Dynamics of roll waves

Shallow-water equations with bottom drag and viscosity are used to study the dynamics of roll waves. First, we explore the effect of bottom topography on linear stability of turbulent flow over uneven surfaces. Low-amplitude topography is found to destabilize turbulent roll waves and lower the critical Froude number required for instability. At higher amplitude, the trend reverses and topography stabilizes roll waves. At intermediate topographic amplitude, instability can be created at much lower Froude numbers due to the development of hydraulic jumps in the equilibrium flow. Second, the nonlinear dynamics of the roll waves is explored, with numerical solutions of the shallow-water equations complementing an asymptotic theory relevant near onset. We find that trains of roll waves undergo coarsening due to waves overtaking one another and merging, lengthening the scale of the pattern. Unlike previous investigations, we find that coarsening does not always continue to its ultimate conclusion (a single roll wave with the largest spatial scale). Instead, coarsening becomes interrupted at intermediate scales, creating patterns with preferred wavelengths. We quantify the coarsening dynamics in terms of linear stability of steady roll-wave trains.