General Theory of the Plasmoid Instability

A general theory of the onset and development of the plasmoid instability is formulated by means of a principle of least time. The scaling relations for the final aspect ratio, transition time to rapid onset, growth rate, and number of plasmoids are derived, and shown to depend on the initial perturbation amplitude $\left({\hat w}_0\right)$, the characteristic rate of current sheet evolution $\left(1/\tau\right)$, and the Lundquist number $\left(S\right)$. They are not simple power laws, and are proportional to $S^{\alpha} \tau^{\beta} \left[\ln f(S,\tau,{\hat w}_0)\right]^\sigma$. The detailed dynamics of the instability is also elucidated, and shown to comprise of a period of quiescence followed by sudden growth over a short time scale.