Gyrokinetic secondary instability theory for electron and ion temperature gradient driven turbulence

The instabilities that drive turbulence and transport in tokamaks themselves become unstable at finite amplitude to secondary instabilities. These "secondaries" are a key part of the nonlinear physics. This work presents a fully gyrokinetic secondary instability theory for electron temperature gradient (ETG) and ion temperature gradient (ITG) driven turbulence. The electrostatic gyrokinetic equation is solved in the local approximation to find "fast" secondary modes that satisfy gamma(s)>gamma(p) and can therefore lead to mode saturation. Finite Larmor radius and other kinetic effects are treated exactly capturing k rho>1 as well as k rho < 1 quasisingular behavior. This theory is therefore well suited to describe the intermediate regime of ITG/ETG coupling. The secondary instability of toroidal (k(parallel to)=0) and slab (L-T/R=0) primary modes is computed along with spectral characteristics and parametric dependence. The results of this paper include a robust secondary growth rate at high k rho and, in the case of ETG, a strong sensitivity to the kinetic form of the primary mode. The convergence properties of the computation of the secondary instability of ETG toroidal modes underscores the need for proper k-space resolution in simulations. The parametric dependence of the secondary mode growth rate reveals a mechanism for the transport suppression near marginal stability that is associated with the Dimits shift. A strengthening of secondary instability at small L-T/R suggests that secondary instabilities may play a role in the formation of electron internal transport barriers. (C) 2007 American Institute of Physics.