Geometric Properties of Mature Drainage Systems and Their Representation in an E⁴ Phase Space

Topographic texture is measured by two distinct parameters: drainage density (D) and channel frequency (F). In maturely dissected regions the functional relation between D and F is (1)$F = 0.694D^{2}$. The dimensionless ratio$F/D^{2}$varies inversely with valley-side slope ($\theta$) and basin relief (R), area and channel length held constant, but is not affected by basin circularity or basin perimeter (P). The ratio$F/D^{2}$is interpreted as a measure of the completeness with which a channel system fills a basin outline, for given values of N (number of channels). For an ideal basin of 1 square mile area, it is postulated that the equation$N = 0.694L^{2}$(L = total channel length) is a growth model, although it is based on essentially instantaneous measurements from many different basins. When relief is considered, regression analysis gives equation (2),$N = 0.8147L^{1.75}/R^{0.25}$, A = 1, which may also be a growth model. If both equations (1) and (2) are valid as growth models, equation (3), R = 1.899/L, must also be true, when A = 1, for mature basins. Equation (1) is true for long-term changes when erosion can reduce basin relief; equation (2) is true when relief is not reduced because of concomitant uplift. A phase space is defined by the co-ordinates$X_{1} = log vA$,$X_{2} = log L$,$ X_{3} = log R$, and$X_{4} = log P$. This is a vector four-space over the field of real numbers. A drainage basin can be represented by an image point in the phase space, and$\theta$,$F/D^{2}$, D, F, etc., are scalar fields, functions of the$X_{i}$. The phase space may be an important means of proving an ergodic theorem for drainage systems, which would allow rigorous substitutions of space averages for time averages, as is done in postulating that equations (1) and (2) are growth models. The function relating valley-side slope to the Xi, found by regression analysis, is$\theta =27.53\frac{L^{0.25}R^{0.25}}{(\sqrt{A})0.75}$(4) showing that area and relief have the greatest effect on 0 and channel length is less important. If this can be considered a growth model, 0 increases as 4vZ when R and A are constant. However, in a basin of 1 square mile, because of equation (3), equation (5),$\theta =37.94/L^{0.25}$, shows that slopes decrease in angle as the drainage net expands, if erosion reduces the relief.