1 F. Zonca, S. Briguglio, L. Chen, G. Fogaccia, G. Vlad ... · ENEA F. Zonca, S. Briguglio, L....

ENEA F. Zonca, S. Briguglio, L. Chen, G. Fogaccia, G. Vlad 1

Energetic particle physics in burning plasmas∗

F. Zonca, S. Briguglio, L. Chen †, G. Fogaccia, G. VladAssociazione Euratom-ENEA sulla Fusione, C.R. Frascati, C.P. 65 - 00044 - Frascati, Italy.

† Department of Physics and Astronomy, Univ. of California, Irvine CA 92697-4575, U.S.A.

May 18.th, 2006

Workshop on ITER SimulationMay 15 – 19, Beijing, China

∗Acknowledgments: G.Y. Fu, S.J. Wang

ITER Simul. Workshop 2006


Nonlinear Dynamics of Burning Plasmas

2 A burning plasma is a complex self-organized system where the crucial process-es to understand are (turbulent) transport and fast ion/fusion product inducedcollective effects.

THERMAL PLASMA

FUSION PRODUCTS

ALPHA PARTICLES

FUSION PRODUCT

PROFILES

(COLLECTIVE EFF.)

FUSION REACTIVITY

(TURBULENT) TRANSPORT

COLLECTIVE EFFECTSEXTERNAL HEATING

CURRENT DRIVE



The roles of simulation and theory

2 Reactor relevant conditions require fast ion (MeV energies) and charged fusionproducts good confinement:





• Identification of burning plasma stability boundaries with respect to ener-getic ion collective mode excitations and their nonlinear dynamic behaviorsabove the stability thresholds






• Obvious impact on the operation-space boundaries, since collective lossesmay lead to significant wall loading and damaging of plasma facing materialsin addition to degrading fusion performance







2 Mutual interactions between collective modes and energetic ion dynamics with driftwave turbulence and turbulent transport should not deteriorate the thermonuclearefficiency:








• MeV ion energy tails introduce a dominant electron heating and differentweighting of the electron driven micro-turbulence w.r.t. present experiments









• They also generate long time-scale nonlinear behaviors typical of self-organized complex systems



The roles of simulation and theory (... continued)

2 These phenomena can be analyzed, at least in part, in present day experimentsand provide nice examples of mutual positive feedbacks between theory, simulationand experiment.





2 In a burning plasma, however, unique features not reproducible in existing exper-iments are:






• energetic ion power density profiles and characteristic wavelengths of thecollective modes







• local power balance dominated by electron heating (fast ions) and self-organization of radial profiles of the relevant quantities: consequence onturbulence spectra and turbulent transport








2 Crucial roles of predictive capabilities based on numerical simulations as well asof fundamental theories for developing simplified yet relevant models, needed forinsights into the basic processes








2 Crucial roles of predictive capabilities based on numerical simulations as well asof fundamental theories for developing simplified yet relevant models, needed forinsights into the basic processes

2 Importance of using existing and future experimental evidences for modeling ver-ification and validation



Energetic particle physics in burning plasmas:

historic background

2 Possible detrimental effect of Shear Alfvén (SA) instabilities on energetic ionsrecognized theoretically before experimental evidence was clear.




historic background


2 Mikhailowskii, Sov. Phys. JETP, 41, 890, (1975)Rosenbluth and Rutherford, PRL 34, 1428, (1975)⇒ resonant wave particle interaction of ≈ MeV ions with SA inst. due to vE ≈ vA(k‖vA ≈ ωE)




historic background



2 Experimental observation of fishbones on PDX – McGuire et al., PRL 50, 891,(1983) – fast ⊥ injected ion losses . . .




historic background



2 Experimental observation of fishbones on PDX – McGuire et al., PRL 50, 891,(1983) – fast ⊥ injected ion losses . . .. . . followed by numerical simulation of mode-particle pumping loss mechanism –White et al., Phys. Fluids 26, 2958, (1983)




historic background



2 Experimental observation of fishbones on PDX – McGuire et al., PRL 50, 891,(1983) – fast ⊥ injected ion losses . . .. . . followed by numerical simulation of mode-particle pumping loss mechanism –White et al., Phys. Fluids 26, 2958, (1983). . . and by theoretical explanation of internal kink excitation –Chen, White, Rosenbluth, PRL 52, 1122, (1984)Coppi, Porcelli, PRL 57, 2272, (1986)



2 Existence of gaps in the SA continuous spectrum (due to lattice symmetry break-ing) ω ≈ vA/2qR – Kieras and Tataronis, J. Pl. Phys. 28, 395, (1982)




2 Existence of discrete modes (TAE) in the toroidal gaps – Cheng, Chen, Chance,Ann. Phys. 161, 21, (1985)





2 Possible excitations of TAE by energetic particles . . .Chen, “Theory of Fusion Plasmas, p.327, (1989)Fu, Van Dam, Phys. Fluids B 1, 1949, (1989).






2 KBM excitation by fast ions: Biglari, Chen, PRL 67, 3681, (1991)







2 Experimental evidence . . .Wong et al., PRL 66, 1874, (1991)Heidbrink et al, Nucl. Fusion 31, 1635, (1991)Heidbrink et al, PRL 71, 855, (1993) ⇒ BAE ω ≈ ωti ≈ ω∗pi







2 Experimental evidence . . .Wong et al., PRL 66, 1874, (1991)Heidbrink et al, Nucl. Fusion 31, 1635, (1991)Heidbrink et al, PRL 71, 855, (1993) ⇒ BAE ω ≈ ωti ≈ ω∗pi

2 TAE modes are predicted to have small saturation levels and yield negligibletransport unless stochastization threshold in phase space is reached: H.L. Berkand B.N. Breizman, Phys. Fluids B 2, 2246, (1990) and D.J. Sigmar, C.T. Hsu,R.B. White and C.Z. Cheng, Phys. Fluids B, 4, 1506, (1992).



2 Excitation of Energetic particle Modes (EPM), at characteristic frequencies ofenergetic particles when free energy source overcomes continuum dampingL. Chen, Phys. Plasmas 1, 1519, (1994). [ . . . also RTAE excitation C.Z. Cheng,N.N. Gorelenkov, C.T. Hsu, Nucl. Fusion 35, 1639, (1995).]




2 Strong energetic particle redistributions are predicted to occur above the EPMexcitation threshold in 3D Hybrid MHD-Gyrokinetic simulations: S. Briguglio, F.Zonca and G. Vlad, Phys. Plasmas 5, 1321, (1998).





2 Nonlinear Dynamics of Burning Plasmas: energetic ion transport in burning plas-mas has two components:






• slow diffusive processes due to weakly unstable AEs and a residual compo-nent possibly due to plasma turbulence (Vlad et al. PPCF 47 1015 (2005);Estrada-Mila et al., submitted to Nucl. Fusion 2006.






• slow diffusive processes due to weakly unstable AEs and a residual compo-nent possibly due to plasma turbulence (Vlad et al. PPCF 47 1015 (2005);Estrada-Mila et al., submitted to Nucl. Fusion 2006.

• rapid transport processes with ballistic nature due to coherent nonlinearinteractions with EPM and/or low-frequency long-wavelength MHD



Overview of shear Alfvén spectra

Energetic Particle Modes (EPM) : resonant modes

TAE – KTAE ⇒ Transition to EPM

Energetic Particle Modes (EPM) : resonant modes

Beta induced Alfvén Eigenmodes (BAE)MHD!!! Kinetic Ballooning Modes (KBM)

KBM ⊕ BAE ⇒ Alfvén ITG (AITG)



Linear Stability Issues: Eigenmodes vs. Resonant

Modes

2 Linear Theory: sound and well understood. However, most codes still do notinclude nonperturbative particle dynamics




Modes


2 Numerical simulations: can address realistic equilibrium/geometries for moderatemode numbers. Must assume model fast ion distribution functions




Modes



2 Key issues: 3-D MHD derived codes and Hybrid MHD gyrokinetic codes not alwaysin agreement with kinetic-fluid closure codes. Difference when wave propagationphysics becomes important (KAW)




Modes




2 Theory vs Experiment comparisons: mostly refer to active (antenna) or passive(spontaneous) mode excitation with weak drive (easier to measure). But whatabout strong excitations? Bursting behaviors




Modes




2 Theory vs Experiment comparisons: mostly refer to active (antenna) or passive(spontaneous) mode excitation with weak drive (easier to measure). But whatabout strong excitations? Bursting behaviors

2 Further developments: global extended MHD codes (NOVA-K, CASTOR-K) orglobal gyrokinetic codes (LIGKA, PENN) need to incorporate SOL model and pos-sibly accurate treatment of plasma free boundary in addition to non-perturbativefast ions



Nonlinear Dynamics: local vs. global processes

2 Mode saturation via wave-particle trapping (H.L. Berk et al. PFB 90, PPR 97)has been successfully applied to explain pitchfork splitting of TAE spectral lines(A. Fasoli et al. PRL 98): local distortion of the fast ion distribution functionbecause of quasi-linear wave-particle interactions.



Pitchfork splitting of TAE in JET

Fasoli, Phys. Rev. Lett. 81, 5564, (1998)

High resolution MHD spectroscopy: Pinches et al, PPCF 46, S47, (2004)





2 Compton scattering off the thermal ions (T.S. Hahm and L. Chen PRL 95): locallyenhance the mode damping via nonlinear wave-particle interactions

2 Mode-mode couplings generating a nonlinear frequency shift which may enhancethe interaction with the Alfvén continuous spectrum (Zonca et al. PRL 95 andChen et al. PPCF 98): locally enhance the mode damping via nonlinear wave-waveinteractions.







2 Role of mode-mode couplings recently confirmed by (Todo et al. 2005 and Fu etal. 2006).







2 Role of mode-mode couplings recently confirmed by (Todo et al. 2005 and Fu etal. 2006).

2 EPM is a resonant mode (L. Chen PoP 94), which is localized where the drive isstrongest: global readjustments in the energetic particle drive is expected to beimportant as well.



Nonlinear self-consistent fast ion dynamics

2 A burning plasma is a self-organized system, where collective effects associated withfast ions (MeV energies) and charged fusion products may alter their confinementproperties and even prevent the achievement of ignition.





2 Simulation results indicate that, above threshold for the onset of resonant EnergeticParticle Modes (EPM) (L. Chen, PoP 94), strong fast ion transport occurs inavalanches (IAEA 02).



Avalanches and NL EPM dynamics (IAEA 02)

|φm,n(r)|

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.05

0.1

0.15

0.2

0.25

x 10-3

r/a

8, 4 9, 410, 411, 412, 413, 414, 415, 416, 4

- 4

- 2

0

2

4

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

δαH

r/a

= 60.00t/τA0 |φm,n(r)|

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.05

0.1

0.15

0.2

0.25

x 10-2

r/a

8, 4 9, 410, 411, 412, 413, 414, 415, 416, 4

- 4

- 2

0

2

4

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

δαH

r/a

= 75.00t/τA0 |φm,n(r)|

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

.001

.002

.003

.004

.005

.006

.007

.008

.009

r/a

8, 4 9, 410, 411, 412, 413, 414, 415, 416, 4

- 4

- 2

0

2

4

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

δαH

r/a

= 90.00t/τA0



Vlad et al. IAEA-TCM, (2003)

Propagation of the unstable front

0.50

0.55

0.60

0.65

0.70

0.75

0.80

0.85

0 50 100 150 200 250 300

rmax

[d(rnH

)/dr]

t/τAlinear

phase convectivephase

diffusivephase

0.025

0.030

0.035

0.040

0.045

0.050

0.055

0.060

0 50 100 150 200 250 300

[d(rnH

)/dr]max

t/τAlinear

phase convectivephase

diffusivephase






2 Such strong transport events occur on time scales of a few inverse linear growthrates – 100 ÷ 200 Alfvén times – and have a ballistic character (R.B. White, PF83) that basically differentiates them from the diffusive and local nature of weaktransport.






2 Such strong transport events occur on time scales of a few inverse linear growthrates – 100 ÷ 200 Alfvén times – and have a ballistic character (R.B. White, PF83) that basically differentiates them from the diffusive and local nature of weaktransport.

2 Observations on JT-60U (and other tokamaks) have confirmed macroscopic andrapid energetic particle radial redistributions in connection with the so calledAbrupt Large amplitude Events (ALE) (K. Shinohara, NF 01).



ALE on JT-60U (K. Shinohara, et al., Nucl. Fus. 41, 603, (2001)

Courtesy of K. Shinohara and JT-60U

ALE = Abrupt Large amplitude Event



Fast ion transport: simulation and experiment I2 Numerical simulations show fast ion radial redistributions, qualitatively similar to

those by ALE on JT-60U.

0

0.01

0.02

0 0.2 0.4 0.6 0.8 1

βH

r/a

Before EPM

After EPM

G. Vlad et al., EPS02, ECA 26B, P-4.088,(2002).

K. Shinohara etal PPCF 46, S31 (2004)Courtesy of M. Ishikawa and JT-60U



Fast ion transport: simulation and experiment II

• ALE in JT60-U:• n = 1 mode, βH0 ≈ 3%;• linear growth rate γ ≈ 0.106τ−1A0 ;• half width of the pulse ∆tALE ≈ 64.5µs;• experimental range ∆tALE ≈ 50 ÷ 200µs;• energetic particle profiles compare well before and

after ALE burst



Nonlinear simulation issues

2 The Hybrid MHD-Gyrokinetic model (Park et al 92) is most suitable for glob-al computations and adequate for the most dangerous class of modes that yieldenergetic particle transports (Zonca and Chen 06).





2 From linear stability considerations: accurate geometry & B.C.’s and realistic fastion sources to be treated non-perturbatively. (e.g. internal kink excitation byenergetic passing ions: Betti 93, Wang 01, Wang 02).






2 Dynamic treatment of the fast ion source build-up: multi-scale approach.







2 Multi-mode simulations: so far mostly one (or a few) toroidal mode numbers.








• Related to stochastic thresholds and losses in particle phase-space








• Related to stochastic thresholds and losses in particle phase-space• Related to non-linear saturation processes and thermal plasma turbulence








• Related to stochastic thresholds and losses in particle phase-space• Related to non-linear saturation processes and thermal plasma turbulence• Generation of Zonal Flows and Fields









• They also generate long time-scale nonlinear behaviors typical of self-organized complex systems








• Related to stochastic thresholds and losses in particle phase-space• Related to non-linear saturation processes and thermal plasma turbulence• Generation of Zonal Flows and Fields

2 Alfvén Eigenmode and EPM NL dynamics is predominantly accompanied by Zon-al Flow generation or fast ion source modulations, depending on proximity tomarginal stability (Zonca et al 00, Chen et al 01).



Avalanches in fusion product transport


n8_9_imirr1_13_light.movMedia File (video/quicktime)


Generalized fishbone-like dispersion relation

2 In general, Chen PRL 94 demonstrated that the mode dispersion relation can bealways written in the form of a fishbone-like dispersion relation (Chen et al. PRL84)

−iΩ + δWf + δWk = 0 ,

where δWf and δWk play the role of fluid (core plasma) and kinetic (fast ion) contributionto the potential energy, while Ω represents a generalized inertia term.



Generalized fishbone-like dispersion relation

2 In general, Chen PRL 94 demonstrated that the mode dispersion relation can bealways written in the form of a fishbone-like dispersion relation (Chen et al. PRL84)

−iΩ + δWf + δWk = 0 ,

where δWf and δWk play the role of fluid (core plasma) and kinetic (fast ion) contributionto the potential energy, while Ω represents a generalized inertia term.

2 The fishbone-like dispersion relation demonstrates the existence of two types ofmodes:

• a discrete gap mode, or Alfvén Eigenmode (AE), for IReΩ2 < 0;

• an Energetic Particle continuum Mode (EPM) (Chen PRL 94) for IReΩ2 > 0.



AITG: a special case of the fishbone-like D.R.2 BAE/KBM and transition to AITG: the fishbone-like dispersion relation has an

interesting application for ω ≈ ω∗pi ≈ ωti, i.e. when accounting for the finitecompressibility due to wave-particle interactions with core plasma ions (Zonca etal. PPCF 96, POP 99):

Ω2 =ω2

ω2A

(

1 − ω∗piω

)

+ q2ωωtiω2A

[(

1 − ω∗niω

)

F (ω/ωti)

−ω∗T iω

G(ω/ωti) −N2(ω/ωti)

D(ω/ωti)

]

,



AITG: a special case of the fishbone-like D.R.2 BAE/KBM and transition to AITG: the fishbone-like dispersion relation has an

interesting application for ω ≈ ω∗pi ≈ ωti, i.e. when accounting for the finitecompressibility due to wave-particle interactions with core plasma ions (Zonca etal. PPCF 96, POP 99):

Ω2 =ω2

ω2A

(

1 − ω∗piω

)

+ q2ωωtiω2A

[(

1 − ω∗niω

)

F (ω/ωti)

−ω∗T iω

G(ω/ωti) −N2(ω/ωti)

D(ω/ωti)

]

,

2 An interesting is ω ≫ ω∗pi, ωti, which yields (Zonca et al. PPCF 96):

Ω2 =1

ω2A

[

ω2 −(

7

4+

TeTi

)

q2ω2ti

]

+ i√

πq2e−ω2/ω2

ti

ω2

ω2A

(

ωtiω

− ω∗T iωti

)

(

ω2

ω2ti+

TeTi

)2

.

2 This expression for Ω2 describes the formation of the low frequency BAE gap(Heidbrink et al. POP 93) and the fact that ion Landau damping decreases expo-nentially with increasing q2.



2 The low frequency shear Alfvén accumulation point is degenerate with the GAMfrequency: but it is an Alfvén wave with (n, m) 6= (0, 0).




2 A new important predicted feature is the excitation of an Alfvénic mode drivenby thermal ion temperature gradients, provided ωω∗T i > ω

2ti. This is the Alfvénic

Ion Temperature Gradient (AITG) driven mode (Zonca et al. POP 99).







2 Excitations by both energetic and thermal ions is in agreement with recent DIIIDexperimental observations of Alfvénic fluctuations in a broad range of wave num-bers. (Nazikian et al. PRL 2006).








2 As for general Alfvénic fluctuations, AITG induced Reynolds and Maxwell stressespartially cancel: slower Zonal flow generation rate Chen et al. NF 2001).








2 As for general Alfvénic fluctuations, AITG induced Reynolds and Maxwell stressespartially cancel: slower Zonal flow generation rate (Chen et al. NF 2001).

2 AITG has even parity, but reconnecting modes can be excited as well: whatrelevance to electron transport?



Conclusions and Outlook

2 A burning plasma is a very exciting nonlinear dynamics system to investigate,with truly peculiar features and many challenging fundamental physics problemsto study

2 Mutual and positive feedbacks between theory, simulation and experiment arenecessary for significant advances in understanding and new discoveries

2 Theory faces the big challenge for providing the fundamental grounds for theunderstanding and interpretation of increasingly complex numerical simulationresults

2 In this respect, plasma theory of burning plasmas can be seen as a natural frame-work for systematically generating interesting and new paradigm problems fornonlinear dynamics. E.g., fusion product avalanches and convective amplifica-tion of wave fronts naturally produce a fractional derivative formulation of theLandau-Ginzburg equation. (Milovanov 2006)


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