Toroidal ion temperature gradient-driven weak turbulence

In this paper, a theory of toroidal ion temperature gradient-driven weak turbulence near the threshold is presented. The model considers gyrokinetic ions and adiabatic electrons in toroidal geometry. The linear theory considers modes with kθρi∼O(1) generally not considered in toroidal theories, giving a toroidal threshold regime dominated by transit resonances as opposed to the more usually considered drift resonances, and with a stability threshold of ηtorthresh congruent 1+2εn[2/τ+1/(1+s)]. It is shown that when 0<ηi-ηtorthresh<2(1+1/τ)1/2 xLn/(RLT)1/2 then 0<γ<ωr, and a weak turbulence expansion can be used to treat the nonlinearity. The instability is saturated via nonlinear ion resonance, and it is shown that this nonlinear process transfers energy directly from the waves to the ion distribution function, and does not conserve wave energy. The saturated spectrum is calculated, and the resulting ion thermal conductivity is found to be χi=(1+1/τ)1/2[(ηi-ηtorth)/ηi] ((LT)1/2/R3/2)ρ3iΩi, which is smaller than typical mixing length estimates by a factor of about LT/R, but in the range of tokamak observations. The diffusive nature of the transport requiring a short radial step size is reconciled with the broad radial extent of the toroidally coupled linear modes by postulating that it is the nonlinear beat wave between linear modes not the radial width of a single linear mode that determines the step length appropriate for transport