Nonlinear and subharmonic stability analysis in film-driven morphological patterns

Abstract

The interaction of a gravity-driven water film with an evolving solid substrate (calcite or ice) results in the formation of fascinating wavy patterns similar both in caves and in ice-falls. Due to their remarkable similarity, we adopt a unified approach in the study of pattern formation of longitudinally oriented organ-pipe-like structures, called flutings. Since the morphogenesis of cave patterns can evolve for millennia, they have an additional value as silent repositories of past climates. Fluting formation is studied with the aid of gradient expansion and center manifold projection. In particular, through gradient expansion, a Benney-type equation accounting for the movable boundary is obtained. The coupling with a wall evolution equation provides a morphodynamic model for fluting formation, explored through linear and nonlinear analyses. In this way, closed relationships for the selected wave number and for the finite amplitude are achieved. However, as finite-amplitude monochromatic waves may be destabilized by nonlinear interactions with other modes, we verify, through center manifold projection, the stability of the fundamental to subharmonic disturbances. Conclusively, we perform numerical simulations of the fully nonlinear equations to validate the theory results.

Hydrodynamic-driven morphogenesis of karst draperies: spatio-temporal analysis of the two-dimensional impulse response

“…2010). In this work, we consider the model used in Bertagni & Camporeale (2017), for the inertialess case, in which the complete curvature is retained (Wilson 1982; Weinstein & Ruschak 2004). The non-dimensional equation for the evolution of the thickness in the presence of substrate variations reads where operates in the plane and is the linear advection velocity (Brun et al.…”

Instability of a thin viscous film flowing under an inclined substrate: the emergence and stability of rivulets

“…Nature provides astonishing examples of the capability of coating flows to modify the topography of the substrate via chemical and thermodynamic reactions. The structures that can be observed in limestone caves, known as speleothems, are characterized by a morphogenesis that is related to the hydrodynamic instability of a coating flow (Short et al 2005;Meakin & Jamtveit 2010;Camporeale 2015;Bertagni & Camporeale 2017). These fascinating structures originate from the interaction between hydrodynamics and chemistry.…”

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

“…For our convenience, we define the saturation time ts , such that A(ts ) = 0.99As and A(0) = 0.01As , and we stipulate that a condition for bar formation is that the formative event must have a duration at least equal to the saturation time-that is, T + Q f ≥ ts . An evaluation of the complex growth rate Ω and the first Landau coefficient σ is here given through the technique of the Center Manifold Projection (CMP), which, unlike the commonly used multiple scaling theory (28), offers a means to find As even away from the critical conditions but close to the neutral curve (31,32). Let us introduce the following scaling quantities: the river half-width B for the horizontal lengths, the depth D0 for the vertical length, U0 = Q f /2BD0 for the velocity, B /U0 for the time scale, and us = (∆ − 1)gd for the sediment velocity (∆, d , and g being the relative density, particle size, and gravitational acceleration, respectively).…”