Transport Bifurcation in a Rotating Tokamak Plasma

The effect of flow shear on turbulent transport in tokamaks is studied numerically in the experimentally relevant limit of zero magnetic shear. It is found that the plasma is linearly stable for all nonzero flow shear values, but that subcritical turbulence can be sustained nonlinearly at a wide range of temperature gradients. Flow shear increases the nonlinear temperature gradient threshold for turbulence but also increases the sensitivity of the heat flux to changes in the temperature gradient, except over a small range near the threshold where the sensitivity is decreased. A bifurcation in the equilibrium gradients is found: for a given input of heat, it is possible, by varying the applied torque, to trigger a transition to significantly higher temperature and flow gradients.

Figure 1

Linear behavior: the time evolution of the heat flux, normalized to its initial value, for different values of ${\gamma }_E$ at $R_0/L_T=11$. Growth is transient for all nonzero flow shears, and the length of the period of transient growth decreases with flow shear. Dashed curve is subcritically stable (no turbulence sustained nonlinearly). Inset: Inverse transient growth time, and the number of exponentiations the heat flux undergoes before switching to decay, both vs flow shear.

Figure 2

Turbulent heat flux (a) and toroidal angular momentum flux (b) vs flow shear for different values of $R_0/L_T$. For lower values of $R_0/L_T$, there is a range of flow shears where the turbulence is fully quenched.

Figure 3

Turbulent heat flux vs $R_0/L_T$ for (a) ${\gamma }_E\le 0.8$ and (b) ${\gamma }_E>0.8$. (c) Nonlinear turbulence thresholds vs flow shear. For each value of $R_0/L_T$, the 1st-threshold curve shows the 2 values of ${\gamma }_E$ at which the turbulence is quenched and then rekindled (cf. Fig. 2). For $R_0/L_T\gtrsim 11.5$ or ${\gamma }_E\gtrsim 1.8$, the turbulence is always present. (d) Profile stiffness vs the flow shear. Low $Q_t$ refers to fluxes between the first and second thresholds; High $Q_t$ to fluxes above both thresholds.

Figure 4

(a) The ratio of the total momentum flux $\Pi$ to the total heat flux $Q$ vs flow shear for a constant value of $Q$. An increase in applied torque (i.e., an increase in $\Pi /Q$) at point $A$ will cause a transition to point $B$. (b) Total heat flux $Q$ vs $R_0/L_T$ for different constant values of $\Pi /Q$. The points $A$ and $B$ on both graphs correspond to the same states. Since neither $Q$ nor $\Pi$ can be specified for a simulation (with the exception of $\Pi =0$), the contours of constant $Q$ and $\Pi /Q$ were interpolated from a large number of data points using radial basis functions with a linear kernel [23]. Also plotted are the neoclassical contributions to $\Pi /Q$ and $Q$. The contours in (b) do not intercept the neoclassical line but curl round and asymptote to it, tending back to the origin, and thus point $B$ is in fact on the same contour as point $A$ (see text and 24). This feature cannot be shown as the contours are too closely spaced for interpolation near the neoclassical line.