On the main channel length‐area relationship for channel networks

Motivated by the empirical relationship between the main channel length and the basin area for river networks, a similar equation is derived analytically using the random model postulates and the additional assumption that link lengths have a common exponential probability distribution. Specifically, it is shown that in networks with large magnitude m , the main channel length l(m) , and the magnitude are related by l(m) ∼ β (2π) ½ (2π) ½ , where β is mean link length. This result is utilized to explain another empirical relationship between the distance to the center of gravity l c (m) and the main channel length; l c (m) ∼ l(m) /2. The observed deviations of the random model prediction regarding the main channel length from empirical observations is used to discuss some important open problems.