Analytical theory of the probability distribution function of structure formation

The probability distribution function (PDF) tails of the zonal flow structure formation and the PDF tails of momentum flux are computed by incorporating the effect of a shear flow in ion-temperature-gradient (ITG) turbulence. The bipolar vortex soliton (modon) is assumed to be the coherent structure responsible for bursty and intermittent events driving the PDF tails. It is found that stronger zonal flows are generated in ITG turbulence than Hasegawa-Mima turbulence, as well as further from marginal stability. This suggests that although ITG turbulence has a higher level of heat flux, it also more likely generates stronger zonal flows, leading to a self-regulating system. It is also shown that shear flows can significantly reduce the PDF tails of Reynolds stress and structure formation.