The breath meter disappears when you are on the surface of the water, but does not reappear if you go underwater unless you go to land first.
Also, it seems it is still possible to drown on the surface of the water, as in the final challenge, I was on the surface of the water while using the Rau for the puzzle, and drowned(i was unaware of the depleting breath meter since it turns invisible when you are on the surface).