I just looked at the trailer, and it's so fast, it's hard to tell. So I don't know how they did it, but I can think of a couple of ways it could be done.
Assuming it was done practically on set, with cotton, you could support the cotton on a sphere or half sphere of clear plastic. You only need the front, or maybe a little more than half the sphere, if there is no back on it you wouldn't know it from this angle. You could hit it with pinspots from behind, to backlight it. (I did that with a planet with cracks and lava which glowed, back in my practical 16mm film days, it was half a sphere with lights inside - I think the clear plastic dome came from an old lampshade. By going close, with a 10mm wide angle lens, you couldn't see the edges, so it looked like a full sphere.) To get those little hotspots, you might use LEDs actually inside the sphere. By adjusting the overall lighting so it's a bit dim, but taking a long exposure, you can make the LEDs come up as bright as you like. There are some pretty bright LEDs, but they don't need to be super-bright, as long as the other lights are less bright. To make the ball grow larger, you might need to change it for larger spheres. That would create quite a change in the pattern the cotton balls, but they did change a lot, it wasn't a slow smooth explosion where you see one shape flowing into another in small increments.
It might be done standing up on set, but it could also be done lying flat on a board, with a hole behind the half sphere and cotton. You light from underneath, and place LEDs inside where you want hotspots. Then you composite it in. That way, you could enlarge the size of the explosion without replacing the support, just rearrange the cotton around it each frame so it boils, and zoom in in post. The outer ring of smoke on the ground probably is on the actual set, it seems lit like the ground. The inner ring might be a mixture. My feeling is, the sea and sky are composited in anyway. There may well have been lights in the sky area, highlighting the edges of the piles of rubble on the sides, which would have been visible if not removed and replaced with a background image.
It could of course be simulated with cgi, but the look of the film overall seems to lean towards practical effects where possible, even allowing a bit of twitch in the puppets' fur like you get from handling them - like Fantastic Mr Fox, it seems to celebrate the artefacts of doing it by hand rather than trying to make it as slick as a cgi film.
It does look like cotton balls on a clear or translucent sphere. The size of the ball stays the same, but because it's such a quick shot, it works. My guess is that it's mounted to a clear acrylic tube for the column and it rises up through the surface of the animation stage/table.
I can't find it, but someone made a scale model diorama of a star destroyer from Star Wars being attacked and a fireball flowing from one side where it was hit from and billowing out the other side and the effect is awesome. Of course, it's a static explosion, but animating a roiling fireball could be possible with a good bit of effort and look very realistic, IMO.
I did find this while looking for an image of the star destroyer:
I managed to find the article, but since the photos were hosted on Photobucket (and photobucket has turned to the dark side) All the images are gone. I found some of them through google images, but it would be great if the the text of the build had the pictures that spoke to what was done were still together in total.
Do a google image search for: Stephan Hacker "David against Goliath" Dreadnaught vs Star Destroyer 1:2256
The build article can be found at this link: http://www.keeperoftheforce.com/viewtopic.php?f=6&t=681
More of the Hoth Battle here: http://www.odditycentral.com/pics/star-wars-fan-turns-living-room-i...
It won't be as nice as the explosion in the trailer but you can put a piece of reflective card within the cotton and shine a flashlight on it to get some glow. That's how I did the explosion at :35 into this sketch https://youtu.be/ZNSovtpOzl8?t=35s
Thank You all for all these tips I really appreciate it!