Drainage-Basin Evolution and Graded River Profiles

Drainage-basin evolution is the long-term development of river networks and their catchments through erosion, capture and changes in base level, and a graded river profile is a smooth, concave long profile in which slope, discharge and sediment load are in balance so that the river neither erodes nor deposits on average. In brief: a river’s long profile typically steepens toward the source and flattens toward the mouth, and its lowest limit is the base level, usually sea level. Rivers adjust to disturbances by cutting down or building up until they return to grade. Networks grow by headward erosion and may be rearranged when one stream captures another, which changes the size of basins.

The structure of a river network and its profile record a basin’s history of uplift, climate and rock resistance. This guide explains drainage patterns and stream order, the concept of the graded profile and base level, knickpoints and stream capture, compares drainage patterns in a table, works through calculations, outlines the history and corrects common mistakes.

How are drainage networks organized?

The drainage pattern is the arrangement of channels, which reflects the geology and the slope.

• Dendritic: a tree-like branching pattern in rock of uniform resistance, the most common.
• Trellis: main streams parallel to one another, with short tributaries joining at right angles, in folded rocks with alternating resistant and weak layers, as in the Appalachians.
• Radial: streams flow outward from a high point, such as a volcano or dome.
• Rectangular: channels with right-angled bends, following joints and faults.
• Parallel: streams flowing in the same direction down a uniform slope.
• Deranged: irregular patterns with many lakes and swamps, in recently glaciated terrain.

Networks are also described by stream order: in the Strahler system, the smallest unbranched streams are first order, when two first-order streams join they form a second-order stream, and so on. The bifurcation ratio, the ratio of the number of streams of one order to the next higher order, is typically between 3 and 5 for natural networks. Drainage density is the total length of channels divided by basin area, and it is high in impermeable, easily eroded rocks and in wet climates. Hack’s law relates the length L of the main stream to the basin area A: L ≈ 1.4 A^0.6 (with L in miles and A in square miles), which means that basins grow longer relative to their width as they get larger.

What is a graded river profile?

The long profile plots the height of the river against distance from the source. In general, it is concave: steep in the headwaters, where the channel is small, and gentle in the lower course, where discharge is large. John Mackin (1948) defined a graded stream as one in which, over a period of years, slope and channel characteristics are adjusted so that the load supplied from upstream can just be transported. A river with too much load for its slope deposits the excess and steepens its bed (aggradation), and a river with spare energy erodes its bed (degradation) and lowers the slope until the balance is restored. This is a self-regulating system with negative feedback. The grade is a balance in time, so real rivers are not perfectly graded, and their profiles have irregularities caused by resistant rock and tributary inputs.

The control on the lowest point is the base level: the level below which a river cannot erode. The ultimate base level is sea level, and local base levels include lakes, resistant rock bands and the confluence with a larger river. A fall in sea level, uplift or removal of a dam lowers the base level and causes renewed down-cutting, which moves upstream as a wave.

How do basins evolve?

• Headward erosion: streams lengthen at their upstream ends by erosion at springs and gullies, and extend their basins into the divides.
• Knickpoints (waterfalls and rapids): steps in the long profile where a resistant layer or a fall in base level creates a steep reach. Knickpoints retreat upstream by erosion. The Niagara Falls have retreated about 11 km from the escarpment where they started after the last ice age, about 12,000 years ago.
• Stream capture (river piracy): a more vigorous stream, with a steeper gradient or weaker rocks, erodes headward and intercepts the flow of a neighboring stream. The captured stream’s upper part is diverted, the point of capture is called an elbow of capture and the abandoned lower valley becomes a misfit or wind gap.
• Divide migration: basins on the steeper side of an asymmetric range gain area at the expense of the gentler side.
• Response to uplift and climate: uplift steepens the profile and rejuvenates the river, and climate change alters discharge and sediment supply.

Models of river incision often use the stream power law, E = K A^m Sⁿ, where E is the rate of erosion, A is the upstream drainage area (a proxy for discharge), S is the channel slope, and K, m and n are constants. It explains why rivers steepen as uplift increases and why knickpoints move at a speed that depends on the drainage area.

Drainage patterns compared

Worked example: stream order, grade and retreat

1. Bifurcation ratio: a basin has 64 first-order streams, 16 second-order, 4 third-order and 1 fourth-order stream. The ratios are 64 ÷ 16 = 4, 16 ÷ 4 = 4 and 4 ÷ 1 = 4, so the mean bifurcation ratio is 4, a typical value.

2. Drainage density: the total stream length in the basin is 520 km, and the basin area is 130 km². The drainage density is 520 ÷ 130 = 4.0 km per km².
3. Hack’s law: for a basin of 100 square miles, L = 1.4 × 100^0.6 = 1.4 × 15.85 = 22.2 miles. For a basin 10 times larger (1,000 square miles), L = 1.4 × 1,000^0.6 = 1.4 × 63.1 = 88.3 miles, which is 4 times longer (not 10 times), showing that long basins are not simply scaled-up copies of short ones.

4. Long profile gradient: the river falls 120 m over a reach of 24 km. The gradient is 120 ÷ 24,000 = 0.005, or 5 m per km. A lower reach that falls 12 m over 24 km has a gradient of 0.0005 (0.5 m per km), ten times gentler, which gives a concave profile.

5. Knickpoint retreat: the Niagara gorge is about 11 km long and formed in about 12,000 years. The average retreat rate is 11,000 ÷ 12,000 = 0.92 m per year. If a waterfall retreats at 0.3 m per year, 3 km of retreat takes 3,000 ÷ 0.3 = 10,000 years.
6. Response to a base-level fall: when sea level falls, a knickpoint forms at the coast and retreats upstream. If it moves at 10 m per year, it reaches a point 20 km (20,000 m) from the coast after 20,000 ÷ 10 = 2,000 years, so the upper river keeps its old profile until then.

How did the ideas develop?

G. K. Gilbert, studying the Henry Mountains in Utah in the 1870s, described the balance between erosion and transport in rivers and the law of uniform slopes. William Morris Davis proposed the cycle of erosion in 1899, where rivers cut down to a base level, and his concept of a “graded stream” was refined by J. Hoover Mackin in 1948. Robert Horton (1945) and Arthur Strahler (1952) developed quantitative laws of drainage composition, and John Hack (1957) proposed the relationship between length and area. From the 1990s, the stream power model and cosmogenic nuclide dating have been used to measure rates of incision and landscape response, and show that rivers adjust over thousands to millions of years. Today, models of river networks explain the geometry of drainage basins with a few simple rules.

Which mistakes are common?

Mistake 1: Thinking that a graded river is a river that does not change

Grade is a balance over time. The river still erodes and deposits locally, and it responds to every change in discharge or load.

Mistake 2: Mixing up base level and sea level

Sea level is the ultimate base level, but lakes, rock bars and main rivers set local base levels for tributaries.

Mistake 3: Assuming that river capture is rare and instant

Capture is a gradual process that occurs repeatedly in the history of a basin, though the final piracy event can be fast.

Mistake 4: Treating drainage patterns as random

Patterns reflect rock structure, slope and history, so they are used by geologists to infer the geology of an area.