This study examines the dynamics of roll waves that spontaneously arise in muddy debris flows, where phase separation is not observed. To this end, a set of bulk mass and momentum balance equations is derived from the depth-averaged model proposed by Meng et al. [J. Fluid Mech. 943, A19 (2022)]. The derived equations are shown to be well-posed, with the threshold Froude number for flow instability slightly exceeding the classical $2/3(1-\mathrm{\Gamma })$, which corresponds to dry granular flows on rough planes. A high-resolution shock-capturing scheme is employed to resolve the evolution of debris roll waves, which emerge from initially steady, uniform flows subjected to sinusoidal perturbations. These waves grow under the influence of inertial forces and eventually saturate into steady, traveling waveforms. A traveling-wave solution is subsequently constructed, accurately capturing the steady traveling waves observed in the simulations. This theoretical framework is further applied to natural debris roll waves observed in the Illgraben torrent, Switzerland. Simulated time series of the flow depth and wave speeds, based on the $\mu \left(I\right)$-dependent basal granular friction law, exhibit excellent agreement with field measurements. Numerical results also reveal that large-amplitude waves, which carry significant mass flux, are dynamically unstable and tend to fragment into multiple smaller-amplitude waves. These smaller waves are overtaken and merged by trailing waves with larger amplitudes and faster propagation speeds, a phenomenon consistent with field observations.

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• Geophysical fluid dynamics
• Granular flows
• Granular fluids