Galerkin-truncated dynamics of ideal fluids and superfluids: cascades, thermalizatión and dissipative effects

Abstract

This thesis is devoted to the study of the thermalization dynamics of several Galerkin-truncated conservatives systems related to the dynamics of ideal fluids and superfluid. A Galerkin truncation is simply obtained (in the homogeneous case) by performing a Fourier transform of a partíal differential equation (PDE) and then keeping only a finite number of Fourier modes. If the truncation is correctly performed the finite-dimensional system inherits a number of conservation laws from the original PDE.These finite-dimensional dynamical systems are interesting in themself because they possess statistically stationary solutions, known as absolute equilibria [1,2,3,4], that are exact solution of the associated Liouville equation.Perhaps the simplest of such truncated systems corresponds to the three-dimensional Euler equation [1,2,3,4,5,6]. The Fourier modesthe first work at-tempting to study in detail the thermalization was done by Cichowlas et al. [5] in 2005.They studied, by using direct numerical simulations at high resolutious, the relaxation dynamics of the truncated Euler equation. Long-lasting transients which behave just as those of high Reynolds-number viscous-flows were reported. In particular they found an approximately k-5/3 Kolmogorov inertial range followed by a dissipative range. The most striking result of this work is that a clear scale separation is exhibited. At high wavenumbers, the Fourier modes themalize following a k-2 equipartition energy spec-trumand this thermalized zone progressively extends to lower wavenumbers and finallycovers the whole spectrum. At intermediate times a very rich behavior was observed: the thermalized modes playing the role of a thermostat create a fictitious microworld that generates an effective dissipation at large scales.