Zonally dominated dynamics and Dimits threshold in curvature-driven ITG turbulence

Abstract

The saturated state of turbulence driven by the ion-temperature-gradient instability is investigated using a two-dimensional long-wavelength fluid model that describes the perturbed electrostatic potential and perturbed ion temperature in a magnetic field with constant curvature (a $Z$ -pinch) and an equilibrium temperature gradient. Numerical simulations reveal a well-defined transition between a finite-amplitude saturated state dominated by strong zonal-flow and zonal temperature perturbations, and a blow-up state that fails to saturate on a box-independent scale. We argue that this transition is equivalent to the Dimits transition from a low-transport to a high-transport state seen in gyrokinetic numerical simulations (Dimits et al., Phys. Plasmas, vol. 7, 2000, 969). A quasi-static staircase-like structure of the temperature gradient intertwined with zonal flows, which have patch-wise constant shear, emerges near the Dimits threshold. The turbulent heat flux in the low-collisionality near-marginal state is dominated by turbulent bursts, triggered by coherent long-lived structures closely resembling those found in gyrokinetic simulations with imposed equilibrium flow shear (van Wyk et al., J. Plasma Phys., vol. 82, 2016, 905820609). The breakup of the low-transport Dimits regime is linked to a competition between the two different sources of poloidal momentum in the system – the Reynolds stress and the advection of the diamagnetic flow by the $\boldsymbol {E}\times \boldsymbol {B}$ flow. By analysing the linear ion-temperature-gradient modes, we obtain a semi-analytic model for the Dimits threshold at large collisionality.

Investigation of tokamak turbulent avalanches using wave-kinetic formulation in toroidal geometry

“…On the contrary, for minima of the flow, , the phase velocity has a saddle point and turbulent structures are expelled. This asymmetry has been observed in numerical simulations for gyro-kinetic toroidal systems (McMillan et al 2011; Dif-Pradalier et al 2015; Zhu et al 2018 b ; Gillot 2020; Ivanov et al 2020). The same generic mechanism exists for a stationary zonal flow (Zhu, Zhou & Dodin 2020): the stability of the zonal flow pattern depends on the sign of the curvature.…”

“…The scale separation allows the treatment of the waves as point particles, neglecting their finite correlation length, and their response to profile evolution is instantaneous. However, the radial scale separation may only be marginal, because of the radial pattern of zonal flow staircases (Dif-Pradalier et al 2015; Ivanov et al 2020). The case of non-ideal waves is plagued by numerous technical and fundamental difficulties that we can avoid here by restricting ourselves to the ideal case (Brillouin 1960).…”

“…They are routinely observed in both gradient-driven and flux-driven nonlinear simulations (Beyer et al 2000; Idomura et al 2009; McMillan et al 2009; Görler et al 2011; Dif-Pradalier et al 2017), although the comparison between gradient-driven and flux-driven dynamics is still an open problem (Rath et al 2016). The zonal flow patterns found in those simulations tend to be asymmetric with sometimes strong radial localisation (McMillan et al 2011; Dif-Pradalier et al 2015; Zhu, Zhou & Dodin 2018 a ; Ivanov et al 2020), which have prompted the nickname of a ‘zonal flow staircase’. Several explanations have been advanced for such avalanches, from self-organised criticality in the sub-marginal regime (Bak, Tang & Wiesenfeld 1987; Hwa & Kardar 1992; Schekochihin, Highcock & Cowley 2012; van Wyk et al 2016; Pringle, McMillan & Teaca 2017; McMillan, Pringle & Teaca 2018), to nonlinear motion of turbulent potential filaments (Beyer et al 2000; Sarazin et al 2010; Gillot 2020).…”

Wave trapping and E × B staircases

“…Zonal flows play a major role in the dynamics of "E × B" staircases, which are self-organised turbulent states observed in a number of gyrokinetic simulations [16,17,18,19,20,21]. Similar patterns were observed in simulations of a reduced 2D model of interchange turbulence [22] and also 2D ion temperature gradient (ITG) turbulence [23]. Some experimental observations were found consistent with a staircase phenomenology [24,25,26].…”

Theory of the tertiary instability and the Dimits shift within a scalar model

“…Understanding of the Dimits shift is already complicated when we study the modified Hasegawa-Wakatani model in Zhu et al (2020), when we observed the presence of avalanche-like structures, which are not supported by the mTHE. Furthermore, the recent paper by Ivanov et al (2020) shows that avalanches themselves can become intricate when additional physics due to finite ion temperature is taken into account. This complicates the problem even further, and more work remains to be done to understand the Dimits shift in the general case.…”