Self-Organized Stationary States of Tokamaks

We demonstrate that in a 3D resistive magnetohydrodynamic simulation, for some parameters it is possible to form a stationary state in a tokamak where a saturated interchange mode in the center of the discharge drives a near helical flow pattern that acts to nonlinearly sustain the configuration by adjusting the central loop voltage through a dynamo action. This could explain the physical mechanism for maintaining stationary nonsawtoothing “hybrid” discharges, often referred to as “flux pumping.”

Figure 1

(a) Comparison of the $q$ profile and toroidally averaged pressure for a stationary state 3D run and 2D run with exactly the same transport coefficients. (b) Poincaré plot of the magnetic field in the final state with (2,1) and (3,1) islands present. Center volume has $q=1$.

Figure 2

The right side shows contours of M3D-C1 velocity stream function $U$ interior to the $q=1.01$ surface at four toroidal angles. Curves on the left side compare midplane values at $\phi =180°$ with linear eigenfunction found in Ref. [13] and given by Eq. (6).

Figure 3

(a) Effective mid-plane toroidal voltage increase in four different 3D stationary states over that in the equivalent 2D case. This voltage is due to dynamo action. (b) Final safety factor profile in the four 3D runs (solid colors) and in the equivalent 2D runs (dashed).

Figure 4

Difference in the final (a) $n=0$ midplane resistivity profiles and (b) midplane $n=1$ pressure profile for each of the four cases from the equivalent 2D case.

Figure 5

Contribution to $[\mathbf{B}·\nabla \mathrm{\Phi }{]}_{n=0}$ for each of the field components on the midplane interior to the $q=1.01$ surface.