River Profiles and Downstream Changes

The Bradshaw Model and Hydrological Changes

It might seem obvious that a steep mountain stream flows much faster than a flat lowland river, but the reality is exactly the opposite. The Bradshaw Model is a geographical framework that predicts how these hydrological variables change with distance downstream.

According to the model, several factors increase downstream: discharge, occupied channel width, channel depth, average velocity, and overall load quantity. Conversely, load particle size, channel bed roughness, and slope gradient all decrease downstream.

Discharge increases continually as water flows downstream. This happens due to the cumulative addition of water from joining tributaries and groundwater flow. We calculate this using a specific formula:

Where:

• $Q$ = Discharge in cumecs ($m^3/s$)
• $A$ = Cross-sectional Area ($Width \times Mean\ Depth$, in $m^2$)
• $V$ = Average Velocity ($m/s$)

Velocity increases downstream despite the flatter gradient due to improved hydraulic efficiency. In the lower course, the channel is deeper and smoother, meaning a smaller proportion of water is in contact with the wetted perimeter (the bed and banks). Because there is less friction dragging the water back, the average velocity reaches its peak.

Changes in Sediment Size and Shape

The rocks found in a river do not stay the same size forever. As sediment moves downstream, it becomes significantly smaller, rounder, and smoother.

This transformation is driven primarily by attrition. This is a process of erosion where transported particles crash into each other over long distances, slowly wearing each other down into smaller fragments. Because heavier particles are deposited as soon as the river loses energy, the sediment also becomes better sorted (more uniform in size) downstream.

The type of sediment transport also shifts. In the upper course, heavy bedload is moved via traction (rolling along the riverbed). By the lower course, the smaller, lighter alluvium is carried effortlessly in suspension within the water column.

Case Study: Downstream Changes on the River Tees

To see these changes in action, we can trace the 137 km journey of the River Tees from the Pennines to the North Sea. The river perfectly illustrates the Bradshaw Model, starting with a tiny discharge of $<5$ cumecs in its headwaters and ending in a massive tidal estuary.

In the upper course (Cross Fell to Middleton-in-Teesdale), the river drops roughly 750m in elevation over a very short distance. Sediment here consists of large, angular boulders often exceeding 1m in diameter. It features the UK's tallest waterfall, High Force Waterfall (21m drop). This formed where a highly resistant igneous rock called Whin Sill (dolerite) sits over softer limestone and sandstone, creating a gorge as it retreats.

In the middle course (near Barnard Castle), channel depth increases to between 0.46m and 1.7m. Discharge rises significantly here because major tributaries, including the River Lune and River Greta, join the main channel. Sediment size reduces to pebbles and gravel, with most bedload visibly rounded by 5km from the source due to attrition.

By the lower course (Stockton-on-Tees to Middlesbrough), the depth reaches between 2.61m and 3.20m. The river carries fine silt and alluvium in suspension, forming extensive mudflats at Seal Sands in the estuary. However, the natural downstream progression has been heavily modified by human intervention, including the Cow Green Reservoir in the upper course and the Tees Barrage near the mouth to regulate flow.