Triple Divide Points and North American Drainage Basins
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Robert N. Andersen
University of Wisconsin-Eau Claire
Department of Mathematics
Hibbard Hall 533
Eau Claire, WI 54701
Email address: andersrn@uwec.edu

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Last Updated: December 7, 2006

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Ph.D., M.S. - Oregon State University
M.Ed., B.A., B.Ed. - Central Washington University

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Triple Divide Points
of
North America


    Anyone who has driven across the United States from coast to coast has most probably seen the signs proclaiming the Continental Divide, that line or arc running north-south where any water falling on one side - the east side - will flow to the Atlantic Ocean (most probably via the Gulf of Mexico), and water falling on the other side - the west side - will flow to the Pacific Ocean. These signs announcing the Continental Divide are usually found in the Rocky Mountain states of Montana, Wyoming, Colorado or New Mexico.  Drivers traveling a northern route may be surprised to see signs in North Dakota announcing the Continental Divide. North Dakotans are not liars nor practical jokers, but indeed a divide does pass through North Dakota, a divide where water on one side will flow north into Canada and into Hudson Bay while water on the other side will flow south into the Gulf of Mexico.   This line will have a generally east-west orientation, and we can well assume
this divide and the more well known divide will have a point of intersection.  This point of intersection is in Glacier National Park and is known as Triple Divide Peak. An `Ideal' raindrop falling on the very summit of Triple Divide Peak will divide into three subdroplets. One subdroplet will flow north into the Saskatchewan River, into Lake Winnipeg and from there down the Nelson River into Hudson Bay. A second subdroplet will flow into the Marias River then into the Missouri and eventually the Mississippi and the Gulf of Mexico.   The last subdroplet will flow west into the Flathead River, then the Clark   Fork and eventually the Columbia and the Pacific Ocean.

    We can generalize this phenomena by saying that any point where an ideal raindrop will split into three parts and flow into three separate drainage basins is a Triple Divide Point. Marty Issacs, a mathematician at the University of Wisconsin at Madison, once told me that while traveling in Canada he saw signs informing him of the North American Triple Divide Point. Indeed a continental triple divide point would occur there as there would be a point where water would divide three ways, one way to Hudson Bay, a second to the Arctic Ocean via the Mackenzie River and the third to the Pacific.  In fact triple divide points are ubiquitous in North America. Wherever there is a confluence of two streams there is a Triple Divide Point uniquely associated with the confluence. Where there is a confluence of two streams, three basins are formed. To see this let us think of Pittsburgh Pennsylvania where the Allegheny and Monongahela Rivers join to form the Ohio. This confluence will define a partition of North America into three disjoint regions, the drainage basin of the Monongahela, the drainage basin of the Allegheny and everything else. If we color the basin of the Monongahela blue and the basin of the Allegheny red leaving the rest white we will see that there are two points that are located on the boundary of the three colored regions. One of
the points is Pittsburgh at the Golden Triangle where the confluence of the Monongahela and Allegheny forms the Ohio. The other point is the triple divide point that is associated with this confluence. An arc passes from the point of confluence to the triple divide point that is defined by the boundary between the red and blue regions. This arc is the Monongahela - Allegheny
divide.

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    The Ohio River, of course, flows into the Mississippi River near Cairo, Illinois. This confluence will have its associated Triple Divide Point. We find this Triple Divide Point in the same fashion as before. In this case we partition North America into the drainage basins of the Ohio (This will include the drainage basin of the Monongahela and Allegheny Rivers as subbasins), the drainage basin of the upper Mississippi with respect to the Ohio - this is the river basin of the Mississippi above the confluence with the Ohio, and lastly everything else. Again using the colors red, white and blue we discover that there are again two points that are on the boundary of all three regions.  One point is the point of confluence and the other is the triple divide point.  The arc that is the boundary between the Ohio and Mississippi river basins is the Ohio - Mississippi Divide.

     Anyone familiar with Chicago Illinois is aware that the boundary of the upper Mississippi basin is somewhat ill defined as the Chicago River flows out of Lake Michigan and into the Illinois River and into the Mississippi. Lake Michigan is connected to Lake Huron and the rest of the Great Lakes which drain into the Atlantic through the St. Lawrence River. Since the flow
of the Chicago River was engineered to flow in this direction we will assume its natural flow when forming the boundary of the upper Mississippi basin.  A similar and natural phenomena exists in Wyoming just south of the South East corner of Yellowstone National Park where Two Ocean Creek divides at a mountain pass into two creeks, the Atlantic Creek that is a tributary to the Yellowstone River and ultimately the Mississippi, and the Pacific Creek that is a tributary to the Snake River and then the Columbia. In this case we will simply let the continental divide follow the stream bed from its source to its point of divergence.  

    The Mississippi River flows into the Gulf of Mexico which we assume to be an arm of the Atlantic Ocean. To find the Triple Divide Point for this confluence we imagine that the oceans surrounding North America is a stream and has a direction of flow, with a point of origin and a point of destination. The point of origin and the point of destination will essentially be the same point. If we choose Panama to be this point and choose the flow to be from the Atlantic side of Panama and counterclockwise around North America to the Pacific side of Panama, then the Upper Ocean Basin with respect to the Mississippi is that area of North America where water flows into the Caribbean Sea and the Gulf of Mexico west of the Mississippi Delta. We thus have a partition of North America into three regions, the Upper Ocean basin with respect to the Mississippi, the Mississippi basin and the remaining region, which would be the lower Ocean basin with respect to the Mississippi. Again there are two points that are on the boundary of the three regions, one is the Mississippi River Delta - the point of confluence, and the Triple Divide Point. This Triple divide point has been identified as a mountain in Colorado that has just been recently named Headwaters Hill where our ideal raindrop will split into three subdroplets, one will flow west into the Colorado River and, if it weren't for human intervention taking water for irrigation, eventually into the Gulf of California, an arm of the Pacific Ocean. Another subdroplet flows into the Rio Grand and into the Gulf of Mexico west of the Mississippi. The third subdroplet flows into the Arkansas River and into the Mississippi.

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    As we have argued a point of confluence defines a unique triple divide point. We can also see that a triple divide point will define a point of confluence. We imagine our ideal rain drop dividing into three subdroplets flowing into three separate basins. Eventually these three subdroplets will be recombined. At the point when the three subdroplets are rejoined, two will have been previously rejoined. We trace the rejoined pair back to their point of recombination and this will be the confluence point associated with the triple divide point.

    A stream may have many tributaries, but it will be a tributary to only one stream or body of water, thus we may identify a confluence uniquely to the tributary at the point of confluence. In the case of the Monongahela and Allegheny we arbitrarily pick one to be the tributary of the other. The other stream is then considered to be the extension of the Ohio river. The triple
divide point that is associated with the confluence can thus be associated with the single stream that is the the tributary and the divide can also be associated with the tributary. In the case of Headwaters Hill, the triple divide point identified with the confluence of the Mississippi River and the Gulf of Mexico, Headwaters Hill can be said to be the triple divide point for the
Mississippi River (The name seems quite appropriate as it can be said to be the headwaters for the Mississippi river, the largest river of North America).  The raindrop falling on Triple Divide Peak in Glacier National Park will divide and flow into the surrounding oceans via the Mississippi River, The Nelson River and the Columbia River. Since the Nelson River is between the Mississippi and Columbia rivers with the previously given orientation of the surrounding oceans, this triple divide point is associated with the Nelson River.

    We have established that every confluence of streams defines a unique triple divide point. It is equally true that any two streams regardless of how remote they are from each other will define a triple define point. To see this let us consider Warm Springs Creek in South Central Idaho and Otter Creek in West Central Wisconsin. We imagine two raindrops flowing
down each of these streams, the first flows from Warm Springs Creek into the Big Wood River and then the Snake and the Columbia to the Pacific Ocean near Astoria Oregon. The latter raindrop will flow from Otter Creek into the Eau Claire River to the Chippewa River and the Mississippi and into the Gulf of Mexico. With our orientation of the surrounding oceans this latter raindrop flows around the continent and joins the first raindrop at the mouth of the Columbia, this will be the confluence of the extensions of the two streams and the triple divide point associated with this confluence is the triple divide point defined by the two streams. In this case it is the triple divide point associated with the Columbia River. This point is Three Waters Mountain located along the continental divide in the Wind River Range in Wyoming near the towns of Jackson on the West side and Dubois on the East side. The Columbia river divide can also be said to be the divide between Warm Springs Creek and Otter Creek. The Columbia River Divide is an arc that begins at Cape Disappointment in the State of Washington,
passes over the summit of Mount Rainier and then North along the Cascade Crest into British Columbia, where it reaches the Canadian Rockies and then follows a southern route into Montana and then Wyoming to Three Waters Mountain, the Columbia River Triple Divide Point. Water falling on Three Waters Mountain flows either into the Columbia River basin via the Snake River, the Colorado River basin via the Green River or the Mississippi River basin via the Wind - Big Horn Rivers.

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    Of course we have overlooked one small issue, and that is there are some basins where the water does not flow to the sea. The Great Salt Lake in Utah is such a place along with Death Valley in California and the Great Divide Basin in Wyoming. We will "whitewash" this problem by the untrue assumption that precipitation will exceed evaporation and percolation, thus all basins will fill to overflowing and all water will then eventually arrive at the surrounding oceans.

     Mathematicians like to abbreviate processes, and we can define an abbreviation for finding triple divide points and divides. Let x and y be two streams that converge to form stream xy. We may assume x is the extension of xy, thus y is the tributary. Let U be the basin for x and V be the basin for y. We now indicate the boundary of Basin of Stream x Boundry and the boundary of V with Basin for stream y boundry. The divide between the two river basins is that part of the boundaries common to both boundaries. We indicate this by The boundary of the basin of x intersected with the boundary of the basin of y. We read this as "The boundary of the basin of x intersected with the boundary of the basin of y." The divide can be thought of as a one dimensional region and it would have boundaries, namely its endpoints, which are the point of confluence and the triple divide point. We indicate these two points by The boundary of the boundary of the basin of x intersected with the boundary of the basin of y Which we read as "The boundary of the boundary of the basin of x intersected with the boundary of the basin of y." One thing to notice here is that the unabbreviated version is ambiguous where the abbreviated version is precise. The difference is the parenthesis. In the unabbreviated version do we take the boundary of the boundary and then intersect the boundary, or do we take the boundary of the intersected boundaries. The parenthesis make it clear that it is the latter that is meant. We summarize the results of our boundary process by The boundary of the boundary of the basin of x intersected with the boundary of the basin of y equals the set containing the points c and t. The notation {c, t} is read "The set containing the points c and t." In this case c represents the point of confluence and t represents the triple divide point.

     We may put an order on streams in North America. We can say stream x is included in stream y if some of the water in stream y originated in stream x. The Ohio river is included in the lower Mississippi, but not in the upper Mississippi, remember we are regarding the upper Mississippi and the lower Mississippi as two separate streams. We may say that a stream can be extended by appending streams in which it is included. The Allegheny is extended by appending the Ohio, and further extended by including the lower Mississippi. Now let x and y be two arbitrary streams in North America, consider Warm Springs Creek and Otter Creek as mentioned above. Let x hat be the maximal extension of x that does not contain any water from y and y hat be the maximal extension of y that does not contain any water from x. Since by our assumption that all water must eventually converge, there must be a confluence between x hat and y hat . Let U hat be the basin for x hat and V hat be the basin for y hat . We can now use the boundary operation to indicate the divide and the confluence and Triple Point for the two streams, Boundry U hat intersects Boundry V hat  is the divide and The boundry of the boundry U hat intersects boundry of V hat  is the triple divide point for streams x and y.

     To determine the triple divide point for the Mississippi river we put an orientation on the surrounding oceans of North America. We imagined the oceans as a stream flowing from the Atlantic side of Panama counterclockwise around North America and terminating on the Pacific side of Panama. If we reverse the flow so that we imagine the stream flowing from the Pacific side to the Atlantic side of Panama the triple divide point for the Mississippi remains the same but its divide is changed. The boundary of the Mississippi basin is a loop that begins and ends at the Mississippi Delta as it enters the Gulf of Mexico and it passes through Headwaters Hill, the Mississippi triple divide point. With the counterclockwise orientation of the ocean stream the Mississippi divide, The boundary of the basin of x intersected with the boundary of the basin of y , would be that portion of the loop that would begin west of the Mississippi Delta and pass through Texas and New Mexico to its endpoint at Headwaters Hill in Colorado. If we choose the clockwise orientation, then The boundary of the basin of x intersected with the boundary of the basin of y  would be that portion of the loop that begins east of the Mississippi Delta and passes through more states (and Canadian Provinces) than we wish to list here before ending at Headwaters Hill. The reason for this change is that the "upper" and "lower" ocean basins have been interchanged. The orientation of the oceans not only included a direction of flow, but also included a point of origin and termination. We chose Panama for that point, as it seemed to be a point most removed from the area of interest that we have examined. Suppose that we move the point of origin and termination from Panama to the Bering Strait, we see that the Mississippi rivers is not between the Rio Grand and the Colorado, but rather the Rio Grand is now the central river and thus inherits Headwaters Hill as its triple divide point.
For the Colorado to inherit Headwaters Hill as its triple divide point we would have to choose a point along the Gulf of Mexico between the mouths of the Rio Grand and the Mississippi rivers, perhaps Galveston Texas.

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THE TRIPLE DIVIDE POINT TOPOLOGY

    The Triple Divide Point Topology is a bit of a misnomer as it is a topology on a geographic region defined by drainage basins.

    We begin by putting an order on streams by the following convention.

1. The point of origin and destination (for lack of a better term we shall call that point the point at infinity, or the point of infinity) will be called the category -1 stream.
2. The ocean is the category 0 stream.
3. A to a category k stream is a category k +1 stream.

    Now let NA be the set of points of North America, and CP be the collection of all confluence points of North America. We define maps the confluence point where a category k stream is a tributary to a category k -1 stream where all the water from x will pass through the confluence point where a category k stream is a tributary to a category k -1 stream where all the water from x will pass through. The topology that we define on North America will be the coarses topology that makes the collection of maps defined above continuous. A basic open set for this topology will be Function of C from K to -1 , where c is a confluence point. In other words a basic open set is the interior of a drainage basin for some stream, or potential stream if water were present. We will assume confluence points and triple divide points will be dense in the Euclidian Metric Topology. To show that this defines a topology we simply verify that the intersection of two drainage basins is a drainage basin. This is clear since drainage basins are either disjoint or nested. Thus

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Euclidian Metric Topology math formula

(or possibly the drainage basin for Two Ocean Creek). We also note that  Function of Infinity from 0 to -1 North America, and Function of K to -1 = 0  ; whenever c is not the confluence of a k category stream and a k - 1 category stream.

    The Triple Divide Point Topology is coarser than the Euclidean Metric Topology. To see this let us focus on the Fire Hole and Gibbon Rivers in Yellowstone National Park. The Fire Hole and Gibbon Rivers converge to form the Madison River near West Yellowstone Montana. The divide between the Fire Hole and Gibbon Rivers is contained in the Madison River Basin. Pick a point on the divide between the Fire Hole and Gibbon and a small open neighborhood, in the Euclidean topology, of that point that does not contain the point of confluence. This neighborhood contains points in both the Fire Hole Basin and the Gibbon Basin, which are disjoint. Since basic open sets in the Triple Divide Point Topology are either disjoint or nested we see that there is no basic open set in the Triple Divide Point Topology that is contained in the Euclidian neighborhood and contains the given point on the divide, hence the Euclidian neighborhood is not open in the Triple Divide Point Topology. Now pick a point in any basin. Since the point is not on the boundary of the basin it is some distance away from the boundary and we simply pick an open disk about the point that has radius less than the distance to the boundary and that disk will lie entirely within the basin, hence the basin is open in the Euclidean Metric Topology. The Triple Divide Point Topology has the property that given any basic open set it is not the union of all its proper open subsets. To see this we consider the Madison basin. Points on the Fire Hole and Gibbon divide belong to no basic open subset but are in the Madison basin. Also we notice that points on the divide belong to only one open set, the Madison basin, hence the topology is not Hausdorf (nor even T0)and thus not metrizable.

THE ALGEBRA OF TRIPLE DIVIDE POINTS

We have established that every stream is uniquely associated with a triple divide point. However for streams that directly flow into the ocean this pairing is dependent on the selection of the point at infinity. By changing the point at infinity we permute the pairings of the streams to the triple divide points, and we will have a group action on this collection of pairings. The group is of course the collection of permutations. Streams that do not flow directly into the ocean will have fixed triple divide points and they will not be permuted by the group action. The group operation is of course composition, but it is induced by moving the point of infinity. Every permutation is induced by the number of miles (or kilometers or light years) the point of infinity is moved counterclockwise around North America modulo the circumference of the continent. The group should be abelian as moving n light years followed by m light years is the same as moving m light years followed by n light years, the net result is moving n+m light years. The triple divide point that is associated with a stream will be cycled through a collection of triple divide points that lie along the boundary of the drainage basin of that stream. As we move the point of infinity counter-clockwise around North America the triple divide point will permute in a counterclockwise direction around the boundary of the basin. In general the nearer (in the Euclidean Metric) the point of infinity is to the confluence of the stream with the ocean the nearer the triple divide point be to the confluence point. It may be amusing to contemplate the number of triple divide points a stream may have in its orbit. It seems mathematically reasonable to say it would have an infinite number of triple divide points, as points of confluence would appear to be dense (again in the Euclidean Metric Topology), but then we could argue that there must be an infinite number of triple divide points on the very nucleus of atoms. It is more reasonable to assume that confluence points have some breadth in the Euclidean Metric Topology, and that there are only a finite number of confluence points, and hence only a finite number of triple divide points. Nevertheless we can assume that the number of triple divide points that are in the orbit of a particular stream is of an indeterminant size.

    If a triple divide point is associated with a stream that flows directly into the ocean, that is, a category 1 stream, then our ideal raindrop that divides into three raindrops must all reach the ocean before they are recombined, since the point of confluence is the first recombination point. As we move the point of infinity around the continent, the stream associated with the triple divide point will be permuted through the three category 1 streams of the triple divide point. Hence the group associated with the permutations of streams will simply be Z3.

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A NOTE ON CONTINENTAL DIVIDES

A continent is essentially a large island in the ocean, a land mass surrounded by water. A continental divide is an arc that connects two points on the boundary, the shoreline, of the continent. The continental divide is defined by the hydrology and topography of the continent. If we label the two points A and B then we have the boundary divided into two arcs that
we can label AB and BA.
Arc line that connects two points
All points in the continent where the water flows to the ocean somewhere along arc AB will be on the AB side of the continental divide and all points in the continent where the water flows to the ocean somewhere along arc BA will be on the BA side of the continental divide. Again we assume that precipitation exceeds evaporation and percolation, hence the continent is divided into two open regions , the AB side and the BA side, the continental divide is the boundary between these regions.

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