Theory of the momentum flux probability distribution function for drift wave turbulence

An analytical theory of the tails of the probability distribution function (PDF) for the local Reynolds stress $(R)$ is given for forced Hasegawa–Mima turbulence. The PDF tail is treated as a transition amplitude from an initial state, with no fluid motion, to final states with different values of $R$ due to nonlinear coherent structures in the long time limit. With the modeling assumption that the nonlinear structure is a modon (an exact solution of a nonlinear Hasegawa–Mima equation) in space, this transition amplitude is determined by an instanton. An instanton is localized in time and can be associated with bursty and intermittent events which are thought to be responsible for PDF tails. The instanton is found via a saddle-point method applied to the PDF, represented by a path integral. It implies the PDF tail for $R$ with the specific form $exp[−cR3/2],$ which is a stretched, non-Gaussian exponential.