Mathematical modeling in the environment: from tsunamis to powder-snow avalanches

Abstract : The present manuscript is devoted to the mathematical modeling of several environmental problems ranging from water waves to powder-snow avalanches. This Habilitation is organized globally in three parts. The first part is essentially introductive and contains also the complete description of my scientific activities. Scientific works dealing with water waves are regrouped in Part II. The spectrum of covered topics is large. We start by proposing in Chapter 3 a generalized Lagrangian for the water wave problem. This generalization allows for easy and flexible derivation of approximate models in shallow, deep and intermediate waters. Some questions of viscous wave damping are also investigated in the same chapter. Chapter 4 is entirely devoted to various aspects of tsunami wave modeling. We investigate the complete range of physical processes from the generation, through energy transformations and propagation up to the run-up onto coasts. The next Chapter 5 is devoted specifically to the numerical simulation and mathematical modeling of the inundation phenomena. This question is studied by various approaches: Nonlinear Shallow Water Equations (NSWE) solved analytically and numerically, Boussinesq-type systems and two-fluid Navier-Stokes equations. In Part III we investigate two important questions belonging to the field of multi-fluid flows. Chapter 6 is essentially devoted to the formal justification of the single-velocity two-phase model proposed earlier for aerated flows modeling. Several numerical results are presented as well. Moreover, similar analytical computations performed in a simpler barotropic setting are provided in Appendix A. These results could apply, for example, to the simulation of violent wave breaking. Finally, in Chapter 7 we propose a novel model for powder-snow avalanche flows. This system is derived from classical bi-fluid Navier-Stokes equations and has several nice properties. Numerical simulations of the avalanche interaction with obstacle are also presented.
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Denys Dutykh. Mathematical modeling in the environment: from tsunamis to powder-snow avalanches. Mathematics [math]. Université de Savoie, 2010. ⟨tel-00542937⟩

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