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Register a new userUnified theory of the semi-collisional tearing mode and internal kink mode in a hot tokamak: implications for sawtooth modelling
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In large hot tokamaks like JET, the width of the reconnecting layer for resistive modes is determined by semi-collisional electron dynamics and is much less than the ion Larmor radius. Firstly a dispersion relation valid in this regime is derived which provides a unified description of drift-tearing modes, kinetic Alfven waves and the internal kink mode at low beta. Tearing mode stability is investigated analytically recovering the stabilising ion orbit effect, obtained previously by Cowley et al. [Phys. Fluids (29) 3230 1986], which implies large values of the tearing mode stability parameter Delta prime are required for instability. Secondly, at high beta it is shown that the tearing mode interacts with the kinetic Alfven wave and that there is an absolute stabilisation for all Delta prime due to the shielding effects of the electron temperature gradients, extending the result of Drake et. al [Phys. Fluids (26) 2509 1983] to large ion orbits. The nature of the transition between these two limits at finite values of beta is then elucidated. The low beta formalism is also relevant to the m=n=1 tearing mode and the dissipative internal kink mode, thus extending the work of Pegoraro et al. [Phys. Fluids B (1) 364 1989] to a more realistic electron model incorporating temperature perturbations, but then the smallness of the dissipative internal kink mode frequency is exploited to obtain a new dispersion relation valid at arbitrary beta. A diagram describing the stability of both the tearing mode and dissipative internal kink mode, in the space of Delta prime and beta, is obtained. The trajectory of the evolution of the current profile during a sawtooth period can be plotted in this diagram, providing a model for the triggering of a sawtooth crash.
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The stabilization of tearing modes with rf driven current benefits from the cooperative feedback loop between rf power deposition and electron temperature within the island. This effect, termed rf current condensation, can greatly enhance and localize current driven within magnetic islands. It has previously been shown that the condensation effect opens the possibility of passive stabilization with broad rf profiles, as would be typical of LHCD for steady state operation. Here we show that this self-healing effect can be dramatically amplified by operation in a hot ion mode, due to the additional electron heat source provided by the hotter ions.
A detailed understanding of island seeding is crucial to avoid (N)TMs and their negative consequences like confinement degradation and disruptions. In the present work, we investigate the growth of 2/1 islands in response to magnetic perturbations. Although we use externally applied perturbations produced by resonant magnetic perturbation (RMP) coils for this study, results are directly transferable to island seeding by other MHD instabilities creating a resonant magnetic field component at the rational surface. Experimental results for 2/1 island penetration from ASDEX Upgrade are presented extending previous studies. Simulations are based on an ASDEX Upgrade L-mode discharge with low collisionality and active RMP coils. Our numerical studies are performed with the 3D, two fluid, non-linear MHD code JOREK. All three phases of mode seeding observed in the experiment are also seen in the simulations: first a weak response phase characterized by large perpendicular electron flow velocities followed by a fast growth of the magnetic island size accompanied by a reduction of the perpendicular electron velocity, and finally the saturation to a fully formed island state with perpendicular electron velocity close to zero. Thresholds for mode penetration are observed in the plasma rotation as well as in the RMP coil current. A hysteresis of the island size and electron perpendicular velocity is observed between the ramping up and down of the RMP amplitude consistent with an analytically predicted bifurcation. The transition from dominant kink/bending to tearing parity during the penetration is investigated.
The role of anisotropic thermal diffusivity on tearing mode stability is analysed in general toroidal geometry. A dispersion relation linking the growth rate to the tearing mode stability parameter, Delta, is derived. By using a resistive MHD code, modified to include such thermal transport, to calculate tearing mode growth rates, the dispersion relation is employed to determine Delta in situations with finite plasma pressure that are stabilised by favourable average curvature in a simple resistive MHD model. We also demonstrate that the same code can also be used to calculate the basis-functions [C J Ham, et al, Plasma Phys. Control. Fusion 54 (2012) 105014] needed to construct Delta.
The RF stabilization of tearing modes with current condensation has the potential to increase stabilization efficiency and loosen power localization requirements. Such benefits stem from the cooperative feedback between the RF deposition and resulting island temperature perturbation governed by diffusion. A self consistent treatment of the damping of an rf ray as it traverses the island shows that low damping scenarios can require unfavorably high powers to overcome initial power leakage and effectively capitalize on the nonlinear effect. In this work it is demonstrated that for such regimes,modulated stabilization schemes can achieve significant improvements in heating and current drive contributions to stabilization for the same average power as a continuous wave scheme. The impact of modulation frequency and duty cycle on the performance is explored, the results of which suggest modulation strategies in which the pulsing periods are kept on the order of a diffusive time.
Calculations of tearing mode stability in tokamaks split conveniently into an external region, where marginally stable ideal MHD is applicable, and a resonant layer around the rational surface where sophisticated kinetic physics is needed. These two regions are coupled by the stability parameter. Pressure and current perturbations localized around the rational surface alter the stability of tearing modes. Equations governing the changes in the external solution and - are derived for arbitrary perturbations in axisymmetric toroidal geometry. The relationship of - with and without pressure flattening is obtained analytically for four pressure flattening functions. Resistive MHD codes do not contain the appropriate layer physics and therefore cannot predict stability directly. They can, however, be used to calculate -. Existing methods (Ham et al. 2012 Plasma Phys. Control. Fusion 54 025009) for extracting - from resistive codes are unsatisfactory when there is a finite pressure gradient at the rational surface and favourable average curvature because of the Glasser stabilizing effect. To overcome this difficulty we introduce a specific pressure flattening function that allows the earlier approach to be used. The technique is first tested numerically in cylindrical geometry with an artificial favourable curvature. Its application to toroidal geometry is then demonstrated using the toroidal tokamak tearing mode stability code T7 (Fitzpatrick et al. 1993 Nucl. Fusion 33 1533) which uses an approximate analytic equilibrium. The prospects for applying this approach to resistive MHD codes such as MARS-F (Liu et al. 2000 Phys. Plasmas 7 3681) which utilize a fully toroidal equilibrium are discussed.
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