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Nonlinear and subharmonic stability analysis in film-driven morphological patterns.

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  • 1Politecnico di Torino, DIATI, Corso Duca degli Abruzzi, 24, 10129 Torino, Italy.

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Summary
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Wavy patterns, called flutings, form on calcite and ice surfaces from water flow. This study develops a unified morphodynamic model to explain fluting formation and stability.

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Area of Science:

  • Geomorphology
  • Fluid Dynamics
  • Pattern Formation

Background:

  • Gravity-driven water films interacting with substrates form wavy patterns.
  • Similar flutings are observed in caves and ice-falls, suggesting a unified formation mechanism.
  • Cave patterns serve as valuable archives of past climate data.

Purpose of the Study:

  • To develop a unified morphodynamic model for fluting formation.
  • To analyze the linear and nonlinear stability of flutings.
  • To validate theoretical findings with numerical simulations.

Main Methods:

  • Gradient expansion to derive a Benney-type equation for movable boundaries.
  • Coupling the Benney-type equation with a wall evolution equation.
  • Center manifold projection to analyze stability against subharmonic disturbances.
  • Numerical simulations of fully nonlinear equations.

Main Results:

  • A morphodynamic model for fluting formation was established.
  • Closed-form relationships for selected wave number and finite amplitude were derived.
  • Stability analysis confirmed the robustness of fundamental modes against subharmonic disturbances.

Conclusions:

  • The study provides a comprehensive theoretical framework for understanding fluting formation.
  • The model successfully explains the development and stability of these geological and glaciological patterns.
  • Numerical simulations validate the theoretical predictions, enhancing confidence in the model's applicability.