The name of a river virtually always takes the definite article. "Drifts down Potomac" is incorrect.
The only exception I've been able to think of is when the word country is appended. You could correctly say, for example, "Colorado River country has become a popular tourist destination."
Interestingly, the appending of other nouns does not have this effect. You must say "the Nile delta" and "the Amazon basin" and "the Thames valley".
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