Nonlinear dynamics and transport processes...10th West Lake International Symposium (WLIS) on...

10th West Lake International Symposium (WLIS) on Magnetic Fusion 1

Nonlinear dynamics and transport processesin the beam-plasma system∗

Fulvio Zonca

http://www.afs.enea.it/zonca

ENEA C.R. Frascati, C.P. 65 - 00044 - Frascati, Italy.

Institute for Fusion Theory and Simulation, Zhejiang University, Hangzhou 310027, P.R.C.

May 10.th, 2016

∗Contributions from: Liu Chen [Rev. Mod. Phys. 88, 015008 (2016)]∗In collaboration with: N. Carlevaro, M. Falessi, A. Milovanov, G. Montani, D. Terzani

10th West Lake International Symposium (WLIS) on Magnetic Fusion,and 12th Asia Pacific Plasma Theory Conference (APTTC)

May 9 – 13 2016, IFTS – ZJU, Hangzhou, China

Fulvio Zonca


Outline

I. Introduction

– Historic background

– The Bump-on-Tail problem and magnetic fusion

– Gyrokinetic transport theory and time scales

II. Applications and numerical simulation results

– Broad spectrum: diffusion & convection

– Toy model for mixed diffusive–convective relaxation

– Narrow spectrum: particle trapping and trapping suppression

III. Summary and Discussion

Fulvio Zonca


Historic background

The physics of the e beam-plasma system, i.e. the relaxation of supra-thermal particle beams in plasmas is a problem of fundamental significance.

• 1960s: it provides a paradigm for the quasi-linear theory of weakplasma turbulence [Vedenov, Drummond], with applications from as-trophysics and geo-physics to fusion plasma

• Seminal work by Bernstein, Greene and Kruskal (BGK): the classic“bump-on-tail” (BoT) problem

• Applications to Landau damping and nonlinear behavior of the beam-plasma instability [O’Neil et al]

• The “perturbation theory of strong plasma turbulence”, proposed byDupree and known as the theory of resonance broadening

Fulvio Zonca


The BoT problem and magnetic fusion

BoT problem for fusion plasma research was revived in the 1990s by Berk,Breizman and coworkers⇒ paradigmatic model for nonlinear interaction of energetic particles (EP)with Alfvn fluctuations: Alfven eigenmodes (AEs), EP modes (EPM) anddrift Alfven waves (DAW)⇒ requires proximity to marginal stability: NL particle motion small w.r.t.mode width (local transport/uniform plasma)

Fluctuation spectrum of AEs, EPM and DAW⇒ disparate spatiotemporal scales

• “broad” features, typical of plasma turbulence

• almost “coherent” (“narrow”), nearly periodic component

• Need to go beyond the local description of fluctuation-induced fluxes,extending the diffusive transport paradigm and accounting for modesof the linear stable spectrum

Fulvio Zonca


Gyrokinetic transport theory and time scales

Fluctuation induced transport in fusion plasmas is due to low frequencyfluctuations (|ω| ≪ |Ω|) ⇒ gyrokinetic transport theory.

Particle dynamics independent of the gyrophase ⇒ reduced phase-spacedescription in terms of an invariant of motion: magnetic moment µ.Gyrokinetic transport theory deals with transport in non-uniform, non-autonomous system with 2 degrees of freedom.

Extended phase space still reflects a system with 3 degrees of freedom:⇒ regular trajectories no longer separate stochastic regions ⇒ Arnold web.

Arnold diffusion may affect the system behavior on long time scales in fusionplasmas.

Additional (nonlinear, adiabatic) constant of motion: reduced description ofnonlinear dynamics may describe short time scale only. Long scale behaviorrequire full accounting of realistic plasma non-uniformity and equilibriumgeometry.

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Illustration of processes of fast dif-fusion across a resonance layer andslow diffusion along the resonancelayer (Lichtenberg and Lieberman,2010).

Reduced dynamic description of a time dependent non-uniform plasma withone degree of freedom in the corresponding reduced phase space

• identification of additional (nonlinear) invariant of motion|ω| ∼ |nωdk| ≪ ωb ⇒ J = const ⇒ fishbone paradigm⇒ neglect finite Larmor and magnetic drift orbit width

The system (e.g., fishbone paradigm) can be further reduced to the descrip-tion of a time dependent uniform plasma with one degree of freedom (e.g.,BoT paradigm) when the nonlinear particle displacement is small comparedto the mode characteristic width [Berk & Breizman 1990].

Fulvio Zonca


Mapping r ↔ v for each (µ, J) = const pair [C&Z NJP 2015, RMP 2016]

−m

e

nc

dψ/dr

∂

∂r↔ k0

∂

∂v, nωdk − ωk ≃ nωdk0

(r − r0)

Ldk0

↔ k0v − ωp .

Use the freedom of choice of beamdensity to preserve overall (inte-gral) wave-particle power exchangeat each (µ, J) = const, keeping each(µ, J) = const pair independent forreconstruction of the relaxed distri-bution function in higher dimension-ality ⇒ finite temporal scale for va-lidity of reduced dynamic descrip-tion.

Fulvio Zonca


Three fundamental time scales enter the theoretical analysis of the BoTparadigm:

• time scale for a particle to diffuse over ∼ k−10 : (Dvk

20/3)

−1/3≡ τnℓ

• autocorrelation time of fluctuations: |k0∆v|−1 ≡ τs, necessary for rel-

ative phase of two waves to randomize due to different phase velocities

• the relaxation time of the particle distribution function: ≡ τNL

Validity limit of quasi-linear approach [Vedenov, Drummond]:

τs ≪ τnℓ ≪ τNL

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The Chirikov parameter is introduced to quantify resonance overlap[Chirikov 1959]

S =∆ωr

∆ωsep

≃ωB

∆ωsep

> 1; (S < 1 isolated resonances)

The Q parameter controls fraction of trapped orbit completed within τs:⇒ Controls coherence of particle response

Q = (2πS)1/3τnℓτs

=∆ℓ

S︸︷︷︸

, ∆ℓ ≡2π

L∆k

︸︷︷︸

Coherence control parameter Spectral width

Fulvio Zonca


Original formulation of quasi-linear theory ⇒ No assumptions on relativeordering of τnℓ and |γk0 |

−1.

The additional condition for validity of quasi-linear theory, expressed as|γk0 |τnℓ ≡ R−1/3 ≫ 1 was introduced by [Laval and Pesme, 1983,1984].

Noting that |γk0 |>∼ ωB [O’Neil et al 1971,1972]

R ≡ (|γk0 |τnℓ)−3 ∼ 2πS(ω3

B/|γk0 |3) , ⇒ |γk0 |τnℓ

>∼ (2πS)−1/3

Mode coupling terms become important for R>∼ 1 ⇒ Assumption that elec-tric field fluctuation behaves as a Gaussian process along the free particleorbits when computing 〈∆v2〉 is broken.

Turbulent trapping model [Laval and Pesme, 1983,1984] ⇒ Correlationlength can be longer than |k0|

−1 due to resonant 2n wave coupling mediatedby the resonant particles.

Nonlinearity vs. RandomnessC&Z [Rev. Mod. Phys. 88, 015008 (2016)]

Fulvio Zonca


In summary:

• BoT paradigm for fusion plasmas requires proximity to marginal sta-bility and describes uniform plasma limit ⇒ |γk| not too large

• Reduced dynamic description breaks down on long time scales ⇒ |γk|not too small

• Variety of spectral features (“broad” vs. “narrow”) and fluctuationintensity require to go beyond the diffusive paradigm of local transport

Applications:

• Illustrate mixed diffusive and convective character in the relaxationof the beam distribution function

• Discuss simple analytic model for self-consistent evolution of EP dis-tribution function and fluctuation intensity for a broad spectrum

• Show the importance of accounting for modes of the stable spectrumfor predicting transport

• Numerical simulations based on Hamiltonian formulation of BoTproblem [Carlevaro et al 2015, 2016]

Fulvio Zonca


Application (I): Broad spectrum

Case (i): S ≃ 2, ∆ℓ = 50 (unst. modes), Q ≃ 30.Normalizations: ξ = 2πξ/L, ℓ = kL/2π, τ = ηωpt, η = (nB/2n)

1/3,ξ′ = dξ/dτ , φj = eϕjk

2j/(mη

2ω2p).

Fulvio Zonca


Dynamics of fluctuation fields

Case (ii): S ≃ 2, ∆ℓ = 50 (unst. modes) +70 (stable modes), Q ≃ 60.Dynamics of fluctuation fields

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Evolution of beam distribution function

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Mixed diffusion–convection relaxation

Broad spectrum case (i) & (ii): beam distribution split into 50 “beamlets”(α = 1, ..., 50) and then into “particles”; test particle transport.

case(i) case(ii)

Signature of meso-scale (spatiotemporal) convective relaxation.

Fulvio Zonca


Toy model for mixed diffusion–convection

Consider Dupree’s quasilinear equations, including trapping in the limit ofsmall broadening w.r.t. the quasilinear growth rate.

Coupled evolution equation of normalized (and smooth) spectral intensityE and beam distribution function G are [Carlevaro et al. 2016]

∂τE = E∂ξ′G , ∂τG = ∂ξ′(E∂ξ′G

)

Note: E is computed at ω/k = v resonance ⇒ ξ′.

System reflects well-known conservation relation between the particle dis-tribution function and the spectral density generated by the instability[A. Hasegawa 1975].

Meso-scale (spatiotemporal) behavior is due to self-consistent evolution offluctuation intensity on the same time scale of particle transport.

Fulvio Zonca


Fulvio Zonca


Near an inflection point of G, solution sat-isfies

∂τG ≃ ∂ξ′E∂ξ′G = (G−G0) ∂ξ′G

with G0(ξ′) the initial distribution function.

NeglectingG0, this equation can be cast intothe form of inviscid Burgers equation

Formal solution ⇒ shock formation and propagation

G = G(Ξ(ξ′, τ)) , ξ′ = Ξ− τG(Ξ)

Further analogy ⇒ heat equation with exponential nonlinearityaway from G0(ξ

′) localization region. Let E ≡ expW ; system reduces to

∂τW = ∂ξ′G0 + ∂ξ′(eW∂ξ′W

)

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Application (II): Narrow spectrum

Case (iii): S ≃ 15, ∆ℓ = 50 (unst. modes), Q ≃ 3.6.

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Dynamics of fluctuation fields

Case (iv): S ≃ 15, ∆ℓ = 50 (unst. modes) +70 (stable modes), Q ≃ 9.Dynamics of fluctuation fields: crucial importance of linear stable spectrum

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Evolution of beam distribution function: diffusive relaxation asymptoticallyrestored accounting for stable fluctuation spectrum.

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Particle trapping and trapping suppression

Narrow spectrum case (iii) & (iv): beam distribution split into 50 “beam-lets” (α = 1, ..., 50) and then into “particles”; test particle transport.

case(iii) case(iv)

Signature of wave-particle trapping with coherent spectrum and partial sup-pression of wave-particle trapping by a sufficiently broad spectrum includinglinear stable modes.

Fulvio Zonca


Summary and discussion

Possible relevance of BoT paradigm for EP transport in fusion plasmas issubject to constraints imposed by

• Reduction of nonlinear dynamics to time-dependent uniform systemwith one degree of freedom

• Transport prediction on long time scales requires comprehensive mod-els fully accounting for system non-uniformity and realistic geometry

Variety of spectral features (“broad” vs. “narrow”) and fluctuation intensityof EP driven fluctuations in burning fusion plasmas require to go beyondthe diffusive paradigm of local transport.

Meso-scale (spatiotemporal) transport events are generally characterized bymixed diffusive–convective relaxation and/or coherent nonlinear behaviors:may be important for fusion plasma operation scenario and performance.

Accounting for fluctuations of the linear stable spectrum may be crucial fora proper prediction of fluctuation intensity and transport processes.

Fulvio Zonca

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