You may need to be using #3 jaws on your chuck for this deep a vase. A
steady rest would also be a great idea. A faceplate with glue block would
hold your vase more solidly.

An endgrain cutter like a hook tool or Oneway Termite would be good for
hollowing. Your typical tools probably won't work for this vase unless you
have some incredible technique. Also, you need to have a handle that is at
least 50% longer than the one on your bowl gouge to be safely cutting so far
over the toolrest. Time to buy a hollowing tool set or make shallower
vases 8" tall or under.

Derek