Not the Black Sea; the Sea of Azov. Here's the scenario:
(1) The Caspian Sea is full to the brim from the overflow from the proglacial Lake Ob. The Black Sea is 100m lower than it is today. The fertile floor of the Avoz "sea" is, thus, exposed and Atlantis flourishes there.
(2) In this EARTHQUAKE prone area, a quake causes a massive underwater landslide in the Caspian Sea triggering a tsunami that sends a huge wall of water over the brim, rushing down to the Don river valley and filling the Avoz plains with water.
(3) The initial overflow of the Caspian continues as the overflow reodes deeper into the brim. The water covering the Avoz plains can only gradually flow into the Balck Sea through the narrow Kirch Strait. When the overflow from the Caspian finally spots, the Black Sea (and Avoz) reached their modern level.
(4) Atlantis has vanished. Forever?
Now you know where to look!