I need help ID'ing this hose.

New guy here. I need help figuring out what this hose is/does. I'm thinking it might have something to do with the transmission but I may be way off. It's my father in laws '89 Crown Vic. The hose ruptured as he was driving on the freeway and he saw white smoke coming from the rear.

I've outlined the hose in red. Anyone know?
Thanks in advance.

http://www.wrxfanatics.com/uploads/post-1-1104271140.jpg

close up
http://www.wrxfanatics.com/uploads/post-1-1104271166.jpg

close up from passenger side
http://www.wrxfanatics.com/uploads/post-1-1104271237.jpg

That is your heater to manifold coolant hose,with this hose busted the car will be undrivable as you will lose all your coolant within about 5-10mi.Luckily it is the easiest of the coolant hoses to change.It costs about $1.00 a foot at most auto parts stores,it is called 3/4in Heater Hose,You should need about 3ft. of hose and 2 new hose clamps.Just loosen the hoses clamps at each end with a flathead screwdriver,if it dos'nt come off when you give it a good pull,put a vertical slice down each end of the hose and pull the sides back.It should then come off with no problem.Measure the new hose against the old,and cut it to the correct length,route the hose in the same position as the old one,install the hose clamps,push each end of the hose onto the coolant hose connectors and tighten the hose clamps down tight.Hope this helps.

Thanks a bunch for the info!

Looks like it will be a fairly simple repair.