The length and spacing of river tributaries

Abstract

River networks are composed of a mainstem and tributaries. These tributaries dissect landscapes, regulate water and habitat availability, and transport sediment and nutrients. Despite the importance of tributaries, we currently lack theory and data describing whether and how tributary length and spacing varies within watersheds, thereby limiting our ability to accurately describe river network geometry. We address this knowledge gap by analyzing 4,696 tributaries across six landscapes with varying climate, tectonic setting, and lithology. Our results show that both tributary length and spacing systematically increase with downstream distance along the mainstem river, following a power-law scaling. This power-law scaling can be modulated by basin shape, with tributaries becoming shorter and, in some cases, more closely spaced as basin elongate. Furthermore, the power-law scaling may break down in cases where river networks have been disturbed by pervasive faulting, raising the possibility that the scaling we observe is not unique to all branching networks, and instead may be universal across undisturbed fluvial networks. These findings can be used to improve predictions of river network geometry and potentially to distinguish fluvial river networks from other branching networks.

Keywords: landscape evolution; relief and habitat distribution; river network geometry; signature of fluvial networks.

Fig. 1.

Examples of variation in tributary length and spacing as a function of mainstem river distance and basin shape for (A) a schematic drainage basin which follows our conceptual model and (B) Dean Creek, Oregon Coast Range, OR. White circles denote confluences of large (ΔN = 1) tributaries with the mainstem river, and blue circles denote the mainstem river outlet. Plots on the Left and Right schematically show variation in tributary length and spacing with mainstem distance, with lines based on our conceptual model prediction for large (ΔN = 1), medium (ΔN = 2), and small (ΔN = 3) streams, and white circles corresponding to large tributaries in the schematic drainage basin and Dean Creek. Gray lines show subtributaries (i.e., river tributaries that do not drain directly into the mainstem). L^*_trib– nondimensional tributary length (Eq. 5), x^* –nondimensional downstream mainstem distance (Eq. 6), and λ* – nondimensional tributary spacing (Eq. 7).

Fig. 2.

(A) Variation in nondimensional tributary length (L^*_trib) and (B) nondimensional tributary spacing (λ^*) as a function of nondimensional downstream mainstem distance (x^*) for data combined data from all field areas, excluding the MFS. Gray, semitransparent circles show all data; filled circles with error bars are binned averages and SD for all data, respectively. Black lines show best-fit power laws with shading denoting confidence bounds (calculated by varying the power-law rate constant and exponent within their 95% CI) (Materials and Methods). (C) Number of tributaries in each x^* bin and percent of tributaries within each bin that reach the basin extent (Materials and Methods).

Fig. 3.

(A) Variation in nondimensional tributary length (L^*_trib) and (B) nondimensional tributary spacing (λ^*) as a function of nondimensional downstream mainstem distance (x^*) for data from tributaries within the MFS. Gray, semitransparent circles show all data; filled circles with error bars show the binned averages with SD, respectively. Black lines show best-fit power laws with shading denoting 95% confidence bounds for the MFS data (Materials and Methods). Due to the limited sample size of large (ΔN = 1) tributaries in MFS, we fit all the data, rather than the binned means. We do not calculate the best-fit power law for the spacing in large tributaries within MFS, due to having only two data points. For visual comparison between tributaries within the MFS and other landscapes, we show best-fit power laws for combined data from all other field areas (excluding MFS, i.e., the power-law fits shown in Fig. 2) as dashed lines. Note increased y-axis extent in the leftmost plot of panel (A).

Fig. 4.

(A) Effect of basin shape on nondimensional tributary length (L^*_trib) and (B) nondimensional tributary spacing (λ^*) as a function of nondimensional downstream mainstem distance (x^*) for data combined data from all field areas (excluding MFS) and grouped into three equally spaced GC bins. Filled circles show the binned averages, and solid lines show the best-fit power laws calculated for each GC bin. For the highest and lowest GC bins, we additionally show 95% confidence bounds on the power-law fits to highlight differences between relatively circular (low GC) and relatively elongated (high GC) basins. (C) The number of tributaries, number of basins, and number of tributaries per basin within each GC bin for large (ΔN = 1) tributaries (Left), medium (ΔN = 2) tributaries (Center), and small (ΔN = 3) tributaries (Right).