Theory of weak ion temperature gradient-driven turbulence near the threshold of instability

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In this paper a theory of weak ion temperature gradient-driven turbulence near the threshold of instability is presented. The model considers kinetic ions and adiabatic electrons in a sheared slab geometry. Linear theory shows that for ηth<ηi≲ηth+(1+Ti/Te) Ln/Ls (where ηth=0.95 is the instability threshold and Ln/Ls≪1) then γ≪ω and a weak turbulence theory applies. The nonlinear wave kinetic equation indicates that ion Compton scattering is the dominant nonlinear saturation process, and it is shown that the energy scatters to the linearly stable low ky modes. The resulting fluctuation spectrum (peaked about k⊥ρi≂1) is much lower than that suggested by naive extrapolation from the strong turbulence regime. The resulting ion thermal conductivity is also extremely low, so that strong ion heating can be expected to drive the ion temperature gradient to a level where this weakly turbulent threshold regime is surpassed. Thus the critical ηi relevant to magnetic confinement experiments is not the linear instability threshold but the point where the diminutions of the weak turbulence regime vanish, and the transport increases to the strong fluid turbulence level.