Tesla coil charging the battery with AC?
The capacitor eventually reaches an AC peak voltage that is enough for the spark gap to arc-over. When AC is initially applied, L1 and Cp form a series resonant tuned circuit. 

At this point the effect of Lp is insignificant to L1 i.e. L1 dominates so ignore this for now.

The AC voltage across Cp starts low and builds and builds rising positively and negatively with the AC power supply frequency. Eventually the peak voltage across Cp (positive or negative) is enough to cause an arc on the spark gap.

This massive dV/dt change in C pushes through a really big current into Lp and this ultimately causes the high voltage stuff we all love (or not)!