as in the case of Energetic Particle Modes (EPM) [3], radial distortions (redistributions) of the fast ion source dominate the mode nonlinear dynamics. In this work ...
35th EPS Conference on Plasma Phys. Hersonissos, 9 - 13 June 2008 ECA Vol.32D, P-5.056 (2008)
Self-consistent energetic particle nonlinear dynamics due to shear Alfvén wave excitations∗ Fulvio Zonca1 and Liu Chen2 1 Associazione 2 Dept.
Euratom-ENEA sulla Fusione, C.P. 65 - I-00044 - Frascati, Rome, Italy
of Physics and Astronomy, University of California, Irvine CA 92697-4575, U.S.A.
Introduction. We adopt the 4-wave modulation interaction model, introduced by Chen et al [1] for analyzing modulational instabilities of the radial envelope of Ion Temperature Gradient (ITG) driven modes in toroidal geometry, extending it to the modulations on the fast particle distribution function due to nonlinear Alfvénic mode dynamics, as proposed in Ref. [2]. In the case where the wave-particle interactions are non-perturbative and strongly influence the mode evolution, as in the case of Energetic Particle Modes (EPM) [3], radial distortions (redistributions) of the fast ion source dominate the mode nonlinear dynamics. In this work, we show that the resonant particle motion is secular with a time-scale inversely proportional to the mode amplitude [4] and that the time evolution of the EPM radial envelope can be cast into the form of a nonlinear Schrödinger equation a la Ginzburg-Landau [5, 6]. Long time-scale fast ion nonlinear behaviors. Three dimensional gyrokinetic simulations show evidence of radial fragmentation of a single n (toroidal mode number) coherent EPM [2, 5, 6]. This radial fragmentation suggests the excitation of low frequency axisymmetric perturbations, for which a nonlinear mechanism is necessary, in analogy with the modulational instability of a single toroidal drift wave [1], where toroidal geometry plays a crucial role [4, 5, 6]. The generalization to e.m. fluctuations [7] of the 4-wave modulation interaction model, introduced by Chen et al [1], is based on the description of the nonlinear dynamics of a single n Alfvén Eigenmode (AE) or EPM given by a “pump” (0) mode, which interacts with a “zonal” toroidally and poloidally symmetric mode generating (±) sidebands: i.e.,
δ φ0 (pump) = ei
R
nθk dq+inϕ
∑ e−imϑ δ φm + c.c m
δ φz (zonal mode) = e
R
i Kz dr
∗ Work
δ φz + c.c.
supported by the Euratom Communities under the contract of Association between EURATOM/ENEA,
by U.S. DOE Contract DE-AC02-CHO-3073, by NSF Grant ATM-0335279 and by the Guangbiao Foundation of Zhejiang University.
35th EPS 2008; F.Zonca et al. : Self-consistent energetic particle nonlinear dynamics due to shear Alfvén wave...
δ φ± (sidebands) =
ei
R
nθk dq
∗ e−i nθk dq
R
e±inϕ +i
R
Kz dr
∑ e∓imϑ δ φm
(±)
+ c.c.
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(1)
m
with δ φEPM = δ φ0 + δ φ+ + δ φ− + c.c. and similarly for the parallel vector potential. Here, we have used (r, ϑ , ϕ ) straight field line toroidal flux coordinates, q is the safety factor, θk = kr /(nq′ ) is the normalized k of the EPM radial envelope, m is the poloidal mode number and other symbols are standard. Given the strong zonal current screening by electrons on the collisionless skin depth, δe = c/ω pe , the nonlinear equation for δ Ak,z is δ jki,z + δ jke,z ≃ δ jke,z ≃ 0. 2 δ 2 ≪, i.e. δ A This readily yields ω /(kk c)(δ Ak,z /δ φz ) ≈ k⊥ k,z can be considered a passive scalar, e
which weakly influences the mode dynamics [7]. Introducing b · ∇δ ψ ≡ −(1/c)∂t δ Ak , pump EPM and sidebands obey the following nonlinear quasineutrality and vorticity equations [7] E D E D Ti ne2 1+ δ φk = eJ0 δ H i − eδ H e (2) k k Ti Te 4π ω∗pi ω2 2 b · ∇δ ψ Bb · ∇ −∇⊥ −∇2⊥ δ φ − 2 ∑heωωd J0 δ H k i = + 2 1− B ω c s vA b · (k′′⊥ × k′⊥ ) c2 2 2 ∂t δ Ak,k′ ∇⊥ δ Akk′′ − 2 δ φk′ ∇⊥ δ φk′′ = (3) cB vA where vA is the Alfvén speed, ω∗pi the thermal ion diamagnetic frequency, ωd the magnetic drift frequency, angular brackets indicate velocity space integration and k = k′ +k′′ . Furthermore, ∑s in Eq. (3) indicates summation on all thermal and energetic particle plasma species, J0 = J0 (λ ) (λ = k⊥ v⊥ /ωc ) is the Bessel function accounting for finite Larmor radius effects, and δ H k satisfies the nonlinear gyrokinetic equation [8]
∂t + vk b · ∇ + iωd
e c ′′ ′ δ H = i QF J ( λ ) δ L − b · k × k J0 (λ ′ )δ Lk′ δ H k′′ , 0 0 k k ⊥ ⊥ k m B
(4)
with δ Lk = δ φk − (vk /c)δ Akk , QF0 = 2ωk ∂ F0 /∂ v2 + k · bˆ × ∇F0 /ωc and where, as usual, the perturbed particle distribution function has been decomposed as δ F = 2(e/m)δ φ ∂ F0 /∂ v2 + ˆ ωc δ H k , F0 being the equilibrium particle distribution function. Finally, ∑k⊥ exp −ik⊥ · v × b/ the evolution of the zonal mode δ φz is given by [7]
h 2
2 ∂t χiz δ φz = (c/B)kϑ Kz Kz2 ρLi α0 − kk vA /ω0 |Ψ0 |2 + 2α0 IRe hh(Φ0 − Ψ0 )∗ Ψ0 ii
∗ (A0 A+ − A0 A− ) . (5) +α0 |Φ0 − Ψ0 |2
Here, χiz ≃ 1.6q2 Kz2 ρi2 /(r/R0)1/2 [9], R0 is the torus major radius, ρi = (Ti /mi )1/2 /ωci is the ion Larmor radius, α0 ≡ 1 + δ P⊥i0 /(neδ φ0 ) [7], (δ φk , δ ψk ) = Ak (φ0 (n0 q − m), ψ0 (n0 q − m)), with k = (0), (±), (Φ0 (θ ), Ψ0 (θ )) = (2π )−1/2
R∞
−∞ exp[i(n0 q − m)θ ](φ0 (n0 q − m), ψ0 (n0 q −
35th EPS 2008; F.Zonca et al. : Self-consistent energetic particle nonlinear dynamics due to shear Alfvén wave...
m))d(n0q − m) and hh...ii =
R∞
−∞ (...)d θ . Thus, Φ0 (θ ) and Ψ0 (θ ) represent
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δ φk and δ Akk paral-
lel mode structures, respectively, while A0 and A± the amplitudes of pump and sideband fluctuations. Equation (5) demonstrates the exact cancellation of zonal flows and currents for a pure shear Alfvén wave, for which the r.h.s. of Eq. (3) also vanishes identically. Thermal particle nonlinearities enter both Eqs. (2) and (3). In Eq. (3), thermal particle nonlinearities are dominated by r.h.s., while fast particle nonlinearities enter via the ballooninginterchange term only (last on the l.h.s.), since they carry pressure but not inertia [2]. Similarly to e.s. case [1], Eqs. (2) to (5) allow the spontaneous excitation of zonal flows via modulational instability of the radial envelope of drift Alfvén waves and AE/EPM [7, 10]. This is the dominant nonlinear dynamics for weak instability drive [5, 6]. For strong unstable conditions, typical of EPM excitations above threshold, nonlinear evolutions are predominantly affected by spontaneous generation of radial modulations in the fast ion profiles [2, 5, 6]. It can be shown that these radial modulations are spontaneously produced with a nonlinear growth rate Γz ∝ δ Bϑ /B and a corresponding frequency shift (line splitting) ∆z ∝ δ Bϑ /B, where [2]
ωz = ∆z + iΓz = ±31/4 e±iπ /4 |∆L ∂ DR /∂ ω0 |−1/2 γM , 2 2 ω0 1 r nE TE v2E 2 2 ′ 4 2 2 eE A0 γM = 2 π ∆ + (kϑ qρE ) Kz vE ∑ ω 2 − 4 + σ |Λ0 | R0 B2 v2A TE ∑ ℓ σ =± A * ! + 2 4 2 2 2 J 2 (λ ) 2 v v ℓ QF0E vE k (lin) d ⊥ J02 (λ ) ℓ 2 J02 (λz ) + 2 , (6) δ (Lℓ,σ /ω0 ) 2 ω0 nE λd 2vE vE with DR the real part of the EPM dispersion function, ∆L = (nq′ )2 (∂ 2 DR /∂ kr2 )(∂ DR /∂ ω0 )−1 [1 − cos (Kz /nq′ )] is the linear frequency shift due to radial modulations, ∆′ is the Shafranov shift, nE , TE and eE the energetic particle density, temperature and electric charge, v2E = TE /mE ,
ωA = vA /qR0 , Λ0 is a measure of EPM continuum damping, λz ≡ Kz q(v2⊥ /2 + v2k )/(ωci vk ) and (lin)
λd ≡ k⊥ q(v2⊥ /2 + v2k )/(ωci vk ). The (inverse) linear propagator, Lℓ,σ , is readily obtained from the (inverse) nonlinear propagator expression Lℓ,σ = (vk /qR0 ) (ℓ + Λ0 + σ /2) − ω0 + (kϑ2 c2 /B2 )Kz2 J02 (λz)(Mℓ,σ /ωz ) |A+ |2 + |A− |2 , (7) DD EE 2 )ℓ2 v2 /(q2 R2 ω 2 ) where Mℓ,σ = (1/2)(|ω02/ωA2 − 1/4| + σ |Λ0 |) J02 (λ0 )Jℓ2 (λd )(kϑ2 /k⊥ . 0 0 k Nonlinear radial envelope equations.
The spontaneous generation of radial modulation in the fast ion profiles are due to “zonal” fast ion responses δ H Ez and are accompanied by frequency splitting of EPM spectral lines ∝ δ Bϑ /B. For strong modulations, the scaling becomes ωz ∝ (δ Bϑ /B)2/3 [2]. Once the “zonal” response δ H Ez is computed, fast ion transport equations are readily derived [2, 5, 6]. Fast particle nonlinearities enter in Eq. (3) via the last term on the l.h.s., through δ H kE ∝ δ Lk′ δ H Ez
35th EPS 2008; F.Zonca et al. : Self-consistent energetic particle nonlinear dynamics due to shear Alfvén wave...
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computed from Eq. (4). This term describes “hole-clump” dynamics in phase space [11], when separation of nonlinear time scales applies, but, more generally, it accounts for phase space structure formation where equilibrium geometry and wave-particle resonances play a major role [4, 5, 6, 12]. Resonant particles are more efficiently scattered (than nonresonant particles) out of the resonance region and dictate the characteristic non-linear time scaling τNL ∝ (lin)
|A0 |−1 [4, 5]. This can be seen from Eq. (7), where Lℓ,σ ≃ −i∂t for resonant particles; meanwhile, ωz ≃ −i∂t . Thus, the nonlinear propagator structure demonstrates radial particle motion with characteristic velocity ∝ |A0 |, consistent with the Mode Particle Pumping process [13], introduced originally for energetic particle transport due to fishbones. Nonlinear partial differential equations (PDE) for the slow radial and time evolution of the EPM and sideband radial envelopes can be derived within this approach. The EPM radial envelope equation can be put in the form of a complex Ginzburg-Landau equation [5] i ∂ 2 DR /∂ kr2 2 2 2 2 ∂t + iω0 + ∂ A0 = −iLNL ∂ξ γL |A0 | A0 , 2 ∂ DR /∂ ω0 ξ
(8)
where ξ = r − vgr t, vgr is the EPM radial group velocity, LNL is a nonlinear characteristic scalelength and γL the energetic particle drive; while nonlinear sideband envelope equations are 2 D+ ∂t2 (A∗0 A+ ) = −iγM (2A∗0 A+ + A0 A− ) , 2 D− ∂t2 (A0 A− ) = iγM (A∗0 A+ + 2A0 A− ) ,
(9)
where D± are the linear EPM sideband dispersion functions, D± ≃ [∂ (DR ± iDI )/∂ ω0 ](∓i∂t ) − (∂ DR /∂ ω0 )∆L and DI is the imaginary part of the EPM dispersion function. Acknowledgments Useful and stimulating discussions with S. Briguglio, G. Fogaccia, G. Vlad and A.V. Milovanov are kindly acknowledged. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13]
L. Chen et al, Phys. Plasmas 7 3129 (2000) F. Zonca et al, Theory of Fusion Plasmas (Bologna: SIF) 17 (2000) L. Chen, Phys. Plasmas 1 1519 (1994) F. Zonca et al, Nucl. Fusion 45 477 (2005) F. Zonca et al, Plasma Phys. Contr. Fusion 48 B15 (2006) L. Chen and F. Zonca, Nucl. Fusion 47 S727 (2007) L. Chen, Z. Lin, R.B. White and F. Zonca, Nucl. Fusion 41 747 (2001) E.A. Frieman and L. Chen, Phys. Fluids 25 502 (1982) M.N. Rosenbluth and F.L. Hinton, Phys. Rev. Lett. 80 724 (1998) P.N. Guzdar, R.G. Kleva, A. Das and P.K. Kaw, Phys. Rev. Lett. 87 015001 (2001) H.L. Berk et al, Phys. Lett. A 234 213 (1997) Y. Elskens and D.F. Escande, Microsc. dynam. of plasmas and chaos (Bristol: IOP) (2003) R.B. White et al, Phys. Fluids 26 2958, (1983)
