As Thierry and Gerry mentioned, Brian Hayes wrote an article "Dividing the Continent" in American Scientist (Volume 88, Number 6, page 481), reprinted in his book, Group Theory in the Bedroom and Other Mathematical Diversions. His focus is algorithms to compute the continental divide, and so does not shed much light on the thrust of your question. But he does mention two interesting connections, which I will mention in the hope that it triggers further associations.

First, there is considerable algorithmic work by those interested in watersheds. For example, he cites the work of Luc Vincent and Pierre Soille, likely this paper: "Watersheds in Digital Spaces: An Efficient Algorithm Based on Immersion Simulations," (IEEE Transactions on Pattern Analysis and Machine Intelligence, Volume 13 Issue 6, June 1991). Generally the algorithms are variants on flooding the surface from minima, preventing the merging of water from different sources. In image processing, this is called the watershed transformation. E.g., see the images here. The watershed transformation is apparently available in MatLab.

Second, the problem was studied in some form by James Clerk Maxwell, although Hayes does not give enough information (in the article—I don't have the book here) for me to locate a precise reference. Perhaps it is related to what I know (from the work of Bob Connelly) as Maxwell-Cremona lifts? I would be interested to learn if anyone knows. Here is what Hayes says:

Maxwell relates the number of topographic peaks, pits and saddles on a surface. In the case of a sphere, the formula is $p+q–s=2$, where $p$ is the number of peaks, $q$ the number of pits and $s$ the number of saddles. Maxwell also outlines a procedure for dividing the landscape into watershed regions.

As Thierry and Gerry mentioned, Brian Hayes wrote an article "Dividing the Continent" in American Scientist (Volume 88, Number 6, page 481), reprinted in his book, Group Theory in the Bedroom and Other Mathematical Diversions. His focus is algorithms to compute the continental divide, and so does not shed much light on the thrust of your question. But he does mention two interesting connections, which I will mention in the hope that it triggers further associations.

First, there is considerable algorithmic work by those interested in watersheds. For example, he cites the work of Luc Vincent and Pierre Soille, likely this paper: "Watersheds in Digital Spaces: An Efficient Algorithm Based on Immersion Simulations," (IEEE Transactions on Pattern Analysis and Machine Intelligence, Volume 13 Issue 6, June 1991). Generally the algorithms are variants on flooding the surface from minima, preventing the merging of water from different sources. In image processing, this is called the watershed transformation. E.g., see the images here. The watershed transformation is apparently available in MatLab.

Second, the problem was studied in some form by James Clerk Maxwell, although Hayes does not give enough information (in the article—I don't have the book here) for me to locate a precise reference. Perhaps it is related to what I know (from the work of Bob Connelly) as Maxwell-Cremona lifts? I would be interested to learn if anyone knows. (See citation in comments.) Here is what Hayes says:

Maxwell relates the number of topographic peaks, pits and saddles on a surface. In the case of a sphere, the formula is $p+q–s=2$, where $p$ is the number of peaks, $q$ the number of pits and $s$ the number of saddles. Maxwell also outlines a procedure for dividing the landscape into watershed regions.