Nonlinear and subharmonic stability analysis in film-driven morphological patterns

Mega riverbed-patterns: linear and weakly nonlinear perspectives

In this paper, we explore the mega riverbed-patterns, whose longitudinal and vertical length dimensions scale with a few channel widths and the flow depth, respectively. We perform the stability analyses from both linear and weakly nonlinear perspectives by considering a steady-uniform flow in an erodible straight channel comprising a uniform sediment size. The mathematical framework stands on the dynamic coupling between the depth-averaged flow model and the particle transport model including both bedload and suspended load via the Exner equation, which drives the pattern formation. From the linear perspective, we employ the standard linearization technique by superimposing the periodic perturbations on the undisturbed system to find the dispersion relationship. From the weakly nonlinear perspective, we apply the centre–manifold-projection technique, where the fast dynamics of stable modes is projected on the slow dynamics of weakly unstable modes to obtain the Stuart–Landau equation for the amplitude dynamics. We examine the marginal stability, growth rate and amplitude of patterns for a given quintet formed by the channel aspect ratio, wavenumber of patterns, shear Reynolds number, Shields number and relative roughness number. This study highlights the sensitivity of pattern formation to the key parameters and shows how the classical results can be reconstructed on the parameter space.

Secondary instability in thin film flows under an inclined plane: growth of lenses on spatially developing rivulets

The response of a thin film flowing under an inclined plane, modelled using the lubrication equation, is studied. The flow at the inlet is perturbed by the superimposition of a spanwise-periodic steady modulation and a decoupled temporally periodic but spatially homogeneous perturbation. As the consequence of the spanwise inlet forcing, the so-called rivulets grow downstream and eventually reach a streamwise-invariant state, modulated along the direction perpendicular to the flow. The linearized dynamics in the presence of a time-harmonic inlet forcing shows the emergence of a time-periodic flow characterized by drop-like structures (so-called lenses) that travel on the rivulet. The spatial evolution is rationalized by a weakly non-parallel stability analysis. The occurrence of the lenses, their spacing and thickness profile, is controlled by the inclination angle, flow rate, and the frequency and amplitude of the time-harmonic inlet forcing. The faithfulness of the linear analyses is verified by nonlinear simulations. The results of the linear simulations with inlet forcing are combined with the computations of nonlinear travelling lenses solutions in a double-periodic domain to obtain an estimate of the dripping length, for a large range of conditions.

Streamwise Dissolution Patterns Created by a Flowing Water Film

The dissolution of rocks by rainfall commonly generates streamwise parallel channels, yet the occurrence of these natural patterns remains to be understood. Here, we report the emergence in the laboratory of a streamwise dissolution pattern at the surface of an initially flat soluble material, inclined and subjected to a thin runoff water flow. Nearly parallel grooves about 1 mm wide and directed along the main slope spontaneously form. Their width and depth increase continuously with time until their crests emerge and channelize the flow. Our observations may constitute the early stage of the patterns observed in the field.

Parametric transitions between bare and vegetated states in water-driven patterns

Conditions for vegetation spreading and pattern formation are mathematically framed through an analysis encompassing three fundamental processes: flow stochasticity, vegetation dynamics, and sediment transport. Flow unsteadiness is included through Poisson stochastic processes whereby vegetation dynamics appears as a secondary instability, which is addressed by Floquet theory. Results show that the model captures the physical conditions heralding the transition between bare and vegetated fluvial states where the nonlinear formation and growth of finite alternate bars are accounted for by Center Manifold Projection. This paves the way to understand changes in biogeomorphological styles induced by man in the Anthropocene and of natural origin since the Paleozoic (Devonian plant hypothesis).