Nonlinear magnetohydrodynamic modeling of ideal ballooning modes in high-$β$ Wendelstein 7-X plasmas

Abstract

We present nonlinear magnetohydrodynamic (MHD) simulations of high-$β$ Wendelstein 7-X plasmas using the stellarator extension of the M3D-$C^1$ code, building on the recent work that shows benign saturation of ideal ballooning modes above the designed $β$ limit in the standard configuration [Y. Zhou et al, Phys. Rev. Lett. 133, 135102 (2024)]. First, we examine the results' sensitivity to the parallel thermal conductivity. It is found that while an increased parallel conductivity reduces the linear growth rate, the saturated pressure profile is barely affected. Second, we consider the dependence on the profile shape. It is shown that an equilibrium with a peaked pressure profile and lower $β$ is subject to more significant change than a broad profile with higher $β$ and a larger growth rate, suggesting that benign saturation, or nonlinear stability, is not guaranteed and not dictated by linear growth. Third, we study the influence of the magnetic configuration, with the equilibrium rotational transform varied by adjusting the planar coil current. With similar growth rates, similar magnitudes of profile change are found regardless of the presence of a low-order resonance, which implies that the saturation mechanism is not specific to a resonant or non-resonant mode. These results indicate that MHD stability should still be treated seriously in stellarator operation and design, for which nonlinear modeling using tools like M3D-$C^1$ can play an instrumental role.

Figures (5)

• Figure 1: (a) The linear growth rate $\gamma$ vs. $\kappa_{\parallel}/\kappa_{0}$ in single-field-period simulations of the standard EIM configuration using a broad pressure profile (green) and a peaked pressure profile (blue and red). The dashed curves show results obtained with higher toroidal resolution (48 planes). The dotted curve shows results obtained with $\kappa_{\perp}=10\kappa_{0}$. (b) The saturated shapes of the pressure profile $\bar{p}$ at $\theta=0$ and $\varphi=\pi$ in full-torus simulations with $\kappa_{\parallel}/\kappa_{0} = 10^6$ (dashed) and $10^7$ (dotted), compared to the respective initial shapes (solid; blue: broad profile with $\beta=5.44\%$; red: peaked profile with $\beta=3.88\%$). The normalized minor radius $r/a=\sqrt{s}$.
• Figure 2: (a) The rotational transform profiles in the simulated EIM equilibria, with the vacuum profile (black solid) shown for comparison. The yellow solid line marks the $\iota=5/6$ resonance. (b) The saturated shapes of the pressure profile in the M3D-$C^1$ simulations, with the initial shapes (solid; blue: broad profile with $\beta=5.44\%$; red: peaked profile with $\beta=3.88\%$ and $4.04\%$) shown for comparison.
• Figure 3: Snapshots of the normalized pressure change in the three simulations in Figure \ref{['fig:prof']}: row 1 shows mode structures at the end of the linear growth phase, while row 2 shows the saturated states at the end of the simulations. Note the different scales in the color bars. Row 3 shows the Poincaré plots of the saturated states, where different colors label different field-lines.
• Figure 4: (a) The rotational transform profiles in the simulated finite-$\beta$ equilibria (colored), with the vacuum profiles (black) shown for comparison. Different values of $I_\mathrm{PC}/I_\mathrm{MC}$ are distinguished by line styles. The yellow line marks the $\iota=5/6$ resonance. (b) The final shapes of the pressure profile in the M3D-$C^1$ simulations, with the initial shape (black solid) shown for comparison.
• Figure 5: Snapshots of the normalized pressure change in the simulations in Figure \ref{['fig:iota']} (except the EIM case): row 1 shows mode structures at the end of the linear growth phase, while row 2 shows the saturated states at the end of the simulations. Note the different scales in the color bars. Row 3 shows the Poincaré plots of the saturated states, where different colors label different field-lines.