First principles model of internal transport barriers in negative central shear discharges

Abstract

Abstract. At and around minima or maxima of the safety factor q, overlapping, and therefore toroidicity induced coupling of drift eigenmodes centred on neighbouring rational surfaces r l,m defined by q(r l,m ) = m/l and r l,m±1 defined by q(r l,m±1 ) = (m ± 1)/l, is generally negligible; l and m(m ± 1) are the toroidal and (main) poloidal mode numbers, respectively. Under these conditions, a careful analysis shows that the growth rate of the ion temperature gradient (ITG) mode is proportional to the absolute value of the magnetic shear parameter; its radial width and frequency (in the E × B rotating frame) are larger in the axisymmetric torus than in the plasma slab or cylinder, especially at small values of k θ ai (k θ is the poloidal mode number and ai the characteristic ion Larmor radius). These results provide a straightforward theoretical interpretation of the origin of internal ion transport barriers (ITBs) in discharges with negative central shear if the ITG instability is indeed the main channel for ion energy transport. In agreement with recent experiments, the mechanism does not degrade for Te ∼ Ti, an important result for extrapolation to reactors. Once initiated, an ITB will locally increase the curvature of the ion temperature profile and, consequently, the gradients of the radial electric field and E × B rotation profiles (until neoclassical (or sub-neoclassical) transport comes into play).The damping associated with velocity shear yields an upper bound on the unstable mode number range, whereas Landau damping is expected to introduce a lower bound. In cases where toroidal effects are subdominant, the ratio of the damping rate associated with velocity shear to the growth rate associated with magnetic shear leads naturally to the Hamaguchi-Horton parameter (divided by the factor ηi − 2/3 which characterizes the departure from the instability threshold (ηi is the ratio of the density and temperature length scales)) when introducing the most unfavourable mode number. Toroidal effects are always dominant for sufficiently weak magnetic shear or sufficiently large values of |LN |/R (where LN is the density gradient length), and therefore before steepening of the profiles has occurred near the zero magnetic shear layer; the counterpart of the Hamaguchi-Horton characteristic parameter appropriate to that case is also obtained, considering again the most unfavourable mode number.

Analytical eigenvalue solution for ηi modes of general modewidth

“…Other models have been developed to treat this case. 15 We finally note that the dependence of the eigenvalue on the FLR parameter k 2 2 has not been studied here. In addition to its local effects studied in Refs.…”

Smoothness of turbulent transport across a minimum-q surface

“…There are numerous papers which discuss the linear theory of weak-shear ITG modes, 1,15,[17][18][19] but none that we are aware of analyze the s→0 limit of the toroidal ITG. In fact, for sϭ0, the toroidal ITG has a finite growth rate, and the corresponding eigenfunction is a complex Mathieu function.…”

Electrostatic Drift Instabilities, Turbulence and Anomalous Transport: Introduction and Basic Theory

“…The growth rate remains essentially unchanged. Results obtained in the framework of this model permit to explain the formation of internal transport barriers in negative central shear discharges 20 .…”