Turbulent cascade, bottleneck, and thermalized spectrum in hyperviscous flows

Abstract

In many simulations of turbulent flows the viscous forces ν∇ 2 u are replaced by a hyper-viscous term −νp(−∇ 2 ) p u in order to suppress the effect of viscosity at the large scales. In this work we examine the effect of hyper-viscosity on decaying turbulence for values of p ranging from p = 1 (regular viscosity) up to p = 100. Our study is based on direct numerical simulations of the Taylor-Green vortex for resolutions from 512 3 to 2048 3 . Our results demonstrate that the evolution of the total energy E and the energy dissipation remain almost unaffected by the order of the hyper-viscosity used. However, as the order of the hyper-viscosity is increased, the energy spectrum develops a more pronounced bottleneck that contaminates the inertial range. At the largest values of p examined, the spectrum at the bottleneck range has a positive power-law behavior E(k) ∝ k α with the power-law exponent α approaching the value obtained in flows at thermal equilibrium α = 2. This agrees with the prediction of Frisch et al. [Phys. Rev. Lett. 101, 144501 (2008)] who suggested that at high values of p, the flow should behave like the truncated Euler equations (TEE). Nonetheless, despite the thermalization of the spectrum, the flow retains a finite dissipation rate up to the examined order, which disagrees with the predictions of the TEE system implying suppression of energy dissipation. We reconcile the two apparently contradictory results, predicting the value of p for which the hyper-viscous Navier-Stokes goes over to the TEE system and we discuss why thermalization appears at smaller values of p.

As a matter of tension -- kinetic energy spectra in MHD turbulence

“…We further demonstrate, using a shell-toshell energy transfer analysis, that this "shallow" kinetic energy spectrum is associated with magnetic tension, which acts to suppress the kinetic energy cascade and provides the major contribution in the energy flux from large to small scales. This result is in marked contrast with incompressible hydrodynamic turbulence, where the kinetic energy cascade is the only means of transferring energy between scales in a self-similar fashion (which in turn leads to the emergence of the k −5/3 scaling) and departures from this expected scaling in hydrodynamic turbulence simulations and experiments have been associated with the existence of "bottlenecks" (Falkovich 1994;Schmidt et al 2006;Frisch et al 2008;Donzis & Sreenivasan 2010;Küchler et al 2019;Agrawal et al 2020). As such, the results presented in this work demonstrate the rich physics phenomenology that can operate even in the simplest scenarios (vanishing mean field, cross-helicity, and magnetic helicity) where mag-netic tension is dynamically important and further serve to highlight the necessary ingredients that MHD turbulence theory and phenomenology should incorporate in order to explain scalings of kinetic and magnetic energy observed in both simulation and observation of magnetized turbulence.…”

Suppressing thermalization and constructing weak solutions in truncated inviscid equations of hydrodynamics: Lessons from the Burgers equation

“…A second possibility is of course the use of a hyperviscous term. This however has the drawback that we would end up solving not the inviscid equation but its viscous form and for higher orders of the hyperviscosity-which is similar in spirit to the idea of purging-the solutions thermalize [10,12,32]. Another approach is due to Pereira et al [18] who showed that a waveletbased filtering technique also leads to a suppression of the resonances leading to tygers.…”

Two-dimensional partially ionized magnetohydrodynamic turbulence

“…In an attempt to better resolve inertial ranges while forcing at intermediate and larger wavenumbers, hyperviscosity, (−1) p+1 ∇ 2p , was used in all runs and for all three fields, replacing the regular viscosity and regular magnetic diffusivity seen in (2.1a)-(2.1c). As long as the value of p is not very large, hyperviscosity has been shown to have no significant effect on the turbulent properties of 3-D turbulence, and we expect the same to be the case for our work (Agrawal et al 2020). The value of p was set to 2 in all runs except for those where k f = 32, in which case p = 4.…”