On the stability of the <i>m</i> = 1 rigid ballooning mode in a mirror trap with high-beta sloshing ions

Stability of the ‘rigid’ ( $m = 1$ ) ballooning mode in a mirror axisymmetric trap is studied for the case of oblique neutral beam injection (NBI), which creates an anisotropic population of fast sloshing ions. Since small-scale modes with azimuthal numbers $m&gt;1$ in long thin (paraxial) mirror traps are easily stabilized by finite-Larmor-radius (FLR) effects, suppression of the rigid ballooning and flute modes would mean stabilization of all magnetohydrodynamic (MHD) modes, with the exception of the mirror and firehose disturbances, which are intensively studied in geophysics, but have not yet been identified in mirror traps. Large-scale ballooning mode can, in principle, be suppressed either by the lateral perfectly conducting wall, or by the end MHD anchors such as the cusp, by biased limiters or by a combination of these two methods. The effects of the wall shape, vacuum gap width between the plasma column and the lateral wall, angle of oblique NBI, radial profile of the plasma pressure and axial profile of the vacuum magnetic field are studied. It is confirmed that the lateral conducting wall still creates the upper stability zone, where the ratio $\beta$ of the plasma pressure to the pressure of vacuum magnetic field exceeds the second critical value $\beta_{\text{cr2}}$ , $\beta &gt;\beta _{\text {cr2}}$ . However, in many cases the upper zone is clamped from above by mirror instability. When the lateral wall is combined with end MHD anchors, a lower stability zone $\beta &lt;\beta _{\text {cr}1}$ appears, where $\beta$ is below the first critical value $\beta_{\text{cr1}}$ . These two zones can overlap in the case of a sufficiently smooth radial pressure profile, and/or a sufficiently low mirror ratio and/or a sufficiently narrow vacuum gap between the plasma column and the lateral wall. However, even in this case, the range of permissible values of beta is limited from above by the threshold of mirror instability $\beta_{\text{mm}}$ , so that $\beta &lt;\beta _{\text {mm}}&lt;1$ , in contrast to the case of transversal NBI, when neutral beams are injected perpendicularly to the magnetic field.