Rampant Rivers (in the wild)
River in the Pentland Hills.
A natural river’s shape is influenced by various mathematical and physical processes, including fluid dynamics, sediment transport, and erosion. One key factor is the meander, the winding curves of a river. Meanders form due to differential flow velocities across a river’s width. Water flows faster on the outer banks of curves and slower on the inner banks, leading to erosion on the outer side and deposition on the inner side, which alters the river’s shape over time. The curvature of these meanders can be mathematically described using differential equations, particularly the sine-generated curve, which approximates the shape of ideal meanders.
In addition, the concept of hydraulic geometry links river width, depth, and velocity through power-law relationships. For example, as the discharge increases, the river’s width and depth also increase, but at different rates, governed by scaling exponents in these equations.
Rivers also seek to minimize energy expenditure, a principle described by the law of least action, which mathematically explains why rivers tend to form meanders instead of flowing in straight lines. Another factor is fractal geometry, where the branching patterns of river networks display self-similarity. The shape and size of river basins can be described using fractal dimensions, providing insights into how rivers carve landscapes into the environment over time.
To find the water flow (in m3/s), multiply the average water speed by the average cross-section. To convert this water flow to litres per second (l/s), and multiply the result (in m3/s) by 1000.
— Ava Brindisi Swan (S3)