Basics of Turbulence II:

Transcription

1 Basics of Turbulence II: Some Aspect of Mixing and Scale Selection P.H. Diamond UC San Diego and SWIP 1 st Chengdu Theory Festival Aug This research was supported by the U.S. Department of Energy, Office of Science, Office of Fusion Energy Sciences, under Award Number DE-FG02-04ER54738 and CMTFO.

2 Outline Prelude: Thoughts on Mixing and Scale Selection Mixing in Pipes and Donuts: Prandtl Kadomtsev Simple, useful ideas Potential Vorticity: Not all mixing is bad... Inhomogeneous Mixing: Corrugations and Beyond Brief Discussion

3 Prelude Why is plasma turbulence hard? 40+ years Modest Re 1) Broad dynamic range: ρρ ee ρρ ii < ll < LL pp (mesoscopic structures galore) 2) Multi-scale bifurcations, bi-stability 3) Ku ~ 1 ( VVττ cc /Δ KKKK) (coherent vs stochastic) 4) Boundary dynamics/dynamic boundaries 5) Dynamic phases

4 Bosch as Metaphor (After Kadomtsev) The Garden of Earthly Delights, Hieronymous Bosch

5 Most Problems: Scale Selection Classic example: Mixing Length Estimate / Rule (Kadomtsev 66) Still used for modelling PP + VV PP = VV rr dd PP /dddd What is Δ? δδδδ PP Δ LL pp N.B. δδδδ PP Δ LL pp KKKK OO(1) ρρ ii linear mode scale, which? shearing modified scale domain scale... what?

6 A Simpler Problem: Drag in Turbulent Pipe Flow - L. Prandtl 1932, et. seq - Prototype for mixing length model

7 Essence of confinement problem: given device, sources; what profile is achieved? ττ EE = WW/PP ii nn, How optimize W, stored energy Related problem: Pipe flow drag momentum flux ΔPP pressure drop a ll Turbulent ΔPPPPaa 2 = ρρvv 2 2ππππππ friction velocity V uu tttttttt Balance: momentum transport to wall (Reynolds stress) vs ΔPP Flow velocity profile Laminar RRRR λλ = 2aaΔPP/ll 1/2ρρuu 2

8 uu (Core) inertial sublayer ~ logarithmic (~ universal) 0 Wall viscous sublayer (linear) Problem: physics of ~ universal logarithmic profile? Universality scale invariance Prandtl Mixing Length Theory (1932) Wall stress = ρρvv 2 = ρρνν TT / eddy viscosity Absence of characteristic scale or: ~ VV xx Spatial counterpart of K41 Scale of velocity gradient? turbulent transport model νν TT VV xx uu VV ln(xx/xx 0 ) xx mixing length, distance from wall Analogy with kinetic theory νν TT = νν xx 0, viscous layer xx 0 = νν/vv

9 Some key elements: Momentum flux driven process ΔPP VV 2 νν TT / Turbulent diffusion model of transport - eddy viscosity Mixing length scale selection ~ xx macroscopic, eddys span system xx 0 < xx < aa ~ flat profile strong mixing νν TT = VV xx Self-similarity x no scale, within xx 0, aa Reduce drag by creation of buffer layer i.e. steeper gradient than inertial sublayer (by polymer) enhanced confinement

10 Without vs With Polymers Comparison NYFD 1969

11 Confinement

12 Primer on Turbulence in Tokamaks I Strongly magnetized Quasi 2D cells, Low Rossby # BB 0 EE θθ * Localized by kk BB = 0 (resonance) - pinning VV = + cc EE zz, BB VV llω cccc ~ RR 0 1 TT ρρ ii aa TT ee, TT ii, nn driven Akin to thermal convection with: g magnetic curvature Re VVVV/νν ill defined, not representative of dynamics Resembles wave turbulence, not high RRRR Navier-Stokes turbulence KK VVττ cc /Δ 1 Kubo # 1 Broad dynamic range, due electron and ion scales, i.e. aa, ρρ ii, ρρ ee

13 Primer on Turbulence in Tokamaks II Correlation scale ~ few ρρ ii mixing TT ρρ ii Key: 2 scales: ρρ gyro-radius aa cross-section ρρ ρρ/aa key ratio ρρ 1 aa length (?!) Characteristic velocity vv dd ~ ρρ cc ss Transport scaling: DD GGGG ~ ρρ VV dd ρρ DD BB DD BB ρρ cc ss TT/BB i.e. Bigger is better! sets profile scale via heat balance (Why ITER is huge ) Reality: DD ~ ρρ αα DD BB, αα < 1 Gyro-Bohm breaking 2 Scales, ρρ 1 key contrast to pipe flow

14 THE Question Scale Selection Pessimistic Expectation (from pipe flow): ll aa DD DD BB Hope (mode scales) ll ρρ ii DD DD GGGG ρρ DD BB Reality: DD ρρ αα DD BB, αα < 1 Why? What physics competition set αα? Focus of a large part of this Festival

15 Correlation function (DBS) exhibits multiple scale behavior (Hennequin, et. al. 2015)

16 Players in Scale Selection Mesoscales: Δ cc < ll < LL pp Transport Events: Enhanced Mixing Turbulence spreading wave emission scattering correlated topplings Avalanching c.f. Hahm, Diamond 2018 (OV), Dif-Pradalier, this meeting Zonal shears regulation Produced by PV mixing c.f. Kosuga this meeting

17 Potential Vorticity and Zonal Flows Not all mixing is bad... A different take on a familiar theme

18 Basic Aspects of PV Dynamics 18

19 Geophysical fluids Phenomena: weather, waves, large scale atmospheric and oceanic circulations, water circulation, jets Geophysical fluid dynamics (GFD): low frequency ( ) We might say that the atmosphere is a musical instrument on which one can play many tunes. High notes are sound waves, low notes are long inertial waves, and nature is a musician more of the Beethoven than the Chopin type. He much prefers the low notes and only occasionally plays arpeggios in the treble and then only with a light hand. J.G. Charney ( Turing s Cathedral ) Geostrophic motion: balance between the Coriolis force and pressure gradient P stream function 19

20 Kelvin s theorem unifying principle Kelvin s circulation theorem for rotating system Ω relative planetary Displacement on beta-plane y z x Quasi-geostrophic eq θ PV conservation Rossby wave t=0 ω<0 t>0 ω>0 G. Vallis 06 20

21 PV conservation. GFD: Quasi-geostrophic system Plasma: Hasegawa-Wakatani system relative vorticity planetary vorticity density (guiding center) ion vorticity (polarization) Physics: ZF Physics: ZF! Charney-Hasewgawa-Mima equation (branching) H-W H-M: Q-G: 21

22 PV Transport Zonal flows are generated by nonlinear interactions/mixing and transport. In x space, zonal flows are driven by Reynolds stress Taylor s Identity PV flux fundamental to zonal flow formation Inhomogeneous PV mixing, not momentum mixing (dq/dt=0) up-gradient momentum transport (negative-viscosity) not an enigma Reynolds stresses intimately linked to wave propagation but: Wave-mixing, transport duality c.f. Review: O.D. Gurcan, P.D.; J. Phys. A (2015) real space emphasis 22

23 How make a ZF? Inhomogeneous PV mixing PV mixing is the fundamental mechanism for zonal flow formation PV PV velocity mixed unmixed McIntyre 1982 Dritschel & McIntyre 2008

24 PV Mixing Wave Propagation How do Zonal Flow Form? Simple Example: Zonally Averaged Mid-Latitude Circulation Rossby Wave: ωω kk = ββkk xx kk 2 vv gggg = 2ββ kk xxkk yy kk 2 2, vv yy vv xx = kk kk xx kk yy φφ kk 2 v~ v~ = k k ˆ φ vv gggg vv ppppp < 0 Backward wave! Momentum convergence at stirring location y x k x y k 2

25 Some similarity to spinodal decomposition phenomena Both negative diffusion phenomena

26 Minimum Enstrophy Relaxation Principle for ~ 2D Relaxation? How Represent PV mixing? Non-perturbative? 26

27 Foundation: Dual Cascade 2D turbulence conservation of energy and potential enstrophy dual cascade (Kraichnan) inverse energy cascading E(k) ~ k -5/3 When eddy turnover rate and Rossby wave frequency mismatch are comparable forward enstrophy cascading E(k) ~ k -3 Rhines scale Rhines scale forcing viscous 27 wave wave zonal flow

28 Upshot : Minimum Enstrophy State (Bretherton and Haidvogel, 1976) -- idea : final state potential enstrophy forward cascades to viscous dissipation -- kinetic energy inverse cascades (drag?!) calculate macrostate by minimizing potential enstrophy Ω subject to conservation of kinetic energy E, i.e. δδ Ω + μμμμ = 0 Minimum Enstrophy Theory [n.b. can include topography ] 28

29 A Natural Question: How exploit relaxation theory in dynamics? (with P.-C. Hsu, S.M. Tobias) 29

30 Further Non-perturbative Approach for Flow! - PV mixing in space is essential in ZF generation. Key: How represent inhomogeneous PV mixing Taylor identity: vorticity flux Reynolds force General structure of PV flux? relaxation principles! most treatment of ZF: -- perturbation theory -- modulational instability (test shear + gas of waves) ~ linear theory based -> physics of evolved PV mixing? -> something more general? non-perturb model: use selective decay principle What form must the PV flux have so as to dissipate enstrophy while conserving energy?

31 Using selective decay for flux minimum enstrophy relaxation (Bretherton & Haidvogel 1976) analogy Taylor relaxation (J.B. Taylor, 1974) turbulence 2D hydro 3D MHD dual cascade conserved quantity (constraint) dissipated quantity (minimized) total kinetic energy fluctuation potential enstrophy global magnetic helicity magnetic energy final state minimum enstrophy state flow structure emergent Taylor state force free B field configuration structural approach (Boozer, 86) flux? what can be said about dynamics? structural approach (Boozer): (this work): What What form form must must the the helicity PV flux flux have have so so as as to to dissipate magnetic-energy enstrophy while conserving while conserving energy? helicity? General principle based on general physical ideas useful for dynamical model 31

32 PV flux PV conservation mean field PV: selective decay energy conserved : mean field PV flux Key Point: form of PV flux Γ q which dissipates enstrophy & conserves energy enstrophy minimized parameter TBD general form of PV flux 32

33 Structure of PV flux transport parameter calculated by perturbation theory, numerics drift and hyper diffusion of PV <--> usual story : Fick s diffusion relaxed state: Homogenization of characteristic scale : zonal flow growth : zonal flow damping (hyper viscosity-dominated) Rhines scale : wave-dominated : eddy-dominated 33

34 What sets the minimum enstrophy Decay drives relaxation. The relaxation rate can be derived by linear perturbation theory about the minimum enstrophy state >0 relaxation The condition of relaxation (modes are damped): Relates with ZF and scale factor ZF can t grow arbitrarily large the minimum enstrophy of relaxation, related to scale 34

35 Role of turbulence spreading Turbulence spreading: tendency of turbulence to self-scatter and entrain stable regime Turbulence spreading is closely related to PV mixing because the transport/mixing of turbulence intensity has influence on Reynolds stresses and so on flow dynamics. PV mixing is closely related to turbulence spreading condition of energy conservation the gradient of y q / v x, drives spreading the spreading flux vanishes when y q / v x is homogenized 35

36 Inhomogeneous Mixing Formation of corrugations, layering, etc Focus: Stratified Fluid See also: Dif-Pradalier Lecture Weixin Guo Robin Heinonen Posters 36

37 Inhomogeneous Mixing Example: Thermohaline Layer Simulation (Radko, 2003) Corrugated Profile Sharp interface formed colors salt concentration Modulation Corrugations Mergers Barrier Single layer Corrugations formed, followed by condensation to single layer Merger events 37

38 Inhomogeneous Mixing Corrugations? How? Bistable Modulations Cf Phillips 72: Instability of mean + turbulence field requiring: δδγ bb /δδδδδδ < 0 flux dropping with increased gradient Γ bb = DD bb bb, RRRR = gg bb/vv 2 (Richardson #) Obvious similarity to transport bifurcation, but now a sequence of layers... 38

39 Mechanism: Inhomogeneous Mixing b buoyancy (density) profile gradient modulation bistability intensity Resembles: Caviton train for Langmuir turbulence εε δδδδ δδδδ/nn εε Intensity εε δδδδ 39

40 Corrugated Layering Kimura, et. al. Contours of temperature fluctuations

41 Kimura, et. al. Corrugated (total) temperature profile PDFs of TT, TT. θθ zz is skewed. 41

42 Is there a simple model encapsulating these ideas N. Balmforth, et al 1998 corrugated profile in stirred, stable stratified turbulence (c.f. A. Ashourvan, P.D.; 17, 18 for drift waves) Idea bistable modulation kinetic energy, mean density evolution DD VV ll mmmmmm ~ εε 1/2 ll mmmmmm ll mmmmmm key

43 What is the Mixing Length (ll mmmmmm )? Stratified fluid: buoyance frequency ( ~ gg/ll ρρ 1/2 ) VV ll ll ~ NN ll oooo Ozmidov scale KKKK ll oooo 1 ~ small stratified scale ~ VV3 ll buoyancy production gg VVVVVV 1 ll oooo turbulent dissipation zzbb εε 1/2 1 ll 2 ~ 1 mmmmmm ll 2 ~ 1 2 ff ll oooo system 2 scales, intrinsically

44 So: 2 ll mmmmmm = ll ff 2 ll2 oooo ll 2 oooo +ll 2 ff 2 ll oooo ~ εε/ zz bb 1/2 ll 2 2 oooo ll ff steep zz bb Feedback loop emerges, as ll mmmmmm drops with steepening zz bb some resemblance to flux limited transport models

45 DD = εε 1/2 ll mmmmmm εε = VV 2 Model tt bb = zz DD zz bb spreading production tt εε = zz DD zz εε ll εε 1 2 zz bb εε 3 2 ll + FF 1 2 = 1 2 ll + 1 ff ll mmmmmm external forcing 2 ll oooo Energetics: dissipation tt εε zzzz = 0 + source, sink

46 Some observations No molecular diffusion branch ( neoclassical H-mode ) Steep zz bb balanced by dissipation, as ll reduced Step layer set by turbulence spreading (N.B. interesting lesson for case when DD nnnnnn feeble i.e. particles) Forcing acts to initiate fluctuations, but production by gradient ( zz bb) is the main driver Gradient-fluctuation energy balance is crucial Can explore stability of initial uniform ee, zz bb field akin modulation problem 46

47 The physics: Negative Diffusion Γ bb bb H-mode like branch (i.e. residual collisional diffusion) need not be input - often feeble residual diffusion - gradient regulated by spreading Instability driven by local transport bifurcation δδγ bb /δδ bb < 0 negative diffusion Negative slope Unstable branch Feedback loop Γ bb bb II Γ b Critical element: ll mixing length Brings a new wrinkle: bi-stable mixing length models 47

48 Some Results Plot of zz bb (solid) and εε (dotted) at early time. Buoyancy flux is dashed near constant in core (not flux driven) zz bb εε Later time more like expected corrugation. Some condensation into larger scale structures has occurred. zz 48

49 Time Evolution zz bb ( 10 4 ) Time progression shows merger process akin bubble competition for steps Suggests trend to merger into tt fewer, larger steps Relaxation description in terms of merger process!? i.e. population evolution zz Predict/control position of final large step? 49

50 Inhomogeneous Mixing - Summary Highly relevant to MFE confinement Theory already extended to simple drift wave systems Bistable inhomogeneous mixing significantly extend concept of modulational instability Natural synthesis of: modulational instability transport bifurcation Defines a new, mesoscopic state

51 Discussion Choppy Profiles TFTR? L-mode hysteresis in QQ vs II, QQ vs TT? Will turbulence spreading saturate ZF for νν 0? Statistical distribution of ll mmmmmm?

52 Shameless Advertising: Mesoscopic Transport Events and the Breakdown of Fick s Law for Turbulent Fluxes T.S. Hahm, P.H. Diamond in press, J. Kor. Phys. Soc., 50 th Ann. Special Issue preprint available

53 This material is based upon work supported by the U.S. Department of Energy, Office of Science, Office of Fusion Energy Science, under Award Number DE-FG02-04ER54738