# Characterize a continental divide

Here is something I've wondered about from time to time: The continental divide in North America is commonly described as the geographic line curve seperating points where a drop of water would drain to the Atlantic from those where it would drain to the Pacific. My question is how to characterize such a curve mathematically given a "reasonable" height function described over a region of the plane. I am not concerned with applied topography but also not interested in exotic pathologies. I'll propose a crude model now but feel free to propose a better one.

MODEL: The domain is the unit disk. A pre-mountain with peak at $(h,k,p)$ is a function $M=M(x,y)=\frac{p}{1+s((x-h)^2+(y-k)^2)}$ where $s>>0$ controls how steep it is and $p>>0$ how high. (note that a sum M_1+M_2 will have local maxima somewhat higher than $p_1$ and $p_2$ and somewhat displaced from $(h_i,k_i)$) The surface will be $b(x,y)(M_1+M_2+\cdots+M_n)$ where the $M_i$ are a large but finite number of pre-mountains and b(x,y) is a function such as $1-x^2$ or $1-x^2-\frac{y^2}{2}$ which is positive except at (-1,0) and (1,0) where it is 0. From each initial point the path of steepest gradient leads somewhere, usually (one might suppose) to $(1,0)$ or $(-1,0).$

Using the crude model as above, or a better one (describe it!) characterize the boundry between the basin of attraction of $(1,0)$ and that of $(-1,0)$

Comments: Of course a ring of mountains could create a pit with a sink in the middle, but that can be ignored or the problem can be changed to "characterize the boundries of the various basins of attraction". At a peak or saddle point the gradient is 0 but usually any direction one goes leads to the same sink. I imagine that there are (useful) applied approximate solutions starting from a grid of sample points with edges joining nearest neighbors. But I'd like some kind of minimax description like the solution of a continuous linear programing problem.

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The boundary of each basin consists of several gradient accents from saddle points to local maxima. In the normal case (finitely many non-degenerate critical points) all you need is to find all saddles and solve the gradient transport equation starting nearby (you'll have two accents from each saddle). You'll get a planar graph that separates the plane into the basins of attraction of local minima. There isn't really much more to say here. – fedja Dec 9 '10 at 5:52
Did you read the chapter on continental divide in Brian Hayes' Group Theory in the Bedroom? – Thierry Zell Dec 9 '10 at 6:13
No, but I actually own the book so I will, in the bedroom. – Aaron Meyerowitz Dec 9 '10 at 6:21
He also discusses it on his blog, bit-player.org/2009/long-division and bit-player.org/2009/distant-shores – Gerry Myerson Dec 9 '10 at 8:14
Fedja: Your last sentence sounds like a challenge. You are assuming that the height function is smooth with finitely many non-degenerate critical points. What about a fractal mountain range? – Bruce Westbury Dec 9 '10 at 9:03

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.