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Statistical theory and transition in multiple-scale-length turbulence in plasmas

The statistical theory of strong turbulence in inhomogeneous plasmas is developed for the cases where fluctuations with different scale lengths coexist. Nonlinear interactions in the same kind of fluctuations as well as nonlinear interplay between different classes of fluctuations are kept in the analysis. Nonlinear interactions are modelled as turbulent drag, nonlinear noise and nonlinear drive, and a set of Langevin equations is formulated. With the help of an Ansatz of a large number of degrees of freedom with positive Lyapunov number, Langevin equations are solved and the fluctuation dissipation theorem in the presence of strong plasma turbulence has been derived. A case where two driving mechanisms (one for the micro mode and the other for semi-micro mode) coexist is investigated. It is found that there are several states of fluctuations: in one state, the micro mode is excited and the semi-micro mode is quenched; in the other state, the semi-micro mode is excited, and the micro mode remains at finite but at a suppressed level. A new type of turbulence transition is obtained, and a cusp-type catastrophe is revealed. A phase diagram is drawn for turbulence which is composed of multiple classes of fluctuations. The influence of the inhomogeneous global radial electric field is discussed. A new insight is given for the physics of the internal transport barrier. Finally, the non-local heat transport due to the long-wavelength fluctuations, which are noise-pumped by shorter-wavelength fluctuations, is analysed and its impact on transient transport problems is discussed.