Ideal ballooning modes, shear flow and the stable continuum

There is a well-established theory of ballooning modes in a toroidal plasma. The cornerstone of this is a local eigenvalue λ on each magnetic surface—which also depends on the ballooning phase angle k. In stationary plasmas, λ(k) is required only near its maximum, but in rotating plasmas its average over k is required. Unfortunately in many cases λ(k) does not exist for some range of k, because the spectrum there contains only a stable continuum. This limits the application of the theory, and raises the important question of whether this 'stable interval' gives rise to significant damping. This question is re-examined using a new, simplified, model—which leads to the conclusion that there is no appreciable damping at small shear flow. In particular, therefore, a small shear flow should not affect ballooning mode stability boundaries.