Turbulent Transport in Tokamak Plasmas with Rotational Shear

Nonlinear gyrokinetic simulations are conducted to investigate turbulent transport in tokamak plasmas with rotational shear. At sufficiently large flow shears, linear instabilities are suppressed, but transiently growing modes drive subcritical turbulence whose amplitude increases with flow shear. This leads to a local minimum in the heat flux, indicating an optimal $\mathbf{E}\times \mathbf{B}$ shear value for plasma confinement. Local maxima in the momentum fluxes are observed, implying the possibility of bifurcations in the $\mathbf{E}\times \mathbf{B}$ shear. The critical temperature gradient for the onset of turbulence increases with flow shear at low flow shears; at higher flow shears, the dependence of heat flux on temperature gradient becomes less stiff. The turbulent Prandtl number is found to be largely independent of temperature and flow gradients, with a value close to unity.

Figure 1

Average linear growth rates (normalized by $R_0/v_{\mathrm{th}}$) vs flow shear. Inset: Critical flow shear vs $R_0/L_T$.

Figure 2

Turbulent heat (top) and toroidal angular momentum (bottom) fluxes vs flow shear for a range of $R_0/L_T$ values. For ${\gamma }_E>{\gamma }_{Ec}$ (see Fig. 1), the turbulence is subcritical, sustained by transiently growing modes.

Figure 3

Heat flux versus time obtained from linear simulations for $R_0/L_T=8.75$ and several ${\gamma }_E$ values at which subcritical turbulence is observed. Inset: Transient amplification factor, A, and growth time, $\tau$, versus ${\gamma }_E$.

Figure 4

Turbulent Prandtl number as a function of flow shear for a range of $R_0/L_T$ values. Unfilled points correspond to linearly stable flow shear values. Zero Pr points represent fully suppressed turbulence.

Figure 5

Turbulent heat flux vs $R_0/L_T$ for various values of the flow shear. Though there is some minimal relaxation of the profile stiffness for $0.3\lesssim {\gamma }_E\lesssim 0.8$, the predominant effect of the shear is to shift the critical temperature gradient.