Ideal Internal Kink Research Articles

The stability of the ideal internal kink mode in realistic tokamak geometry

The ideal stability of the internal kink mode is analysed for realistic tokamak geometry. Accurate numerical results demonstrate convergence with Bussac's toroidal solution (Bussac M N, Pellat R, Edery D and Soulé J L 1975 Phys. Rev. Lett. 35 1638) for inverse aspect ratio ϵ1 ∼ 0.01 at the q = 1 rational surface. For realistic inverse aspect ratio (e.g. ϵ1 ∼ 0.1) the growth rate is found to scale linearly with the poloidal beta and to be smaller than the analytical prediction of the toroidal growth rate. The effect of the shaping of the plasma cross-section is also analysed. Analytical results are found to disagree with numerical results at realistic inverse aspect ratio primarily because of the toroidal nature of the Mercier mode shaping terms. Furthermore, in addition to the Mercier shaping terms, there are quasicylindrical contributions to the kink mode which are quadratic and stabilizing in both triangularity and ellipticity. To verify this empirically and to further demonstrate the importance of the ideal internal kink stability boundary, discharges in the tokamak à configuration variable are shown to display longer sawteeth for both very positive and negative triangularity. Finally, a parameter scan in the triangularity, elongation, aspect ratio and poloidal beta has been undertaken using the code KINX. Versatile predictions of the ideal internal kink stability in future tokamaks should be assisted by the functional fitting presented here of the parameter scans.

Effects of nonresonant hot ions with large orbits on Alfvén cascades and on magnetohydrodynamic instabilities in tokamaks

The effects of nonresonating hot ions on the spectrum of magnetohydrodynamic (MHD) waves and instabilities in tokamaks are studied in the limit when the width of the hot ion drift orbits is much larger than the radial scale length of the MHD perturbations. Due to the large magnetic drift velocities the hot ions cannot contribute to the MHD perturbations directly, but two main effects of the hot ions, the hot-ion density-dependent effect and the hot-ion pressure-dependent effect, influence the MHD perturbations indirectly. The physics of both effects is elucidated and it is shown that both these effects can be described in MHD approach. A new code, MISHKA-H (MISHKA including the hot-ion indirect effects), is developed as an extension of the ideal MHD code MISHKA-D [Huysmans et al., Phys. Plasmas 8, 4292 (2002)]. Analytical benchmarks for this code are given. Results of the MISHKA-H code on Alfvén spectrum in a shear-reversed discharges with ion-cyclotron resonance frequency (ICRF) heating are presented. Modeling of Alfvén cascades and their transition into toroidal Alfvén eigenmodes in shear-reversed tokamak equilibrium is considered. The hot-ion effect on the unstable branch of the MHD spectrum is studied for the test case of an n=1 ideal MHD internal kink mode, which is relevant to short-period sawteeth in low-density plasmas observed in Joint European Torus (JET) [Rebut et al., Proceedings of the 10th International Conference, Plasma Physics and Controlled Nuclear Fusion, London (International Atomic Energy Agency, Vienna, 1985), Vol. I, p. 11] experiments with high-power ICRF heating.

Aspect ratio dependence of the ideal internal kink mode stability in a toroidal plasma with circular cross section

The theory by Bussac et al. [Phys. Rev. Lett. 35, 1638 (1975)] on the stability of the ideal, internal kink mode in a toroidal plasma with circular cross section is extended to finite values of the inverse aspect ratio εa. The aspect ratio dependence comes from the term following the Bussac term in a large aspect ratio expansion of the potential energy δW of the mode. Although formally of higher order in the inverse aspect ratio, this term is of the same order of magnitude as the Bussac term when the safety factor q is close to (but below) unity at the plasma center and εa is finite. The new term is always stabilizing, but affects δW only marginally when the poloidal beta value βp is small and/or εa is small. For values of εa relevant for most tokamaks and βp larger than 0.3–0.5, however, the new term strongly modifies δW and leads to a second stability regime of the ideal, internal kink mode, in agreement with earlier numerical calculations.

Internal Kink Instability in Shaped Tokamaks

A criterion of an ideal internal kink mode is derived for a shaped tokamak configuration in which q-profile is very flat in the core region. A combining criterion is obtained including the necessary criterion of Mercier and the sufficient criterion of Lortz. The new criterion makes progress compared with the necessary criterion of Mercier. In the elongated plasma, a poloidal beta can cause instability, while the triangularity has a stabilizing effect. The result is applicable for DIII-D and SUNIST.

Effect of combined triangularity and ellipticity on the stability limit of the ideal internal kink mode in a tokamak

The effect of combined triangular and elliptical shaping of the plasma cross section on the stability limit of the ideal, internal kink mode in a tokamak is analyzed by means of a computer algebra expansion of the magnetohydrodynamic equations. The work extends the result of a previous investigation of the effect of ellipticity alone on this mode in toroidal plasmas [Wahlberg, Phys. Plasmas 5, 1387 (1998)]. It is shown that, in contrast to the strongly destabilizing effect of (positive) ellipticity alone, the effect of combined positive ellipticity and positive triangularity is stabilizing. The effect is of the same importance as the effect of ellipticity alone if the triangularity of the q=1 surface is of the same order of magnitude as the inverse aspect ratio. When the safety factor at the magnetic axis is close to (but below) unity, the geometrical factor governing the stability of the ideal, internal kink mode is found to be the same as the corresponding factor in the Mercier criterion for shaped tokamaks.

Plasma shape effects on sawtooth/internal kink stability and plasma shaping using electron cyclotron wave current profile tailoring in TCV

The effect of plasma shaping (triangularity and elongation) on sawtoothstability is addressed as well as the technique of current profile broadening using off-axis ECHto enlarge the operational range towards higher elongations. The plasma shape strongly influencesthe sawtooth period and amplitude. This effect is emphasized by ECH, with the sawtooth period becoming shorter at low triangularity or at high elongation; for these plasma shapes, the pressure profile inside the q = 1 surface remains essentiallyflat throughout the sawtooth cycle. A comparison of the sawtooth response with marginal Mercier stability shows that the critical pressure gradient at q = 1 is particularly low for plasma shapes where the increased sawtooth repetition frequency prevents the peaking of the pressure profiles. For these shapes, stability calculations show that the ideal internalkink is also unstable. The stability of highly elongated plasmas depends largely on the current profiles. The operational range at low currents has been extended towards higher elongations using ECH discharges. Far off-axis second harmonic X mode ECHpower deposition proves to be an efficient tool for current profile tailoring, allowing a significant elongation increase at constant quadrupole field.

The effects of sheared toroidal plasma rotation on the internal kinkmode in the banana regime

The stability of the ideal internal kink mode is calculated, taking intoaccount the kinetic response of thermal ions in the external region and thesingular layer. By extending the collisionless dispersion relation to includethe equilibrium radial electric field it is found that the stability of theinternal kink mode depends sensitively on sheared toroidal plasma rotation.The sheared toroidal plasma rotation can increase the critical pressure forinternal kink mode displacements typically by a factor of two.

Critical magnetic shear for the tokamak ideal internal kink mode in the ion kinetic regime

It is shown that for the ideal internal kink mode of tokamak plasma, there exists a critical magnetic shear s0c for its excitation when the finite ion Larmor radius effect is considered in the mode singular layer. This can be related to the sudden onset of the sawtooth and also to the stabilization of the sawtooth by the trapped hot ions observed experimentally. The alpha particle effect on this mode is revisited and some new results are reported.

Sawtooth evolution during JET ion-cyclotron-resonance-heated pulses

The recent Joint European Torus deuterium-tritium campaign has yielded ion-cyclotron-resonance-heated (ICRH) pulses during which both the sawtooth characteristics and the ICRH minority ion population both evolve substantially. At multiple times during each pulse, the evolution of the kinetic-fluid MHD energy is calculated from measurement of the energetic ions and compared with the evolving sawtooth duration. There is strong correlation between sawtooth duration and minority ion stabilization of the ideal internal kink.

Stability of negative central magnetic shear discharges in the DIII-D tokamak

Discharges with negative central magnetic shear (NCS) hold the promise of enhanced fusion performance in advanced tokamaks. However, stability to long wavelength magnetohydrodynamic modes is needed to take advantage of the improved confinement found in NCS discharges. The stability limits seen in DIII-D [J. L. Luxon and L. G. Davis, Fusion Technol. 8, 441 (1985)] experiments depend on the pressure and current density profiles and are in good agreement with stability calculations. Discharges with a strongly peaked pressure profile reach a disruptive limit at low beta, βN=β(I/aB)−1⩽2.5 (% m T/MA), caused by an n=1 ideal internal kink mode or a global resistive instability close to the ideal stability limit. Discharges with a broad pressure profile reach a soft beta limit at significantly higher beta, βN=4 to 5, usually caused by instabilities with n>1 and usually driven near the edge of the plasma. With broad pressure profiles, the experimental stability limit is independent of the magnitude of negative shear but improves with the internal inductance, corresponding to lower current density near the edge of the plasma. Understanding of the stability limits in NCS discharges has led to record DIII-D fusion performance in discharges with a broad pressure profile and low edge current density.

Gyroscopic stabilization of the internal kink mode

Toroidal rotation is shown to have a strongly stabilizing effect on the ideal internal kink mode. In the gyroscopically stabilized regime, the internal kink is a neutrally stable discrete eigenmode with a finite rotation velocity relative to the plasma at the resonant surface. In the unstable regime, the mode frequency is equal to the plasma rotation frequency at the resonant surface, so that reconnection will prevent nonlinear saturation.

Fluctuations and low-frequency stability

Electromagnetic fluctuations can cause a modification of time-averaged plasma dynamics. Their presence manifests itself through ponderomotive and magnetization forces. For incompressible transverse plasma motions, the ponderomotive force is mainly responsible for an alteration of the momentum balance while the magnetization causes primarily a modification of Ohm’s law. These modifications can substantially affect the low-frequency stability. Specifically, it can be shown that the ponderomotive force of fluctuations driven by velocity-space instabilities can stabilize the ideal internal kink mode in a tokamak and the flute mode in a tandem mirror reactor. Magnetization force of fluctuations can also affect the resistive modes. For the resistive internal kink, thought to be responsible for sawtooth oscillations, this effect can be stabilizing.

Stability of internal kink modes with energetic trapped particles

An analysis of the stability of ideal internal kink modes in high temperature tokamaks is presented. The theory includes kinetic effects associated with energetic trapped particles, bulk ion diamagnetic drifts as well as transit resonances. A corresponding dispersion relation is derived and analysed both analytically and numerically. The roles and implications of various kinetic effects are delineated and comparisons with previously obtained results made.

Nonlinear evolution of ideal internal kink modes in tokamaks with nonmonotonic q profiles

A calculation of the nonlinear state of the m=n=1 internal kink mode within the ideal magnetohydrodynamic approximation is presented. The nonlinear problem is divided into a layer around the q=1 surface and an outer region. An examination of the nonlinear terms from the two regions shows that within the optimal ordering of the linear theory, nonlinear terms from the outer region are formally bigger than the corresponding terms of the layer. However, because of the rigid shift nature of the linear radial displacement, these terms are actually weaker than the terms from the layer. The contribution from the nonlinear terms from the outer region has been calculated and it is shown that it produces a small correction in the final expression. This is essential for the calculation of Avinash et al. [Phys. Rev. Lett. 59, 2647 (1987)] and Bussac and Pellat [Phys. Rev. Lett. 59, 2650 (1987)] to be correct. Further, the case of arbitrary resonant q profiles has been considered and the relevance of these results to recently observed sawtooth activity in tokamaks is discussed.

Energetic particle effects on global magnetohydrodynamic modes

The effects of energetic particles on magnetohydrodynamic (MHD) type modes are studied using analytical theories and the nonvariational kinetic-MHD stability code (nova-k) [Workshop on Theory of Fusion Plasmas, (Societa Italiana di Fisica, Bologna, 1987), p. 185]. In particular, the problems of (1) the stabilization of ideal MHD internal kink modes and the excitation of resonant ‘‘fishbone’’ internal modes and (2) the alpha particle destabilization of toroidicity-induced Alfvén eigenmodes (TAE) via transit resonances are addressed. Analytical theories are presented to help explain the nova-k results. For energetic trapped particles generated by neutral beam injection or ion cyclotron resonant heating, a stability window for the n=1 internal kink mode in the hot particle beta space exists even in the absence of core ion finite Larmor radius effect. On the other hand, the trapped alpha particles are found to resonantly excite instability of the n=1 internal mode and can lower the critical beta threshold. The circulating alpha particles can strongly destabilize TAE modes via inverse Landau damping associated with the spatial gradient of the alpha-particle pressure.

Influence of an energetic ion population on tokamak plasma stability

The stability of a tokamak plasma to ideal and resistive internal kink modes in the presence of an energetic trapped-particle population is analyzed. Domains of stability are found for both slowing-down and Maxwellian particle distributions. The stable domains are at moderately high plasma beta and involve a significant, but easily obtainable high-energy trapped-particle density.

The effect of compressibility on magnetohydrodynamic instabilities in toroidal tokamak geometry

The effect of compressibility on the stability of ideal internal kink modes, ballooning modes, and infernal modes is studied. Most characteristics of these modes and their thresholds are not affected by compressibility. However, in general, the linear growth rates obtained from calculations that include compressibility are larger than those from calculations that do not. In the calculations with compressibility, the sound wave contribution to δW, Γ|∇⋅V|2, is negligible. The increase in the linear growth rate with compressibility is caused mainly by the change in the parallel component of the displacement vector.

Magnetohydrodynamic stability and nonlinear evolution of the m=1 mode in toroidal geometry for safety factor profiles with an inflection point

Magnetohydrodynamic (MHD) stability and nonlinear evolution of the m=1 mode are studied in toroidal geometry for safety factor profiles having an inflection point at the q=1 surface. For ideally stable cases, linear growth rates of the resistive m=1 mode are decreased by any combination of the following variations: decreasing the aspect ratio, decreasing shear at the q=1 surface, increasing S, decreasing q0, and/or decreasing beta. Transitions from resistive kink to tearing mode to complete stability are accompanied by increasing localization as the mode is stabilized. The ideal stability of the m=1 mode for low-shear inflection-point profiles is broken at very low, but finite, beta by the appearance of a radially localized ideal interchange mode. As beta is increased, the m=1 mode becomes a global ideal internal kink. These ideal modes are stabilized by decreasing the aspect ratio and, to a lesser extent, by increasing shear at the q=1 surface. Nonlinear evolution of the m=1 mode is found to follow the Kadomtsev complete reconnection process or to be stable, depending on the linear stability of the mode. However, all nonlinear calculations were carried out at Lundquist number S=105 and hence contain substantial resistive kink contributions. Consequently, nonlinear evolution in the m=1 tearing mode regime is not examined. Even for cases in which ideal effects dominate the linear mode, Kadomtsev reconnection is observed. The nonlinear stability of linearly stable modes was also uniformly observed, even for several cases having large initial perturbations. Hence complete linear stabilization resulting from a combination of toroidal effects and low shear at the q=1 surface can suppress sawteeth, as described by the Kadomtsev reconnection process, in tokamaks having safety factor profiles with q0<1 at the magnetic axis; but, for the S values considered, nonlinear calculations of unstable modes result in complete Kadomtsev reconnection.

Toroidal internal kink stability in tokamaks with ultra flat q profiles

Linear stability properties of the ideal internal kink mode in toroidal geometry are calculated for equilibria in which the q(r) profile is very flat and q ≈ 1 in the core region of the plasma. Marginal stability criteria and growth rates are calculated analytically in the large aspect ratio limit and compared with numerical results from the FAR code. The theory is developed for m = n = 1 modes and for higher m (= n) modes. The temperature perturbation due to adiabatic expansion in the linear phase of the 1/1 mode is calculated and compared with experimental data, showing that the predictions from ideal MHD theory cannot account for the observed temperature increase on JET. Comparison with collisionless theory suggests that the observed temperature increase can be accounted for by compression of the trapped particles.

Numerical simulations of ideal internal kink modes with flat central q-profile

The paper describes non-linear numerical simulations of ideal internal kink modes in cylindrical geometry with a flat central q-profile as a model for the rapid sawtooth collapse seen in large tokamaks. The resulting plasma motion is consistent with that obtained experimentally from X-ray tomographic reconstruction. Attention is drawn to the effects of limited poloidal resolution in this diagnostic.