Nonlinear Saturation of Ballooning Modes in Stellarators

Abstract

Ballooning mode saturation is investigated in realistic stellarator configurations using the flux tube approach of Ham et. al. [1] [2]. The method is adapted to account for the lack of exact force balance in stellarator equilibrium solvers that assume existence of nested flux surfaces. A variational approach for calculating flux tube energy is developed to overcome this force error problem in stellarator numerical equilibria. Saturated (equilibrium) flux tube states that cross 10-20% of the plasma minor radius are shown to exist for linearly ballooning unstable profiles. It is shown that several features of the displaced flux tube structure in a full nonlinear MHD simulation of Wendelstein 7X are reproduced by our model. Saturated states are found in a compact stellarator equilibrium close but below the marginal ballooning linear instability, i.e. the unperturbed equilibrium is metastable. This suggests that Edge-Localized-Mode-like explosive MHD behavior may be possible in stellarators.

Figures (21)

• Figure 1: An illustration of the geometry of the flux tube model using a compact QAnelson_design_2002 equilibrium. Flux surfaces are plotted in grey. The surface of constant $\alpha$, which is tangential to the equilibrium field at all points, is depicted in blue. The perturbed flux tube (orange color) moves on the $\alpha$ surface with its elliptical cross-section elongated along the $\alpha$ surface. A field line inside the flux tube is plotted in black -- note that it is bent with respect to the equilibrium field. We assume that the flux tube does not intersect itself after going around the device.
• Figure 2: Typical shape of the saturated flux tube states, plotted as radial displacement in normalized minor radius $\Delta \rho$ v.s. cylindrical toroidal angles. This case is calculated on a compact QAnelson_design_2002 equilibrium on a stellarator symmetric $\alpha$ surface in the domain $\zeta \in [-12 \pi, 12 \pi]$. Two saturated states are presented, each corresponding to a stationary point of the energy curve. The procedure for generating such energy curve will be presented in section \ref{['sec:4']}. Four solutions of different resolutions are plotted for each saturated state, showing good convergence.
• Figure 3: Benchmark against COBRAVMEC. Here, $\rho$ is normalized minor radius and $\gamma$ is linear ballooning growth rate normalized to the Alfvén time. The growth rates calculated from our nonlinear model with small initial condition are shown with solid dots. The initial values $Y$ have the unit of meters$^{-1}$
• Figure 4: Pressure perturbation on the $\zeta=\pi/5$ (left subfigure) and $\zeta=0$ (right subfigure) cross sections at $t=2500 \tau_A$ of the M3D-C1 simulation reported in zhou_benign_2024. Red solid circles are the intersection of the same field line with the cross sections. The three orange lines are the $\alpha$ surfaces $\theta - \iota \zeta = -2\pi/5, 0, 2\pi/5$.
• Figure 5: Comparison of relative pressure perturbation $\delta p / p_0$ and relative total pressure perturbation $\delta p_{total} / p_0$, at $\zeta=0, \,t=2500\tau_{A}$. Notice that the small scale structures in $\delta p / p_0$ are not present in $\delta p_{total} / p_0$, justifying the assumption of total pressure continuity in our model.