Rotary spark gaps are self-quenching, since the gap is only small for a very short period of time.
Also, a rotary gap is significantly self-cooling, since one half of the gap is rotating at a very high rate.
Last, and probably most significantly, a rotary spark gap basically lets you control the arc rate, since you generally only get one spark per gap closure. Therefore, you can control the repetition rate by varying the rotor speed.
Brief answer: Resonance (oscillations) of the primary circuit only occur while the spark gap is conducting. However, even at zero current, the spark still remains ionized and conducting for tens of microseconds. That means it won't quench, even though the high-freq oscillations repeatedly pass through zero amperes.
But, as the oscillations slowly decay, and the average current decreases, eventually the spark will quench. When the spark disappears, the oscillations in the primary circuit sudddenly halt.
Also note that when the primary coil halts, the secondary coil's oscillations still continue for a much longer period.
Overall sequence:
- The HV power supply (DC or AC 60Hz) rapidly charges the capacitor.
- The spark gap fires, connecting the capacitor to the primary coil.
- The coil/capacitor oscillates at high frequency (the spark stays lit.)
- Over several cycles the secondary coil begins oscillating as well.
- The primary coil/capacitor rapidly loses energy (EM energy is moving to the secondary coil.) The tighter the coupling between coils, the faster this occurs.
- Over several high-freq cycles, the average primary current falls to zero, the average secondary coil simultaneously rises to maximum, the spark gap then quenches, and the primary coil/capacitor stops oscillating.
- The secondary coil now continues to ring, and its oscillations slowly die away as energy is lost: to wire-heating, to heating of sparks and corona, and to a very small amount of radio waves.
The critical adjustments are:
- Matching the LC frequency of the primary coil and capacitor to the secondary coil's natural frequency.
- Adjusting the coupling between primary and secondary, in order to transfer the LC oscillations into the secondary much faster than the RLC decay time of the primary, but not overly rapidly (since then the oscillations may then transfer back again before the spark quenches, leaving the secondary with low voltage.)
- Adjusting the spark gap so it quenches just as the primary coil oscillations have fully transferred to the secondary. Cooling of the gap may be required (by large solid electrodes, by multiple small electrodes in series, by a rotary gap, or a combination of all three.)
Interesting triva: if instead of a spark gap we used a switch, then when the switch is suddenly closed, the capacitor and primary coil oscillate. But then the oscillations all "slosh into" the secondary coil, and primary-circuit oscillations are halted. Next, they "slosh back" again, and the primary oscillates, while the secondary coil stops. Then, forward again from primary to secondary, then back again, then forward, over and over. This "slow slosh" is an example of "Line Splitting" or "Coupled Pendulums" or coupled oscillators, and it has a low frequency determined by the amount of coupling between the two coils, as well as the pri/sec resonant frequency. Ideally we want our switch (or spark gap) to open just as the RF voltage on the secondary coil reaches maximum, and before the RF voltage on the primary coil starts rising again during the first "reverse slosh."
When driving the system with raw AC (neon sign transformer,) all of the above must be adjusted to occur in a few hundred microseconds: a little less than 1/4 of one AC cycle. That way the AC supply has time to fully charge the main capacitor, and then the total energy in the capacitor can be deposited into the secondary coil before the next half-cycle of 60Hz begins. This maximizes the output wattage of the system, where the secondary puts out enormous high-freq surges at 120Hz.
For more info, see figs 2.12 and 2.13 here.
Also, here's a classic 1991 paper from AJP journal, with oscilloscope photos of the primary-secondary energy transfer.
Best Answer
DC gas-discharge voltage (post-breakdown spark-gap voltage) isn't simple, and depends on the spark-length, value of current, the type of metals, the time-length of the spark, the temperature of the metal surface, the gas mixture, initial gas temperature, dust contamination, etc. Spark-plasma is a fairly good conductor. Roughly expect the voltage along the spark to be somewhere between ten and a few hundred.
So, the voltage across the resistor will briefly be about 40KV, minus the small voltage which appears across the spark. VERY briefly.
After some nanoseconds, the voltage of the Tesla Coil will have collapsed. Its stored energy was discharged. (A Tesla Coil's secondary coil is something like a capacitor, also something like an energy-storage inductor.) The energy which had been stored in the Tesla Coil will mostly end up inside the spark-plasma, with some being deposited into your 1-ohm resistor. Use a much large resistor value if you want the energy to mostly end up inside the resistor.
Find books on DC gas-discharge at one atmosphere. It's an ancient topic, so not much exists online. Here's the V-I curve for a fairly long discharge in a neon tube: