An effective physics-based rule for the evolution of bedrock channel cross sections is required for quantitative modeling of the roles of climate, tectonics, and sediment supply in setting bedrock longitudinal profiles and landscape form. Here we propose a modeling strategy in which the spatial pattern of erosion rates in a channel cross section is calculated, allowing exploration of the origin of the channel cross-sectional profile, and of the dependence of channel width on flow discharge and channel slope. Our approach reproduces many of the scaling relationships observed in natural systems, including power-law width-discharge (W~Q0.4) and width-slope (W~S-0.2) relationships. Models of channel cross-sections linked in series and subject to varying rock uplift (baselevel lowering) rates produce concave-up longitudinal profiles with power-law slope-uplift (S~B1.31) and width-uplift (W~B-0.24) relationships. Our modeling strategy can easily be adapted to handle i) better representations of erosional processes, ii) better approximations of the flow structure, and iii) the role of non-uniform sediment mantling of the bed.