 Single-azimuthal-mode current-driven instability threshold

In the uniform-current rotating magnetohydrodynamic wave dispersion, exponential growth requires $J^2(2m\delta -m^2)/(4{\mu }_0\rho )>{\delta }^2{\Omega }^2$. For integer $m$ and $|\delta |<1$ this is possible only for $|m|=1$ and $m\delta >1/2$. Positive and negative $m$ branches are related by complex conjugation. The equality case is a repeated frequency and requires separate analysis of possible algebraic growth.