Statistical mechanics of Beltrami flows in axisymmetric geometry: Equilibria and bifurcations

Abstract

We characterize the thermodynamical equilibrium states of axisymmetric Euler-Beltrami flows. They have the form of coherent structures presenting one or several cells. We find the relevant control parameters and derive the corresponding equations of state. We prove the coexistence of several equilibrium states for a given value of the control parameter like in 2D turbulence [Chavanis and Sommeria, J. Fluid Mech. 314, 267 (1996)]. We explore the stability of these equilibrium states and show that all states are saddle points of entropy and can, in principle, be destabilized by a perturbation with a larger wavenumber, resulting in a structure at the smallest available scale. This mechanism is therefore reminiscent of the 3D Richardson energy cascade towards smaller and smaller scales. Therefore, our system is truly intermediate between 2D turbulence (coherent structures) and 3D turbulence (energy cascade). We further explore numerically the robustness of the equilibrium states with respect to random perturbations using a relaxation algorithm in both canonical and microcanonical ensembles. We show that saddle points of entropy can be very robust and therefore play a role in the dynamics. We evidence differences in the robustness of the solutions in the canonical and microcanonical ensembles. A scenario of bifurcation between two different equilibria (with one or two cells) is proposed and discussed in connection with a recent observation of a turbulent bifurcation in a von Karman experiment [Ravelet et al., Phys. Rev. Lett. 93, 164501 (2004)].

arXiv comment25 pages; 16 figures

References in corpus (9)

• Statistical mechanics and dynamics of solvable models with long-range interactions
• Stability criteria of the Vlasov equation and quasi-stationary states of the HMF model
• Multistability and memory effect in a highly turbulent flow: experimental evidence for a global bifurcation
• Dynamics and thermodynamics of a simple model similar to self-gravitating systems: the HMF model
• A maximum entropy principle explains quasi-stationary states in systems with long-range interactions: the example of the Hamiltonian Mean Field model
• Statistical theory of high-gain free-electron laser saturation
• Fingerprints of nonextensive thermodynamics in a long-range Hamiltonian system
• Dynamics and thermodynamics of axisymmetric flows: I. Theory
• Statistical mechanics of Beltrami flows in axisymmetric geometry: Theory reexamined

Cited by in corpus (8)

• Statistical mechanics of two-dimensional Euler flows and minimum enstrophy states
• Statistical mechanics of the 3D axi-symmetric Euler equations in a Taylor-Couette geometry
• Additional Invariants and Statistical Equilibria for the 2D Euler Equations on a spherical domain
• Susceptibility divergence, phase transition and multistability of a highly turbulent closed flow
• Statistical mechanics of quasi-geostrophic flows on a rotating sphere
• Statistical mechanics of two-dimensional point vortices: relaxation equations and strong mixing limit
• Strong vorticity fluctuations and antiferromagnetic correlations in axisymmetric fluid equilibria
• Statistical equilibrium principles in 2D fluid flow: from geophysical fluids to the solar tachocline