The author obtains analytical expressions for the collisional losses of electrons and their energy from an adiabatic trap when the plasma has a high positive potential. The expressions are derived on the basis of an approximate solution of the Fokker-Planck equation, which yields more reliable results than those obtained with the estimative formulas used in a number of other works. In addition, an expression is obtained for the mean energy of the electrons escaping from the trap.

The role of magnetohydrodynamic nonlinearities in precessional mϭnϭ1 fishbone oscillations has been analyzed analytically and numerically. The work is based on the reduced magnetohydrodynamic ͑MHD͒ equations including a linear energetic particle drive model. When the energetic particle pressure is close to the instability threshold, the top-hat linear eigenmode profile of the ideal MHD mϭ1 radial displacement splits up into a two-step structure around the qϭ1 flux surface, due to the finite frequency of the mode. The width of the individual steps is a factor ␥/ smaller than the distance between them, where ␥ is the growth rate of the mode. We find that the MHD nonlinearities modify the mode structure further, and produce explosive nonlinear growth, accompanied by frequency chirping, for modes that are near the instability threshold. The results are quite different for fishbone oscillations that are excited well above the stability threshold. The growth rates of these linearly fast growing modes decreases nonlinearly and the MHD nonlinearities are stabilizing in this limit. The nonlinear MHD effects are important when the plasma displacement is comparable to, or larger than, the scale length of the fishbone structure.

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