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

A statistical theory and transition in multiple-scale-length turbulence in plasmas is developed where fluctuations with different scale lengths coexist. Langevin equations were solved and the fluctuations dissipation theorem in presence of strong plasma turbulence has been derived. It was found that several states of fluctuations in one state the 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 of composed of multiple classes fluctuations.