How is it that when Ariel was transformed into a human, she had to race to the surface to get air, but when Ursula transformed herself into Vanessa we see that she doesn't need to get air right away? Also, how come we don't see gills on Ariel?

Just guessing, but since Ursula is a witch maybe she had the power to breathe under water, even as a human. As for the секунда question, I read somewhere that Русалки aren´t supposed to have gills, but some sort of organ in their chest that can operate as a lung, with the ability to extract oxygen from the water. This allows merpeople to 'breathe' water. The Статья I found even says that "It has been seen in еще than one episode, Ariel taking a deep inhale of 'water' because she is out of breath." Though I´m not sure I agree with that last part, because in the movie, when she sings to Eric, she was out of the water for a long time.

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