Ideal MHD Limited Electron Temperature in Spherical Tokamaks

It is well documented that the central electron temperature in the national spherical torus experiment (NSTX) remains largely unchanged as the external heating power, and hence the normalized volume averaged plasma pressure $\beta$ increases [Stutman, Phys. Rev. Lett. 102, 115002 (2009)]. Here we present a hypothesis that low $n$, pressure driven ideal magnetohydrodynamic (MHD) instabilities that are nondisruptive, can break magnetic surfaces in the central region and thereby flatten the electron temperature profiles. We demonstrate this mechanism in a 3D resistive MHD simulation of a NSTX discharge. By varying the toroidal magnetic field strength, and/or the heating power, we show that there is a critical value of $\beta$, above which the central temperature profile no longer peaks on axis.

Figure 1

Surface $q$ profile and pressure profiles for NSTX shot 124379 at 640 ms.

Figure 2

Normalized growth rates for unstable modes with toroidal mode numbers 1–18 for NSTX shot 124379 640 ms. Also shown (in green online) are the growth rates with the resistivity increased by 10, indicating that these are ideal MHD instabilities.

Figure 3

Linear eigenmodes showing pressure contours for modes with $n=3$,9, and 15 from Fig. 2. Other modes have similar structure, with poloidal mode number $m\sim (4/3)n$.

Figure 4

Poincaré plots of the 3D M3D-C1 simulation at times (a) $t=0$, (b) $t=500{\tau }_A$, (c) $t=600{\tau }_A$, and (d) $t=6000{\tau }_A$.

Figure 5

Midplane temperature profiles for the initial state and for the results of the calculation at $t=1200{\tau }_A$ using (left) Grid $A$, Grid $B$, and Grid $C$ with 24 planes, and (right) Grid $B$ with 24, 36, and 48 planes.

Figure 6

(a) Experimental midplane electron temperature for the three equilibria considered in Refs. [12, 30]. (b) M3D-C1 midplane electron temperature at time $t=1200{\tau }_A$ starting from the three Bateman scaled equilibria corresponding to $F_S=1.1$, $F_S=1.0$ (the original), and $F_S=0.9$.

Figure 7

Midplane electron temperature at the initial and five additional times for the 3D calculation (right) and for a 2D calculation with the same transport coefficients and heat source (left).