This was my first sequence. I did only one streak and simply animated the curve from which the particles were emitted.

I added multiple streaks and made them all green to prevent the colors from clashing too much. I only animated a couple curves and the movement is very subtle.

I had three streaks of the same particle type. I animated the movement of the curves more than in the previous video and added a very small vortex field. I animated the rate to start at 0 and increase to 1000 halfway through the animation.

I strongly increased the magnitude of the vorticity and animated its source to sweep from left to right. I changed the colors slightly from the previous animation and moved the curves further apart. There is a lot of movement in this one.