I assume that what happened here is that some fraction of the input flow was drained to a lower level, and the liberated potential energy captured and used to lift the remaining fraction to a higher level. That's not what's shown in the video, though.
There's a description starting on page 369 here, although it's not completely clear how it works (the author is an archaeologist, not an engineer!): https://sci-hub.se/10.1093/jis/etw016
Subsequent prior art (IYSWIM) is the hydraulic ram, which does this a different way: https://en.wikipedia.org/wiki/Hydraulic_ram
This mechanism is ingenious in the best meaning of the word.
My understanding was that the water was aerated, and so less dense meaning it could rise higher.
Værsgo:
https://youtube.com/shorts/BkVUQQnfrDg
Ideally I’d like to see an experiment to explicitly confirm it.
new to me: https://en.wikipedia.org/wiki/Trompe
https://en.wikipedia.org/wiki/Airlift_pump
edit: the text version of the video refers a study that I don't have access to.
https://primalnebula.com/the-engineering-of-the-alhambras-wa...
The Mastery in Hydraulic Techniques for Water Supply at the Alhambra García-Pulido, Luis José Journal of Islamic Studies, Volume 27 (3) – Sep 1, 2016 https://www.deepdyve.com/lp/oxford-university-press/the-mast...