BENEATH THE ARCTIC OCEAN—Five hundred feet below the Arctic ice cap, the USS New Mexico's crew filled two torpedo tubes. "Match sonar bearings and shoot," ordered the skipper, Cmdr. Todd Moore. The air pressure rose sharply as a simulated torpedo headed toward its simulated target: a Russian Akula-class submarine.

The Arctic exercise, one of two over this past weekend, was intended as a show of U.S. force for the benefit of America's allies, defense officials said. The drills were arranged before Russia's annexation...