transformer
For Tesla coil, eventually the capacitor voltage becomes so high that the air in the spark gap is unable to hold-off the high electric field and breakdown occurs.
But usually from what I know DC can charge up the capacitor, not AC. What am I missing?
Modelling the spark gap is the "fun" bit. I'd recommend downloading a student edition of microcap and using their sparkgap model: -
As for the transformer, this is modelled as two inductors with a coupling coefficient (k in microcap and m in spice I believe). The coupling coefficient dictates the amount of primary and secondary leakage inductance. You'll also need to model exteranl resistors that represent the winding losses and quite probably a few hundred pF across the secondary winding.
Here's the spectrum (suppliers of microcap) explanation of the spark gap. Maybe you can make use of this article in other simulators?
A car ignition coil is far too small output current to be loaded by a 13nF MMC. As you mention, even a multimeter drags the output down.
Usually we use a Neon Sign Transformer (NST) with 10s of mA output current in this sort of setup. I notice you started with NST, which explains a lot! If you hired a U-Haul trailer, then found your car didn't have a tow-ball, would you tie it with string to your bicycle? You need a car for the trailer, you need an NST for 13nF. You have the reasons written on your diagram, coil output \$600\mu A\$.
If you want to continue with an iggy coil, then suggest you rectify the output so it can take its time to charge the capacitor to a high enough voltage for a decent spark. DC coils can work, though a rotary spark gap (RSG) is often necessary for extinguishing the arc after firing. Otherwise, tune the MMC down to work with the iggy coil. Either way, you need to change the design.
Unfortunately GFI-less NST's are becoming rarer, they're not being made these days. But that's what you need to make an unfussy TC that will just work.
SO is a very general forum, Tesla coiling has a very small active community. You would be far better off asking these questions on a dedicated TC forum, you'd get a much better signal to noise ratio. Start with the pupman lists (archive) or 4HV (active) for instance, where you can find the community that has the stuff, and the domain knowledge.
Best Answer
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)!