Quantitative classification of roll-wave development via shape-based metrics

Roll waves play critical roles in influencing overland flow hydrodynamics and erosion. However, the existing criteria offer limited insights into their morphological evolution trends. The quantitative characterization of roll-wave development is essential for understanding the underlying mechanisms. In this study, experiments were conducted in a laboratory flume with a smooth bed across five slopes (3°–15°), six unit-width discharges (0.02–0.20 m2/min), and eight measurement sections. A laser-based system was applied to record free-surface fluctuations along the flume. Two shape-based metrics were introduced: the shape factor (α) and the RMS fluctuation amplitude (σh). These metrics effectively captured roll-wave evolution trends. With increasing discharge and downstream development, the central waveform generally shifted from weak, relatively regular undulations toward sharper and more asymmetric profiles. This evolution was reflected by an overall decrease in the α values and an increase in σh. Conversely, traditional single-parameter criteria (e.g., the Vedernikov number Ve and instability parameter δ/h) exhibited significant limitations under these conditions. Accordingly, a joint classification framework using α and the minimum-amplitude threshold σh,min was established. Conditions with α ≤ 0.532 and σh ≥ σh,min = 0.397 were identified as roll-wave states. Additionally, the mean number of waves detected per measurement section served as a supplementary indicator for near-onset cases. This framework should be considered an exploratory indicator of the morphology of free-surface fluctuations. Nonetheless, it enables evaluations of the link between hydrodynamic processes and morphological evolution, from small-amplitude undulations to roll-wave structures. In future research, the applicability of this method across varying bed surface roughness levels should be evaluated.