Yes, literally, this is the border river in a gorge beneath the site. You can make out the remains of a bridge also. The border is closed and only a few years ago, visiting Ani meant getting permission from local security authorities.
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The next hill is Armenia! Returning to Kars after the visit, I had an Armenian phonenetwork for part of the time.
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Ths was reported to be a (Russian?) military base and a stone quarry, both across the border, of course.
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Actually, the years since the fall of the Soviet Union are the first in many centuries in which there is no direct Turkish-Russian border.