Fig. 4: Allometric scaling for first-order processes.

Cumulative biogeochemical function vs. watershed area when aquatic processes (as υ_f) are first-order, wherein local areal process rates are influenced by concentration of substrate in the water column. Colors represent flow conditions as a proportion of mean annual flow: 0.1x mean annual flow (gray), 0.5x mean annual flow (black dashed), and 2x mean annual flow (orange). Red dashed line indicates linear scaling. Each line represents a distinct scenario combining variation in river hydraulics and scaling relationships of local areal process rate with increasing river size (m.local) for a rectangular network (n = 81 for each flow category; parameters described in Supplementary Table 1). Model results reflect average process rate (υ_f) in a 1 km² headwater watershed = 500 m yr⁻¹ (b in Eq. (2)). Scaling of cumulative function becomes increasingly superlinear and variable at higher flows.