Dimits transition in three-dimensional ion-temperature-gradient turbulence

We extend our previous work on the two-dimensional (2-D) Dimits transition in ion-scale turbulence (Ivanov et al., J. Plasma Phys., vol. 86, 2020, 855860502) to include variations along the magnetic field. We consider a three-field fluid model for the perturbations of electrostatic potential, ion temperature, and ion parallel flow in a constant-magnetic-curvature geometry without magnetic shear. It is derived in the cold-ion, long-wavelength asymptotic limit of the gyrokinetic theory. Just as in the 2-D model, a low-transport (Dimits) regime exists and is found to be dominated by a quasistatic staircase-like arrangement of strong zonal flows and zonal temperature. This zonal staircase is formed and maintained by a negative turbulent viscosity for the zonal flows. Unlike the 2-D model, the three-dimensional (3-D) one does not suffer from an unphysical blow up beyond the Dimits threshold where the staircase becomes nonlinearly unstable. Instead, a well-defined finite-amplitude saturated state is established. This qualitative difference between the 2-D and 3-D models is due to the appearance of small-scale `parasitic' modes that exist only if we allow perturbations to vary along the magnetic field lines. These modes extract energy from the large-scale perturbations and provide an effective enhancement of large-scale thermal diffusion, thus aiding the energy transfer from large injection scales to small dissipative ones. We show that in our model, the parasitic modes always favour a zonal-flow-dominated state. In fact, a Dimits state with a zonal staircase is achieved regardless of the strength of the linear drive, provided the system is sufficiently extended along the magnetic field and sufficient parallel resolution is provided.