Bifurcation theory of the transition to collisionless ion-temperature-gradient-driven plasma turbulence

The collisionless limit of the transition to ion-temperature-gradient-driven plasma turbulence is studied using a dynamical systems approach. A model with ten degrees of freedom is used to identify the difference between the bifurcation patterns of collisional and collisionless systems. The importance of systematic bifurcation analysis for understanding the resulting difference in the dynamics of linearly damped and undamped systems is emphasized. A four-dimensional collisionless center manifold (CM) is studied and fixed points of its dynamics are identified and used to predict a "Dimits shift" of the threshold for turbulence due to the excitation of zonal flows. The exact value of that shift in terms of physical parameters is established for the model and the effects of higher-order truncations on the dynamics are studied. Possible effects of long-wavelength envelope modulations on the transition to turbulence scenarios in both collisional and collisionless cases are studied via application of multiple-scale analysis of the CM equations. The modulational effects on the dynamics are used to show that the system can undergo the transition to turbulence via the Benjamin-Feir mechanism. A collisionless version of the Ginzburg-Landau equation which captures both the Dimits shift phenomenon and the transition to turbulence above the upshift is derived.