Zonal flows and ITG saturation in the presence of magnetic perturbations

At large $\beta$ but below the KBM threshold, gyrokinetic simulations of ITG turbulence may fail to saturate through zonal flows. It is shown that, if background gradients are sufficiently strong, a threshold exists for a transition to a non-zonal ITG regime with extremely large transport levels, an effect sometimes referred to as ``runaway.'' Resonant (i.e., tearing-parity) radial magnetic fluctuations $B_x$ are usually seen as the primary source of stochasticity, but non-resonant $B_x$ may also become stochastic once the field line displacement after half a poloidal turn exceeds the correlation length. This rapidly shorts out the zonal flows, and they are no longer able to saturate the ITG mode. To examine how flux-surface-breaking magnetic fluctuations affect zonal flows, the Rosenbluth-Hinton residual flow scenario is studied: In the presence of a superimposed, constant resonant $B_x$, initial flows do not decay to the theoretical residual, but vary quadradically in time, reaching zero potential in the time $t_{\Phi=0} \propto q_0 (1 - \epsilon_{\mathrm{t}}) B_x^{-1}$. The flow evolution is calculated analytically from the Laplace-transformed, bounce-averaged ion gyrokinetic equation with a $B_x$ of the above type, and compared with the simulation results.