In this paper, to explore the origin of the onset of meandering of a straight river, we, first, analyse the linear stability of a straight river. We discover that the natural perturbation modes of a straight river maintain an equilibrium state by confining themselves to an onset wavenumber band that is dependent on the flow regimes, aspect ratio, relative roughness number and Shields number. Then, we put forward a phenomenological description of the onset of meandering of a straight river. Its mechanism is governed by turbulent flow, with counter-rotation of neighbouring large-scale ormacroturbulent eddies in succession to generate the processes of alternating erosion and deposition of sediment grains of the riverbed. This concept is explained by a theorem (universal scaling law) stemming from the phenomenology of a turbulent energy cascade to provide a quantitative insight into the criterion for the onset of meandering of a straight river. It is revealed from this universal scaling law that, at the onset of meandering of a river, the longitudinal riverbed slope is a unique function of the river width, flow discharge and sediment grain size. This unique functional relationship is corroborated by the data obtained from the measurements in natural and model rivers. © 2017 The Author(s) Published by the Royal Society. All rights reserved.