AEA-CN-116/TH1-4 Gyrokinetic Studies of Turbulence Spreading T.S. Hahm, Z. Lin, a P.H. Diamond, b G. Rewoldt, W.X. Wang, S. Ethier, O. Gurcan, b W. Lee, and W.M. Tang Princeton University, Plasma Physics Laboratory, USA a Univ. of California, rvine, USA b Univ. of California, San Diego, USA Nov. 3, 24, Vilamoura, Portugal UCrvine
Outline and Conclusions Turbulence spreading into linearly stable zone is studied using global gyrokinetic particle simulations and theory. Motivation from experiments to study spreading of Edge Turbulence into Core. Results Fluctuation amplitude in the linearly stable zone can be significant due to turbulence spreading. Sometimes Spreading of Edge Turbulence into Core can exceed local turbulence in connection region. t is likely to affect the edge boundary conditions used in core modeling, and predictions of pedestal extent.
Determination of Fluctuation Amplitude γ = γ lin k 2 D turb Nonlinear coupling induced dissipation leads to saturation (B. Kadomtsev 65) γ amplitude (log scale) e φ /Te amplitude Local Balance in Space for a mode k Conceptual Foundation of Most Transport Models Missing: Meso-scale Phenomena: Barrier Dynamics, Avalanches,... Anomalous transport in the region γ lin < Turbulence Spreading into Less Unstable Zone t
Excitation of Linearly Damped Modes Nonlinear Saturation from Balance between: γ lin vs. Spectral Transfer from Nonlinear Mode Coupling e φ /Te n nonlinear/1 linear k θ ρ e Non-zero Amplitude for Linearly Damped Modes Sagdeev and Galeev, Nonlinear Plasma Theory (1969) Gang-Diamond-Rosenbluth, Phys. Fluids B 3, 68 (1991) Hahm-Tang, Phys. Fluids B 3, 989 (1991) Horton, Rev. Mod. Phys. (2) for more references Lin et al., AEA/TH/8-4 (24), this Friday
Nonlinear Coupling Leads To Radial Diffusion Nonlinear interactions of modes must spread fluctuation energy in radius due to: i) ik x x ii) poloidal harmonics at q(r) = m/n iii) with different radial extents iv) Numerical Studies with both Linear Toroidal Coupling and Nonlinear Coupling [Garbet-Laurent-Samain-Chinardet, NF 1994] E B nonlinearity local turbulent damping and radial diffusion : (k k b) 2 R k,k k k x D r() x + k2 θ D θ(). [eg., Kim-Diamond-Malkov-Hahm et al., NF 23]
Simple Model of Turbulence Spreading [Hahm, Diamond, Lin, toh, toh, PPCF 46, A323 4] 2 = γ(x) α + χ ( ) t x x γ(x) is local growth rate, α: a local nonlinear coupling χ = χi is a turbulent diffusivity : turbulence intensity, Σk Modes Σ Eddys t1 t2 + - γ lin x+ dx (x, t) χ ]x+ +... x x t x Profile of Fluctuation ntensity crucial to its Spatio-temporal Evolution
Turbulence Spreading after Local Saturation 4 3 2 1 r/a..2.4.6.8 unstable From Gyrokinetic (GTC) simulations, turbulence spreads radially ( 25ρi) into the linearly stable zone, causing deviation from GyroBohm scaling. γ(x) [Lin et al., Phys. Rev. Lett. (22)] -x x Local excitation rate γ(x) as a function of radius x 1.
Propagation and Saturation of Fluctuation Front The nonlinear diffusion, in the absence of dissipation, will make the front propagate beyond x indefinitely. (x,t) t = t > -x o x x Front propagation stops when radial flux due to propagation is balanced by dissipation: T prop /U x T damp ( γ ) 1 (x,t) growth due to spreading o x x + Δ dissipation x 2 12χ γ x, using the values from simulation 18ρ i From GK simulation for a profile considered: 25ρ i
Connection Region between Edge and Core TFTR Fonck, Mazzucato, et al. Vermare, et al., 24 Profile of Turbulence ntensity crucial in turbulence spreading: Γ = χ() x Core confinement improvement after L-H transition: JET, ASDEX, D-D, C-mod,... Connection Region: Local Turbulence + ncoming Edge Turbulence
Turbulence Spreading from Edge to Stable Core Nonlinear GTC Simulations of on Temperature Gradient Turbulence: R L T =5.3 atcore (within Dimits shift regime) R L T =1.6 at edge: nitial Growth at Edge Penetration into stable Core (Lin-Hahm-Diamond, PRL 2, PPCF, PoP 4) Saturation Level at Core: eδφ T e 3.6 ρ i a Γ >> γ local 2 15 1 5 time(l T /c s )= 1 (a) 2..2.4.6.8 1. 15 1 5 time(l T /c s )= 2 (b) 2..2.4.6.8 1. 15 1 5 time(l T /c s )= 3 (c) 2..2.4.6.8 1. 15 1 5 time(l T /c s )= 4 (d)..2.4 r/a.6.8 1. 1.6 5.3 R/L Ti
Spreading in Unstable Zone [Gurcan, Diamond, Hahm, and Lin, Submitted to Phys. Plasmas 4] Turbulence spreading with constant γ and γ d 1 t= t=5 t=1 t=15 t = γ(x) α2 + χ x ( x ) 1 1 When γ(x), α, χ are constant in radius, the Fisher- Kolmogorov equation with nonlinear diffusion exhibits propagating front solutions. The spreading can beat local growth and a solution exhibits ballistic propagation d(t) =U x t with U x = γ 1/2 ( χ 2 )1/2 U x geometric mean of local growth and turbulent diffusion, faster than transport time scale. v=1/2
Edge Turbulence Spreading to Unstable Core Nonlinear Gyrokinetic Simulations of on Temperature Gradient Turbulence: R L T =6.9 at core (Cyclone value) R L T =13.8 at edge nitial Growth at Edge followed by Ballistic Front Propagation into Core Saturation Level at Core 2 Core (only) Result Γ γ local 25 2 15 1 5 time(l T /c s )= 1 (a) 25..2.4.6.8 1. 2 15 1 5 time(l T /c s )= 15 (b) 25..2.4.6.8 1. 2 15 1 5 time(l T /c s )= 2 (c) 25..2.4.6.8 1. 2 15 1 5 time(l T /c s )= 25 (d)..2.4 r/a.6.8 1. 13.8 6.9 R/L Ti
Front Propagation Speed ncreases with R/L T U x (in (ρ/r)v Ti ) 5 4 3 2 8 6 4 (r/a=.4) (in (ρ/a) 2 ) 2 1 6.1 6.9 9. 2 4 6 8 1 R/L T (core) From Simulation, U x and increase with ( L R ) T Nonlinear Diffusion Model: U x (γ) 1/2 by [Gurcan-Diamond-Hahm-Lin, submitted to PoP 4] 3 x 1 3 Numerical Solution of the Nonlinear Theoretical Model Equation Turbulence ntensity 2 1 Region A Region B v B v A.1.2.3.4.5.6.7.8.9 1 x/a D =2.61*.8*a*v T Weak Turbulence Case Toroidal Linear Coupling dominant Regime: U x ρ i R v Ti by [Garbet-Laurent-Samain-Chinardet, NF 94] Four Wave Model: Complex Bursty Spreading by [Zonca-White-Chen, PoP 4]
Summary Turbulence spreading has been widely observed in global gyrokinetic particle simulations: t can be responsible for deviation of transport scaling from GyroBohm. Fluctuation ntensity in the linearly stable region can be significant due to turbulence spreading. Sometimes Spreading of Edge Turbulence into Core can exceed local turbulence in connection region. t is likely to affect the edge boundary conditions used in core modeling, and predictions of pedestal extent. UCrvine
Turbulence Spreading has been widely observed From Most Global Gyrokinetic/Gyrofluid Simulations: X. Garbet et al., NF 94 (Mode-coupling in Torus) R. Sydora et al., PPCF 96 (Torus with Zonal Flows) Y. Kishimoto et al., PoP 96 (Torus with Zonal Flows) S. Parker et al., PoP 96 (Torus without Zonal Flows) W.W. Lee et al., PoP 97 (Torus without Zonal Flows) Y. domura et al., PoP (Sheared Slab with Zonal Flows) Z. Lin et al., PRL 2 (Torus with Zonal Flows) L. Villard et al., AEA 2 (Cylinder with Zonal Flows) R. Waltz et al., PoP 2 (Torus with Zonal Flows) Y. Kishimoto et al., H-mode 3 (Sheared Slab with ZF) Neither Zonal Flows nor Toroidal Coupling necessary for Turbulence Spreading.
Anomalous Transport where γ lin < Core of Reversed Shear Plasmas where profiles are nearly [Rewoldt et al., NF, ʻ2] JT-6U E29728, t = 6. s flat (JT-6U, TFTR, D-D,...) 3 5 γ (1 sec 1 ) 2.5 2 1.5 1.5 FULL code (with rotation) k =.53 θ ρ i 1x1 5 total pressure (pascals) 1 χ e χ i Nonlinearly Unstable? (Self-sustained Turbulence (B. Scott)) χ e,i (m 2 / s) 1.1.2.4.6.8 1 Spreading from the Linearly Unstable Zone r / a
Distinction between Core and Edge blurred Researchers have frequently divided the tokamak into three zones a central sawtoothing zone, a middle confinement zone, and an edge zone... Goldston-U.S.A. Kyoto AEA (1986) the edge..., often used as a boundary condition for core transport modeling V. Parail, Plasma Phys. Control. Fusion, 44, A63 (22) x 2 γ(x) x2p : large at the top of pedestal C-mod, Hughes et al., 22
Long Term Behavior: Sub-Diffusion Self-similar Variable: l(t) 2 χ β t (t)l(t) = ()l() ɛ, up to dissipation l(t) [χ ɛ β t] 1 2+β t 1/3 : Weak Turbulence t 2/5 : Strong Turbulence Previous numerical mode coupling study: X. Garbet et al., NF 1994 Linear toroidal coupling usually dominates t 1 : convective Without linear toroidal mode coupling t 1/2 : diffusive
Short Term Behavior: Ballistic Propagation x front =(x 3 +6ɛχ t) 1/3 U x = d dt x front 2ɛχ /x 2 : for small t (consequence of << x ) t 2/3 : for large t (sub-diffusion) Note: ɛ, turbulence intensity Scaling of U x drastically different from V gr of linear drift (TG) wave contrast our theory from others relying on linear dispersion [eg., Garbet et al., PoP 96; Zonca et al., PoP 4]
Simple theory captures ρ dependence of spreading (x, t) = ɛ (6ɛχ t + W 3 ) 1/3 ( 1 (x x i ) 2 (6ɛχ t + W 3 ) 2/3 H ( (6ɛχ t + W 3 ) 1/3 x x i ) ) 4 3.2 (x,t) t = 3 a=5ρ i χ i 2.4 t > 2 a=25ρ i 1.6 x i -W x i x i +W x 1 a=125ρ.8 i r/a..2.4.6.8 1.
Spreading of Self-sustained Turbulence [toh, toh, Hahm, and Diamond, submitted to J. Phys. Soc. Jpn. 4] t =Γ NL(,x) + χ x ( x ) Model self-sustained sub-critical turbulence [eg., B. Scott, PRL 9]: Γ NL (,x) > for crit << γ α, Γ NL (,x)=for< crit, > γ α,at x <L,and Γ NL (,x) < at x >Laccording to local linear and nonlinear damping Due to turbulence spreading, there exists a minimum size system (L) that can sustain the self-sustained turbulence