Nonlocal Theory of Trapped Particle Instability

A nonlocal theory for the low‐frequency trapped particle instability in toroidal systems has been developed in a model field. The stability of the trapped particle mode is studied by using normal mode analyses with radial boundary conditions. In the absence of magnetic shear, the finite‐orbit effect is shown to be incapable of stabilizing the mode; large magnetic shear is needed for stabilization. The ratio of the localization width of the mode to the characteristic density length is of the order of ${\mathrm{(\Lambda /r;)}}^{\text{1/2}}{\epsilon }^{\text{1/4}}$ (Λ is the charactistic “banana” size, r is the characteristic density length, and $\epsilon$ is the aspect ratio). An estimate of the nonlinear diffusion coefficient for the mode is given.