Rotation of Tokamak Equilibria

It is found that resistivity causes the self-consistent Tokamak equilibria to be unstable towards rotation about the magnetic axis. The growth rate is $\sim \frac{2{\pi }^2\eta R^2\beta }{{\iota }^2{a}^4}$, with $\eta$ the resistivity, $R\left(a\right)\mathrm{}$ the major (minor) radius, and $\iota$ the rotational transform. This is comparable with the skin penetration rate $\sim \frac{\eta }{a^2}$ of the poloidal field. We employ simple magnetohydrodynamic theory with resistivity, without considering possible stabilization by other dissipative effects. The quantities $\eta$, $\beta$, and $\frac{a}{R}$ are regarded as small. Our result agrees with one obtained previously in the case of an externally fixed field.