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Entropy and relaxation processes Motivation

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HAL Id: cea-02479438

https://hal-cea.archives-ouvertes.fr/cea-02479438

Submitted on 14 Feb 2020

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Entropy and relaxation processes Motivation

Xavier Garbet

To cite this version:

Xavier Garbet. Entropy and relaxation processes Motivation. Third Asia pacific conference on plasma physics, Nov 2019, Hefei, China. �cea-02479438�

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Entropy and

relaxation processes

Xavier Garbet

IRFM

CEA Cadarache

Acknowledgements: C. Bourdelle, P.H. Diamond, G. Dif-Pradalier, P. Ghendrih, A. Samain, Y. Sarazin

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Motivation

• Open systems: find constitutive relations that link

fluxes to gradients  transport matrix.

• Relaxation processes  transport equations.

Consistency with second principle?

• Principle of minimum entropy production  second

principle, relaxation processes, Onsager reciprocal relations. Can be done with quasi-linear theory - not always valid.

• Present status in magnetised plasmas?

• What can be done when this procedure fails? X. Garbet, AAPPS-DPP, Hefei, Nov. 5 2019

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Entropy production and Onsager symmetry

• Entropy production 𝑆ሶ vs “forces” 

and “fluxes” 

𝑆 = 𝚲 ⋅ 𝚪

• Linear constitutive relationships

𝚪 = ധ𝑳 ⋅ 𝚲

• ധ𝑳 symmetric B-B Onsager 1931

• Compact form

𝑆 = 𝚲 ⋅ ധ𝑳 ⋅ 𝚲

 minimum of entropy production Prigogine 47

T =-T x Λ = 𝛻 1 𝑇 ሶ𝑆 = න 𝑑𝑥 𝜅 𝛻𝑇 𝑇 2

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Boltzmann-Gibbs statistical mechanics

• Distribution function 𝐹 𝒙, 𝒗, 𝑡  entropy 𝑆 = − ׬ 𝑑𝜏 𝐹 𝑙𝑛𝐹

• Maximum of entropy under

constraints  local Maxwellian

𝐹 = 𝑒𝑥𝑝 − 𝐻

𝑇 + 𝑈(𝒙)

• F solution of a kinetic equation

𝑑𝐹

𝑑𝑡 = 𝐶(𝐹)

• Collisional transport well

documented OVs Braginskii 65, Hinton 76, Balescu 87, Shaing 88  Turbulent

transport

X. Garbet, AAPPS-DPP, Hefei, Nov. 5 2019

v F(x,v) M(x) 𝑇(𝒙) 𝑚 න 𝑑3𝒗𝐹 𝒙, 𝒗 = 𝑁(𝒙)

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x y z B(x) M// k,k 𝚲 = 𝑑𝑁 𝑁𝑑𝑥 , 𝑑𝑇 𝑇𝑑𝑥 , 𝑑𝑀 𝑑𝑥 x A= (N, T, M//)

Quasi-linear theory provides fluxes for a given

spectrum of fluctuations

Sheared B field, drift waves, chaotic Hasegawa-Mima 77, Hasegawa-Wakatani 83

Drummond 62, Vedenov 62, OVs Krommes 02, Diamond 10

Flux Γ = σ𝒌 ׬ 𝑑3𝒗 𝐹𝒌𝑣𝐸𝒌∗ Fluctuations of ExB drift velocity 𝒗𝐸 = 𝑬×𝑩

𝐵2

Plasma linear response

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Entropy production is explicit

| PAGE 6 X. Garbet, AAPPS-DPP, Hefei, Nov. 5 2019

Numata 06 𝑽𝐸 = 𝑬 × 𝑩 𝐵2 𝐷𝒌 = 2𝜋 𝑣𝐸𝒌 2𝑅 𝜔𝑘 − 𝑘𝑣 x y B df>0 df<0

• Entropy production rate Horton 80, Itoh 82, Sugama 96, XG 13

wave/particle energy & momentum transfer ሶ 𝑆 = 1 2 ෍ 𝒌 න 𝑑𝜏 𝐹𝐷𝒌 𝜕𝑈 𝜕𝑥 − 𝑒𝐵 𝑘𝑦𝑇 𝜔𝒌 − 𝑘∥𝑀∥ 2

Diffusion due to ExB velocity fluctuations Forces 𝚲 = 𝑑𝑁 𝑁𝑑𝑥 , 𝑑𝑇 𝑇𝑑𝑥 , 𝑑𝑀 𝑑𝑥

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• Evolution of thermodynamical variables A=(N,T,M//)

𝜕𝑨

𝜕𝑡 + 𝛻 ⋅ 𝚪 = 𝚺 𝚪 = ധ𝑳 ⋅ 𝚲 + 𝚪res

• Transport matrix ധ𝐿 Onsager symmetric. However :

- “Residual” momentum and energy fluxes res

- Sources = turbulent heating and acceleration Rudakov 71, Ott 72

Transport equations bear a puzzling shape

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Fluxes possess pinch and residual components

• Diffusion/convection flux structure

Γ = −𝐷 𝑑𝐴

𝑑𝑥 + 𝑉𝐴 + Γ𝑟𝑒𝑠

• Diffusion = average of σ 𝒌 𝐷𝒌.

• Pinch velocity due forces other

than 𝑑𝐴

𝑑𝑥.

• Residual momentum and heat

fluxes 0  requires symmetry

breaking 𝑘𝑘𝑦 ≠ 0.

X. Garbet, AAPPS-DPP, Hefei, Nov. 5 2019

A=(N,T,M//) x A Diffusion −𝐷 𝑑A 𝑑𝑥 Pinch VA Inward V<0  =0

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Plasma sped up and heated by turbulence

Waltz 11 • Source terms Itoh 88, Hinton 06, Lu Wang 13, XG 13

Σ𝑀∥ = 𝑁𝑒𝐸 Σ𝑇 = 𝑁𝑒𝑽. 𝑬

• Charge conservation  global conservation

𝑠𝑝𝑒𝑐𝑖𝑒𝑠

𝑠𝑜𝑢𝑟𝑐𝑒𝑠 = 0

• Momentum and energy

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Residual fluxes and sources are Onsager symmetric

X. Garbet, AAPPS-DPP, Hefei, Nov. 5 2019

x T, M// Wave propagation T ran sfer T ran sfer

• Define new forces 1

𝑇 , 𝑀

𝑇 

transport matrix is symmetric

Horton 80, Itoh 82, Sugama 96

• Total field + particle momentum/ energy is conserved  extended thermodynamics Boozer 92,

Krommes 93, Watanabe 06, XG 12

• Sources = fluxes of momentum/

energy carried by waves Diamond

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Boxer 10 - LDX

Magnetic drift contributes to pinch velocities

• Resonance 𝜔 = 𝑘𝑣  𝜔 =

𝑘𝑣 + 𝒌 ⋅ 𝒗𝐷  introduces

magnetic drift  pinch velocities proportional to B/B.

• Related to Lagrangian invariants

→ compressibility vE≠0 Yankov 94, Isichenko 95&97,Baker 01, XG 04, Gürcan 10 𝑑 𝑑𝑡 𝑁 𝐵2

=0

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0 0.5 1 1.5 2 2.5 3 0 0.2 0.4 0.6 0.8 1 1.2 Normalized radius n e ( x 10 19 m -3 ) Particle source (a.u)

Hoang 04 - Tore Supra

0 • Particle flux Γ = −𝐷 𝑑𝑁

𝑑𝑥 + 𝑉𝑁

• Pinch velocity XG 04, Angioni 04 & 06, Camenen 09

• Onsager symmetry  thermal

pinch Luce 92, Itoh 96, Mantica 05,

Lu Wang 11

Turbulent pinch theory

successfully tested in tokamaks

X. Garbet, AAPPS-DPP, Hefei, Nov. 5 2019

Thermo diffusion compression Roto diffusion 𝑉 = 𝑉𝑇 𝑑𝑇 𝑇𝑑𝑥 + 𝑉𝑀∥ 𝑑𝑀 𝑑𝑥 + 𝑉𝐵 𝑑𝐵 𝐵𝑑𝑥

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• Particle flux Γ = −𝐷 𝑑𝑁

𝑑𝑥 + 𝑉𝑁

• Pinch velocity XG 04, Angioni 04 & 06, Camenen 09

• Onsager symmetry  thermal

pinch Luce 92, Itoh 96, Mantica 05,

Lu Wang 11

Turbulent pinch theory

successfully tested in tokamaks

Thermo diffusion compression Roto diffusion 𝑉 = 𝑉𝑇 𝑑𝑇 𝑇𝑑𝑥 + 𝑉𝑀∥ 𝑑𝑀 𝑑𝑥 + 𝑉𝐵 𝑑𝐵 𝐵𝑑𝑥 Luce 95 – DIII-D

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Momentum flux has both pinch

and residual components

• Plasma spin-up in tokamaks without external torque.

• Pinch and residual stress Γ = −𝐷 𝑑𝑀∥

𝑑𝑥 + 𝑉𝑀∥ + Γ𝑟𝑒𝑠

OVs Diamond 09, Peeters 11, Ida 13, Tynan 19 CD-I9 entropy production Kosuga 10

X. Garbet, AAPPS-DPP, Hefei, Nov. 5 2019

Solomon 09 - DIII-D Experimental torque Model with res. stress

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Thermodynamics of non local transport

is an open issue

• Non local transport Van Milligen 04, Del-Castillo-Negrete 05,

Dif-Pradalier 10, OV Ida 15

Γ x = − න 𝑑𝑥′𝜅 𝑥 − 𝑥′ 𝛻N (x′)

• Pinch effect with single force

Del-Castillo-Negrete 05, Bouzat 05. • Second principle  Tsallis

entropy Tsallis 88, Anderson 18

Onsager symmetry ? Van Milligen 04 Source Source Density x =0

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• Fast relaxation of a system with

long range interactions. Entropy

Lynden-Bell 67

• Maximum entropy with

conservation constraints 

Quasi-Stationary States Robert 92, Antoniazzi

08, Chavanis 06, Carlevaro 13

• Relaxation? Maximum entropy production principle? Martyushev 06

Violent relaxation theory predict coherent states

X. Garbet, AAPPS-DPP, Hefei, Nov. 5 2019

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Conclusions

• Minimum of entropy production principle coupled to

quasi-linear theory predicts fluxes vs forces in

turbulent magnetised plasmas.

• Predicts pinches, residual contributions to energy and momentum fluxes, turbulent heating and

acceleration.

• Onsager symmetry respected under conditions.

• May fail, typically in systems with long range

interactions, with memory effects. Tsallis and Lynden

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