Large time behavior and asymptotic stability of the two-dimensional Euler and linearized Euler equations

Freddy Bouchet (1, 2) , Hidetoshi Morita (2)

1 Phys-ENS - Laboratoire de Physique de l'ENS Lyon
2 INLN - Institut Non Linéaire de Nice Sophia-Antipolis

We study the asymptotic behavior and the asymptotic stability of the two-dimensional Euler equations and of the two-dimensional linearized Euler equations close to parallel flows. We focus on spectrally stable jet profiles $U\left(y\right)$ with stationary streamlines $y_{0}$ such that $U'(y_{0})=0$, a case that has not been studied previously. We describe a new dynamical phenomenon: the depletion of the vorticity at the stationary streamlines. An unexpected consequence, is that the velocity decays for large times with power laws, similarly to what happens in the case of the Orr mechanism for base flows without stationary streamlines. The asymptotic behaviors of velocity and the asymptotic profiles of vorticity are theoretically predicted and compared with direct numerical simulations. We argue on the asymptotic stability of these flow velocities even in the absence of any dissipative mechanisms.

Mots clés: 2D Euler equations; Large scales of turbulent flows; 2D turbulence; geophysical turbulence; asymptotic behavior; asymptotic Stability

Domaines: Mécanique statistique [cond-mat.stat-mech]; Mécanique des fluides [physics.class-ph]; Mécanique des fluides [physics.class-ph]; Physique mathématique [math-ph]; Physique mathématique [math-ph]

Freddy Bouchet, Hidetoshi Morita. Large time behavior and asymptotic stability of the two-dimensional Euler and linearized Euler equations. 2009. ⟨hal-00382670v2⟩