I don’t think the actual battle has begun yet. The current incursions are more an intimidation measure by the Americans, to show that they can’t be stopped, and can come and go at will. The real battle, the one in which there will be a huge exchange of fire, will likely occur in the next day or two.
Tonite on CNN they’ve done a lot of talking about how it appears that the command structure of the Iraqi armies has been essentially shredded.
If an effective commander can step up and put together a cohesive defense, the “Battle of Baghdad” could get much more heated.
Failing that, I think we’re going to see sporadic guerilla fighting and a sharp spike in terrorist style attacks, until Coalition forces sercure the remainder of Iraq.