Gyrokine)c Phase space Turbulence and Energy Flows

Transcription

1 Gyrokine)c Phase space Turbulence and Energy Flows Review, Progress and Open Problems Gabriel G. Plunk 2 August, 2010, Newton Ins)tute Tomo Tatsuno, M. Barnes, S. Cowley, W. Dorland, G. Howes, R. Numata, A. Schekochihin

2 Energy flow in mul)scale gyrokine)cs Mul)scale gyrokine)cs: Sugama, et al., (1996) Scale separa)on Gyrokine)cs as a theory of turbulence (and transport)

3 Energy flow in mul)scale gyrokine)cs Mul)scale gyrokine)cs: Sugama, et al., (1996) Scale separa)on Turbulent fluctua)ons Instability drives: Gyrokine)cs as a theory of turbulence (and transport)

4 Energy flow in mul)scale gyrokine)cs Mul)scale gyrokine)cs: Sugama, et al., (1996) Scale separa)on Turbulent fluctua)ons Energies of the fluctua)ons: Instability drives: injec)on Gyrokine)cs as a theory of turbulence (and transport)

5 Energy flow in mul)scale gyrokine)cs Mul)scale gyrokine)cs: Sugama, et al., (1996) Scale separa)on Turbulent fluctua)ons Energies of the fluctua)ons: Instability drives: injec)on The importance of W ( Free Energy or Entropy): Krommes and Hu, (1994) Actually, E is part the exact GK invariant: Dubin and Krommes (1983) Gyrokine)cs as a theory of turbulence (and transport)

6 Why be interested? Simula)ons Ques)ons: Resolu)on: When is enough enough? Are there basic (nonlinear) problems on which all codes can agree? Fusion Physics: What are the kine)c nonlinear mechanisms which damp turbulence? How can small scales and large scales interact? Under what condi)ons are interac)ons local or nonlocal? When and how does a cascade go inverse (i.e. zonal flows) Reduced fluid models: How can we create simple physical models which are robust across parameter ranges, and dras)cally improve computa)onal efficiency? Gyrofluid Models Sub grid scale models, large eddy simula)ons

7 Outline Phase mixing: Drive, Damping or Transfer? 2D Gyrokine)cs: Navier Stokes eqn for magne)zed plasma turbulence Adver)sement: Sub Larmor cascade Cascade through phase space Fjortoc and the new flavors of the dual cascade Linear + nonlinear phase mixing: a step forward? Free energy balance in a torus: preliminary results from Brussels

8 Linear phase mixing in gyrokine)cs Parallel phase mixing and linear Landau damping Perpendicular phase mixing and interchange instability f(v ) v z f(v ) v z

9 Physics of nonlinear phase mixing

10 2D Gyrokine)cs: A minimal model of magne)zed plasma turbulence Gen. Free Energy : Hankel & Fourier Transform:

11 Sub Larmor free energy cascade

12 Sub Larmor free energy cascade

13 Nonlinear Free Energy Transfer: Very local 100 K shell transfer ()" P shell transfer ("#""$ K Q Pv *+ (&" ('" () (& (' (' (& () ('" (&" ()" Sv *+ ("#""% ("#""& ("#""' ("!"#""'!"#""&!"#""%!"#""$ Tatsuno, et al., J. Plasma Fusion Res. SERIES (2010); arxiv:

14 Dual cascade in phase space Free Energy : Electrosta)c Energy : Constraint:

15 Inverse cascade of E

16 Flavors of dual cascade: Local forward, Local inverse

17 Flavors of dual cascade: Local forward, Nonlocal inverse

18 Flavors of dual cascade: Dual forward Sub Larmor damping

19 Nonlinear + linear phase mixing? Parallel phase mixing can be phenomenologically treated with a cri)cal balance assump)on it is probably weak! Difficult to inves)gate numerically because it requires an addi)onal dimension Idea: Include perpendicular phase mixing via magne)c dric:

20 Coexistence of fast weak and slowstrong phase mixing fast wave )mescale : slow turbulent )mescale :

21 Phenomenological Cascade Scaling Effec)ve poten)al depends on NL turnover )me The usual iner)al range assump)on of constant nonlinear flux gives: E ϕ k!!"#$ k! % 1 k!

22 Preliminary Results (Tomo Tatsuno)

23 Free Energy Cascade in ITG turbulence A. Bañon Navarro, et. al ITG Cyclone base case 32 8 grid points in v μ spaial pts Hyperviscosity in z and v Injection rate Dissipation rate!%#!%"!$#!$"!#!"!!"!#!$"!$#!%"!%#!#"!'"!("!%"!$"!" &$" &%" &(" &'" &#" (a) ln(k) Injec)on occurs at kρ i < 1 Self similar from kρ i ~ 1 to kρ i ~ 4 Local and directed forward (larger k) Sta)onary: dw/dt = 0

24 Electrosta)c Energy in ITG turbulence A. Bañon Navarro, et. al ITG Cyclone base case 32 8 grid points in v μ spaial pts Hyperviscosity in z and v #, #+ #* #) #( #' #& #% #" #$!" Injection rate Dissipation rate #$ #( #"$ #"( #%$ #%( ln(k) Electrosta)c Energy cascade is weak Not yet reached steady state Zero dissipa)on (looks 2D!) Mixture of local, nonlocal, inverse and forward Energy re circula)ng in k space

25 Highlights Gyrokine)c turbulence fits the fluid turbulence mold In the absence of injec)on, damping, or dissipa)on, nonlinear interac)ons conserve two posi)ve definite quan))es A local cascade transfers free energy from large scales to small scales to be dissipated The nonlinear cascade of these quan))es is constrained in the sense of 2D fluids and the dual cascade can take on different forms depending on the injec)on and damping.

26 Open Ques)ons Mixture of linear and nonlinear phase mixing What sets k? Mixture of weak and strong processes? Physical regimes for different inverse cascade behavior What determines injec)on rates of W/E Energy flows at the injec)on scale (zonal flows, fluid equa)ons, phase mixing channels)

27 Papers [A. A. Schekochihin, S. C. Cowley, W. Dorland, G. W. Hammex, G. G. Howes, G. G. Plunk, E. Quataert, and T. Tatsuno,.Plasma Phys. Control. Fusion, 50:124024, (2008) arxiv:] [T. Tatsuno, W. Dorland, A. A. Schekochihin, G. G. Plunk, M. Barnes, S. C. Cowley, and G. G. Howes, Phys. Rev. Lex. 103, (2009) arxiv: ] [T. Tatsuno, M. Barnes, S. C. Cowley, W. Dorland, G. G. Howes, R. Numata, G. Plunk and A. A. Schekochihin, accpeted J. Plasma Fusion Res. (2010), arxiv: ] [G. G. Plunk, S. C. Cowley, A. A. Schekochihin, T. Tatsuno, accepted JFM (2010) arxiv: ] [G. G. Plunk, T. Tatsuno, submixed POP (2010) arxiv: ]

28 Gyrokine)c Equa)on