Simula'ng Plasma Turbulence D. R. Hatch ICTP Oct 2018 1 - PowerPoint PPT Presentation

Simula'ng Plasma Turbulence D. R. Hatch ICTP Oct 2018 1

Turbulence 2

Turbulence: Enduring Fascina'on and Challenge Da Vinci `Turbulence is the most important unsolved problem of classical physics.' - Richard Feynman - 3

Turbulence: Enduring Fascina'on and Challenge ‘When I die and go to Heaven there are two maOers on Da Vinci which I hope for enlightenment. One is quantum electrodynamics and the other is the turbulent mo'on of fluids. And about the former I am really rather op'mis'c.’ -Horace Lamb- 4

Turbulence: Enduring Fascina'on and Challenge Millenium Prize: Existence and Smoothness of Da Vinci Navier Stokes ‘This is the equa'on which governs the flow of fluids such as water and air. However, there is no proof for the most basic ques'ons one can ask: do solu'ons exist, and are they unique? Why ask for a proof? Because a proof gives not only cer'tude, but also understanding.’ 5

Turbulence: Why is it Important? Dye in Pipe Flow How long can rolling waters remain impure? • Turbulence makes things happen: • Incredibly effec've at mixing / Re+ transpor'ng par'cles, momentum, heat, etc. • Much more effec've than laminar flow or molecular diffusion (typically factor of ~Re faster—i.e. 10 4 -10 7 !) 6

Fundamental Turbulence Paradigm Kolmogorov Turbulence: ε k ∝ k − 5/3 Energy 1. Injec'on 2. Redistribu'on 3. Dissipa'on Large scales Small scales 7

Moments of Distribu'on Func'on . . 8

How To Solve for Distribu'on Func'on + Maxwell’s Equa'ons 9

Kine'c Theory = Infinite Hierarchy of Moment Equa'ons 10

Kine'c Theory = Infinite Hierarchy of Moment Equa'ons 11

Kine'c Theory = Infinite Hierarchy of Moment Equa'ons 12

Navier Stokes = Limi'ng Case of Plasma Fluid Equa'ons q è 0, incompressibility, etc. 13

Fluid Dynamics Described by Single Fluid Equa'on 14

Plasma: Challenge and Opportunity 15

Many Non-Fusion Applica'ons Founda'on for space and astrophysical turbulence • – G. G. Howes et al. ApJ , (2006). – A. A. Schekochihin et al. ApJS , (2009). Solar wind turbulence • – G. G. Howes, et al. Phys. Rev. Le/. , (2011). – J. M. TenBarge et al. Physics of Plasmas , (2012). – D. Told et al. Phys. Rev. Le/. (2015). Magne'c reconnec'on • – J. M. TenBarge, et al. Physics of Plasmas ( 2014). – M. J. Pueschel,, et al. ApJS , (2014). Fundamental turbulence • – Tatsuno et al. Phys. Rev. Le/. (2009) – Banon-Navarro et al. Phys. Rev. Le/. (2011) – Teaca et al. Phys. Rev. Le/. (2012) – Hatch et al. Phys. Rev. Le/. (2011,2013) Codes • – AstroGK (based on fusion code GS2) – GENE – Others 16

Compare / Contrast with Fluid Turbulence 17

What Drives Turbulence in a Tokamak? • Several drive mechanisms for fluid turbulence • Kelvin-Helmholtz (shear flow) • Rayleigh Taylor (density gradient) • There exist a whole zoo of instabili'es (ITG, ETG, TEM, MTM, KBM, etc.), each with both instability and wave- like proper'es • (Jonathan Citrin will discuss further in later lecture) • Driven by extreme gradients in fusion plasmas (usually gradients in temperature, density) • I will briefly introduce the ion temperature gradient (ITG) instability • Perhaps the most important instability for tokamak transport 18

Ion Temperature Gradient Instability Ion Temperature Gradient Outboard Side of Torus Instability (ITG) 1 2 m v + B B j ⊥ × ∇ ↑ 2 v = ↓ d 2 q B B − j v E = E × B B 2 Mike Beer - Thesis 19

Ion Temperature Gradient Instability Ion Temperature Gradient Outboard Side of Torus Instability (ITG) + + + + + + + + + - - - - - - - - - - - + + + + + + + + + 1 2 m v B B + j × ∇ 2 ⊥ ↑ v = ↓ d 2 q B B - - - - - - - - - - - − j v E = E × B B 2 20

Ion Temperature Gradient Instability Ion Temperature Gradient Outboard Side of Torus Instability (ITG) + + + + + + + + + - - - - - - - - - - - 1 2 m v B B j + + + + + + + + + ⊥ × ∇ 2 v = + ↑ d 2 q B B j ↓ − - - - - - - - - - - - v E = E × B B 2 21

Fundamental Turbulence Paradigm Kolmogorov Turbulence: ε k ∝ k − 5/3 Energy 1. Injec'on 2. Redistribu'on 3. Dissipa'on Large scales Small scales 22

Fluid Turbulence - Saturation Hydrodynamic turbulence: Navier Stokes equation è Kolmogorov picture: Drive Dissipation Inertial Range k 1. Energy drive (stresses) at large scales. 2. Conservative nonlinear energy transfer through inertial range of scales. 3. Dissipation at small scales. Saturation è Energy drive at large scales balances with dissipation at small scales. 23

Dissipation mechanisms 1 ∇ 2 u Dissipa'on in fluid turbulence: Re Re=large è small scale dissipa'on 2 Dissipa'on in kine'c plasma turbulence: f ∝ υ ∇ v =small è small scale dissipa'on in velocity space υ 24

Dissipation in gyrokinetic ITG turbulence Q=gradient drive C=collisional dissipa'on Contrast: Drive Dissipation Inertial Range 25 k

Dissipation in gyrokinetic ITG turbulence Q=gradient drive C=collisional dissipa'on Small scales develop in velocity space even at drive scales in real space. è Scale range of drive and dissipaHon overlap! D. R. Hatch et al. PRL, 2011. (Also a cascade at higher k Banon Navarro, PRL, 2011.) 26

Model—`Reduced Gyrokine'cs’ Hermite representa'on: Equa'ons: ⇤ f n H n ( v ) e � v 2 ⌦ ˆ f ( v ) = ∂ ˆ k 2 ∂ t = η i ik y f n φδ n, 0 � ik y φδ n, 0 � η i ik y ¯ ¯ ¯ ⌅ ⇧ φδ n, 2 n =0 1 1 1 2 2 π π π 4 4 4 ⇧ ⇤ ⇧ n ˆ � ik z ⌅ ¯ n + 1 ˆ φδ n, 1 � ik z f n � 1 + f n +1 1 π 4 ⇥ ¯ � ν n ˆ ⌦ φ k � ˆ k ⇥ x k y � k x k ⇥ � f n + f n, k � k � . y k � Simple rela'ons between moments and Hermites DNA code: Reduced GK model: φ = D ( k ⊥ ) ˆ f 0 As simple as possible while s'll capturing dynamics of interest. u = π 1 / 4 ˆ f 1 Hatch et al PRL ’13 √ 2 Hatch et al JPP ’14 p = π 1 / 4 f 2 + π 1 / 4 Hatch et al NJP ‘16 ˆ ˆ f 0 √ 2 2 27

Injec'on, Redistribu'on, and Dissipa'on of Energy in Phase Space Gyrokine'c energy in Hermite space: k ,n = 1 ε ( f ) 2 π 1 / 2 | ˆ f k ,n | 2 Energy evolu'on equa'on: ∂ε ( f ) k ,n = η i Q k δ n, 2 − C k ,n − J ( φ ) n, 2 � ν n ε ( f ) k ,n − ν ⊥ ( k x , y / k max ) 8 k δ n, 1 − ν ⊥ ( k x , y / k max ) 8 ∂ t + J k ,n − 1 / 2 − J k ,n +1 / 2 + N ( f ) k ,n , 28

Descrip'on of Hermite Energy Spectra √ n ε n = − ν n ε n − αε n ∂ ε = c 0 n − 1 − α e − n/n c ∂ n ⟨ k z ⟩ n Vary collisionality: Varying driving gradients and collisionality constant power law, changing n c n c = k z n ν n 1/2 Compare Schekochihin et al. JPP 2016 29