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.

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 # ¼ 6p=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 (k ¼ Çp=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.

A theoretical model for investigating the effect of the trapped energetic particles ͑EPs͒ on the resistive wall mode ͑RWM͒ 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.

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 βc*, which is sensitive to the pitch angle of trapped EPs. The dependence of βc* 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.

Kinetic effects of both trapped thermal and energetic particles on the resistive wall mode (RWM) and on the fishbone-like mode (FLM) are investigated in theory. Here, the trapped thermal particles include both ions and electrons. The FLM is driven by trapped energetic particles. The results demonstrate that thermal particle collisions can either stabilize or destabilize the RWM, depending on the energetic particle pressure βh. Furthermore, the critical value of βh for triggering the FLM is increased when the thermal particle contribution is taken into account. The critical value sensitively depends on the plasma collision frequency. In addition, the plasma inertia is found to have a negligible influence on the FLM.