Bed form evolution remains dynamic even in the special case of steady, uniform flow. Data from the sandy, braided North Loup River, Nebraska, show that roughness features on the channel bottom display a statistical steady state and robust scaling that are maintained through the collective interactions of transient (short-lived) bed forms. Motivated by such field data, and laboratory observations of bed form growth, we develop a nonlinear stochastic surface evolution model for the topography of bed load dominated sandy rivers in which instantaneous sediment flux explicitly depends on local elevation and slope. This model quantitatively reproduces laboratory observations of initial growth and saturation of bed forms from a flat surface, and also generates long-term dynamical behavior characteristic of natural systems. We argue that the variability in geometry and kinematics of bed forms in steady flow, and the existence of roughness at all wavelengths up to the largest dunes, are a consequence of the nonlinear relationship between sediment flux and topography, subject to noise.