Quasisymmetric Optimization of Nonaxisymmetry in Tokamaks

Predictive 3D optimization reveals a novel approach to modify a nonaxisymmetric magnetic perturbation to be entirely harmless for tokamaks, by essentially restoring quasisymmetry in perturbed particle orbits as much as possible. Such a quasisymmetric magnetic perturbation (QSMP) has been designed and successfully tested in the KSTAR and DIII-D tokamaks, demonstrating no performance degradation despite the large overall amplitudes of nonaxisymmetric fields and strong response otherwise expected in the tested plasmas. The results indicate that a quasisymmetric optimization is a robust path of error field correction across the resonant and nonresonant field spectrum in a tokamak, leveraging the prevailing concept of quasisymmetry for general 3D plasma confinement systems such as stellarators. The optimization becomes, in fact, a simple eigenvalue problem to the so-called torque response matrices if a perturbed equilibrium is calculated consistent with nonaxisymmetric neoclassical transport.

Figure 1

Quasisymmetric $n=1$ configurations tested in the KSTAR (left) $\#22972$ and the DIII-D (right) $\#178620$ high-$\beta$ tokamak plasmas using their 3 rows of versatile control coils, with the perturbed flux surfaces (amplified visually by $\times 25$ for KSTAR and $\times 10$ for DIII-D), and the illustrated distribution of nonaxisymmetric magnetic fields and coil currents in colors (in red for $+$ and blue for $-$ with contrast proportional to their amplitudes.)

Figure 2

Comparison of the (a-1) predicted torque density, (a-2) integrated torque profiles in the KSTAR, and (b) DIII-D discharges by QSMPs (blue), NRMPs (green), and RMPs (red). The RMP produces torque mainly through the field resonances, whereas the NRMP induces torque and transport broadly across the region. The QSMP optimization leads to the suppression of both effects as well as the kinetic orbit-resonant effect, as pronounced in the (a-2) RMP application. In very high-$\beta$ plasmas like the studied DIII-D case, the field resonance is shifted and split due to the kinetic effects which can strongly change the originally optimized spectrum as implied by the revisited QSMP (light blue).

Figure 3

Comparison of the (a) field strength averaged on magnetic surfaces (up) and the (b) same but normalized by overall response (down), by the QSMP, NRMP, and RMP applied to the KSTAR discharge. Note that here the RMP and NRMP have the same $|\overrightarrow{C}|$, i.e., the rms sum of 3 rows of coil currents, whereas the QSMP is based on a higher $|\overrightarrow{C}|$ up to the maximum allowed in the experiments.

Figure 4

Impact of QSMP, NRMP, and RMP shown by the time traces in a single dedicated KSTAR discharge; (a) 3D coil configurations with 3 amplitudes and phases (rad) for top, middle, bottom row of in-vessel coils, (b) ${\overline{n}}_e$ measured by two-color interferometer (TCI) and $T_e$ measured by electron-cyclotron emission (ECE) at $R\sim 1.98\mathrm{m}$, and (c) ${\omega }_{\phi }$ measured by charge-exchange recombination spectroscopy (CERS) at $R\sim 2.0\mathrm{m}$.

Figure 5

Impact of QSMP, NRMP, and RMP fields on the $n_e$ and ${\omega }_{\phi }$ profiles as measured before (solid in black) and after (dash in blue, and dashed-dot in red) the perturbations. Note that the QSMP or NRMP was ramped to its desired maximum at $t=3.05\mathrm{s}$ but RMP was slowly ramped until it caused a locking and disruption. Note the profiles were taken from OMFIT profile toolkit [47].

Figure 6

Comparison of the predicted and measured total torques by QSMP, NRMP, RMP in the (a) KSTAR and (b) DIII-D discharges.

Figure 7

Structure of the integrated torque response matrix $\overleftrightarrow{\mathcal{T}}$ calculated at the plasma boundary for the (a) KSTAR (${10}^{-3}\mathrm{Nm}/{\mathrm{G}}^2$) and (b) DIII-D (${10}^{-2}\mathrm{Nm}/{\mathrm{G}}^2$) plasmas. The solid circle indicates the dominant mode group with $m>nq$, and the second circle represents the second nonresonant mode group with $m<nq$.