Here is something I was playing with last night before I had to get some sleep. I did a 30 frame render as an example
Sorry for bad gif results, I lost my chart for ffmpeg and couldn't find the flags I had saved for "nice" gifs, so I hit google but couldn't find one with better colour settings.
Z stretch on PFs may be a tad too strong. But this at least incorporates your idea of 4D noise, with simple translation on Z. and some added "channel warping" and isolation of water displacements. Basically water in the center of the river is, and should be moving faster and rougher, while water on along the shore is a little slower and calmer.
