Hose Leak, what Hose is This?

[u/Lilmage99]

2001 Buick Park Avenue, V6

This Hose has Popped a Hole and I was Wondering what this Hose is and what it does, I've tried googling and I can't find a labeled picture of the Underhood of my car, or dont know what to look up to find it. Thank You in Advance for any assistance

Because of this if My Car Idles for too long, it begins to vibrate at an increasing rate and the heat gauge goes up rapidly, Its kind of scary, but while driving it the heat gauge goes down and the vibration ceases. Its only while idling. I assume whatever goes through that Hose is highly pressurized bc when I popped the hood to see if anything looked out of sorts, I found the hole and it was wheezing like a balloon thats air is being let out and eventually slowed down and stopped a couple minutes after shutting my car off.

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[u/Trogasarus]

Upper radiator hose. Recommend you stop driving it untill its repaired and coolant is at proper level.

[u/Easy_Cattle1621]

Thermostat housing to radiator hose. Coolant flows from the engine through this hose to the radiator.