In this study, at first, we analyse the linear stability of a straight river. We find that the natural perturbation modes maintain an equilibrium state by confining themselves to a threshold wavenumber band. The effects of river aspect ratio, Shields number and relative roughness number on the wavenumber band are studied. Then, we present a phenomenological concept to probe the initiation of meandering of a straight river, which is governed by the counter-rotational motion of neighbouring large-scale eddies in succession to create the processes of alternating erosion and deposition of sediment grains of the riverbed. This concept is deemed to have adequately explained by a mathematical framework stemming from the turbulence phenomenology to obtain a quantitative insight. It is revealed that at the initiation of meandering of a river, the longitudinal riverbed slope obeys a universal scaling law with the river width, flow discharge and sediment grain size forming the riverbed. This universal scaling law is validated by the experimental data obtained from the natural and model rivers. © The Authors, published by EDP Sciences, 2018.