It's taken me longer than it should to realise you mean air conditioning rather than alternating current haha. I was thinking wtf, they had alternating current back then? I thought that was Tesla's work!

I'm thinking that they probably don't need to do anything to ensure the air flow is consistent.

In a hot arid environment, the river will be evaporating relatively quickly. This will cause a cooling of the surrounding air, which will mean the air around the river itself is denser (and more humid) than the air above the plains. This will mean there is a constant outward air pressure, which is exploited by their tunnels. The air moves up the tunnel to the pool, where it evaporates the water, cooling and humidifying the air, which continues upward thanks to the lower air pressure at the top of the tunnel.

Simple but very clever.