Critically Balanced Ion Temperature Gradient Turbulence in Fusion Plasmas

Abstract

Scaling laws for ion temperature gradient driven turbulence in magnetized toroidal plasmas are derived and compared with direct numerical simulations. Predicted dependences of turbulence fluctuation amplitudes, spatial scales, and resulting heat fluxes on temperature gradient and magnetic field line pitch are found to agree with numerical results in both the driving and inertial ranges. Evidence is provided to support the critical balance conjecture that parallel streaming and nonlinear perpendicular decorrelation times are comparable at all spatial scales, leading to a scaling relationship between parallel and perpendicular spatial scales. This indicates that even strongly magnetized plasma turbulence is intrinsically three-dimensional.Introduction. Microscale turbulence is a ubiquitous feature of the plasmas used for magnetic confinement fusion. It is driven by kinetic instabilities feeding predominantly off a strong mean gradient in the ion temperature, and it is responsible for the majority of particle and heat transport observed in experiment. As with neutral fluid and magnetohydrodynamic turbulence, exact analytical results for kinetic plasma turbulence are rare, and numerical simulations are costly. Phenomenological scaling laws are thus useful for guiding simulation and providing gross predictions of plasma behavior in a multi-dimensional parameter space.Experimental, numerical, and analytical results (cf. [1-4]) have long been used to predict the dependence of turbulent fluxes on the mean plasma gradients and on the magnetic field configuration. However, scalings based on empirical observations provide limited physical insight, and the theoretical predictions, which are predominantly based on linear or quasilinear arguments, are not sufficiently detailed to be easily falsifiable. A more detailed examination of the properties of kinetic plasma turbulence has been conducted for scales smaller than the ion Larmor radius [5][6][7], but it is the ion temperature gradient (ITG) driven turbulence above the Larmor scale that is most relevant for heat transport in fusion devices (cf. [8]). Recent advances in plasma fluctuation measurements [9, 10] have provided turbulence spectra in this scale range; direct numerical simulations have also calculated spectra [11] and studied energy injection, transfer, and dissipation [12,13].In this Letter, we propose a phenomenological scaling theory of ITG turbulence. A number of simple, physically-motivated conjectures about the nature of this turbulence are formulated and applied to obtain fluctuation spectra from the driving scale to the ion Larmor scale. We then derive predictions for the dependence of heat flux on plasma current and ion temperature gradient. Numerical results are presented to support our predictions and justify our conjectures.Gyrokinetic turbulence. Plasma fluctuations in a strong mean magnetic field are anisotropic with respect to the mean field direction and have typical frequencies

Suppression of phase mixing in drift-kinetic plasma turbulence

“…Recent work leads to some disquieting observations. Firstly, a fluid-like theory for the nonlinear cascade [11] predicts power-law spectra for the electrostatic potential in good agreement with those found in gyrokinetic simulations, but its derivation neglects free-energy transfer by phase mixing [12], contrary to what might be expected on the basis of linear theory. Including a constant flux of free energy into velocity space leads to non-universal spectra that tend to be steeper than those empirically observed [12][13][14][15][16][17].…”

“…This captures the essential feature of ITG turbulence: that the nonlinear turnover rate eventually dominates the injection rate as k ⊥ increases. However, as the nonlinear and linear characteristic rates are respectively τ −1 nl ∼ k 4/3 ⊥ and ω * ∼ k y , [11,12], it is necessary to limit the free-energy injection artificially to a narrow range of small wavenumbers to allow an inertial range to develop with the available resolution. Now only low wavenumbers can grow, and only for very large temperature gradients.…”

Experimental Signatures of Critically Balanced Turbulence in MAST

“…This means that the density fluctuations we are measuring in MAST are not isotropic in the perpendicular plane, but rather elongated in the poloidal direction ℓ y /ℓ x ∼ R/L * (∼ 5 in our data). Interestingly, this clashes with the reported approximate isotropy (ℓ x ∼ ℓ y ) both in Cyclone Base Case simulations [5] and in measured DIII-D turbulence (where ℓ y /ℓ x ∼ 1.4 [36] and ℓ x does not appear to depend on B p [37]). Whether this is a difference between spherical and conventional tokamaks is not as yet clear.…”

“…Some of these time scales and, consequently, the corresponding physics may be irrelevant, while others play a crucial role for the saturation of the linearly unstable fluctuations. There has been a growing understanding [5], driven largely by theory [6][7][8][9], observations [10][11][12] and simulations of magnetohydrodynamic [13][14][15] and kinetic [7,16] plasma turbulence in space, that if a medium can …”

“…This cannot be checked directly because there are no diagnostics capable of measuring ℓ on MAST [33]. Considering that the inboard side of the torus is a region of "good" (stabilizing) curvature, not much turbulence is expected there, so we assume that, at the energy injection scale, ℓ ∼ Λ [5], where the distance along the field line that takes a particle from the outer to the inner side of the torus is Λ = πrB/B p (r is the minor radius at the BES position on the outer side and B p the poloidal component of the magnetic field) [34]. Then critical balance means that τ c should be comparable to…”