Stochastic wave growth

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Abstract
Evolution of waves subject to a randomly varying growth rate is considered and the statistical properties of the waves are calculated in terms of the mean, variance, and correlation time of the growth rate. This enables stochastic growth to be studied without needing full knowledge of the microphysics. However, where the microphysics is understood, this approach also allows it to be easily incorporated into studies of larger‐scale phenomena involving stochastic growth. Stochastic differential equations and Fokker–Planck equations are obtained, which describe the wave evolution in the presence of a variety of linear and nonlinear processes and boundary conditions, and it is shown that these phenomena can be diagnosed observationally through their effects on the statistical distribution of the wave field strengths. The results are particularly useful for waves with small dispersion, where they explain the strong wave clumping often observed in nature and emphasize the role of marginal stability in setting the level about which fluctuations occur and in determining their magnitude. Application to type III solar radio bursts illustrates many of the main results and verifies and generalizes earlier conclusions reached using a less rigorous approach. In particular, a new condition for marginally stable propagation of type III solar electron beams is found.