Circuit Model for Gun Driven Spheromaks

In this note we derive circuit equations for sustained spheromaks, in the phase after a spheromak is detached from the gun and sustained in a flux conserver. The impedance of the spheromak during the formation and ''bubble burst'' phase has been discussed by Barnes et. al. We assume here that the spheromak is formed and helicity is being delivered to it from the gun, currents are above the threshold current, and the λ-gradients are outward (λ decreasing inward). We follow an open field line that begins and ends at the gun electrodes, encircling the closed flux surfaces of the spheromak, and apply power and helicity balance equations for this gun-driven system. In addition to these equations one will need to know the initial conditions (currents, stored energies) after the ''bubble burst'' in order to project forward in time