Gyrokinetic Transport Driven by Energetic Particle Modes

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1 Gyrokinetic Transport Driven by Energetic Particle Modes by Eric Bass (General Atomics) Collaborators: Ron Waltz, Ming Chu GSEP Workshop General Atomics August 10, 2009

2 Outline I. Background Alfvén (TAE/EPM) modes are destabilized by an energetic particle (EP) radial density gradient. Increased EP transport expected. EP destabilized modes in GYRO vs. Berk-Breizman model: different transport and saturation mechanisms, different dynamical limits GYRO code used to study transport enhancement caused by steady-state Alfvén microturbulence. II. Linear stability analysis Identification of multiple, coexisting, low-k θ drift-alfvén modes driven unstable by a sparse EP population. A new eigenvalue solver within GYRO maps these overlapping modes. III. Non-linear transport Saturated states with finite, low-k θ Alfvén drive are elusive, but can be found. Preliminary studies show transport enhancement in most species channels, especially in EPs

3 EPs can de-stabilize Alfvén turbulence. Multiple Alfvén eigenmodes, created or destabilized by an EP plasma component, are predicted by high-n ballooning mode theory. Toroidal Alfvén Eigenmode (TAE): MHD mode existing in the toroidally induced gap in the Alfvén continuum. De-stabilized by EPs. TAE gap: ω TAE 1 with 1+ 2 r R ω ω TAE 1 ω TAE = v A 2qR 1 2 r R Energetic Particle Mode (EPM): Exist in the Alfvén continuum when kinetic EP drive exceeds continuum damping. Don t exist without EPs. Both modes are fed by the free energy in a radial density gradient in the EPs. F. Zonca, L. Chen, PoP 3, 323 (1996) G. Y. Fu and C. Z. Cheng, Phys. Fluids B 2, (1990)

4 Two Alfvén-induced transport pictures have different mechanisms. Global, Berk-Breizman High-n microturbulence, GYRO ν ν d Wave-trapped particles are constrained by an adiabatic invariant to move perpendicular to the field as they slow. Transport behavior depends sensitively on velocity space diffusion regime: ω ω b Fig. 1 from Berk- Breizman, 1990 : slowing down rate : velocity diffusion rate ω ω b ν ν d 2 ω ω b : mode frequency : trapped particle bounce frequency Self-consistent local fluctuations induce particle and heat flux. Q = δv E B δe Γ = δv E B δn heat flux particle flux Strictly analogous to collisionless ITG/ TEM and ETG turbulent transport. Particle trapping by the wave is not invoked in this picture. Saturation amplitude is generally much lower than the point where wave particle trapping becomes significant. H. L. Berk and B. N. Breizman, Phys. Fluids B 2 (9), 1990

5 GYRO microturbulence saturates by mode-mode interaction. Berk-Breizman model One-mode saturation: Saturation occurs when finite-amplitude power transfer to the wave falls below the background plasma damping or when local profile gradient relaxes. Saturation amplitude varies with collisional regime. GYRO microturbulence Multi-mode saturation: Total mode growth rate (including damping) is determined self-consistently. Saturation is primarily from interaction with n=0 zonal flow. n EP =0.007n e, al nep -1 =4, all others GA standard

6 GYRO microturbulence assumes steady-state dynamics. In B-B model, radially overlapping modes create a conveyor belt to the edge. GYRO presumes a steady-state with finite instability drive. Radial gradient is fixed. nep nep with overlap no mode overlap r r A burst cycle develops as the driving radial gradient is reduced by wave transport and restored by the EP source. Berk, TTF modes, nep=0.007ne with alnep-1=4 and all others GA standard. Only long-time average of saturated state is physically relevant in this view.

7 The GYRO code GYRO is a versatile, parallel initial-value solver of the gyrokinetic equations (electrostatic or electromagnetic) in toroidal geometry. Tracks up to four kinetic species (electrons and three ions), each with a Maxwellian velocity distribution and independent temperature. Local (flux tube) or global simulations. Treats one toroidal number mode number n at a time (linear) or a spectrum of interacting n numbers (non-linear). Linear operation Growth rate, frequency, and eigenfunction of the leading mode. Linear diffusion for one toroidal n. Non-linear operation Saturated amplitude of linearly driven turbulence. Total non-linear diffusion and n-dependence. Well benchmarked against several linear, nonlinear, and electromagnetic flux tube gyrokinetic codes (e.g. GS2, GENE, GEM). J. Candy, R.E. Waltz, JCP (2003)

8 GYRO has been used to study fusion α transport by ITG turbulence. An equivalent Maxwellian F M (v) at temperature T is defined for a given slowing-down distribution F S (v) by equating pressure. v 2 F M ( v)v 2 dv = v 2 F S ( v)v 2 dv T = 2I 1 4 E α I n dx 0 3I 2 v c /v α x n ( ) 3 + x 3 Electrostatic simulations give transport of α particles by ITG turbulence. GA standard case with: n α /n e = L Tα -1 = 0.5 L nα -1 = 5 Normalized energy and particle flux for α particles as a function of α temperature T α. C. Estrada-Mila, J. Candy, R.E. Waltz, PoP 13, (2006)

9 Maxwellian and slowing-down distributions give similar results. Slowing-down and Maxwellian distributions weighted by v 2 for T e = 15 kev. C. Estrada-Mila, J. Candy, R.E. Waltz, PoP 13, (2006) Diffusion coefficient for α particles in a slowingdown and equivalent Maxwellian distribution. C. Angioni, A.G. Peeters, PoP 15, (2008) Maxwellian EP distribution required by GYRO is qualitatively justified for studying interaction with spatially driven turbulence.

10 Parameters of the simulations Electromagnetic, flux-tube simulations to study destabilized Alfven turbulence Simulation requirements: Include a Maxwellian EP species with a sufficiently large density gradient to drive Alfvén turbulence. Background species gradients to drive the usual ITG-TEM turbulence. Physically relevant parameters that can be easily compared to previous simulations. Care must be taken to stay below the MHD critical β gradient. A GYRO subcritical β requires an even lower β in non-linear simulations. Choose a deuterium plasma (GA standard case) with sparse, hot α particles: β e = T i = T e al ne -1 = al ni -1 = 1 al Te -1 = al Ti -1 = 3 q = 2 s = 1 R = 3a r = 0.5a T EP =100T e al nep -1 = 4 al TEP -1 = n EP Energetic particle parameters Note: v EP c s = 7.07 << v A c s = 31.6

11 EPs drive instability at low k θ. Frequency ω and growth rate γ of leading mode for n EP = GA standard case with EP density gradient GA standard case with no EP gradient Ballooning space potential eigenfunction for k θ ρ s = k θ ρ EP = 50k θ ρ s =1 Drift-like frequency dependence on k θ. φ Usual ballooning mode representation: Φ(ψ,θ,ζ) = φ(ψ,θ 2πl,δ)e in ( ζ q(ψ )θ + θ k (ψ )dq) l= θ/π

12 New GYRO eigensolver reveals TAE and EPM existing side-by-side. ω, γ vs. k θ ρ s, n EP = ω, γ vs. n EP, k θ ρ s = 0.03 Drive increases with n EP for leading high-frequency modes. Frequency scaling changes from Alfvénlike to drift-like at cross point. Frequency and growth rate for unstable modes found by the GYRO eigensolver.

13 Low-k θ Alfvén drive dominates the turbulent spectrum as n EP increases Potential power spectrum for 16-mode nonlinear simulations TAE/EPM Low density spectrum is consistent with ITG/TEM turbulence. At higher n EP, low k θ drive kicks in. Low k θ, Alfvén component clearly dominates the turbulent spectrum at sufficient EP density. Zonal flows Greater mode density is required to fully resolve the low-k θ peak.

14 40 mode cases underway show less noise and better saturation. Saturation above n EP =0.007n e is still problematic. Runs with E B shear will hopefully push this boundary to higher density. 16 mode cases (6 cheaper) are still instructive and give qualitative physics.

15 EP density below the stability threshold does not affect background plasma transport. Energy flux per particle Q for all species vs. time at n EP = (16 modes). Density flux per particle Γ for all species vs. time at n EP = (16 modes). Q i /T i n i Region of average Γ ι /n i Γ e /n e Region of average Q e /T e n e Γ EP/n EP Q EP /T EP n EP n EP = : Q EP /T EP n EP = 0.93 Γ EP /n EP = 2.23 Q i /T i n i = 40.4 Γ i /n i = 0.84 Q e /T e n e = 21.2 Γ e /n e = 0.86 Results consistent with previously observed transport 1,2. 1 Ron Waltz, private communication 2 C. Estrada-Mila, J. Candy, R.E. Waltz, PoP 13, (2006)

16 Transport k θ ρ s dependence at n EP =0.005 Energy: Density: At n EP =0.005, electron and ion results are consistent with simulations where no EPs are present. EPs are a passive tracer.

17 Initial unsaturated time trace shows increased transport in most species channels with higher EP density. Energy flux per particle Q for all species vs. time at n EP = (16 modes). Density flux per particle Γ for all species vs. time at n EP = (16 modes). Region of average Γ EP /n EP Region of average Q e /T e n e Q i /T i n i Γ e /n e Γ ι /n i Q EP /T EP n EP n EP = : Q EP /T EP n EP = 4.45 Γ EP /n EP = 4.27 Q i /T i n i = 49.0 Γ i /n i = 0.65 Q e /T e n e = 24.1 Γ e /n e = 0.71 Oscillation at ωa/c s =2.2 corresponds to linearly unstable k θ ρ s =0.05 TAE.

18 Transport k θ ρ s dependence above threshold Energy: Density:

19 Summary Electromagnetic GYRO simulations have been run with an EP component. The TAE and EPM can be simultaneously destabilized at k θ ρ EP < 1 by the EPs. Linear flux tube studies with initial value and spectral solvers show multiple such modes coexisting with large growth rates at modest EP density. When n EP is below the linear stability threshold, transport is the same as in finite β simulations without EPs. At higher n EP, destabilized, low-k θ Alfvén modes enhance transport across most channels. Alfvén drive rises rapidly and saturated states can be elusive. Preliminary results show TAE/EPM turbulence, like ITG/TEM, can achieve a finite-drive steady state. Additional physics, such as E B drift, will be required to get well saturated states above n EP 0.007n e.

20 Unanswered Questions What role do velocity space instabilities play in transport? An inverted EP distribution can excite n=0 EGAMS. Will EGAMS suppress finite k θ turbulence and reduce transport? How much does the transport picture change in a global simulation? Certain Alfvén eigenmodes such as RSAEs are absent in the flux tube model. Do high-n approximations made in GYRO accurately describe very long wavelength Alfvén turbulence? Does large TAE/EPM incremental transport create a soft limit on the EP profile gradient?

21 Medium density n EP =0.007 is an intermittent case. Energy flux per particle Q for all species vs. time at n EP = Density flux per particle Γ for all species vs. time at n EP = Q i /T i n i Region of average Γ e /n e Γ ι /n i Region of average Γ EP /n EP Q e /T e n e Q EP /T EP n EP n EP = : Q EP /T EP n EP = 1.24 Γ EP /n EP = 2.45 Q i /T i n i = 42.8 Γ i /n i = 0.35 Q e /T e n e = 20.8 Γ e /n e =

22 Transport k θ ρ s dependence at n EP =0.007 Energy: Density:

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