This page describes an idea for a triggered spark gap intended for tesla coil use. The basic concept is to use two copper pipe caps facing each other as the basic gap. A hole bored in one of the caps provides a path for air to flow, and for the trigger electrode to be in. The air flow cools the gap electrodes.
Two ideas for triggering. The first is to put the trigger potential between the trigger and Main Electrode 1. This breaks down the gap, irradiating the main gap (as well as injecting ions and making a hot spot on Main Electrode 1). The main gap would then start conducting. The other is to trigger between the Trigger Electrode and Main Electrode 2. The annular spark would irradiate the main gap, causing it to breakdown.
There doesn't need to be a whole lot of air flow. I think that one of the small "car vacuum cleaner" type blowers would work, or perhaps, even an appropriately sized muffin fan, although "propeller" type fans generally don't work as well against high static pressure. A centrifugal type blower would be more appropriate.
Getting the air flow into the pipe cap might be as easy as using a PVC T, where one arm of the T provides a path and support for the trigger electrode, and the side arm is used for the air flow. Some form of locking or clamping bushing (that is reasonably airtight) would be useful to hold the trigger electrode. Some testing would need to be done to see how hot the copper pipe cap gets.
If we make an assumption that 10% of the electrical input power for a typical TC is lost in the spark gap, then the gap is going to be dissipating somewhere around 50-100W, a substantial amount of heat. Most of that will go into heating the air in the gap, of course. Some significant fraction will go to heating the electrodes, either by conduction from the arc, or radiation. Copper is a pretty good conductor, so IR losses are probably quite low.
However, consider that small CPU fans are moving enough air over a 30-40W source to keep their temperature in the 50C range, so clearly the cooling airflow requirements for this application aren't huge.
Supersonic flow through nozzle probably isn't necessary. This isn't a Marx type blast gap, where high flow rates are used to insure rapid quenching at very high (kilowatt) power levels.
more on triggered gaps