Low-n Modes Research Articles

Improved plasma performance in tokamaks with negative magnetic shear.

A tokamak plasma configuration is reported that simultaneously improves on the maximum stable plasma pressure, the bootstrap current contribution, and kinetic stability to temperature and density gradient driven modes in toroidal geometry. It is characterized by negative magnetic shear in the plasma interior and a peaked pressure profile. Stability to the ideal low-n external kink modes requires a conducting shell at 1.3 times the plasma minor radius. This novel plasma configuration is promising for improved plasma performance in advanced tokamak experiments.

Two-fluid equations for low-n singular modes in the low-frequency regime

Braginskii’s two-fluid equations are employed to obtain the eigenmode equations governing low-n singular modes in toroidal geometry. Collisional effects are neglected. The equilibrium electric field is taken into account. The frequency of the modes is assumed to be lower than that of the ion acoustic wave along the magnetic field lines. In this low-frequency regime it is shown that the pressure of the ion fluid tends to be constant along the magnetic field lines, as also does that of the electron fluid, and the parallel electric field tends to vanish. The finite-gyroradius (FGR) effect is recovered by taking into consideration the diamagnetic drift velocity and the gyroviscosity of the ion fluid. The results show that the toroidal effect gives rise not only to the so-called apparent mass effect, but also to an enhancement of the FGR and equilibrium electric field effects. The toroidicity-induced FGR and equilibrium electric field effects are shown to be even more important than those exhibited in the cylindrical equilibrium model.

Fully three-dimensional ideal magnetohydrodynamic stability analysis of low-n modes and Mercier modes in stellarators

The terpsichore three-dimensional linear ideal magnetohydrodynamic (MHD) stability code [Theory of Fusion Plasmas, Proceedings of the Joint Varenna–Lausanne International Workshop, Chexbres, Switzerland, 1988 (Editrice Compositori, Bologna, Italy, 1989), p. 93; Controlled Fusion and Plasma Heating, Proceedings of the 17th European Conference, Amsterdam, The Netherlands (European Physical Society, Petit-Lancy, Switzerland, 1990), Vol. 14B, Part II, p. 931; Theory of Fusion Plasmas, Proceedings of the Joint Varenna–Lausanne International Workshop, Valla Monastero, Varenna, Italy, 1990 (Editrice Compositori, Bologna, Italy, 1990), p. 655] has been extended to the full MHD equations. The new code is used to calculate the physical growth rates of nonlocal low-n modes for l=2 torsatron configurations. A comprehensive investigation of the relation between the Mercier modes and the low-n modes has been performed. The unstable localized low-n modes are found to be correlated with the Mercier criterion. Finite growth rates of the low-n modes correspond to finite values of the Mercier criterion parameter. Near the Mercier marginal stability boundary, the low-n modes tend to be weakly unstable with very small growth rates. However, the stability of global-type low-n modes is found to be decorrelated from that of Mercier modes. The low-n modes with global radial structures can be more stable or more unstable than Mercier modes.

MHD stability analysis of high- beta JET discharges

The theoretical stability of high- beta JET discharges is discussed. For discharges near the beta limit, a significant fraction of the core is found to be marginally stable to high-n ballooning modes. For low-n internal modes (n=1 and 2) the theoretical predictions are found to correlate well with the experimentally observed soft X-ray behaviour, and provide a basis for predicting the central q-profile.

Magnetic fluctuation studies in Extrap T1

Magnetic field fluctuations in Extrap T1 plasma discharges are studied with internal and external arrays of pick-up coils. Statistical methods are employed to determine the spatial structure of the fluctuating modes. The discharge evolution is divided into two phases with different fluctuation characteristics. During start-up, dominatly m=0,1, low-n modes are seen. The poloidal harmonics are well correlated and correspond to horizontally-polarized perturbations. These fluctuations modes are attributed to variations of the toroidal equilibrium position. In the subsequent quasi-stationary stage mainly turbulent activity is observed. The experimental mode spectra are compared with a circular cylinder geometry, ideal MHD model. The stability calculation for the quasi-stationary phase gives comparable growth rates for many m-modes, which is qualitatively consistent with the measurements.

Experimental study of the relaxation cycle of a decaying spheromak in an external magnetic field

For high electron temperature plasma discharges in the S-1 spheromak device [Plasma Physics and Controlled Nuclear Fusion Research, 1984 (IAEA, Vienna, 1985), Vol. 2, p. 535], ‘‘sawtooth’’- like oscillations appear on signals of magnetic field, flux, q value, and electron temperature. Based on the internal magnetic field profiles measured by magnetic probe scans, the mechanisms and causes of these oscillations are revealed. The cycle of one oscillation consists of a toroidal current peaking phase and a subsequent relaxation phase. In the peaking phase, resistive current decay at the edge causes the spheromak to deviate from the initial minimum-energy Taylor state. The deviation was revealed experimentally by the preferential decay of toroidal flux over poloidal flux. A simple calculation shows that a peaking of the electron temperature profile is the most probable cause for the preferential decay of toroidal flux over poloidal flux. During the peaking phase, q decreases so low as to make the configuration unstable to low-n magnetohydrodynamic (MHD) modes (mainly n=2 mode). In the relaxation phase, these modes invoke current redistribution (relaxation), restoring the Taylor state. A significant finding in the relaxation phase is that a reversed toroidal field similar to that of the reversed-field pinch (RFP) configuration is sometimes measured at the edge of the plasma for a brief period. The disappearance or resistive decay of the reversed toroidal flux is attributed to a flux conversion through the magnetic reconnections caused by the low-n modes.

Equilibrium, Stability, and Deeply Trapped Energetic Particle Confinement Calculations for l = 2 Torsatron/Heliotron Configurations

This paper studies confinement properties of 1 = 2 torsatron/heliotron configurations with number of toroidal field periods, M, in the range of 10 to 14. This involves the calculation of zero-current and flux-conserving equilibria; stability against Mercier modes and low-n ideal modes, with n denoting the toroidal mode number; and orbit confinement of deeply trapped energetic particles. Optimization of both magnetohydrodynamic (MHD) and transport properties is pursued under the condition of plasma aspect ratio A = R/a {ge} 7, with R denoting the major radius and a the average plasma radius. For configurations with M, {le} 12, an average MHD beta limit of 4 to 5% is possible. The addition of a quadrupole field improves the confinement of trapped particles at zero pressure, but particle losses increase with increasing beta. This loss is less severe if the vacuum magnetic axis is shifted slightly inward. A configuration with M = 10, a coil pitch parameter p{sub c} in the range 1.25 to 1.30, and an added quadrupole field satisfies the beta and energetic particle confinement requirements for the next generation of large torsatron/heliotron devices.

Ideal MHD stability of pellet fuelled plasmas in JT-60

Stability against ideal infinite-n ballooning and low-n internal kink modes is analysed for pellet fuelled plasmas in JT-60, which have strongly peaked pressure profiles within the q = 1 resonance surface. The observed pressure gradient in pellet fuelled plasmas locally reaches the marginal value for the infinite-n ballooning mode limit just inside the q = 1 surface. After reaching the marginal value on the magnetic flux surface, the marginally stable region grows inside the q = 1 surface. However, the poloidal beta within the q = 1 surface, βp1, does not reach the limit imposed by the ideal ballooning mode. The results of the analysis of the low-n kink mode using experimental observations show that the maximum βp1 attained is consistent with the beta limit imposed by the n = 1 internal kink mode.

Ideal low-n and Mercier mode stability boundaries for ℓ = 2 torsatrons

The relationship between the stability properties of ideal low-n internal modes and the threedimensional (3-D) ideal Mercier criterion for ℓ = 2 torsatron configurations is discussed. A broad range of aspect ratios and pressure profiles are considered. It is found that the low-n modes with resonant surfaces lying in a Mercier unstable region are unstable in most of the cases studied and that the critical beta given by the Mercier criterion is always lower than the critical beta for the lowest-n unstable mode. This is verified even in the cases where global n = 1 modes are unstable. Therefore, the 3-D Mercier criterion is a useful guide in mapping the ideal stability beta limits for these torsatron configurations.

Transition to the second region of ideal MHD stability

Transition to the second region of MHD stability has been examined in circular cross-section boundary tokamak geometry. Values of ⟨β⟩ and ∊βp, corresponding to equilibria which are marginally stable in the second region on each flux surface, are presented. The aspect ratio scaling of these functions is shown to be sensitive to the value of the on-axis safety factor qo. Equilibria evolving from the first to the second region of stability have been generated by a transport code which calculates the two-dimensional evolution of the magnetic configuration self-consistently with the thermal and particle diffusion. The transport model enhances the diffusion coefficients on flux surfaces which are unstable to high-n modes, thereby altering the pressure profile locally. The q-profile, modified by neutral beam driven current so that q0 > 1, reduces the size of the unstable region sufficiently to allow transition to the second region. Auxiliary power requirements for this transition are estimated. Stability of the transition equilibria to low-n external and internal modes is also examined. All ideal modes restabilize as ∊βp approaches and exceeds unity. A close fitting conducting wall is needed to stabilize low-n external kink modes.

MHD stability of vertically asymmetric tokamak equilibria

The ideal-MHD stability properties of a special class of vertically asymmetric tokamak equilibria, with the plasma boundary a rotated ellipse, are examined. The calculations confirm that no major new physical effects are introduced and the modifications can be understood by conventional arguments. The results indicate that significant departures from up-down symmetry can be tolerated before the reduction in β becomes important for reactor operation. They show that the second stable region for internal modes can be extended to low-n modes and small-aspect-ratio devices by rotating the cross-section from a standing ellipse.

Ideal magnetohydrodynamic stability of the spheromak configuration

Results are presented of a parametric study of the ideal magnetohydrodynamic stability properties of the spheromak, or compact torus, configuration. In the absence of a nearby conducting wall, the spheromak is always unstable to at least one current-driven mode. With a conducting wall at the surface, the spheromak can be unstable to current-driven modes if the current is too peaked, i.e. q0(R/a) < 2/3, or if the shear is too low at the origin. The Mercier criterion sets an upper limit on the pressure gradient everywhere, but configurations that are everywhere Mercier-stable can be unstable to pressure-driven low-n modes. Stable toroidal configurations exist with a spherical wall separated by half a minor radius, and with βθ = 30%.

Stabilité magnétohydrodynamique des modes ballooning dans un Tokamak

We implanted in the ERATO spectral code [1] the stability criterion for ballooning modes (toroidal wave numbern→∞) [2]. The study of ballooning waves located around a given magnetic surface reduces itself to the investigation of the sign of the potential energy on this surface; this method avoids the determination of eigenvalues which is expensive in computing time. The stability criterion obtained by this way has been applied to a high-beta equilibria family proposed for JET. The stability limit found is more constraining than the limit for low-n internal modes.

Dependence of ideal-MHD kink and ballooning modes on plasma shape and profiles in tokamaks

Extensive numerical studies of ideal-MHD instabilities have been carried out to gain insight into the parametric dependence of critical β's in tokamaks. The large number of interrelated equilibrium quantities involved in establishing a critical β has demanded a careful, systematic survey in order to isolate this dependence. The results of this survey establish the scaling with geometrical quantities including aspect ratio, elongation, and triangularity in the parameter regimes appropriate to both current and reactor-sized plasmas. A moderate dependence on the pressure profile and a strong variation with the current profile is found. The principal result is that, for aspect ratio R/a ≈ 3, critical β's are of the order of 2% for circular cross-sections and 5% for plasmas with elongation K ≈2; somewhat higher values could be achieved with more optimal shaping. Finally, sequences of equilibria have been analysed to compare critical β as a function of toroidal mode number n. It is concluded that the infinite-n analytic ballooning theory provides a sufficient condition for ideal-MHD internal-mode stability. Low-n free-boundary modes appear to set a lower lim̀it.

High-βTokamaks

A model allowing the calculation of very high-..beta.. tokamak equilibria with choice of ..beta.., ..beta../sub p/, and q/sub 0/ is described. One of these equilibria having ..beta.. = 12% has been shown to be stable to low-n internal modes.