Basics of Turbulence II: Some Aspect of Mixing and Scale Selection

P.H. Diamond

UC San Diego and SWIP 1st Chengdu Theory Festival Aug. 2018

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. Outline

• Prelude: Thoughts on Mixing and Scale Selection

• Mixing in Pipes and Donuts:

– Prandtl Simple, useful ideas – Kadomtsev

• Potential Vorticity: Not all mixing is bad...

• Inhomogeneous Mixing: Corrugations and Beyond

• Brief Discussion Prelude

• Why is plasma turbulence “hard”?

– 40+ years

– Modest ‘Re’

1) Broad dynamic range: < < (mesoscopic

structures galore) 𝜌𝜌𝑒𝑒 → 𝜌𝜌𝑖𝑖 𝑙𝑙 𝐿𝐿𝑝𝑝 2) Multi-scale bifurcations, bi-stability

3) Ku ~ 1 ( / ) (coherent vs stochastic)

4) Boundary𝑉𝑉�𝜏𝜏 𝑐𝑐dynamics/dynamicΔ ≡ 𝐾𝐾𝐾𝐾 boundaries 5) Dynamic phases Bosch as Metaphor (After Kadomtsev) “The Garden of Earthly Delights”, Hieronymous Bosch • Most Problems: Scale Selection

• Classic example:

– Mixing Length Estimate / “Rule” (Kadomtsev ‘66)

– Still used for modelling

+ = / • What is ?

𝜕𝜕𝑃𝑃� 𝑉𝑉� ⋅ 𝛻𝛻𝑃𝑃� −𝑉𝑉�𝑟𝑟 𝑑𝑑〈𝑃𝑃〉 𝑑𝑑𝑑𝑑 – Δ – 𝜌𝜌linear𝑖𝑖 mode scale, which? 𝛿𝛿𝛿𝛿 Δ 𝑃𝑃 ∼ 𝐿𝐿𝑝𝑝 – shearing modified scale  (1) N.B. – domain scale 𝛿𝛿𝛿𝛿 Δ 𝑃𝑃 ∼ 𝐿𝐿𝑝𝑝 𝐾𝐾𝐾𝐾 ∼ 𝑂𝑂 – ... what? A Simpler Problem:  Drag in Turbulent Pipe Flow - L. Prandtl 1932, et. seq - Prototype for mixing length model • Essence of confinement problem:

– given device, sources; what profile is achieved?

– = / , How optimize W, stored energy

𝐸𝐸 𝑖𝑖 𝑛𝑛 • Related𝜏𝜏 𝑊𝑊problem:𝑃𝑃 Pipe flow  drag ↔ flux

 pressure drop

Δ𝑃𝑃 a Turbulent 𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝟏

𝑙𝑙 = Laminar 2 2 Δfriction𝑃𝑃𝑃𝑃𝑎𝑎 velocity𝜌𝜌𝑉𝑉∗ 2𝜋𝜋𝜋𝜋𝜋𝜋 V Balance: momentum transport∗ to↔ 𝑢𝑢wall𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 () vs / = 𝑹𝑹𝑹𝑹  Flow velocity profile 1/ Δ𝑃𝑃 2𝑎𝑎Δ𝑃𝑃 𝑙𝑙 𝜆𝜆 2 2𝜌𝜌𝑢𝑢 (Core)

inertial sublayer  ~ logarithmic (~ universal) 𝑢𝑢 viscous sublayer (linear) 0 • Problem: physics of ~ universal logarithmic profile? Wall • Universality  scale invariance

• Prandtl Mixing Length Theory (1932) Spatial counterpart – Wall stress = = / or: ~ of K41 2 𝜕𝜕𝜕𝜕 𝑉𝑉∗ ∗ 𝑇𝑇 𝜌𝜌𝑉𝑉 −𝜌𝜌𝜈𝜈 𝜕𝜕eddy𝜕𝜕 𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕 𝑥𝑥 Scale of velocity gradient? – Absence of characteristic scale 

turbulent mixing length, distance from wall transport 𝑇𝑇 ∗ln( / ) Analogy with kinetic theory … model 𝜈𝜈 ∼ 𝑉𝑉 𝑥𝑥 𝑥𝑥 ≡ 𝑢𝑢 ∼=𝑉𝑉∗ 𝑥𝑥 𝑥𝑥, 0viscous layer  = /

𝜈𝜈𝑇𝑇 𝜈𝜈 → 𝑥𝑥0 𝑥𝑥0 𝜈𝜈 𝑉𝑉∗ Some key elements:

• Momentum flux driven process / 2 • Turbulent diffusion model of transportΔ𝑃𝑃 ↔ -𝑉𝑉∗eddy→ 𝜈𝜈viscosity𝑇𝑇𝜕𝜕𝜕𝜕 𝜕𝜕𝜕𝜕

• Mixing length – scale selection

~  macroscopic, eddys span system < <

0 = 𝑥𝑥  ~ flat profile – strong mixing 𝑥𝑥 𝑥𝑥 𝑎𝑎 𝜈𝜈𝑇𝑇 𝑉𝑉∗𝑥𝑥 • Self-similarity  x ↔ no scale, within ,

• Reduce drag by creation of buffer layer𝑥𝑥 i.e.0 𝑎𝑎 steeper gradient than

inertial sublayer (by polymer) – enhanced confinement Without vs With Polymers Comparison  NYFD 1969 Confinement Primer on Turbulence in Tokamaks I

• Strongly magnetized

– Quasi 2D cells, Low Rossby # 𝐵𝐵0 * – Localized by = 0 (resonance) - pinning 𝐸𝐸𝜃𝜃 𝑘𝑘 ⋅ 𝐵𝐵 • = + × , ~ 1 𝑐𝑐 𝑉𝑉⊥ 𝑇𝑇 𝜌𝜌𝑖𝑖 𝑉𝑉⊥ 𝐵𝐵 𝐸𝐸 𝑧𝑧̂ 𝑙𝑙Ω𝑐𝑐𝑐𝑐 𝑅𝑅0 ≪ • , , driven 𝑎𝑎 • 𝛻𝛻Akin𝑇𝑇𝑒𝑒 𝛻𝛻to𝑇𝑇 𝑖𝑖thermal𝛻𝛻𝑛𝑛 convection with: g  magnetic curvature  • Re / ill defined, not representative of dynamics

• Resembles≈ 𝑉𝑉𝑉𝑉 𝜈𝜈 wave turbulence, not high Navier-Stokes turbulence

 • / 1  Kubo # 1 𝑅𝑅𝑅𝑅

 • 𝐾𝐾Broad∼ 𝑉𝑉�𝜏𝜏 dynamic𝑐𝑐 Δ ∼ range, due≈ electron and ion scales, i.e. , ,

𝑎𝑎 𝜌𝜌𝑖𝑖 𝜌𝜌𝑒𝑒 Primer on Turbulence in Tokamaks II

• Correlation scale ~ few  “mixing

length”(?!) 𝜌𝜌𝑖𝑖 Characteristic velocity ~ 𝑇𝑇 • 𝜌𝜌𝑖𝑖 𝑣𝑣𝑑𝑑 𝜌𝜌∗𝑐𝑐𝑠𝑠 • Transport scaling: ~ Key: 𝑎𝑎 𝐷𝐷𝐺𝐺𝐺𝐺 𝜌𝜌 𝑉𝑉𝑑𝑑 ∼ 𝜌𝜌/∗ 𝐷𝐷𝐵𝐵 2 scales: • i.e. Bigger is better!𝐷𝐷 𝐵𝐵 ∼sets𝜌𝜌 𝑐𝑐 𝑠𝑠profile∼ 𝑇𝑇 𝐵𝐵 scale via heat gyro-radius balance (Why ITER is huge…) 𝜌𝜌 ≡ cross-section • Reality: ~ , < 1  ‘Gyro-Bohm 𝑎𝑎 ≡ /  key ratio 𝛼𝛼 𝜌𝜌∗ ≡ 𝜌𝜌1 𝑎𝑎 breaking’𝐷𝐷 𝜌𝜌∗ 𝐷𝐷𝐵𝐵 𝛼𝛼 ∗ 𝜌𝜌 ≪ • 2 Scales, 1  key contrast to pipe flow

𝜌𝜌∗ ≪ THE Question ↔ Scale Selection

• Pessimistic Expectation (from pipe flow):

–

– 𝑙𝑙 ∼ 𝑎𝑎 • Hope𝐷𝐷 ∼ (mode𝐷𝐷𝐵𝐵 scales)

–

– 𝑙𝑙 ∼ 𝜌𝜌𝑖𝑖 • Reality:𝐷𝐷 ∼ 𝐷𝐷 𝐺𝐺𝐺𝐺 ∼ 𝜌𝜌∗𝐷𝐷𝐵𝐵 , < 1 𝛼𝛼 𝐷𝐷Why?∼ 𝜌𝜌∗ 𝐷𝐷 What𝐵𝐵 𝛼𝛼physics competition set ?

 Focus of a large part of this Festival𝛼𝛼 Correlation function (DBS) exhibits multiple scale behavior (Hennequin, et. al. 2015) Players in Scale Selection “wave emission • Mesoscales: < < scattering”

Δ𝑐𝑐 𝑙𝑙 𝐿𝐿𝑝𝑝 • Transport Events: Enhanced Mixing “correlated topplings” – Turbulence spreading

– Avalanching

– c.f. Hahm, Diamond 2018 (OV), Dif-Pradalier, this meeting

• Zonal shears – regulation c.f. Kosuga this meeting  Produced by PV mixing Potential Vorticity and Zonal Flows  Not all mixing is bad...  A different take on a familiar theme Basic Aspects of PV Dynamics

18 Geophysical fluids

• Phenomena: weather, waves, large scale atmospheric and oceanic circulations, water circulation, jets… • Geophysical (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 (“Turing’s arpeggios in the treble and then only with a light hand.“ – J.G. Charney Cathedral” ) • Geostrophic motion: balance between the Coriolis force and pressure gradient

P stream function

19 Kelvin’s theorem – unifying principle

• Kelvin’s circulation theorem for rotating system Ω

relative planetary z y • Displacement on beta-plane x

θ • Quasi-geostrophic eq

PV conservation

 Rossby wave t=0 ω<0

t>0 ω>0 G. Vallis 06 20 PV conservation .

GFD: Plasma: Quasi-geostrophic system Hasegawa-Wakatani system

relative planetary density ion vorticity vorticity vorticity (guiding center) (polarization)

Physics: ZF Physics: ZF!

(branching) • Charney-Hasewgawa-Mima equation

H-W  H-M:

Q-G: 21 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

Wave-mixing, transport but: duality

c.f. Review: O.D. Gurcan, P.D.; J. Phys. A (2015) real space emphasis 22 How make a ZF?  Inhomogeneous PV mixing

• PV mixing is the fundamental mechanism for zonal flow formation

PV PV velocity

unmixed mixed

McIntyre 1982 Dritschel & McIntyre 2008  PV Mixing  Wave Propagation  How do Zonal Flow Form? Simple Example: Zonally Averaged Mid-Latitude Circulation

Rossby Wave:

= 2 v~ v~ = − k k φˆ 𝛽𝛽𝑘𝑘𝑥𝑥 y x ∑ x y k 𝜔𝜔𝑘𝑘 − 2 k 𝑘𝑘⊥ = , = 𝑘𝑘𝑥𝑥𝑘𝑘𝑦𝑦 𝑔𝑔𝑔𝑔 2 2 𝑦𝑦 𝑥𝑥 𝑣𝑣 2𝛽𝛽 𝑘𝑘⊥ 𝑣𝑣� 𝑣𝑣�

2 ∑𝑘𝑘 −𝑘𝑘𝑥𝑥𝑘𝑘𝑦𝑦 𝜙𝜙<�𝑘𝑘0  Backward wave!

∴𝑣𝑣Momentum𝑔𝑔𝑔𝑔𝑣𝑣𝑝𝑝푝𝑝𝑝 convergence at stirring location Some similarity to spinodal decomposition phenomena  Both ‘negative diffusion’ phenomena Minimum Enstrophy Relaxation

 Principle for ~ 2D Relaxation?  How Represent PV mixing? Non-perturbative?

26 Foundation: Dual Cascade

• 2D turbulence conservation of energy and potential enstrophy inverse energy -5/3 cascading E(k) ~ k  dual cascade (Kraichnan)

forward enstrophy cascading • When eddy turnover rate -3 and Rossby wave frequency E(k) ~ k mismatch are comparable

Rhines viscous scale forcing 27  Rhines scale wave wave

zonal flow  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 subject to conservation of kinetic energy E, i.e. Ω + = 0 [n.b. can include topography ]  “Minimum𝛿𝛿 Ω 𝜇𝜇𝜇𝜇 Enstrophy Theory”

28 A Natural Question:

How exploit relaxation theory in dynamics? (with P.-C. Hsu, S.M. Tobias)

29 Further Non-perturbative Approach for Flow!

- PV mixing in space is essential in ZF generation. Taylor identity: vorticity flux Reynolds force

Key: How represent inhomogeneous General structure of PV flux? PV mixing relaxation principles! most treatment of ZF: -- perturbation theory -- modulational instability non-perturb model: use selective decay principle (test shear + gas of waves) What form must the PV flux have so as to ~ linear theory based dissipate enstrophy while conserving energy? -> physics of evolved PV mixing? -> something more general? Using selective decay for flux

minimum enstrophy Taylor relaxation analogy relaxation (J.B. Taylor, 1974) (Bretherton & Haidvogel 1976) turbulence 2D hydro 3D MHD conserved quantity total kinetic energy global magnetic helicity dual (constraint) cascade dissipated quantity fluctuation potential magnetic energy (minimized) enstrophy minimum enstrophy state Taylor state final state flow structure emergent 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 fluxflux havehave soso as as to to dissipate magneticenstrophy-energywhile conservingwhile conserving energy? helicity ?

General principle based on general physical ideas  useful for dynamical model 31 PV flux  PV conservation

mean field PV: Key Point: form of PV flux Γq which : mean field PV flux dissipates enstrophy & conserves energy selective decay

 energy conserved

 enstrophy minimized

general form of PV flux

parameter TBD 32 Structure of PV flux

transport parameter calculated by drift and hyper diffusion of PV perturbation theory, numerics… <--> usual story : Fick’s diffusion

relaxed state: Homogenization of

characteristic scale Rhines scale

: zonal flow growth : wave-dominated : zonal flow damping : eddy-dominated (hyper viscosity-dominated) 33 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 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 / vx , drives spreading  the spreading flux vanishes when ∂ q / v is homogenized ⟨ ⟩ ⟨ ⟩ y x ⟨ ⟩ ⟨ ⟩

35 Inhomogeneous Mixing

 Formation of corrugations, layering, etc  Focus: Stratified Fluid

See also: Dif-Pradalier Lecture Weixin Guo Posters Robin Heinonen

36 Inhomogeneous Mixing Example: Thermohaline Layer Simulation (Radko, 2003) Corrugated Profile

Sharp interface formed colors  salt concentration Single layer Corrugations formed, followed Modulation  Corrugations by ‘condensation’ to single layer   Mergers  ”Barrier” Merger events 37 • Inhomogeneous Mixing  Corrugations? How?  Bistable Modulations • Cf Phillips’72:

• Instability of mean + turbulence field requiring: / < 0  flux dropping with increased gradient

𝛿𝛿Γ𝑏𝑏=𝛿𝛿𝛿𝛿𝛿𝛿 , = / (Richardson #) ′2 • ObviousΓ𝑏𝑏 −𝐷𝐷 similarity𝑏𝑏𝛻𝛻𝑏𝑏 𝑅𝑅𝑅𝑅 to𝑔𝑔𝛻𝛻 transport𝑏𝑏 𝑣𝑣 bifurcation, but now a sequence of layers... 38 Mechanism: Inhomogeneous Mixing

b modulation gradient buoyancy (density) profile bistability

intensity Intensity Resembles: Caviton train for Langmuir / turbulence 𝛿𝛿𝛿𝛿 𝑛𝑛 ≈ −𝜀𝜀

𝜀𝜀 𝜀𝜀 𝛿𝛿𝛿𝛿 𝛿𝛿𝛿𝛿 39 Corrugated Layering Kimura, et. al.

Contours of temperature fluctuations Kimura, et. al.

Corrugated (total) PDFs of , . temperature profile is skewed. 𝑇𝑇 ∇𝑇𝑇 41 𝜃𝜃𝑧𝑧 • 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

– ~ / 1 2 – 𝐷𝐷 ∼ 𝑉𝑉� 𝑙𝑙key𝑚𝑚𝑚𝑚𝑚𝑚 𝜀𝜀 𝑙𝑙𝑚𝑚𝑚𝑚𝑚𝑚

𝑙𝑙𝑚𝑚𝑚𝑚𝑚𝑚 • What is the Mixing Length ( )?

𝑚𝑚𝑚𝑚𝑚𝑚 𝑙𝑙 / • Stratified fluid: buoyance frequency ( ~ / ) 1 2 𝑔𝑔 𝐿𝐿𝜌𝜌 • ~  Ozmidov scale 1 𝑉𝑉 𝑙𝑙 𝑜𝑜𝑜𝑜 𝑜𝑜𝑜𝑜 ~ 𝑙𝑙 small𝑁𝑁 “stratified”𝑙𝑙 scale 𝐾𝐾𝐾𝐾 𝑙𝑙 →

buoyancy production / ~  3 𝑉𝑉 1 𝜕𝜕𝑧𝑧𝑏𝑏 1 2 turbulent dissipation 𝑙𝑙 ∼ 𝑔𝑔〈𝑉𝑉𝑉𝑉𝑉𝑉〉 𝑙𝑙𝑜𝑜𝑜𝑜 ∼ 𝜀𝜀 • ~ ~  1 1 1 system 2 2 2 2 scales, intrinsically 𝑙𝑙𝑚𝑚𝑚𝑚𝑚𝑚 𝑙𝑙𝑓𝑓 𝑙𝑙𝑜𝑜𝑜𝑜 • So:

= 2 2 𝑜𝑜𝑜𝑜 2 𝑙𝑙𝑓𝑓𝑙𝑙 2 2 𝑙𝑙𝑚𝑚𝑚𝑚𝑚𝑚 𝑙𝑙𝑜𝑜𝑜𝑜 +𝑙𝑙𝑓𝑓  2 2 2 𝑜𝑜𝑜𝑜 steep𝑜𝑜𝑜𝑜 𝑓𝑓 ~ 𝑙𝑙/ / 𝑙𝑙 ≪ 𝑙𝑙 1 2 𝜕𝜕𝑧𝑧𝑏𝑏  Feedback loop emerges,𝜀𝜀 𝜕𝜕𝑧𝑧𝑏𝑏 as drops with steepening

 some resemblance to flux limited𝑙𝑙𝑚𝑚𝑚𝑚𝑚𝑚 transport models 𝜕𝜕𝑧𝑧𝑏𝑏 = / 1 2 𝐷𝐷= 𝜀𝜀 𝑙𝑙𝑚𝑚𝑚𝑚𝑚𝑚 2 � • Model 𝜀𝜀 1 𝑉𝑉 1 1 = + = 2 2 2 𝑚𝑚𝑚𝑚𝑚𝑚 𝑓𝑓 𝑜𝑜𝑜𝑜 𝜕𝜕𝑡𝑡𝑏𝑏 𝜕𝜕𝑧𝑧 spreading𝐷𝐷𝜕𝜕𝑧𝑧𝑏𝑏 production 𝑙𝑙 𝑙𝑙 𝑙𝑙

= 3 + external forcing 1 2 2 𝜀𝜀 𝜕𝜕𝑡𝑡𝜀𝜀 𝜕𝜕𝑧𝑧𝐷𝐷𝜕𝜕𝑧𝑧𝜀𝜀 − 𝑙𝑙 𝜀𝜀 𝜕𝜕𝑧𝑧𝑏𝑏 − 𝑙𝑙 𝐹𝐹 Energetics: dissipation

= 0 + source, sink

𝜕𝜕𝑡𝑡∫ 𝜀𝜀 − 𝑧𝑧𝑧𝑧 • Some observations – No molecular diffusion branch (“neoclassical H-mode”) Steep balanced by dissipation, as reduced

– Step layer𝜕𝜕𝑧𝑧𝑏𝑏 set by turbulence spreading𝑙𝑙 (N.B. interesting lesson for case when feeble – i.e. particles)

– Forcing acts to initiate𝐷𝐷 𝑛𝑛fluctuations,𝑛𝑛𝑛𝑛 but production by gradient ( ) is the main driver

– Gradient-fluctuation∼ 𝜕𝜕𝑧𝑧𝑏𝑏 energy balance is crucial – Can explore stability of initial uniform , field 

akin modulation problem 𝑒𝑒 𝜕𝜕𝑧𝑧𝑏𝑏 46 • The physics: Negative Diffusion

𝑏𝑏 Γ “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𝛻𝛻𝑏𝑏 bifurcation • / < 0 Negative slope 𝑏𝑏 𝛿𝛿Γ‘negative𝛿𝛿𝛻𝛻𝑏𝑏 diffusion’ Unstable branch • Feedback loop ↓  ↑  ↓  ↓ Critical element: mixing length Γ𝑏𝑏 𝛻𝛻𝑏𝑏 𝐼𝐼 Γb 𝑙𝑙 → Brings a new wrinkle: bi-stable mixing length models 47 • Some Results

• Plot of (solid) and (dotted)

at early𝜕𝜕 time.𝑧𝑧𝑏𝑏 Buoyancy𝜀𝜀 flux is dashed  near constant in core (not flux driven)

• Later time  more like

𝜕𝜕𝑧𝑧𝑏𝑏 expected “corrugation”. Some condensation into larger 𝜀𝜀 scale structures has occurred.

48 𝑧𝑧 • Time Evolution (× 10 ) 4 𝜕𝜕𝑧𝑧𝑏𝑏 • Time progression shows merger process – akin bubble competition for steps

• Suggests trend to merger into fewer, larger steps

• Relaxation description in terms of 𝑡𝑡 merger process!? i.e. population evolution

• Predict/control position of final large step?

49

𝑧𝑧 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 Discussion

• “Choppy Profiles” – TFTR?

• L-mode hysteresis in vs , vs ?

• Will turbulence spreading𝑄𝑄 saturate𝐼𝐼̃ 𝑄𝑄 ZF𝛻𝛻𝑇𝑇 for 0 ?

• Statistical distribution of ? 𝜈𝜈 →

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of Fick’s Law for Turbulent Fluxes”

T.S. Hahm, P.H. Diamond

– in press, J. Kor. Phys. Soc., 50th Ann. Special Issue

– preprint available • 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