Ok, now that I'm actually doing it, I can learn more about the Kratky method of hydroponics.
It's a system that doesn't use any electricity or pumps (air or water) for the hydroponic system (note: you'll still use electricity for LED grow lights... none if you use the sun).
I've found a great resource that breaks it down really well. Including details on growing fruiting plants (which take a lot more water and may necessitate the use of a passive float valve to maintain water from a separate reservoir).
