Stability of Highly Shifted Equilibria in a Large-Aspect-Ratio Tokamak

High beta poloidal tokamaks can confine plasma pressures an order of magnitude higher than their low beta poloidal counterparts. The theoretical stability of these high beta poloidal magnetohydrodynamics equilibria was left unresolved for many years. Using modern computational tools, such configurations are now found stable to Mercier, resistive and high-$n$ (ideal and resistive) ballooning criteria as well as fixed and free-boundary modes for a wide range of current density profiles in the framework of a low field large-aspect-ratio machine.

Figure 1

(a) Flux surfaces for a circular plasma cross section. The $Z$ axis is the axis of axisymmetry (actually located at $R=0$). Because of the up-down symmetry, only one half of the configurations are shown. The top set of surfaces is for a 15% Shafranov shift ($\beta =1.8\%$, ${\beta }_p=1.15$) and the lower set of surfaces represents an equilibrium with a 90% shift ($\beta =35\%$, ${\beta }_p=50$). The magnetic axis is located on the plane of symmetry ($Z=0$) at $R=5.15\mathrm{m}$ (15% shift) and $R=5.90\mathrm{m}$ (90% shift). (b) Pressure and (c) normalized current density profiles for the same shifts.

Figure 2

Current profile scans used in the stability study of high ${\beta }_p$ equilibria. (a) Peak position, (b) peaking factor, (c) current profile transition scans. The plasma extends from $R=4\mathrm{m}$ to $R=6\mathrm{m}$. Part of the null region of the current profile has been truncated from the plots to enlarge the section where shape changes occur. The magnetic axis is located around $R=5.80\mathrm{m}$.

Figure 3

GGJ resistive and high-$n$ ideal ballooning criteria of profiles A (solid lines), B (dashed lines), C (dotted lines), and D (dash-dot lines) for (a) peak location, (b) peaking factor, and (c) transition scans. The different criteria are plotted as a function of $\rho$ ($=r/a$). $\rho =0$ is at the magnetic axis and $\rho =1$ at the plasma edge. The ordinates of all curves are in arbitrary absolute units. In the left column, the part of the profiles that is positive shows instabilities to resistive interchange instabilities. In the right column, the profiles are stable if the criterion is positive.

Figure 4

$s\mathrm{\text{−}}\alpha$ diagrams for (a) peak, (b) peaking factor, and (c) transition scans. The majority of profiles are in the second stability regime for high-$n$ resistive ballooning modes. The diagrams exclude the region above $q_{95}$.