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

Abstract

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 $U(y)$ with stationary streamlines $y_0$ such that $U^{\prime }(y_0)=0$, 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.

To be published in Physica D, nonlinear phenomena (accepted January 2010)

References in corpus (3)

• Random changes of flow topology in two dimensional and geophysical turbulence
• Simpler Variational Problem for Statistical Equilibria of the 2D Euler Equation and Other Systems with Long Range Interactions
• 2-D constrained Navier-Stokes equation and intermediate asymptotics

Cited by in corpus (27)

• Enhanced dissipation and inviscid damping in the inviscid limit of the Navier-Stokes equations near the 2D Couette flow
• Inviscid dynamical structures near Couette flow
• Kinetic theory of jet dynamics in the stochastic barotropic and 2D Navier-Stokes equations
• Stability threshold of the 2D Couette flow in Sobolev spaces
• Linear inviscid damping in Gevrey spaces
• Algebraic damping in the one-dimensional Vlasov equation
• Enhanced dissipation and axisymmetrization of two-dimensional viscous vortices
• Linear inviscid damping and enhanced dissipation for the Kolmogorov flow
• Invariant measures of the 2D Euler and Vlasov equations
• A dynamical approach to the study of instability near Couette flow
• Nonlinear inviscid damping near monotonic shear flows
• Linear inviscid damping and vorticity depletion for shear flows
• Axi-symmetrization near point vortex solutions for the 2D Euler equation
• Theoretical prediction of Reynolds stresses and velocity profiles for barotropic turbulent jets
• Linear vortex symmetrization: the spectral density function
• Uniform linear inviscid damping and enhanced dissipation near monotonic shear flows in high Reynolds number regime (I): the whole space case
• Long time behavior of Alfvén waves in a flowing plasma: mathematical analysis on the generation of the magnetic island
• Linear inviscid damping for a class of monotone shear flow in Sobolev spaces
• Inviscid damping of monotone shear flows for 2D inhomogeneous Euler equation with non-constant density in a finite channel
• Direct Statistical Simulation of a Jet
• Fluctuations and large deviations of Reynolds stresses in zonal jet dynamics
• Metastability and rapid convergence to quasi-stationary bar states for the 2D Navier-Stokes Equations
• Dynamical pattern formations in two dimensional fluid and Landau pole bifurcation
• Fluctuations of large-scale jets in the stochastic 2D Euler equation
• Linear damping and depletion in flowing plasma with strong sheared magnetic fields
• Capillary gravity water waves linearized at monotone shear flows: eigenvalues and inviscid damping
• On the long-time behavior of a perturbed conservative system with degeneracy