Start with a carvers' chisel to chop out some of that waste in the trough. Be sure to 'follow the form - in other words, keep a feel for the convex shape of the bine and the way it winds around the cylinder at all times, even as you carve in with the straight edge.
'Following the form', as I express it, is a fundamental idea in carving. The other way of putting it is: 'Even though you are removing wood, it's what you leave that matters...'