The primary is resonant tank circuit working on frequency 1
The secondary is resonant tank circuit working on frequency 2
The secondary contains also antenna and because of that there is also 3 resonant frequency for longitudinal 1/4 wavelength so the secondary is efficient radio transmitter.
You spark primary tank circuit not frequently, but it oscillate freely, secondary oscillate on some harmonics and the resonant LC and the length of antenna is correlated (L= 1/4 wavelength). Antenna instead of transmitting radio around is matched to the output part so all energy from ambient is stored in capacitors.
I do not pretend it works but the essence is something like above.
The secondary is resonant tank circuit working on frequency 2
The secondary contains also antenna and because of that there is also 3 resonant frequency for longitudinal 1/4 wavelength so the secondary is efficient radio transmitter.
You spark primary tank circuit not frequently, but it oscillate freely, secondary oscillate on some harmonics and the resonant LC and the length of antenna is correlated (L= 1/4 wavelength). Antenna instead of transmitting radio around is matched to the output part so all energy from ambient is stored in capacitors.
I do not pretend it works but the essence is something like above.
) this is a reactive electric power transformed into active power in L1, L1 has its own
some details can be found in my thread , the capacitor remain charged for near 30 min and after short circuit its leg several time for a long period, but when trying to recharge it the phenomena gone 
i think it's possible to work with less 20KHZ in primary side where it's possible to put a kind of variable inductance to catch the real resonance point ..
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