Session PI3: MHD and Rotation

Collapse of core toroidal angular momentum due to the coupling of rotating magnetic islands in tokamaks

The dynamic, nonlinear evolution of tearing instabilities on DIII-D reveals a coupling of rational surfaces that can lead to phase-locking amongst multiple rotating magnetic island chains.\footnote{B. Tobias, et. al, Rev. Sci. Instrum. 85, 11D847 (2014).} This loss of flow shear increases disruptivity, particularly at the low level of rotation expected in ITER. Bifurcation of differential mode frequency and fluid rotation in hybrid scenario discharges has been interpreted by comparison to a recently developed theory of nonlinear mode coupling.\footnote{R. Fitzpatrick, Phys. Plasmas 22, 042514 (2015).} Magnetic islands of different toroidal mode number couple to flatten the toroidal rotation profile, and the resulting phase-locked state is similar to the so-called ``slinky'' mode observed in reversed field pinch devices. Reduction of the edge safety factor increases the momentum transport, easily overwhelming the local torque density available from neutral beam injection. In discharges with $q_{95}\sim$ 4.5, however, the participating modes do not remain phase-locked. In these cases, ECE-Imaging data have been used to show that the poloidal rotation of the composite, multi-helicity structure exceeds that of the measured carbon (and estimated deuterium) fluid flow. The present model of nonlinear 3-wave mode coupling does not generate the forces required to drive this rotation. Therefore, flow shear inversion represents a transition from phase-locking to a new regime of convective momentum transport in which additional mechanisms become important. These results highlight the importance of controlling multi-mode interactions in order to maintain stabilizing flow shear.   

Effects of the q Profile on Toroidal Rotation in Alcator C-Mod LHCD Plasmas

Changes in the core toroidal rotation velocity profiles following injection of lower hybrid (LH) waves have been documented in Alcator C-Mod plasmas. Shot by shot scans of LH input power have been performed at fixed magnetic field and electron density for several plasma currents. If the input power is low enough that there are still sawtooth oscillations, the change in the core rotation is in the counter-current direction, consistent in sign and magnitude with direct momentum input from the LH waves. If the power level is high enough that there are significant changes to the q profile, the change in the toroidal rotation is in the co-current direction, consistent with changes in the residual stress through its dependence on the current density profile. The direction of the rotation changes depends on the whether q$_{\mathrm{0}}$ is below or above unity, and seemingly not on the magnetic shear.   

Torque-consistent 3D force balance and optimization of non-resonant fields in tokamaks

A non-axisymmetric magnetic perturbation in tokamaks breaks the toroidal symmetry and produces toroidal torque, which is well known as neoclassical toroidal viscosity (NTV) effects. Although NTV torque is second order, it is the first-order change in the pressure anisotropy that drives currents associated with local torques and thereby modifies the field penetration in force balance. The force operator becomes non-Hermitian, but can be directly solved using parallel, toroidal, and radial force balance, leading to a modified Euler-Lagrange equation. The general perturbed equilibrium code (GPEC), which has been successfully developed to solve the modified Euler-Lagrange equation, gives the torque-consistent 3D force balance as well as self-consistent NTV torque. The self-shielding of the torque becomes apparent in the solutions in high $\beta $, which was implied in recent MARS-K applications [1]. Furthermore, the full response matrix including the torque in GPEC provides a new and systematic way of optimizing torque and non-resonant fields. Recently the optimization of 3D fields for torque has been actively studied using the stellarator optimizing tools [2], but the efficiency and accuracy can be greatly improved by directly incorporating the torque response matrix. There are salient features uncovered by response with the torque, as the response can become invisible in amplitudes but only significant in toroidal phase shift. A perturbation in backward helicity [3] is an example, in which NTV can be induced substantially but quietly without measurable response in amplitudes. A number of other GPEC applications will also be discussed, including the multi-mode responses in high-$\beta $ tokamak plasmas and the new non-axisymmetric control coil (NCC) design in NSTX-U. This work was supported by DOE Contract DE-AC02-09CH11466. \\[4pt] [1] Z. R. Wang, M. Lanctot, Y. Liu, J.-K. Park et al., Phys. Rev. Lett. \textbf{114,} 145005 (2015)\\[0pt] [2] S. Lazerson, J.-K. Park et al., Plasma Phys. Controlled Fusion, in Press (2015)\\[0pt] [3] J.-K. Park, Y. M. Jeon et al., Phys. Rev. Lett. \textbf{111}, 095002 (2013)   

Effect of Resonant Magnetic Perturbations on 3D equilibria in the MST RFP

The orientation of 3D, stellarator-like equilibria in the MST RFP can now be controlled with application of an m $=$ 1 RMP. This has led to greatly improved diagnosis, revealing enhancements in both the central electron temperature and density. Coupled to a recent advance in the V3FIT code, reconstructions of the 3D equilibria have also been dramatically improved. The RMP also inhibits the generation of high-energy \textgreater 20 keV electrons that is otherwise common with the 3D state. This state occurs when the normally broad spectrum of core-resonant m $=$ 1 tearing modes condenses, with the innermost resonant mode growing to large amplitude, reaching $\sim$ 8{\%} of the axisymmetric field strength. This occurs in plasmas of sufficiently large Lundquist number $\sim$ I$_{\mathrm{p}}$T$_{\mathrm{e}}$$^{3/2}$, and the duration of the state is maximized with zero applied Bt (infinite toroidal beta). As the dominant mode grows, eddy current in MST's conducting shell slows the mode's rotation. This leads to locking of the 3D structure, but with an orientation that varies randomly shot to shot, making diagnosis difficult. An m $=$ 1 RMP can now be applied with an array of saddle coils at the vertical insulated cut in the shell. With an amplitude br/B $\sim$ 10{\%} and a tailored temporal waveform, the RMP can force the 3D structure into any desired orientation relative to MST's diagnostics. A recent advance in V3FIT allows calculation of the substantial helical image current flowing in MST's shell, which has in turn allowed self-consistent utilization of both external and internal (Faraday rotation) measurements of the magnetic field. The ORBIT code predicts reduced stochasticity and improved confinement of high-energy electrons within the 3D structure. The suppression of these electrons by the m $=$ 1 RMP may reflect a change to the central magnetic topology. The generation of these electrons is unaffected by non-resonant perturbations, such as m $=$ 3. Supported by the US DOE.   

Gyrokinetic Simulation of Low-n Tearing Modes

Direct gyrokinetic simulation of the low-n tearing mode in a tokamak plasma has been a great computational challenge, for two reasons. First, low-n tearing modes, unlike the micro-tearing modes, have very small growth rates and very fine mode structure in the tearing layer, which requires a large number of radial grid cells and fine control of numerical dissipation. Second, kinetic electron effects are needed in the tearing layer. Here, we first present linear gyrokinetic simulation of the low-n tearing mode in cylindrical geometry. Ions are gyrokinetic and electrons are either drift kinetic or fluid. New field solvers have been developed in the gyrokinetic code GEM [Chen and Parker, J. Comput. Phys. 220, 839 (2007)] to simulate low-n modes. For the fluid electron model, an eigenmode analysis with finite Larmor radius effects has been developed to study the linear resistive tearing mode. Excellent agreement between eigenmode analysis and initial value gyrokinetic simulation is obtained. The mode growth rate is shown to scale with resistivity as $\eta^{1/3}$, the same as the semi-collisional regime in previous kinetic treatments. Simulation of the collisionless and semi-collisional tearing mode with drift kinetic electrons has been carried out with GEM's direct split-weight control-variate algorithm. It is found that a full torus simulation of the m=2, n=1 tearing mode in a present day large tokamak is still difficult with kinetic electrons, but a generalized matching technique can be used to ameliorate the problem. The radial dimension is divided into an external region and the tearing region, with the external region described by a reduced model that gives the boundary condition for the tearing region. The size of the tearing region is small compared with the minor radius, but not arbitrarily small as done in the standard asymptotic matching approach. Gyrokinetic simulation verifies the collisionless tearing mode growth rate with finite electron mass, the semi-collisional linear growth rate, as well as the nonlinear scaling of particle trapping frequency with linear growth rate in the collisionless regime. Nonlinear results including collisions show that the mode initially saturates at low amplitude, followed by an algebraic growth period (the Rutherford regime). Current progress on simulating the low-n toroidal tearing mode in toroidal geometry will be reported.   

Quasi-steady multiple flux tubes induced by localized current perturbation in toroidal plasma

Quasi-steady helical modes with dual, triple, or more flux tubes are easily produced by localized current drive in the core of \textit{sawtoothing} plasma on the KSTAR tokamak [1, 2]. Individual flux tubes have $m/n=1/1$ helicity, co-rotate around the magnetic axis, and later merge into a single $m=1$ mode. The merged mode eventually crashes with rapid collapse of the core pressure and the next cycle repeats the same pattern, exhibiting sawtooth-like oscillations in the core pressure. The generation mechanism of multiple flux tubes (MFTs) has been studied in two different approaches to understand the observed trend that the number of flux tubes increases as the current drive location moves away from the magnetic axis up to about the magnetic surface of the safety factor $q=1$ at the mode collapse: (1) nonlinear reduced MHD simulation with a localized current source modeling the time-varying interaction between the current source and flux tubes [3] and (2) linear MHD simulation with a prescribed $q$ profile with a radially localized current blip. Both studies show that MFTs can be produced only in plasmas with nearly flat $q$ profile close to unity, \textit{suggesting the collapse of the $m=1$ mode (i.e., sawtooth crash) is complete.} Recent observation of long-lived MFTs induced by localized current drive in non-sawtoothing plasma suggests that $q$ profile evolution toward lower-$m$ instability is required for the merging and crash of MFTs.\\[4pt] [1] G.S. Yun et al., Phys. Rev. Lett. 109,145003 (2012)\\[0pt] [2] G.H. Choe et al., Nucl. Fusion 55, 013015 (2015)\\[0pt] [3] A. Bierwage et al., Nucl. Fusion 55, 013016 (2015)