Zero-Turbulence Manifold in a Toroidal Plasma

Sheared toroidal flows can cause bifurcations to zero-turbulent-transport states in tokamak plasmas. The maximum temperature gradients that can be reached are limited by subcritical turbulence driven by the parallel velocity gradient. Here it is shown that $q/ϵ$ (magnetic field pitch/inverse aspect ratio) is a critical control parameter for sheared tokamak turbulence. By reducing $q/ϵ$, far higher temperature gradients can be achieved without triggering turbulence, in some instances comparable to those found experimentally in transport barriers. The zero-turbulence manifold is mapped out, in the zero-magnetic-shear limit, over the parameter space (${\gamma }_E$, $q/ϵ$, $R/L_T$), where ${\gamma }_E$ is the perpendicular flow shear and $R/L_T$ is the normalized inverse temperature gradient scale. The extent to which it can be constructed from linear theory is discussed.

Figure 1

(a) The simulations used to find the point on the manifold ${\gamma }_E=1.8$, $R/L_T=15$, $q/ϵ=6.3$, showing the heat flux vs $q/ϵ$ at (${\gamma }_E=1.8$, $R/L_T=15$). The point on the manifold is the point where the heat flux drops to zero. (b)–(d) Sections through the critical manifold with parameters as indicated. Turbulence cannot be sustained for $R/L_T<R/L_{Tc}$ in (b),(c), or for ${\gamma }_E<{\gamma }_{Ec}$ in (d). The data points were found as illustrated in (a), and used to generate the manifold shown in Fig. 2.

Figure 2

The zero-turbulence manifold. Turbulence can be sustained at all points outside the manifold (that is, at all points with a higher temperature gradient and/or higher value of $q/ϵ$ than the nearest point on the manifold). This plot is made up from the sections shown in Figs. 1b, 1c, 1d (heavy lines) and the manifold interpolated from them (thin grey mesh).

Figure 3

Contours of the zero-turbulence manifold plotted against the toroidal flow shear $u^{\prime }=dR\omega /dr/(v_{\mathrm{thi}}/R)={\gamma }_E/(q/ϵ)$. The contours indicate the value $R/L_T=R/L_{Tc}$ below which turbulence is quenched.

Figure 4

(a) $N_{\max }$, the transient amplification exponent maximized over all wave numbers, vs the flow shear ${\gamma }_E$, at different values of the ion temperature gradient $R/L_T$ and at constant $q/ϵ=6.0$. Also shown is the critical amplification exponent $N_c({\gamma }_E,q/ϵ)$ for the same value of $q/ϵ$, and the fitted critical amplification exponent $N_c^f({\gamma }_E)$. (b) The critical temperature gradient $R/L_{Tc}$ vs $q/ϵ$ for different values of ${\gamma }_E$, showing $R/L_{Tc}$ obtained both from the interpolated manifold (Fig. 2), and from the equation $N_{\max }({\gamma }_E,q/ϵ,R/L_T)=N_c^f({\gamma }_E)$, with $N_c^f({\gamma }_E)$ [shown in (a)] chosen to produce the best fit.