Subdominant Modes in Zonal-Flow-Regulated Turbulence

K. D. Makwana, P. W. Terry, M. J. Pueschel, D. R. Hatch
2014 Physical Review Letters  
From numerical solutions of a gyrokinetic model for ion temperature gradient turbulence it is shown that nonlinear coupling is dominated by three-wave interactions that include spectral components of the zonal flow and damped subdominant modes. Zonal flows dissipate very little energy injected by the instability, but facilitate its transfer from the unstable mode to dissipative subdominant modes, in part due to the small frequency sum of such triplets. Although energy is transferred to higher
more » ... ve numbers, consistent with shearing, a large fraction is transferred to damped subdominant modes within the instability range. This is a new aspect of regulation of turbulence by zonal flows. Turbulence driven by ion gyroradius-scale microinstabilities limits confinement in magnetically confined fusion plasmas [1] . The turbulence saturates when the rate of energy injection by the instability is balanced by the rate of energy dissipation. In the traditional view, dissipation arises at small scales. However, plasma instabilities represent one root (or a few) of dispersion relations that typically have many other stable roots at any given wave number of the spectrum. These modes dissipate energy at large scales, within the wave number range of the instability. They are excited to finite amplitude by three-wave interactions that couple unstable and stable modes of a wave number triplet [2]. Such modes have been shown to be the dominant saturation mechanism in a variety of fusion plasma turbulence models [3] . The energy dissipation rate in the wave number range of the instability is comparable to the instability energy injection rate, creating a saturation process that is fundamentally different from a cascade to collisionally dissipative, large wave numbers. Ion temperature gradient (ITG) driven turbulence, an important contributor to ion heat transport in tokamaks, is regulated by zonal flows, i.e., self-generated, toroidally and poloidally symmetric, radially sheared E × B flows [4]. Zonal flows are driven to high amplitude in ITG turbulence as a result of turbulent interactions [5, 6] . Conventionally, the regulation of ITG turbulence by zonal flows is explained by zonal flow shearing [6]: zonal flows shear apart turbulent eddies, causing enhanced energy transfer to small dissipative scales and saturating ITG turbulence. Given that stable modes have also been found to be important in the saturation of ITG in both fluid and gyrokinetic analyses [7] , it is crucial to investigate the role of stable modes in zonal-flow-regulated ITG turbulence. In Ref. [8], analysis of a simple two-field fluid ITG model uncovered evidence for a different mechanism of zonal flow regulation. Zonal flows provide a high-efficiency energy-transfer channel from the unstable mode to a large-scale stable mode that saturates the turbulence. With efficient transfer, the turbulence level is low. Because the zonal flow absorbs only a small fraction of the energy transferred, it acts as a catalyst. However, the fluid model treats the enhanced nonlinear excitation of zonal flows artificially, includes only one stable mode, and its modeling of dissipation has only crude collisional diffusivities with no kinetic resonance effects. Thus, the essential analysis of the role of stable modes in zonal flow regulation using gyrokinetics [9, 10] , which overcomes the deficiencies of the fluid model, is given here. We use the gyrokinetic code GENE [11], developing diagnostics that trace energy transfer between zonal flows, the unstable mode, and stable modes. These measure the partition of energy flow in the joint space of perpendicular wave number and the phase space of parallel motion spanned by stable modes. We probe the efficiency of energy transfer channels by measuring nonlinear triplet correlation times and amplitude dependence. GENE describes plasma dynamics in terms of the modified perturbed ion distribution function gðk x ; k y ; z; v ∥ ; μ; tÞ (see Ref. [12]). Here the radial and binormal wave numbers are denoted by k x and k y , normalized to the ion sound gyroradius ρ s . Also, z is the equilibrium magnetic-fieldline-following coordinate, v ∥ is the velocity in z direction, μ is the magnetic moment, and t is time. We simulate Cyclone base case parameters with adiabatic electrons and only electrostatic perturbations [13], using resolutions of (128, 16, 16, 32, 8) for (k x , k y , z, v ∥ , μ).
doi:10.1103/physrevlett.112.095002 pmid:24655261 fatcat:k5ircy6qk5fn7hl24dosqy5xzm