When water comes out of a faucet at the bottom of a tank, it comes out quickly first and then it tapers off. Is there an adverb for such a case?
Let's face it, there is no unclumsy adverb for this. Go with a verb. The stream dies off, loses momentum, dries up are all possible. So is your very own tapers off, and the peters out suggested in a comment. The pressure drops down. Custom-tailored, more colorful metaphors are possible as well, like loses dedication, gets lazy, gives up, or any number of others. Try as you might, you won't get that kind of flexibility with an adverb.
There are a few phrases that get close to this meaning but the more accurate terms are somewhat technical.
decreasingly — becoming less or fewer; diminishing.
logarithmically — something that changes according to logarithmic growth
The major issue with these terms is that they sound weird and the meanings are not very clear:
The decreasingly escaping water.
The water is escaping logarithmically.
A more understandable phrasing would specifically refer to the rate being described:
The water pressure decreases quickly.
The faucet releases water at a decreasing rate.
Diminishingly seems to fit the bill.