During my Navy days aboard USS Nimitz, we never encountered what I'd call really bad seas. However, as this video shows, it can happen. This is a Nimitz-class carrier rounding the horn of South Africa.
Oh yes, carriers don't travel alone. There are usually two guided missile destroyers or cruisers along for the trip. In fact, before I was posted to the Nimitz I was on one of her usual escorts - the former USS Texas CGN-39.
Those Texas days were fun. That was where I first experienced 40 feet of green water crashing over the bridge as the front third of the ship went underwater every so often. Try doing that and keep two nuclear reactors running! Those were some amazing watches. I recall getting 37 pressurizer low-level alarms in one 4 hour watch. But the reactors stayed critical and we plowed on through the seas.
The original USS Texas is at Seawolf Park between Houston and Galveston. If anyone books a cruise from there and wants something interesting to do before or after their cruise, there are still guided tours of her. Very interesting tour, it's been many years since I did it, I think I'll head back down there soon and tour her again.
In the great traditions of the Navy, we had a silver tea service aboard the nuclear cruiser Texas which had been on the original battleship. I love that kind of thing since it keeps a connection from the past with the new. I remember the memorabilia was in a couple glass showcases aft of the mess deck outside the Chief Engineers office.