Momentum Injection in Tokamak Plasmas and Transitions to Reduced Transport

The effect of momentum injection on the temperature gradient in tokamak plasmas is studied. A plausible scenario for transitions to reduced transport regimes is proposed. The transition happens when there is sufficient momentum input so that the velocity shear can suppress or reduce the turbulence. However, it is possible to drive too much velocity shear and rekindle the turbulent transport. The optimal level of momentum injection is determined. The reduction in transport is maximized in the regions of low or zero magnetic shear.

Figure 1

Schematic dependence of the turbulent energy flux $Q_t$ on the velocity shear ${\gamma }_E$ and the temperature gradient $\kappa$.

Figure 2

Energy flux $Q$ vs temperature gradient $\kappa$ for a constant ratio $\Pi /Q$ of momentum and energy input.

Figure 3

(a) Curves of constant $Q$ (dashed lines) and constant $\Pi /Q$ (solid lines). The thin red line is the critical temperature gradient ${\kappa }_c$ below which there is no turbulence. (b) Sketch of the intersection between a curve of given $\Pi /Q$ and curves of constant $Q$ with $Q_1>Q_2>Q_3$. The black dash-dotted line is Eq. (2). (c),(d) Similar sketches for higher $\Pi /Q$.

Figure 4

(a) Asymptotic approximation to the curves of constant $Q$ (dashed lines) and constant $\Pi /Q$ (solid lines) in the transition region between neoclassical and turbulent regimes. (b) Graphical representation of condition (5) for type $B$ solutions to exist. (c),(d) The cases in which type $B$ solutions no longer exist for small and large $\Pi /Q$.

Figure 5

(a) Critical temperature gradient ${\kappa }_c$ for zero (solid line) and finite (dashed line) magnetic shear $\widehat{s}$ based on 6, 7. (b) Region in the ($\Pi /Q$, $Q$) space where abrupt transitions can happen.