Drift stabilization of ballooning modes in a high-$\langle \beta \rangle$ LHD configuration

Ideal MHD yields at best inconclusive predictions about the stability of the LHD heliotron for $\langle \beta \rangle \geq 3\%$. We investigate the impact of the drift stabilization of ballooning modes for the inward shifted LHD configuration (vacuum magnetic axis $R_0 \sim 3.5m$). The background equilibrium is considered anisotropic in which the neutral beam ions contribute about $1/4$ fraction of the total diamagnetic beta, $\langle \beta_{dia} \rangle$. A drift corrected ballooning mode equation obtained from the linearized gyrokinetic equation is expanded assuming that the hot particle drifts are much larger than the mode frequency. The fast particle pressure gradients contribute weakly to both the instability drive and the diamagnetic drift stabilization (which is dominated by the thermal ion diamagnetic drifts) for $\langle \beta_{dia} \in [0,4.8] \%$. In the single fluid limit (diamagnetic drifts ignored), the thermal pressure gradients drive ballooning modes in a broad region encompassing the outer $60-90 \%$ of the plasma volume at $\langle \beta_{dia} \rangle \approx 4.8 \%$. To stabilize these modes, we find that diamagnetic drift corrections must be invoked (mainly due to the thermal ions). The energetic ion diamagnetic drifts play a role only for low wave number values, $k_{\alpha}\leq 8$. It has been verified that the fast particle drift ordering imposed by the model is amply satisfied for on-axis hot particle to thermal density $N_{h0}/N_{i0} \approx 1\%$ even at high $\langle \beta_{dia} \rangle$.