I've now seen the Alcubierre drive repeatedly mentioned and fail to see what's so peculiar new or innovative about it. Back in 1979, when they were trying to put the motion picture on science factual footing, I remember NASA's Jesco von Puttkamer (RIP) joined as a scientific consultant and the warp drive ("space bubble") was considered to be working basically the same as this Alcubierre drive (so maybe Puttkamer drive would be more accurate? Or A & P drive?
Here is one of my favorite TOS tech quotes from Spock (in "By Any Other Name"):
The Enterprise is propelled by matter-antimatter reactors.
(a very concise statement that the ship is driven by energy and not particle thrust).
Please bear in mind, that according to "The Doomsday Machine" and the USS Constellation
the caps don't seem to have any function that's essential to achieve warp drive.
As for the spheres at the aft of the nacelles I guess it's not too farfetched to assume that those are the nacelles' antimatter "pods" that had been repeatedly mentioned / suggested in the series.