[en] A stability analysis for the resistive wall mode is studied in the presence of trapped energetic particles (EPs). When the EPs' beta exceeds a critical value, a fishbonelike bursting mode (FLM) with an external kink eigenstructure can exist. This offers the first analytic interpretation of the experimental observations [Phys. Rev. Lett. 103, 045001 (2009)]. The mode-particle resonances for the FLM and the q=1 fishbone occur in different regimes of the precession frequency of EPs. In certain ranges of the plasma rotation speed and the EPs' beta, a mode conversion can occur between the resistive wall mode and FLM.

[en] A theoretical model for investigating the effect of the trapped energetic particles (EPs) on the resistive wall mode (RWM) instability is proposed. The results demonstrate that the trapped EPs have a dramatic stabilizing effect on the RWM because of resonant interaction between the mode and the magnetic precession drift motion of the trapped EPs. The results also show that the effect of the trapped EPs depends on the wall position. In addition, the stabilizing effect becomes stronger when the plasma rotation is taken into account. For sufficiently fast plasma rotation, the trapped EPs can lead to the complete stabilization of the RWM. Furthermore, the trapped EPs can induce a finite real frequency of the RWM in the absence of plasma rotation.

[en] The 2-D ballooning transform, devised to study local high toroidal number (n) fluctuations in axisymmetric toroidal system (like tokamaks), yields a well-defined partial differential equation for the linear eigenmodes. In this paper, such a ballooning equation of the second kind is set up for ion temperature gradient driven modes pertinent to a 2-D non-dissipative fluid plasma; the resulting partial differential equation is numerically solved, to calculate the global eigenvalues, and the 2-D mode structure is presented graphically along with analytical companions. The radial localization of the mode results from translational symmetry breaking for growing modes and is a vivid manifestation of spontaneous symmetry breaking in tokamak physics. The eigenmode, poloidally ballooned at θ=±π/2, is radially shifted from associated rational surface. The global eigenvalue is found to be very close to the value obtained in 1-D parameterized (λ=±π/2) case. The 2-D eigenmode theory is applied to estimate the toroidal seed Reynolds stress [Y. Z. Zhang, Nucl. Fusion Plasma Phys. 30, 193 (2010)]. The solution obtained from the relatively simplified ballooning theory is compared to the solution of the basic equation in original coordinate system (evaluated via FFTs); the agreement is rather good.

[en] The MARS-F code [Y. Q. Liu et al., Phys. Plasmas 7, 3681 (2000)] is applied to numerically investigate the effect of the plasma pressure on the tearing mode stability as well as the tearing mode-induced electromagnetic torque, in the presence of a resistive wall. The tearing mode with a complex eigenvalue, resulted from the favorable averaged curvature effect [A. H. Glasser et al., Phys. Fluids 18, 875 (1975)], leads to a re-distribution of the electromagnetic torque with multiple peaking in the immediate vicinity of the resistive layer. The multiple peaking is often caused by the sound wave resonances. In the presence of a resistive wall surrounding the plasma, a rotating tearing mode can generate a finite net electromagnetic torque acting on the static plasma column. Meanwhile, an equal but opposite torque is generated in the resistive wall, thus conserving the total momentum of the whole plasma-wall system. The direction of the net torque on the plasma is always opposite to the real frequency of the mode, agreeing with the analytic result by Pustovitov [Nucl. Fusion 47, 1583 (2007)]. When the wall time is close to the oscillating time of the tearing mode, the finite net torque reaches its maximum. Without wall or with an ideal wall, no net torque on the static plasma is generated by the tearing mode. However, re-distribution of the torque density in the resistive layer still occurs

[en] A new damping model on the resistive wall mode (RWM) instability is studied, via the turbulence induced plasma viscosity (in short, turbulent viscosity). In a cylindrical plasma, the synergistic effect on suppressing the RWM is investigated between this new damping mechanism and the plasma flow. An eigenmode formulation is derived based on magneto-hydrodynamic (MHD) theory, where the momentum equation is extended by including the turbulent viscosity term with a proportionality coefficient χ. Numerical results show that, in the absence of plasma flow, increasing χ decreases the RWM growth rate but does not fully stabilize the mode. However, in the presence of sufficiently fast plasma flow, turbulent viscosity can lead to full suppression of the RWM, when χ exceeds a critical value. Similarly, at a given χ value, the plasma flow can fully suppress the mode when the flow velocity exceeds a threshold value. In particular, turbulent viscosity significantly reduces the threshold value of the flow velocity required for full stabilization of the RWM. (paper)

[en] The kinetic effects of thermal particles and fast ions on internal kink (IK) mode are numerically investigated by the MHD-kinetic hybrid code MARS-K. It is shown that either thermal particles or fast ions have stabilizing influence on IK. However, the former can not fully stabilize IK, and the later can suppress the IK. In addition, the synergistic effect from thermal particles and fast ions induces more stronger damping on IK. The kinetic effects from particles significantly raise the critical value of poloidal beta (${\mathrm{\beta <!--\; ??\; -->}}_{\mathrm{p}}^{\mathrm{c}\mathrm{r}\mathrm{i}\mathrm{t}}$) for driving IK in the toroidal plasma. This implies a method of controlling IK or sawtooth in the high-β _p discharge scenario of tokamak. It is noted that, at the q = 1 rational surface, mode structure becomes more sharp due to the self-consistent modification by particles’ kinetic effect. (paper)

[en] The kinetic effect of trapped energetic particles (EPs), arising from perpendicular neutral beam injection, on the stable low-n peeling modes in tokamak plasmas is investigated, through numerical solution of the mode's dispersion relation derived from an energy principle. A resistive-wall peeling mode with m/n=6/1, with m and n being the poloidal and toroidal mode numbers, respectively, is destabilized by trapped EPs as the EPs' pressure exceeds a critical value β_c^*, which is sensitive to the pitch angle of trapped EPs. The dependence of β_c^* on the particle pitch angle is eventually determined by the bounce average of the mode eigenfunction. Peeling modes with higher m and n numbers can also be destabilized by trapped EPs. Depending on the wall distance, either a resistive-wall peeling mode or an ideal-kink peeling mode can be destabilized by EPs

[en] The magnetohydrodynamic-kinetic hybrid theory has been extensively and successfully applied for interpreting experimental observations of macroscopic, low frequency instabilities, such as the resistive wall mode, in fusion plasmas. In this work, it is discovered that an analytic version of the hybrid formulation predicts a bifurcation of the mode dynamics while varying certain physical parameters of the plasma, such as the thermal particle collisionality or the ratio of the thermal ion to electron temperatures. This bifurcation can robustly occur under reasonably large parameter spaces as well as with different assumptions, for instance, on the particle collision model. Qualitatively similar bifurcation features are also observed in full toroidal computations presented in this work, based on a non-perturbative hybrid formulation

[en] The neoclassical toroidal plasma viscosity torque and electromagnetic torque, generated by tearing mode (TM) in a toroidal plasma, are numerically investigated using the MARS-Q code [Liu et al., Phys. Plasmas 20, 042503 (2013)]. It is found that an initially unstable tearing mode can intrinsically drive a toroidal plasma flow resulting in a steady state solution, in the absence of the external momentum input and external magnetic field perturbation. The saturated flow is in the order of 0.5%ω_A at the q=2 rational surface in the considered case, with q and ω_A being the safety factor and the Alfven frequency at the magnetic axis, respectively. The generation of the toroidal flow is robust, being insensitive to the given amplitude of the perturbation at initial state. On the other hand, the flow amplitude increases with increasing the plasma resistivity. Furthermore, the initially unstable tearing mode is fully stabilized by non-linear interaction with the self-generated toroidal flow

[en] The toroidal magnetohydrodynamic (MHD) code MARS-F (Liu et al 2000 Phys. Plasmas 7 3681) is applied to numerically investigate multiple MHD instabilities in high-β _N (β _N is the beta normalized) toroidal plasmas with reversed magnetic shear, and with different radial separations $\mathrm{\Delta <!--\; ??\; -->}{r}_s$ between the two q = 2 rational surfaces. A resistive wall is also taken into account. In the small $\mathrm{\Delta <!--\; ??\; -->}{r}_s$ regime, it is found that a finite β _N leads to multiple branches of the double tearing mode (DTM). The beta normalized has a stabilizing effect on the most unstable branch. There exists a critical value β _Nc, above which the real frequency of the most unstable mode becomes finite due to the favorable average curvature effect (Glasser et al 1975 Phys. Fluids 18 875). Moreover, the critical value β _Nc decreases with increasing plasma resistivity $\mathrm{\eta <!--\; ??\; -->}.$ In the large $\mathrm{\Delta <!--\; ??\; -->}{r}_s$ regime, on the other hand, finite beta normalized can help to transform the two DTM branches into an external kink mode (EKM). Increasing β _N can also couple two single tearing modes, forming a DTM. In the intermediate $\mathrm{\Delta <!--\; ??\; -->}{r}_s$ regime, interestingly, a new branch with EKM structure appears, which successively couples with the other two branches as $\mathrm{\Delta <!--\; ??\; -->}{r}_s$ increases, recovering the EKM found in the large $\mathrm{\Delta <!--\; ??\; -->}{r}_s$ limit. Characteristics of the eigenmode structures in different $\mathrm{\Delta <!--\; ??\; -->}{r}_s$ regimes are compared and analyzed in detail. Furthermore, the properties of the high-β _N MHD instabilities, with higher toroidal mode number n, are also investigated. It is found that, in the small $\mathrm{\Delta <!--\; ??\; -->}{r}_s$ limit, the growth rate always first increases and then decreases with n, forming a broad n spectrum. The critical value ${\mathrm{\beta <!--\; ??\; -->}}_{Nc}$ decreases with n. In the large $\mathrm{\Delta <!--\; ??\; -->}{r}_s$ limit, however, the growth rate of the n = 2 mode is strongly reduced with increasing β _N.  (paper)