Hyperviscosity and statistical equilibria of Euler turbulence on the torus and the sphere

Abstract

Coherent structures such as jets and vortices appear in two-dimensional (2D) turbulence. To gain insight into both numerical simulation and equilibrium statistical mechanical descriptions of 2D Euler flows, the Euler equation with added hyperviscosity is integrated forward in time on the square torus and on the sphere. Coherent structures that form are compared against a hierarchy of truncated Miller-Robert-Sommeria equilibria. The energy-circulation-enstrophy MRS-2 description produces a complete condensation of energy to the largest scales, and in the absence of rotation correctly predicts the number and polarity of coherent vortices. Perturbative imposition of the quartic Casimir constraint improves agreement with numerical simulation by sharpening the cores and transferring some energy to smaller-scale modes. MRS-2 cannot explain qualitative changes due to rotation, but descriptions that conserve higher Casimirs beyond enstrophy have the potential to do so. The result is in agreement with the somewhat paradoxical observation that hyperviscosity helps to remedy the non-conservation of the third and higher Casimirs in numerical simulation. For a rotating sphere, numerical simulation also demonstrates that coherent structures found at late times depend on initial conditions, limiting the usefulness of statistical mechanics.

57 pages, 16 figures, 86 references. This version (v3) is accepted for publication in J. Stat. Phys. Th. Exp

References in corpus (11)

• Hyperviscosity, Galerkin truncation and bottlenecks in turbulence
• Random changes of flow topology in two dimensional and geophysical turbulence
• Direct Statistical Simulation of Out-of-Equilibrium Jets
• Simpler Variational Problem for Statistical Equilibria of the 2D Euler Equation and Other Systems with Long Range Interactions
• Strange magnetic form factor of nucleon in heavy baryon chiral effective approach at next to leading order
• Invariant measures of the 2D Euler and Vlasov equations
• Statistical mechanics of 2D turbulence with a prior vorticity distribution
• Phase transitions and marginal ensemble equivalence for freely evolving flows on a rotating sphere
• Statistical mechanics of quasi-geostrophic flows on a rotating sphere
• Energy extremals and Nonlinear Stability in a Variational theory of Barotropic Fluid - Rotating Sphere System
• A complete theory of low-energy phase diagrams for two-dimensional turbulence steady states and equilibria

Cited by in corpus (11)

• Mathematical and Physical Ideas for Climate Science
• Cumulant expansions for atmospheric flows
• A Meshfree Lagrangian Method for Flow on Manifolds
• Direct Statistical Simulation of a Jet
• An Introduction to Large Deviations and Equilibrium Statistical Mechanics for Turbulent Flows
• Statistical equilibrium principles in 2D fluid flow: from geophysical fluids to the solar tachocline
• Onsager vortex clusters on a sphere
• A localized dynamic closure model for Euler turbulence
• A statistical mechanics framework for the large-scale structure of turbulent von K{á}rm{á}n flows
• Direct statistical simulation of Lorenz96 system in model reduction approaches
• Statistical equilibria of two-dimensional turbulent flows for generic initial vorticity fields on a sphere, calculated on the basis of the original Miller-Robert-Sommeria theory