Anders Bondeson

Chapter 3: MHD stability, operational limits and disruptions

Progress in the area of MHD stability and disruptions, since the publication of the 1999 ITER Physics Basis document (1999 Nucl. Fusion 39 2137-2664), is reviewed. Recent theoretical and experimental research has made important advances in both understanding and control of MHD stability in tokamak plasmas. Sawteeth are anticipated in the ITER baseline ELMy H-mode scenario, but the tools exist to avoid or control them through localized current drive or fast ion generation. Active control of other MHD instabilities will most likely be also required in ITER. Extrapolation from existing experiments indicates that stabilization of neoclassical tearing modes by highly localized feedback-controlled current drive should be possible in ITER. Resistive wall modes are a key issue for advanced scenarios, but again, existing experiments indicate that these modes can be stabilized by a combination of plasma rotation and direct feedback control with non-axisymmetric coils. Reduction of error fields is a requirement for avoiding non-rotating magnetic island formation and for maintaining plasma rotation to help stabilize resistive wall modes. Recent experiments have shown the feasibility of reducing error fields to an acceptable level by means of non-axisymmetric coils, possibly controlled by feedback. The MHD stability limits associated with advanced scenarios are becoming well understood theoretically, and can be extended by tailoring of the pressure and current density profiles as well as by other techniques mentioned here. There have been significant advances also in the control of disruptions, most notably by injection of massive quantities of gas, leading to reduced halo current fractions and a larger fraction of the total thermal and magnetic energy dissipated by radiation. These advances in disruption control are supported by the development of means to predict impending disruption, most notably using neural networks. In addition to these advances in means to control or ameliorate the consequences of MHD instabilities, there has been significant progress in improving physics understanding and modelling. This progress has been in areas including the mechanisms governing NTM growth and seeding, in understanding the damping controlling RWM stability and in modelling RWM feedback schemes. For disruptions there has been continued progress on the instability mechanisms that underlie various classes of disruption, on the detailed modelling of halo currents and forces and in refining predictions of quench rates and disruption power loads. Overall the studies reviewed in this chapter demonstrate that MHD instabilities can be controlled, avoided or ameliorated to the extent that they should not compromise ITER operation, though they will necessarily impose a range of constraints.

Feedback stabilization of nonaxisymmetric resistive wall modes in tokamaks. I. Electromagnetic model

Active feedback stabilization of pressure-driven modes in tokamaks is studied computationally in toroidal geometry. The stability problem is formulated in terms of open-loop transfer functions for fluxes in sensor coils resulting from currents in feedback coils. The transfer functions are computed by an extended version of the MARS stability code [A. Bondeson et al., Phys. Fluids B 4, 1889 (1992)] and can be accurately modeled by low order rational functions. In the present paper stability is analyzed for a system with an ideal amplifier (current control). It is shown that feedback with modest gain, and a single coil array poloidally, gives substantial stabilization for a range of coil shapes. Optimum design uses sensors for the poloidal field, located inside the resistive wall, in combination with rather wide feedback coils outside the wall. Typically, the feedback does not strongly modify the plasma-generated magnetic field perturbation. A future companion paper [C. M. Fransson et al., Phys. Plasmas (ac...

Stable FEM-FDTD Hybrid Method for Maxwell's Equations

In this thesis edge elements are applied to solve several problems in computational electromagnetics. In particular, a hybrid scheme joining the Finite Element Method (FEM) and the Finite-Difference Time-Domain (FDTD) algorithm is developed, tested and exploited. The hybrid scheme combines the efficiency of FDTD with the ability of the FEM to model complex geometry. The hybrid scheme is rigorously proven to be stable up to the maximal FDTD time step without added dissipation and it is free from spurious solutions. The reflection from the FEM-FDTD interface is low. The hybrid scheme has been tested by computing the Radar Cross Section (RCS) for a Perfect Electric Conducting (PEC) sphere and the NASA almond. It has also been used for extensive parameter studies of patch antennas and transitions from a waveguide to a microstrip.

Quasiperiodically forced dynamical systems with strange nonchaotic attractors

Abstract We discuss the existence and properties of strange nonchaotic attractors of quasiperiodically forced nonlinear dynamical systems. We do this by examining a particular model differential equation, φ = cos φ + e cos 2φ + ⨍(t) , where ⨍ is a two-frequency quasiperiodic function of t. When ϵ = 0 the analysis of the equation is facilitated since then it can be related to the Schrodinger equation with quasiperiodic potential. We show that the equation does indeed exhibit strange nonchaotic attractors, and we consider the important question of whether these attractors are typical in the sense that they exist on a set of positive Lebesgue measure in parameter space. (The equation also exhibits two- and three-frequency quasiperiodic behavior.) We also show that the strange nonchaotic attractors have distinctive frequency spectrum; this property might make them experimentally observable.

Stabilization of resistive wall modes in ITER by active feedback and toroidal rotation

Two approaches are examined for stabilization of the resistive wall mode (RWM) of toroidal mode number n = 1 in an advanced ITER scenario. Active feedback control, with the present coil design and poloidal sensors placed just inside the inner wall, can be very efficient in stabilizing the RWM. Within the voltage limit of the present design for the feedback coils and conservative constraints on performance, the plasma pressure can be increased up to at least 70% between the no-wall and ideal-wall beta limits. Stabilization of the RWM by toroidal plasma rotation depends on the rotation profile as well as on the model for ion Landau damping. Feedback control of rotating plasmas for the advanced scenario is considered. The effect of the blanket is also studied using a simplified model.

Resistive wall stabilized operation in rotating high beta NSTX plasmas

The National Spherical Torus Experiment (NSTX) has demonstrated the advantages of low aspect ratio geometry in accessing high toroidal and normalized plasma beta, and βN ≡ 10 8〈βt〉 aB0/Ip. Experiments have reached βt = 39% and βN = 7.2 through boundary and profile optimization. High βN plasmas can exceed the ideal no-wall stability limit, βNno-wall, for periods much greater than the wall eddy current decay time. Resistive wall mode (RWM) physics is studied to understand mode stabilization in these plasmas. The toroidal mode spectrum of unstable RWMs has been measured with mode number n up to 3. The critical rotation frequency of Bondeson-Chu, Ωcrit = ωA/(4q2), describes well the RWM stability of NSTX plasmas when applied over the entire rotation profile and in conjunction with the ideal stability criterion. Rotation damping and global rotation collapse observed in plasmas exceeding βNno-wall differs from the damping observed during tearing mode activity and can be described qualitatively by drag due to neoclassical toroidal viscosity in the helically perturbed field of an ideal displacement. Resonant field amplification of an applied n = 1 field perturbation has been measured and increases with increasing βN. Equilibria are reconstructed including measured ion and electron pressure, toroidal rotation and flux isotherm constraint in plasmas with core rotation ω/ωA up to 0.48. Peak pressure shifts of 18% of the minor radius from the magnetic axis have been reconstructed.

Inertia and ion Landau damping of low‐frequency magnetohydrodynamical modes in tokamaks

The inertia and Landau damping of low‐frequency magnetohydrodynamical modes are investigated using the drift‐kinetic energy principle for the motion along the magnetic field. Toroidal trapping of the ions decreases the Landau damping and increases the inertia for frequencies below (r/R)1/2vthi/qR. The theory is applied to toroidicity‐induced Alfven eigenmodes and to resistive wall modes in rotating plasmas. An explanation of the beta‐induced Alfven eigenmode is given in terms of the Pfirsch–Schluter‐like enhancement of inertia at low frequency. The toroidal inertia enhancement also increases the effects of plasma rotation on resistive wall modes.

Application of stable FEM-FDTD hybrid to scattering problems

A recently developed, stable, finite-element method (FEM), finite-difference time-domain (FDTD) hybrid that eliminates the staircase approximation of complex geometries is tested by convergence studies for radar cross-sections. For a conducting sphere, 1 dB accuracy in all directions is obtained with nine cells per wavelength, whereas the NASA almond requires a higher resolution of about 15 cells per wavelength. For scatterers with a smooth boundary, the results converge quadratically with the mesh size, but for a horizontally polarized wave incident on the NASA almond, the order of convergence is lower because of singular fields at the tip.

Resistive tearing modes in the presence of equilibrium flows

A combined numerical and analytical study is presented of the linear stability problem for resistive tearing modes in the presence of perpendicular and parallel equilibrium flows. Detailed information is given about growth rates and the shape of the marginal stability curve. In the case of purely perpendicular flows, a region in parameter space is found where the growth rate is complex, and the region of instability is larger than previously recognized. An exact analytic solution is found for parallel flows when the plasma response is dominated by viscous forces.

Energy balance of the collisional tearing mode

The energy balance of the collisional tearing mode is examined within linear theory. It is found that in an asymmetric case the quadratic form given by Furth for the net release of magnetic energy must be completed with a term connected with the current gradient in the resistive layer. The growth‐rate and the inner‐layer solution are calculated in the limit where viscosity dominates over inertia. The amounts of energy going into Joule heating and either kinetic energy or viscous dissipation are calculated analytically. In the inertial regime 1/4 of the net decrease in magnetic energy goes into kinetic energy and (3)/(4) into Joule heating, while, in viscous regime, (1)/(6) goes into viscous dissipation and (5)/(6) into Joule heating. The analytical results, based on the constant‐ψ approximation, are in good agreement with numerical simulations when the resistive layer is sufficiently narrow.