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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�
Entropy and
relaxation processes
Xavier Garbet
IRFM
CEA Cadarache
Acknowledgements: C. Bourdelle, P.H. Diamond, G. Dif-Pradalier, P. Ghendrih, A. Samain, Y. Sarazin
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
Entropy production and Onsager symmetry
• Entropy production 𝑆ሶ vs “forces”
and “fluxes”
ሶ
𝑆 = 𝚲 ⋅ 𝚪
• Linear constitutive relationships
𝚪 = ധ𝑳 ⋅ 𝚲
• ധ𝑳 symmetric B-B Onsager 1931
• Compact form
ሶ
𝑆 = 𝚲 ⋅ ധ𝑳 ⋅ 𝚲
minimum of entropy production Prigogine 47
T =-T x Λ = 𝛻 1 𝑇 ሶ𝑆 = න 𝑑𝑥 𝜅 𝛻𝑇 𝑇 2
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)
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
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 − 𝑘∥𝑣∥ 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
• 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
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 𝑘∥𝑘𝑦 ≠ 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
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
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
Boxer 10 - LDX
Magnetic drift contributes to pinch velocities
• Resonance 𝜔 = 𝑘∥𝑣∥ 𝜔 =𝑘∥𝑣∥ + 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
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
• 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
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
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
• 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
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