1 IFTSIntensiveCourseonAdvancedPlasmaPhysics ...IFTSIntensiveCourseonAdvancedPlasmaPhysics-Spring2012 Lecture2– 4 Resonanceconditionsandequilibriumparticlemo-tion in toroidal geometry

IFTS Intensive Course on Advanced Plasma Physics-Spring 2012 Lecture 2 – 1

Lecture 2

Hamiltonian mappings for toroidal systems: numerical implementation for

synthetic diagnostics and analytic forms of Hamiltonian mappings for

theoretical studies

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.

May 3.rd, 2012

IFTS Intensive Course on Advanced Plasma Physics-Spring 2012,Nonlinear charged particle dynamics in tokamaks: theory and applications

2–12 May 2012, IFTS – ZJU, Hangzhou

F. Zonca


General notion of mapping(A short review of Spring 11 Lecture 5)

✷ Visualization of multiply periodic motions is important for increased under-standing.

✷ Topological considerations that can be made, naturally yield to a set of dif-ference equations, that can be seen as mapping of the dynamical trajectoryonto a subset of the system phase space.

✷ Not only visualization (and understanding) are often simpler via mappingequations. Physics properties, such as numerical calculation of stochasticbehaviors, are often simpler when considered for mappings.

✷ Conversely, regular trajectories are very conveniently described in terms ofdifferential equations.

✷ Spring 11 Lecture 5 is concerned with the conversion of differential equa-tions into mappings and vice-versa.

F. Zonca

http://www.afs.enea.it/zonca/references/seminars/IFTS_spring11/lecture2.pdf



Problems that yield to mapping

✷ Mapping arise in various ways. Notably as:

• as physical formulation of a dynamical problem, as in the case of Fermiacceleration or the twist mapping/standard mapping

• considering successive intersections of a dynamical trajectory with asurface of section (see Spring 11 Lecture 2)

• as approximation of the motion, valid for a limited time scale or rangeof the phase space; e.g., the periodic perturbed pendulum (see p. 26Spring 11 Lecture 5)

✷ Spring 11 Lecture 5 focuses on systems with two degrees of freedom, butcontains useful extensions to more general cases.

F. Zonca





Resonance conditions and equilibrium particle mo-

tion in toroidal geometry

✷ This section is aimed at deriving the formalism for constructing simplifiedanalytic expressions for Hamiltonian mapping, to be used for comparisonwith the numerical diagnostics from simulation results and for gaining fur-ther insights into the transport processes.

✷ 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



✷ 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


✷ Recall the definition the canonical angle conjugate to J (see Lecture 1),obtained from integration along the particle orbit as

η = ωb

∫ θ

0

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 (see p.15 ofLecture 1) show that the particle trajectories in the (ψ, θ) plane are closed anddiscuss how this implies the 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

ωd =ωb2π

∮

(

ξ + θq) dθ

θ,

given on p.18 of Lecture 1, is justified.

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

F. Zonca


✷ Recall that (see p.5)

f(r, θ, ξ) =∑

m,n

einξei(nq−m)θFm,n(r)

✷ Using the trajectory parametrization summarized on p.10, 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 and resonance

detuning

✷ 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. (seealso Lecture 4 and Lecture 5).

✷ 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.

✷ From Lecture 1 we know the expressions for δψ, δθ, δξ (see p.23) and forH0, Pφ and J (see pp.25-26 and 28-29).

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: Derive by yourself the expressions for (r, θ, ξ) in the presence of fluctuations,following the details of the algebra step by step.

E: Show that it is indeed ∆ζ that should appear on the right hand side andnot ∆φ or ∆ξ. From the expressions on p.23 of Lecture 1 derive the followingexpression for δζ

δζ = δφ−∂ν

∂θδθ −

∂ν

∂ψδψ

F. Zonca



δφ = −cB0

B∗‖

|∇ψ|2

B20R

2

[(

∂

∂ψ〈δφ〉 −

v‖c

∂

∂ψ

⟨

δA‖

⟩

)

−

(

∂ν

∂ψ+ qψθ

)(

∂

∂ξ〈δφ〉 −

v‖c

∂

∂ξ

⟨

δA‖

⟩

)]

−cB0

B∗‖

∇ψ ·∇θ

B20R

2

[(

∂

∂θ〈δφ〉 −

v‖c

∂

∂θ

⟨

δA‖

⟩

)

−FJ

R2

(

∂

∂ξ〈δφ〉 −

v‖c

∂

∂ξ

⟨

δA‖

⟩

)]

.

E: Using the chain rules, interpret the partial derivatives above in the toroidalcoordinate system (ψ, θ, φ).

E: Explain why the last term in the equation for ξ is connected with the definitionof λm,n on p.14. (Hint: see the following and look at the generalization of λm,nincluding 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;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


✷ Note that the correction due to the δr dependence in the Fm,n function –see p.12 – has not been considered, for it would yield nonlinear behaviorsthat are second order in the fluctuation amplitudes and, thus, are beyondour present scopes.

✷ This means that the Fm,n,ℓ(r;µ, J, Pφ) functions remain the same and al-ready contain all essential ingredients for computing wave-particle interac-tions to the lowest order.

✷ In fact, when nonlinear physics enter, the whole new part to be considered ishow the wave-particle phase is shifted nonlinearly (resonance-detuning) byΘNLn,m,ℓ and Λn,m. Neglecting resonance-detuning, the transport processdescribed is mode-particle pumping (White 83).

✷ 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;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

F. Zonca


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. (see also Lecture 6 and the work byZhichen Feng and Xiao and Lin 11).

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.

E: Write your own code for numerical calculations of the equations for the chargedparticle motions discussed in Lectures 1 and 2.

F. Zonca



Hamiltonian forms and mappings(from Spring 11 Lecture 5; to be continued in Lecture 3)

✷ In two dimensions, the functions f and g in the perturbed twist mappingare determined from integration of the Hamilton’s equations

dJ1dt

= −ǫ∂H1

∂θ1;

{

Jn+1 = Jn + ǫf(Jn+1, θn)θn+1 = θn + 2πα(Jn+1) + ǫg(Jn+1, θn)

✷ The jump in action over one period in the θ2 motion is ∆J1 = ǫf(Jn+1, θn),using the unperturbed motion for the integration

∆J1 = −ǫ

∫ 2π/ω2

0

∂H1

∂θ1(Jn+1, J2, θn + ω1t, θ20 + ω2t)dt

E: Discuss the analogy of using the unperturbed motion for obtaining the jumpin action, used here, and the use of the projection operators on p.22.

F. Zonca




✷ The jump is phase can be obtained from the area preserving condition

g(J, θ) = −

∫ θ ∂f

∂Jdθ′

✷ For computation of ∆J1, the second degree of freedom can be expressed ingeneric (not necessarily action-angle) canonical coordinates.

∆J1 = −ǫ

∫ 2π/ω2

0

∂H1

∂θ1(Jn+1, θn + ω1t, p2(t), q2(t))dt

✷ This method can be generalized to N degrees of freedom

∆J = −ǫ

∫ 2π/ω2

0

∂H1

∂θ(Jn+1, JN , θn + ωt, θN0 + ωN t)dt

✷ One can then derive the jumps of N − 1 actions ǫf = ∆J . One then findsg = ∂G/∂Jn+1, knowing G from direct integration of f = −∂G/∂θn.

F. Zonca


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).

F. Zonca

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1 IFTSIntensiveCourseonAdvancedPlasmaPhysics ...IFTSIntensiveCourseonAdvancedPlasmaPhysics-Spring2012 Lecture2– 4 Resonanceconditionsandequilibriumparticlemo-tion in toroidal geometry