Lecture 5Max-Planck-Institutfu¨rPlasmaphysikLectureSeries-Winter2013 Lecture5– 6 E: From the equations of motions in the equilibrium magnetic ﬁeld show that the particle trajectories

Max-Planck-Institut fur Plasmaphysik Lecture Series-Winter 2013 Lecture 5 – 1

Lecture 5

From local to global nonlinear behaviors

Fulvio Zonca

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

Associazione Euratom-ENEA sulla Fusione, C.R. Frascati, C.P. 65 - 00044 - Frascati, Italy.

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

February 21.st, 2013

Max-Planck-Institut fur Plasmaphysik Lecture Series-Winter 2013,Kinetic theory of meso- and micro-scale Alfvenic fluctuations in fusion plasmas

19–22 February 2013, IPP, Garching

F. Zonca


Resonance conditions and equilibrium particle mo-

tion in toroidal geometry

✷ Particle motions in toroidal geometry depend on constants of motion andreflect the equilibrium magnetic configuration. Meanwhile, resonance con-ditions also depend on geometry and breaking of constants of motion bywave-particle resonances reflects fluctuation induced transport.

✷ Consider a generic fluctuation f(r, θ, ξ ≡ ζ − q(r)θ) in Clebsch coordinates(r, θ, ξ) (see Lecture 1 and Z.X. Lu etal 12), leaving time dependences im-plicit. We can generally write

f(r, θ, ξ) =∑

n

einξFn(r, θ) ,

F. Zonca

http://www.afs.enea.it/zonca/references/seminars/IPPGarching_winter13/lecture1.pdf


✷ While periodicity in ξ is maintained, periodicity in θ is substituted byFn(r, θ + 2π) = e2πinqFn(r, θ). In (r, ϑ) mapping space, the transform cor-responding to using Clebsch coordinates is obtained by the periodizationoperator

Fn(r, θ) = 2π∑

ℓ

e2πiℓnqFn(r, θ − 2πℓ)

=∑

m

ei(nq−m)θ

∫

ei(m−nq)ϑFn(r, ϑ)dϑ .

The governing equations are generally expressed either for Fn(r, θ) or forFn(r, ϑ), as discussed by (Z.X. Lu etal 12). All together,

f(r, θ, ξ) =∑

m,n

einξei(nq−m)θFm,n(r) ,

Fm,n(r) =

∫

ei(m−nq)ϑFn(r, ϑ)dϑ .

F. Zonca


✷ Define as η the canonical angle conjugate to J (second invariant), obtainedfrom integration along the particle orbit as

J = m

∮

v‖dℓ , η = ωb

∫ θ

0

dθ′

θ′, ωb =

2π∮

dθ/θ.

✷ For circulating particles, recalling that ωb is the transit/bounce frequency,

θ = η +ΘC(η) = ωbτ +ΘC(η) ,

where ΘC(η) is a periodic func-tions of η, which depends alsoon H0, µ, Pφ or, equivalently, ofµ, J, Pφ; and τ is a time-like pa-rameter describing the particlemotion along the particle trajec-tory.

F. Zonca


✷ For magnetically trapped particles

θ = ΘT (η) ,

where ΘT (η) is a periodic func-tion of η, which depends alsoon H0, µ, Pφ or, equivalently, ofµ, J, Pφ.

✷ From the periodicity of the radial particle motion in the equilibrium fields,we also have

r = r + ρ(η) ,

with ρ(η) a periodic function of η, different for trapped and circulatingparticles, which depends also on H0, µ, Pφ or, equivalently, of µ, J, Pφ.

F. Zonca


E: From the equations of motions in the equilibrium magnetic field show that theparticle trajectories in the (ψ, θ) plane are closed and discuss how this impliesthe results above for the radial particle motion.

✷ The connection of ξ with η = ωbτ is more subtle to obtain. Recall thedefinition

ξ = ζ − q(ψ)θ = φ− ν(ψ, θ)− q(ψ)θ

and that the particle orbit follows the magnetic field line and is accompaniedby a periodic motion about it, of period 2π/ωb, which is therefore a periodicfunction of η.

✷ Keeping this in mind we can conclude that, except for periodic dependencesin η (indicated by the ≃ sign and the . . .)

ζ ≃ ωdτ +

∫

qdθ + . . . ≃ ωdτ + qθ + . . .

with q(r + ρ(η)) = q(r) +Q(r + ρ(η)) and∮

Qdθ =∮

(θ/ωb)Qdη = 0.

F. Zonca


✷ The representation for ξ is then obtained as

ξ = ωdτ + (q − q) θ + Ξ(η) ,

with Ξ(η) a periodic function of η, which depends also on H0, µ, Pφ or,equivalently, of µ, J, Pφ.

✷ Given the above representation for ξ, the definition of the precession fre-quency is

ωd =ωb

2π

∮

(

ξ + θq) dθ

θ,

E: Demonstrate the last statement. Hint: take the total time derivative of the ξparametric expression and integrate on one full periodic orbit in η.

F. Zonca


✷ Summarizing: introduce the periodic functions Θ(η), Ξ(η) and ρ(η), i.e.without distinction between trapped and circulating particles but keepingin mind that they are different for each class of particles, which depend alsoon H0, µ, Pφ or, equivalently, of µ, J, Pφ

Circulating particles

θ = η +Θ(η) = ωbτ +Θ(η)

r = r + ρ(η)

ξ = ωdτ + (q − q) θ + Ξ(η)

Trapped particles

θ = Θ(η)

r = r + ρ(η)

ξ = ωdτ + (q − q) θ + Ξ(η)

with the definition

q(r + ρ(η)) ≡ q(r) +Q(r + ρ(η))

∮

Qdθ =

∮

(θ/ωb)Qdη = 0

✷ Here, τ is a time-like parameter tracking the position along the trajectory(η ≡ ωbτ), but no time dependence has been assumed so far.

F. Zonca


✷ With the above parametrization of particle trajectories in terms of action-angle variables and characteristic particle frequencies in the equilibriummagnetic configuration, we can systematically decompose the fluctuatingfield in action-angle coordinates, for given constants of motion.

✷ This procedure, which is not the usual practice, corresponds to decomposea mode structure in elements that are actually describing the wave-particleinteractions, once the Fourier decomposition is known. This is somewhattedious, but systematic, and has the advantage of:

• identifying the resonance condition in general geometry and suggest-ing a practical way of computing it

• describing wave-particle resonances and their effect on the particlemotion in the most natural coordinates describing the unperturbedparticle motion

• lifting of f to the particle phase space

F. Zonca


✷ Recall that (see p.3)

f(r, θ, ξ) =∑

m,n

einξei(nq−m)θFm,n(r)

✷ Using the trajectory parametrization summarized on p.8, we have

f(r, θ, ξ) =∑

m,n,ℓ

λn,mei(nωd+ℓωb)τFm,n,ℓ(r) ,

Fm,n,ℓ(r;µ, J, Pφ) =1

2π

∮

exp {inΞ(η) + i [nq(r)−m] Θ(η)}Fm,n(r + ρ(η))e−iℓηdη .

✷ Note that, here, we have made explicit the dependences of Fm,n,ℓ(r;µ, J, Pφ)on µ, J, Pφ. Note also that no specific time dependence has been assumedhere, so far, and that the time-like parameter τ parameterizes (r, θ, ξ) alongthe particle trajectory, as noted before.

F. Zonca


✷ Note also that, because of the finite orbit width ρ(η), the effective modewidth – that of Fm,n,ℓ(r;µ, J, Pφ) – can be significantly larger than that ofthe Fourier harmonic Fm,n(r).

E: Explain qualitatively why the effective mode width can be larger than that ofthe Fourier components and discuss which physics this may impact.

✷ The mode structure decomposition in action angle variables correspondsto a further Fourier decomposition of the amplitudes Fm,n(r) in bounceharmonics (η ≡ ωbτ), as effectively experienced by the particle moving onthe phase space surface given by its constants of motion.

E: Can you recognize this further Fourier decomposition? Can you explain whyit is induced by the periodic functions Θ(η), Ξ(η) and ρ(η)?

F. Zonca


✷ The quantity λn,m is λn,m = 1 for trapped particles, while, for circulatingparticles, it is given by

λn,m = exp [i (nq(r)−m)ωbτ ] .

E: Derive this last equation step by step.

✷ The resonance condition is obtained when, for some partial amplitudes inthe mode structure decomposition, the wave-particle phase is constant alongthe unperturbed orbits, so that, assuming fluctuations ∝ exp(−iωt)

ω = ω(µ, J, Pφ) = nωd + ℓωb

for trapped particles, while, for circulating particles,

ω = ω(µ, J, Pφ) = nωd + ℓωb + (nq(r)−m)ωb .

E: Discuss the physics underlying each term in the resonance conditions above.

F. Zonca


Perturbed particle motion, resonance detuning

and radial decoupling

✷ When considering the effect of fluctuations, for every bounce/transit - i.e. ashift of 2π in the angle η, a resonance detuning will be accumulated becauseof radial nonuniformities and the dependence of ωd and ωb on µ, J, Pφ thatare not conserved anymore in the nonlinear regime.

E: Discuss the role of radial nonuniformities using qualitative arguments.

✷ Moreover, the wave-particle phase in the mode structure decomposition inaction-angle variables will be also shifted, since Θ(η), Ξ(η) and ρ(η) willnot be simple periodic functions of η any longer.

F. Zonca


✷ Fundamental assumption: Assume also that for every bounce/transit theeffect of the nonlinear dynamics is small compared with the equilibriumtrajectory; thus, the nonlinear interaction does not destroy the wave-particleresonances, nor the particle orbit is changed significantly with respect tothat at equilibrium.

✷ This means that the periodic functions Θ(η), Ξ(η) and ρ(η) will be weaklymodified by fluctuations on a single bounce/transit, so that we can neglectthis change at the lowest order.

✷ Note, however, that the cumulative effect of the nonlinear dynamics on manybounce/transit motions can be large and even connected with a secularmechanism, i.e. not bounded in time.

F. Zonca


✷ For trapped particles, we have

θ = Θ(η) + ∆θ ,

∆θ =

∫

δθdθ

θ,

where, it must be kept in mind that the periodic function Θ(η) is differentfor trapped and circulating particles, for it also depends on (µ, J, Pφ).

✷ For circulating particles, we have

θ = ωbτ +Θ(η) + ∆θ +∂ωb

∂Pφ

∫ τ

0

δPφdτ′ +

∂ωb

∂J

∫ τ

0

δJdτ ′ ,

where we have considered η =∫ τωbdτ

′. Meanwhile, δθ, δPφ and δJ de-note changes in the respective quantities that are formally linear in thefluctuation fields, which is the level of accuracy needed here.

F. Zonca


✷ Obviously, rather than computing the variation of ωb with respect to J , wecould take the corresponding variation with respect to H0.

✷ The modification of r is obtained as follows:

r = r + ρ(η) + ∆r ,

∆r =

∫

δrdθ

θ.

✷ Finally, the modification of ξ is:

ξ = ωdτ + (q − q) θ + Ξ(η) + ∆ζ −∆(qθ) +∂ωd

∂Pφ

∫ τ

0

δPφdτ′ +

∂ωd

∂J

∫ τ

0

δJdτ ′

+q

(

∂ωb

∂Pφ

∫ τ

0

δPφdτ′ +

∂ωb

∂J

∫ τ

0

δJdτ ′)

+ ωbdq

dr

∫ τ

0

δrdτ ′ ,

∆ζ =

∫

δζdθ

θ.

F. Zonca


✷ Similarly as above, rather than computing the variation of ωd with respectto J , we could take the corresponding variation with respect to H0.

✷ Note that ∆ζ appears on the right hand side. This is connected with thedefinition of ξ = ζ − qθ = φ− ν(ψ, θ)− qθ (White 89).

✷ The second line in the equation for ξ is to be used for circulating particlesonly. The last term comes from the fact that, when considering the ∝ qηterm in the nonlinear phase, one should see this as ∝

∫ τqωbdτ

′.

E: Explain why the last term in the equation for ξ is connected with the definitionof λm,n on p.12. (Hint: see the following and look at the generalization of λm,n

including fluctuations).

✷ With these results, we can extend the mode structure decomposition inaction-angle variables, including the resonance detuning effect in the pres-ence of fluctuations.

F. Zonca


✷ Summarizing:

f(r, θ, ξ) =∑

m,n,ℓ

λn,mΛn,mei(nωd+ℓωb)τ+iΘNLn,m,ℓFm,n,ℓ(r +∆r;µ, J, Pφ) ,

ΘNLn,m,ℓ = n∆ζ −m∆θ + n

(

∂ωd

∂Pφ

∫ τ

0

δPφdτ′ +

∂ωd

∂J

∫ τ

0

δJdτ ′)

+ℓ

(

∂ωb

∂Pφ

∫ τ

0

δPφdτ′ +

∂ωb

∂J

∫ τ

0

δJdτ ′)

.

✷ Λm,n is the nonlinear extension of λm,n, i.e. Λm,n = 1 for trapped particles,while for circulating particles

Λn,m = exp

[

i (nq(r)−m)

(

∂ωb

∂Pφ

∫ τ

0

δPφdτ′ +

∂ωb

∂J

∫ τ

0

δJdτ ′)

+ inωbdq

dr

∫ τ

0

δrdτ ′]

.

F. Zonca


✷ Thus, given a fluctuation decomposed in Fourier components Fm,n, for com-puting the nonlinear response it is useful to introduce the projection oper-ators Pm,n,ℓ : Fm,n(r) 7→ Fm,n,ℓ(r +∆r;µ, J, Pφ)

Pm,n,ℓ◦Fm,n = Fm,n,ℓ(r;µ, J, Pφ) =1

2π

∮

e{inΞ(η)+i[nq(r)−m]Θ(η)−iℓη}Fm,n(r+ρ(η))dη

✷ In fact, when nonlinear physics enter, the whole new part to be consideredis:

• the wave-particle phase is shifted nonlinearly (resonance-detuning) byΘNLn,m,ℓ and Λn,m.

• the particle can be radially decoupled from the effective mode struc-ture Fm,n,ℓ(r +∆r;µ, J, Pφ)

F. Zonca


✷ Neglecting resonance-detuning (phase locking), the transport process de-scribed is mode-particle pumping (White 83). Thus, the nonlinear dynamicsis dominated by radial decoupling.

E: Discuss the structure of the term Λm,n and comment about possible reasonswhy trapped and circulating particles should behave differently when consideringnonlinear behaviors in nonuniform systems.

E: Using the results above, provide a qualitative estimate of the drive strength,which is needed to expect a transition of the system behavior from local to non-local.

✷ Due to the ordering |ω0τNL|−1 ∼ |γL/ω0|, the processes involved in nonlinear

wave-particle resonances can be understood as cumulative effects of bounce-averaged responses on linear particle motions.

F. Zonca


Circulating vs. trapped particle resonances

✷ From energetic particles equations of motion one generally has δr/r ∼δPφ/Pφ ∼ (ω∗E/ω)δH0/H0. Thus, not only radial decoupling but also res-onance detuning for energetic particles is expected to be dominated by thenonlinear radial displacement, since |ω∗E/ω| ≫ 1.

✷ Specializing, for simplicity, to the case of shifted circular magnetic surfaces

Θm,n,l ≃ ndq

drωt∆r −∆ω

for circulating particles, having denoted ωb as ωt and neglected ωd, whereas,for trapped particle resonance

Θm,n,l ≃

(

n∂ωd

∂r+ ℓ

∂ωb

∂r

)

∆r −∆ω .

F. Zonca


✷ These relations allow estimating the finite interaction time, τNL, and finiteinteraction length, ∆rL, for resonant particles

τNL ∼ Θ−1m,n,ℓ ∼ (3γL)

−1

3γL ∼ nq′ω∆rL , 3γL ∼ ω(∆rL/r) ,

(circulating) (trapped)

✷ Here, the characteristic scale length of ωd and ωb radial profiles as ∼ r.

E: For high mode numbers, the finite interaction length for magnetically trappedparticle is significantly larger than for circulating particles. Explain why and howthis is connected with magnetic shear.

E: Explain why, in these conditions, we can expect that circulating particle trans-port is diffusive, while trapped particle transport is convective. Is this true alsoat low mode number?

F. Zonca


✷ Finite interaction time and finite interaction length can be further increasedfor modes which readjust their frequency to preserve resonance as particleare redistributed radially (|ω∗E/ω| ≫ 1) ⇒ phase-locking and mode particlepumping [White 83].

∆ω ≃ ndq

drωt∆r ,

∆ω ≃

(

n∂ωd

∂r+ ℓ

∂ωb

∂r

)

∆r .

✷ The extended interaction lengths are

3γL ∼ nq′ωǫω∆rL , 3γL ∼ ωǫω(∆rL/r) ,

(circulating) (trapped)

with ǫω < 1 depending on the wave dispersive properties.

F. Zonca


Local saturation (weak drive)

✷ The typical ∆rL (separatrix width of particles trapped within the wave)is small compared with the AE mode structure at short scales (Berk andBreizman 90), e.g., the fine TAE structure at ∆r/r ≈ δ/(nrq′) ≈ δ/(nqs),δ ≈ r/R0 (Cheng, Chen and Chance 85). (ω = ωd for simplicity)

|∆rL|<∼ |∆r| ⇒γLωdk

ǫ−1ω<∼

δ

nqs⇒

γLω<∼

ǫωδ

nqs

✷ Local mode structure becomes important as AE mode structure of the singlepoloidal harmonics: theories at marginal stability (Berk and Breizman 90)need refinement. Note ǫω ∼ (ω/ω∗E) ≈ r/R0 for EPM, while ǫω ≈ 1 for AE.

|∆rL|<∼ |∆r| ⇒γLωdk

ǫ−1ω<∼

1

nqs⇒

γLω<∼

ǫωnqs

F. Zonca


E: Discuss the validity limits of (Berk and Breizman 90) analyses using theseestimates and quantify how strong the drive should be to make improper theuse of simplified 1D description of nonlinear AE dynamics. What happens forcirculating particles? What happens for strong drive?

F. Zonca


Global saturation (strong drive)

✷ Assume that local saturation is not possible, i.e. (from previous page)ǫ−1ω (γL/ω) ≈ (γL/ω)(ω∗E/ω)>∼ (nqs)−1

✷ Fast ion avalanches (see Lecture 6) and EPM convective amplification is trig-gered when local saturation is not possible (Zonca et al 05). Still |∆rL| isless than the EPM radial envelope width ≈ (LpE/kθ)

1/2, with LpE the char-acteristic energetic particle pressure scale length (Zonca and Chen 96/00)

r

nqs<∼ |∆rL|<∼

∣

∣

∣

∣

LpE

kθ

∣

∣

∣

∣

1/2

⇒1

nqs<∼γLω

ω∗E

ω<∼

LpE/r

|kθLpE|1/2

✷ For even stronger drive, strong avalanching is expected, with macroscopicdistortions of the EP distribution function (Zonca et al 05)

γLω

ω∗E

ω>∼

LpE/r

|kθLpE|1/2

F. Zonca

http://www.afs.enea.it/zonca/references/seminars/IPPGarching_winter13/lecture6.pdf


References and reading material

A. J. Brizard and T. S. Hahm, Rev. Mod. Phys. 79, 421 (2007).

R. G. Littlejohn, J. Plasma Phys. 29, 111 (1983).

T. G. Northrop, Adiabatic Motion of Charged Particles (Wiley, New York) (1963).

Z.X. Lu, F. Zonca and A. Cardinali, Phys. Plasmas 19, 042104 (2012).

R. B. White, Theory of Tokamak Plasmas, North Holland (1989).

R. B. White, R. J. Goldston, K. McGuire, A. H. Boozer, D. A. Monticello andW. Park, Phys. Fluids 26, 2958 (1983).

Y. Xiao and Z. Lin, Phys. Plasmas 18, 110703 (2011).

A. J. Lichtenberg and M. A. Lieberman, Regular and Stochastic Motion, Springer-Verlag (1983).

A. J. Lichtenberg and M. A. Lieberman, Regular and Chaotic Dynamics, Second

F. Zonca


Edition, Springer-Verlag (2010).

H.L. Berk and B.N. Breizman, Phys. Fluids B 2, 2226 (1990).



C.Z. Cheng, L. Chen and M.S. Chance, Ann. Phys. 161 21 (1985).

F. Zonca and L. Chen, Phys. Plasmas 3, 323 (1996).

F. Zonca and L. Chen, Phys. Plasmas 7, 4600 (2000).

F. Zonca and L. Chen, Plasma Phys. Control. Fusion 48 537 (2006).

L. Chen and F. Zonca, Nucl. Fusion 47, S727 (2007).

L. Chen and F. Zonca, Phys. Scr. T60, 81 (1995).

F. Zonca, S. Briguglio, L. Chen, G. Fogaccia and G. Vlad, Nucl. Fusion 45, 477(2005).

F. Zonca

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Lecture 5Max-Planck-Institutfu¨rPlasmaphysikLectureSeries-Winter2013 Lecture5– 6 E: From the equations of motions in the equilibrium magnetic ﬁeld show that the particle trajectories