So without a single external twitch to create ripples, they can ascend or descend.
Their spines are also highly specialized for undulation, with ball & socket joints ("procoelous") in the caudal (tail) vertebrae that make up 50-60% of the bones in their full spine.
Alt: Petting a Wild Alligator: A man in baseball cap and blue sweatshirt reaches down over the edge of a swamp boat or air boat and scratches an American alligator behind the ridge of the head. It closes its eyes satisfied-looking, and releases from the boat, sinking back into the dark waters.