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PhD Defense

Romain Nguyen van yen (LMD)

Title : Wavelet-based study of dissipation in plasma and fluid flows

Date and time : The 08-12-2010 at 13h30

Type : thèse

Université qui délivre le diplôme :

Location : Salle Dussane, École Normale Supérieure, 45 rue d'Ulm, Paris
Members of jury :

Claude Bardos, Examinateur

Jean-Marcel Rax, Examinateur

Olivier Talagrand, Examinateur

Rupert Klein, Rapporteur

Marie Farge, Directrice de thèse

Kai Schneider, Co-directeur de thèse

Summary :

The topic of dissipation by macroscopic flows is approached by considering two of its most representative occurences, namely dissipation by plasma flows in the vanishing collisionality limit and dissipation by fluid flows in the vanishing viscosity limit. It is argued that dissipation can arise either due to the residual effect of a microscopic coupling parameter, or due to purely macroscopic nonlinear mixing effects. The combination of these two phenomena puts the problem out of reach of the most successful statistical methods that have been developed in the context of conservative systems, and also raises fundamental mathematical questions. Moreover, explicit computations that would resolve all scales of such flow are still unfeasible in this context, because of the present limitations in memory size and number of operations. It is thus widely recognized that new ways have to be found to make progress.

For that purpose, we explore the applicability of a multiscale wavelet framework. First, the partial differential equations which describe the flow must be recast into a discrete wavelet representation, while preserving consistency with the dissipative mechanisms we have outlined. This step, which we call regularization, is the subject of two chapters in this thesis, concerning the special cases of the one-dimensional Vlasov-Poisson equations on the one hand, and of the two-dimensional incompressible Euler equations on the other hand. The possibilities to develop these schemes for the practical simulation of flows is assessed, and they are compared with other existing regularizations mechanisms.

To proceed further, the origin of the residual dissipation must be tracked down and linked to mathematical properties of the solutions. We obtain some elements in this direction by studying the collision of a vorticity dipole with a wall in the vanishing viscosity limit. If the solutions are known behave well mathematically, one can readily move to the next step, which is the definition of macroscopic dissipation in the wavelet representation. This is the case for two-dimensional homogeneous turbulent flows which we subsequently address. Finally, as a perspective for future work, we perform a preliminary wavelet analysis of a three-dimensional turbulent boundary layer flow.

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