Bottlenecks: an interplay of equilibrium statistical mechanics and turbulence

Transcription

1 Bottlenecks: an interplay of equilibrium statistical mechanics and turbulence Parisi Fest, Roma Sept. 8-, 28 Uriel Frisch Observatoire de la Côte d Azur, Nice with W. Pauls (Goettingen), A. Wirth (Grenoble), S. Kurien and J.-Z. Zhu (Los Alamos), R. Pandit and S.S. Ray (Bangalore) Phys. Rev. Lett. in press

2 Hyperviscous equations Burgers t v + v v = µk G 2α ( 2 ) α v µ >, k G >, α = dissipativity Here α >.. N-S t v + v v = p µk G 2α ( 2 ) α v, v = Dissipation rate Abstract form µ(k/k ) 2α G or when α t v = B(v, v) + L α v Galerkin truncation t u = P kg B(u, u), u o = P kg v Projector P kg : low-pass filter at wavenumber k G

3 Large dissipativity limit and thermalization For α, and fied µ and k G, the solution of the hyperdissipative equations tend to the solution of the Galerkin-truncated equations True for: Burgers, Navier-Stokes, MHD, DIA and EDQNM. False for: MRCM and resonant wave interaction theory. Galerkin-truncation thermalization (Lee, 952; Hopf, 952; Kraichnan, 958) Galerkin-truncated Burgers first studied by Majda and Timofeyev 2 Galerkin-truncated 3D incompressible Euler first studied at high resolution by Cichowlas, Bonaiti, Debbasch and Brachet 25 E(k) D Euler Galerkin-truncated 6 3 Same resolution; different times -6-2 Cichowlas et al. Same time; different resolutions -4-6 k

4 k Cichowlas et al. (25) reproduced by Bos and Bertoglio(26) with EDQNM

5 Eddy-Damped Quasi-Normal Markovian spectrum ( t + 2νk2) E(k, t) = k dpdqθ kpq b(k, p, q) k pq E(q, t) [ k 2 E(p, t) p 2 E(k, t) ] QN --- Chou(94), Millionshtchikov(94): realizability problem N --- Lee (952), Hopf(952): statistics of absolute equilibria of truncated Euler DIA (Kraichnan): tractability problem ED, M --- Orszag(97, 977)

6 & '() *+,&-!" #!"!!" " ".!' ".!/ ".!/#!" #!" )!" $!" '!!!#!)!!#2!/!!2# & #3'() *!"!!!! & 4!" '!"!# *!" "!" #!" $!" % & Hyperviscous EDQNM: convergence to Galerkin truncation and secondary bottleneck...

7 Bottleneck, thermalization, depletion of intermittency, etc Large α produces a huge thermalized bottleneck The standard α = bottleneck may be viewed as an aborted thermalization Kaneda et al. 23 (Earth Simulator). Compensated energy spectrum Thermalization is accompanied by Gaussianization and isotropization Spurious effects are epected: depletion of intermittency and isotropization

8 Hyperviscosity and Galerkin truncation for the Burgers equation t v + v v = µk G 2α ( 2 ) α v µ >, k G >, α = dissipativity

9 Hyperviscous and Galerkin-truncated Burgers *+,--.//!!$!%!&!) Burgers resolution 2 4 k G $ 23% 23) 23$! 23%! 23&!.!% 456*78, Burgers equation with random initial condition u () = sin + sin(2 + φ) φ uniformly distributed in [ π, π]!#!(!3$!!! hyperviscous Energy spectrum averaged over 2 realizations!'!!"# $ $"# % %"# & *+,-./ Evolution of Galerkin-truncated initial condition sin +,*- ")& "!)&!!!)&!".//! sin!")&! " # $ % & ' ( * Galerkin-truncated Burgers first studied by Majda and Timofeyev 2.5.5! t =! !!.5! t = 2 t = !!.5!

10 The shock acts as a black hole.5.5! t =.99! ! t =.! resolution 2 k G = t =..5 t =.2.5.5!!! ! t =.3 t =.4.5.5!!! ! ! t =.5! Are these genuine shocks? Mathematical question: do the solutions of the inviscid truncated Burgers eq. converge to the entropy solution when? k G! t =.!