How does rotating water affect turbulence?
The dye diffuses vertically because it has a higher density. So when you put the dye in from the top, it wants to go to the bottom, albeit slowly. If it were more buoyant, it would float. It does diffuse horizontally, though much slower. This can be shown that, after a long time, the water will turn the color of the dye.

As far as rotation affecting the dye, that is complicated. If you start the rotation before you put the dye in, you can get Taylor Columns. Throw in different variables/change different things about the tank, and you can find different phenomenon (see the Weather in a tank experiments).
What you see in your experiment is called &quot;shear-induced diffusion.&quot; The movement of your pen makes shear, and that shear increases the diffusion rate. I would also think that the eddies generated help diffuse the dye.