Another joint exercise, known as Foal Eagle, has been under way since the beginning of March.
The ongoing annual drills, called Foal Eagle, are to finish at the end of April.
The two countries also are running separate Foal Eagle drills in South Korea through April 30.
They will be followed later in the week by another set of joint military exercises called Foal Eagle.
BBC: North Korea warning as US and S Korea begin joint drill
Pyongyang has been angered by tightened UN sanctions imposed after its 12 February nuclear test, as well as the "Foal Eagle" drills.
BBC: US and South Korea wrap up "Foal Eagle" military drills
Moves such as keeping nuclear-capable military assets in the region could unnecessarily prolong the risk of miscalculation after the Foal Eagle exercises end.
Thurman also oversees the annual military maneuvers under way, known as Foal Eagle, as well as an accompanying war game, called Key Resolve.
The Pentagon struggled over the decision to deploy the bombers as part of the war games, called Foal Eagle, according to U.S. officials.
Though the maneuvers, called Foal Eagle, are held regularly, North Korea was particularly angry over their inclusion this year of nuclear-capable B-2 stealth bombers and F-22 fighters.
The scripted actions included an increased show of U.S. military force -- such as the flying of B-2 bombers -- during the annual U.S.-South Korea military exercise, the Foal Eagle.
During ongoing Foal Eagle military maneuvers in South Korea, two U.S. B-2 strategic stealth bombers, flying from their base in Missouri, conducted a mock bombing run on a South Korean range.
"Eyebrows started to go up when it was clear Foal Eagle was going to be protected from the budget cuts of sequestration, " the official said, referring to the forced federal spending cuts that went into effect in March.
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