Asymptotic behavior of guiding-center diffusion in a model of electrostatic turbulence

To compare with computer simulations of the diffusion of a test guiding center in a given electrostatic turbulence, a nonlinear theory is applied to the ‘‘randomly phased waves’’ model, with a single frequency ω and an arbitrary wave number spectrum. The asymptotic behavior of the diffusion coefficient D is determined in both limits of large and small turbulence amplitude a. For a→∞, the classical ‘‘frozen turbulence’’ scaling D∝a is found. For a→0, an unusual quadratic scaling is obtained: for all isotropic models, D goes to the same limit (√2 /ω)${\mathit{a}}^2$. This behavior originates in the ‘‘two scales’’ character of this asymptotic problem. It is examined in detail on a simple form of the equation where the exact asymptotic solutions are obtained.