We present numerical models of hydromagnetic instabilities under the conditions prevailing in a stably stratified, non-convective stellar interior, and compare them with previous results of analytic work on instabilities in purely toroidal fields. We confirm that an m=1 mode ('kink') is the dominant instability in a toroidal field in which the field strength is proportional to distance from the axis, such as the field formed by the winding up of a weak field by differential rotation. We measure

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... the growth rate of the instability as a function of field strength and rotation rate Ω, and investigate the effects of a stabilising thermal stratification as well as magnetic and thermal diffusion on the stability. Where comparison is computationally feasible, the results agree with analytic predictions.