Water Science and the Environment HWRS 201 Dr. Marek Zreda

Water Science and the Environment

HWRS 201

Dr. Marek Zreda

[email protected] 621-4072 Harshbarger 230 A watershed Characterizing watersheds (stream networks)

Hydrologists want to understand how the physical characteristics of watersheds influence the water balance within them.

If possible, we want to characterize watersheds by simple indicators because frequently we don’t have detailed information about them.

We’ve already discussed three ways of characterizing watersheds:

1. Discharge at a point (e.g., feet3/sec, km3/year) 2. Drainage basin area 3. Stream length

But none of these quantify the shape and connectedness of the streams in the network. The fourth way does that:

4. Strahler ordering Sketch of a stream network

Note that confluences are always higher in elevation than the main stem descending from them.

All streams flow downhill (due to gravity) Sub-basins and self-similarity

All large watersheds are composed of sub- basins, which are watersheds within watersheds.

The Rio Grande is a good example. It comprises 7 major sub-basins, which are themselves composed of sub-sub-basins, etc. Each sub-basin is like a miniature of its parent basin.

For this reason, we say watersheds are self-similar and that they scale. Elements and properties of stream networks

The concept of self-similarity has a particularly nice representation in the concept of stream networks. Stream networks are composed of stream segments, which are sections of the network between confluences. A confluence is a place where 2 streams come together and form a new stream that continues on downhill. The 2 upper streams are tributaries of the new stream. The smallest streams in a network are headwater streams. The smallest don’t have tributaries. They’re the furthest upstream. All other streams in the network have tributaries. Sketch of a stream network

Note that confluences are always higher in elevation than the main stem descending from them.

All streams flow downhill (due to gravity) Stream segments and reaches

A stream reach is a channel between 2 confluences. A stream segment is the entire channel of a stream between the confluence where its order is initially set and the next confluence where its order increases (if it ever does). Strahler ordering

Strahler ordering is a systematic way to identify the place of a stream segment in a stream network based on 4 simple rules: 1. The order (Strahler number) of headwater streams is always 1. 2. When 2 streams of order ω join, a stream of order ω + 1 is created 3. When 2 streams of different order join, the downstream channel has the higher order of the 2 combining streams. 4. The order of a stream network is the Strahler number of its outlet. Strahler ordering

When streams of the same order combine, the downstream segment has twice as much flow (discharge) as its tributaries. Strahler rationale

When streams of the same order combine, the downstream segment has twice as much flow (discharge) plus 1 as its tributaries. This isn’t very different from the last slide. Sparseness of the network

Some stream networks are “bushier” than others. Sparseness of the network: bifurcation ratio

The number of segments of order ω in a network is symbolized by N(ω). We can construct a table of these numbers for a stream, for instance the one to the right. Horton’s Law of Bifurcations

Now that you understand the notation, all you need is a little algebra to see that: (ω) = (ω + 1) 푁 푅퐵 implying that: 푁

ω = × (ω + 1)

푁 푅퐵 푁 so N(1) = RB × N(2), N(2) = RB × N(3) and N(3) = RB × N(4) there’s a pattern here. Horton’s Law of Bifurcations

Here’s the pattern again, N(ω) = RB × N(ω +1) Suppose the total order of the stream system is 4, i.e., the Strahler number of the outlet is Ω = 4. The pattern above says

N(1) = RB × N(2), N(2) = RB × N(3) and N(3) = RB × N(4)

But N(1) = RB × N(2) and N(2) = RB × N(3) means 2 N(1) = RB × (RB × N(3)) = R B × N(3) and N(3) = RB × N(4) means 2 2 3 N(1) = R B × N(3) = R B × (RB × N(4)) = R B × N(4)

Also N(2) = RB × N(3) and N(3) = RB × N(4) means 2 N(2) = R B × N(4)

We already have N(3) = RB × N(4)

Try this out and see if you can understand it. Horton’s Law of Bifurcations

Here’s the pattern again, N(ω) = RB × N(ω +1) This leads to a general formula:

Ω-ω N(ω) = RB × N(Ω) Horton’s Law of Bifurcations

If RB(ω) is pretty much the same for all stream orders, i.e., if the stream network is about equally bushy for all orders, then:

RB (1) = RB (2) = RB (3) = … = a constant = RB and we can conclude that

(ω) = (ω + 1) 푁 푅퐵 for every pair of orders. So: 푁

( ) ( ) ( ) = = = = ( ) ( ) ( ) … 푁 1 푁 2 푁 3 푅퐵 푁 2 푁 3 푁 4 Sparseness of the network: Bifurcation ratio

(ω) = The bifurcation ratio is: (ω + 1) 푁 푅퐵 푁

Bushy streams have lower bifurcation ratios than linear streams (streams that aren’t so bushy). The bifurcation ratio is an indicator of the likelihood that a stream will flood.

Streams with low RB are more likely to flood than streams with higher RB. Overall bifurcation ratio of a stream network

The overall bifurcation ratio of a stream network is the average of all the values of RB(ω):

RB = (RB(1) + RB(2) + … + RB(Ω - 1)) / (Ω - 1) Here Ω is the order of the stream (the Strahler number of its outlet). Bifurcation ratio table Bifurcation ratio and flooding

The risk of flooding within a basin increases as the bifurcation ratio goes down.

RB also indicates the flood risk for parts of a basin. Sub-basins with a lower RB are more likely to flood than those with a higher RB. Bifurcation ratio: Complete binary stream network Horton’s Law of Stream Lengths

Horton found a similar “law” for stream lengths: (ω + 1) (ω) = (ω) 퐿 푅퐿 In this case, L(ω) is the average length퐿 of a ωth order stream, and

RL(ω) is the stream length ratio. Note that in this case the ratio is

ℎ푖푖�푖푖 − 표�표𝑜 푠푠𝑠푠푠 But for bifurcations the ratio푙�푙𝑙 was− 표 �표𝑜 푠푠𝑠푠푠

푙�푙𝑙 − 표�표𝑜 푠푠𝑠푠푠 ℎ푖푖�푖푖 − 표�표𝑜 푠푠𝑠푠푠 Horton’s Law of Stream Lengths

Once again, if there isn’t too much variation from one order to the next, RL(ω) is approximately constant:

(ω + 1) (ω) = (ω) 퐿 푅퐿 for all ω, so in general: 퐿

ω-1 L(ω) = RL × L(1) Horton’s Law of Stream Lengths

ω-1 This “law” L(ω) = RL × L(1) is less abstract than the bifurcation law. It has practical use, especially in basins where we don’t have a lot of data about stream lengths, but we know RL and the average length of the 1st order stream. For instance, suppose the average 1st order stream has length ½ mile and RL = 2, i.e., the average 2nd order stream is twice as long as the average 1st order stream, and, the average 3rd order stream is twice as long as the average 2nd order stream, and so on. Then approximately L(2) = 1, L(3) = 2, L(4) = 4, L(5) = 8, … Main stems and Strahler order

The main stem of a river starts at its outlet and has the same overall Strahler order as the outlet. The main stem of the Mississipi River has a Strahler number of 10. Eight rivers in the US have order 9. Headwater streams (Strahler orders 1-3) are most common. There are more than 1.5 million of these small streams in the US. Strahler order of US rivers The world’s largest drainage basins

Drainage basins can be ranked by: • Area • Average annual discharge • Length • Strahler number

The Amazon has Strahler number 12, while the Mississippi has number 10.

Possible presentation topic 18:

Find (or compute) Strahler numbers for the rivers in the two graphs. Correlate the Strahler numbers with discharge and, separately, with area and length.