle effects [1]. The active stabilization system employs a mode control algorithm using RWM sensor input analyzed to distinguish the amplitude and phase of the n = 1 mode. During n = 1 stabilization, the n = 2 mode amplitude increases and surpasses the n = 1 amplitude, but the mode remains stable. By varying the system gain, and relative phase between the measured n = 1 RWM phase and the applied control field, both positive and negative feedback were demonstrated. Contrary to past experience in moderate aspect ratio tokamaks with poloidally continuous stabilizing structure, the RWM can become unstable in certain cases by deforming poloidally, an important consideration for feedback system sensor and control coil design in future devices such as ITER and KSTAR. \newline **In collaboration with R.E. Bell, J.E. Menard, D.A. Gates, A.C. Sontag, J.M. Bialek, B.P. LeBlanc, F.M. Levinton, K. Tritz, H. Yuh. \newline [1] W. Zhu, S.A. Sabbagh, R.E. Bell, et al., \textit{Phys. Rev. Lett}. \textbf{96}, 225002 (2006).   

Feedback Stabilization of Resistive Wall Modes in RFX-mod

The search of efficient strategies for active control of MHD instabilities is one of the main missions of existing devices and one active field of research where important contributions can come not only from tokamak devices but also from alternative configurations. Resistive Wall Mode (RWM) instabilities are known in particular to limit plasma performances in all toroidal devices with plasma duration exceeding the penetration time of the resistive magnetic boundary surrounding the plasma. RWMs are the main limit for tokamak high-beta advanced scenarios, where a high fraction of non-inductive current is requested to study long (in the limit steady state) operations. Historically RWMs were first observed in Reversed Field Pinch (RFP) devices, where the current gradient plays the role of the drive, typically with multi-mode spectrum whose composition depends on magnetic equilibrium field profiles. RFX-mod device is a large RFP (R=2 m, a=0.46 m) where active control of MHD instabilities is intensively studied by means of a system of 192 active saddle coils placed outside the resistive shell (50 ms for Bv diffusion time) fully covering its external surface and driven by a digital controller. This system provides a very powerful and flexible environment where the study of RWMs physics and their active stabilisation under different experimental conditions is possible. Recent results from RFX-mod show that the complete stabilisation of multi-mode RWM spectrum at high plasma currents (Ip=1 MA) is possible allowing discharges longer than 6 times the diffusion time of the shell. Different control schemes are tested as well, such as open loop operations (intrinsic error field correction and Resonant Field Amplification studies), feedback operations using different measurement systems or incomplete set of coils to simulate systems with partial coverage by active coils.   

Resistive Stability of 2/1 Modes Near 1/1 Resonance

The stability of multiple coupled resistive modes is examined using reconstructions of experimental equilibria in the DIII-D tokamak, revealing the important physics in mode onset as discharges evolve to instability. Experimental attempts to access the highest beta in tokamak discharges, including hybrid discharges, are typically terminated by the growth of a large 2/1 tearing mode. In hybrid discharges the plasma current is significantly noninductive, intended to lengthen the discharge time, while sustaining the baseline dimensionless parameters of a burning plasma experiment. Model equilibria, based on experimental reconstructions from these discharges, are generated varying $q_{min}$ and pressure. For each equilibrium the PEST3 code is used to determine the ideal MHD solution including both tearing and interchange parities at all resonant surfaces. This outer region solution must be matched to the resistive inner layer solutions at each rational surface to determine resistive mode stability. From this analysis we find that the approach to $q=1$ resonance simultaneously causes the 2/1 mode to become unstable and the nonresonant 1/1 displacement to become large, as the ideal beta limit rapidly decreases toward the experimental value. Here the 2/1 mode grows to a large size, leading to loss of confinement. However, this nonresonant 1/1 component is strongly coupled to the 2/2 harmonic of the unstable 3/2 mode, which is thought to contribute to the current drive sustaining $q_{min}$ above 1 in hybrid discharges. Thus, the approach to $q=1$ resonance is self-limiting. Nonlinear coupling to the n=2 mode is computationally investigated using the NIMROD code. This work suggests that sustaining qmin slightly above 1 will help avoid the 2/1 and allow access to significantly higher beta values in these discharges.   

Sawtooth Behavior in Bean and Oval Shapes

Experiments and modeling of sawtoothing plasmas on DIII-D show that plasma shape has a severe effect on core transport, leading to significant differences in sawtooth behavior. We compare discharges with different cross-sectional shapes: an indented bean shape and a low-triangularity oval. Results in the two shapes are quite different. The oval has a small electron but a large ion sawtooth with very small changes in safety factor at the crash. The bean has a large electron and somewhat smaller ion sawtooth with a large change in safety factor at the crash. In both cases, the axial safety factor is found to be near unity following the crash. The two shapes are designed so that the Mercier instability threshold is reached when the axial safety factor is below unity for the bean and above unity for the oval cross-sections. This allows the role of interchange modes to be differentiated from that of the kink-tearing mode. A feature of the experiment is that the two shapes can be alternated in successive plasmas, minimizing the effect of systematic errors. $T_e$, $T_i$, and $B_\theta$ profiles are resolved to 5, 160, and 500~$\mu$s respectively. The differences in the nature of the sawtooth oscillations in the bean and oval discharges are determined primarily by extreme differences in the electron heat transport during the reheat. Although the electron transport rate is effectively infinite in the oval, the ion confinement is excellent, near neoclassical. By contrast, in the bean the electron transport rate is low. Modeling the sawtooth evolution with TRANSP, the current evolution during the ramp is found to be neoclassical; the very different results arise from the very different $T_e$ profiles through the plasma resistivity. The bean and oval collapses are internal kink and quasi-interchange respectively.   

Intermediate Nonlinear Development of a Line-tied $g$-Mode

The nonlinear gravitational instability ($g$-mode) of a line-tied plasma flux tube is a prototypical model for edge localized modes (ELMs) in tokamaks and magnetotail substorms. Earlier theory predicted the explosive nonlinear growth of these modes near marginal stability~[S.~C. Cowley and M. Artun, Phys. Rep., {\bf 283}, 185-211 (1997)]. Recent direct MHD simulations with both a finite-difference code and NIMROD indicate that the mode remains bounded in magnitude throughout, from early to intermediate nonlinear phases~[P. Zhu, A. Bhattacharjee, and K. Germaschewski, Phys. Rev. Lett. {\bf 96}, 065001 (2006); P. Zhu, C.~C. Hegna, and C.~R. Sovinec, submitted to Phys. Plasmas (2006)]. The mode grows nonlinearly at a rate near or smaller than the linear growth rate, producing shock-like discontinuities and large sheared flows. To understand these simulation results, a new theoretical framework has been developed. The theory is based on an expansion using two small parameters, $\epsilon\sim |{\mbox{\boldmath $\xi$}}|/L_{\rm eq}\ll 1$, and $n^{-1}\sim k_\parallel/k_\perp\ll 1$, where ${\mbox{\boldmath $\xi$}}$ denotes the plasma displacement, $L_{\rm eq}$ is the characteristic equilibrium scale, and $k_\parallel$ and $k_\perp$ are the dominant wavenumbers of the perturbation parallel and perpendicular to equilibrium magnetic field lines, respectively. When $\epsilon\sim n^{-1}$, the Cowley-Artun regime is recovered where the plasma is incompressible to the lowest order and the Lagrangian compression is very small [$\nabla_0\cdot{\mbox{\boldmath $\xi$}}\sim{\cal O}(n^{-1})$]. In this regime, the nonlinearities only modify the development of the global mode envelop across field lines whereas the local eigenmode structure along field lines remains intact. The detonation regime where the nonlinear growth of the mode tends to a finite-time singularity is a narrower subset of the Cowley-Artun regime. However, this regime is not generic and breaks down when $\epsilon\gg n^{-1}$. In the intermediate nonlinear phase when $\epsilon\sim n^{-1/2}$, the lowest order Lagrangian compression is significant [$\nabla_0\cdot{\mbox{\boldmath $\xi$}}\sim{\cal O}(1)$]. During this phase, the nonlinearities directly influence the growth of the local eigenmodes, and couple the global mode structure across and along field lines. The corresponding governing equations for this intermediate nonlinear phase are derived. Comparison of the predictions of analytic theory and numerical simulations will be discussed.   

Effect of Peeling Ballooning Stability on Steady-State ELM-Free

Recent stability modeling provides insight into two operating regimes that maintain the good energy confinement properties associated with the edge H-mode transport barrier (ETB), but without edge-localized modes (ELMs), or the buildup of plasma impurities characteristic of other ELM-free regimes. The steady state, low toroidal mode number, edge localized mode observed in Q(quiescent)H-mode, the edge harmonic oscillation (EHO), is consistent with a model in which the low collisionality and high rotational shear, associated with the low density and counter neutral beam injection, leads to a low n peeling-kink mode becoming the most unstable. This model suggests that this mode is saturated by the stabilizing effects of the conducting wall, both through the wall currents canceling the mode magnetic field perturbation, and through the reduced rotational shear associated with drag of the mode on the wall which reduces the linear growth rate at low n. In addition transport associated with the mode itself may help to keep it near the marginal stability point as observed experimentally. In resonant magnetic perturbation (RMP) H-mode, an external n=3 coil provides the non-axisymmetric perturbation. Complete ELM suppression is limited to low collisionality where the RMP generated stochastic field is effective at increasing transport in the ETB. The ETB pressure gradient is kept below the stability limit and its value controlled with the RMP coil. In the case of QH-mode the EHO may play the role of the RMP field in enhancing particle transport, avoiding the buildup of impurity or main ion density. Planned experiments, using the new counter neutral beam injector on DIII-D, will address the importance of rotation in these two regimes, and the effect of plasma shaping on RMP H-mode will also be reported.