Hovering over the Trident Juncture exercise in Norway and among the roughly 15,000 American troops — most of them Marines — who are participating, is a consuming narrative about the alliance’s next possible war.

The next, more immediate adversary is closer to northern Europe: Russia.

The country’s president, Vladimir V. Putin, has made no secret of his distaste for the Atlantic alliance’s encroachment into territory he considers part of his sphere of influence — particularly in the Baltics and in the Balkans.And since Moscow seized Crimea from Ukraine in 2014, Western officials have worried that NATO states could be next.

Warfare in northern Europe is entirely different. It requires attention to small details, like carrying cold-weather lubricant for machine guns, as well as to seismic decisions, like moving thousands of men and women, and their heavy machinery and weapons, across fields packed with snow — something American troops have not had to worry much about for more than a half-century.