Field statistics and correlation functions for stochastically growing waves

Bursty waves are common in laboratory and space plasmas. This paper simulates the generation of bursty waves using stochastic differential equations, calculating the field statistics and correlation functions with and without thermal effects, linear instability, nonlinear processes, intrinsic spatiotemporal inhomogeneities (clumps), and different sampling techniques. Driven thermal waves are shown to have field statistics that agree very well with an analytic prediction (typically power-law above a low field peak near the thermal level, but whose peak can be moved to high fields with appropriate fine tuning of parameters) and are robust against changes in sampling and inclusion of clumping effects. Purely stochastically growing waves, expected to have the log normal statistics observed in multiple applications, only do so under stringent conditions and inclusion of spatiotemporal clumping effects. These conditions have similar forms to ones derived previously using analytic arguments. Deviations from a log normal can be due to sampling and clumping effects, rather than due to the nonlinear and convolution effects inferred previously. Correlation functions are predicted and observed to have an exponential decrease at small lags, with time constant equal to the inverse effective growth rate, provided stochastic effects are relatively small and sufficient averaging is possible. Extraction of the wave, stochastic, and clump parameters from observed field statistics and correlation functions appears viable. An evolutionary transition must exist between driven thermal waves and stochastically driven waves, since their field statistics have different functional forms, dependencies, and sensitivity to clump effects, but still requires identification. (c) 2007 American Institute of Physics.