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IFTS Intensive Course on Advanced Plasma Physics-Spring 2019 Lecture 4 – 1

Lecture 4

The linear gyrokinetic equation and the global dispersion

relation of Alfven waves in toroidal geometry

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.

June 12.th, 2019

Kinetic theory and global dispersion relationof Alfven waves in tokamaks

June 4 – 20, 2019, IFTS – ZJU, Hangzhou

Fulvio Zonca


Gyrokinetic orderings

So far, we have investigated SAW spectral properties based on the simplifiedideal MHD fluid description.

This is, however, inadequate for reliable and accurate understanding of SAWdynamics in high-temperature collisionless fusion plasmas:

• the “singular” behaviors associated with the ubiquitous SAW contin-uum is a manifestation that ideal MHD description ignores micro-scopic scales such as the finite ion Larmor radius

• microscopic-scale dynamics ⇒ SAW is modified into the so-called ki-netic Alfven wave (KAW) and the SAW continuum becomes a densebut regular discrete spectrum

• crucial roles of wave-particle interactions: instability drive by ener-getic particles and damping/drive by thermal electrons/ions

• crucial roles of plasma compression in the low-frequencySAW spectrum

Fulvio Zonca


As the SAW frequency is generally much less than the charged particle’scyclotron frequencies, the cyclotron motion can be suitably averaged out,and the kinetic theory can, therefore, be based on a reduced dynamic de-scription.

Gyrokinetic theory, and the corresponding orderings for temporal and spa-tial scales:

|ρi/LB| ∼ ǫB , |ρi/Lp| ∼ ǫF ;

|ω/Ωi| ∼ ǫω , |k⊥ρi| ∼ ǫ⊥ ,∣

∣k‖/k⊥

∣

∣ ∼ ǫω/ǫ⊥ .

Physically, the smallness of ǫB and ǫF are obviously based on the argumentthat the plasmas of interest here are well confined.

Fulvio Zonca


That ǫ⊥ can be O(1), instead, is based on the argument that SAW insta-bilities are typically excited via plasma inhomogeneities (i.e., similar to thewell-known drift waves) and, thus, the growth rates tend to increase withk⊥ and are limited by the orbital averaging due to the finite ion Larmorradii.

Finally, the |k‖/k⊥| ordering enters because:

• thermal ion Landau damping tends to limit the maximum excitable|k‖| to |k‖| ∼ |ω|/vti

• finite k‖ tends to stabilize SAW instabilities via magnetic field-linebending

Fulvio Zonca


The linear gyrokinetic equation

Introduce the notion of “guiding-center” of the particle trajectory in theequilibrium magnetic field B0.

Particles guiding-center orbit can be described by the energy per unit mass,E = v2/2, and the magnetic moment adiabatic invariant, µ, which, at theleading order, can be expressed as µ = v2⊥/(2B0) + . . ..

Particle response can be obtained from the guiding-center response bythe “pull-back” transformation from guiding-center to particle coordinates,e−ρ·∇, with ρ ≡ Ω−1b × v denoting the particle position in the guiding-center frame and Ω = eB0/(mc) the cyclotron frequency.

Separation of adiabatic response of the particle distribution function (allterms that are not acted upon by e−ρ·∇) from the non-adiabatic responseof the guiding-center distribution.

Fulvio Zonca


Within this framework, one can derive the following expression for the linearperturbed distribution function, δf

δf = e−ρ·∇

[

δg −e

m

1

B0

∂F0

∂µ〈δLg〉

]

+e

m

[

∂F0

∂Eδφ+

1

B0

∂F0

∂µδL

]

.

Here, F0 is the equilibrium guiding-center particle distribution function,

δLg = δφg −v‖cδA‖g = eρ·∇δL = eρ·∇

(

δφ−v‖cδA‖

)

,

and 〈· · · 〉 denotes gyrophase averaging.

Note here that, consistent with the β ≪ 1 and suppression of compressionalAlfven wave (CAW) approximations, we have neglected the δB‖ term in δL.

Fulvio Zonca


The non-adiabatic response δg is obtained from the solution of the lineargyrokinetic equation [Antonsen 80, Catto 81, Frieman & Chen 82]

(

∂

∂t+ v‖∇‖ + vd ·∇⊥

)

δg = −

(

e

m

∂

∂t〈δLg〉

∂F0

∂E+

c

B0

b×∇ 〈δLg〉 ·∇F0

)

.

Here, the magnetic drift velocity vd is

vd =b

Ω×(

µ∇B0 + κv2‖)

≃

(

µB0 + v2‖

)

Ωb× κ ,

where ∇B0 ≃ κB0 in the low-β limit.

As in the ideal MHD analysis, this is consistent with well-known cancel-lations in the linear vorticity equation, arising from plasma equilibriumcondition and the perpendicular pressure balance

∇⊥

(

B0δB‖ + 4πδP⊥

)

≃ 0 ,

when anisotropic pressure response is considered [Chen 91].

Fulvio Zonca


In the long wavelength limit the last term on the right hand side of the GKEequation has to be slightly modified to accurately account for the perturbedparticle motion in the perpendicular fluctuating magnetic field[Chen RMP 16]

c

B0

b×∇ 〈δLg〉 →c

B0

b×∇ 〈δLg〉+v‖B0

κ⟨

δA‖g

⟩

=c

B0

b×∇ 〈δφg〉+ v‖〈δB⊥g〉

B0

,

with 〈δB⊥g〉 = ∇× b⟨

δA‖g

⟩

.

The field equations, meanwhile, are the quasineutrality condition and thegyrokinetic vorticity equation, which are derived below.

Fulvio Zonca


Quasi-neutrality condition

Due to the gyrokinetic orderings, k2λ2D ∼ λ2D/ρ2i ∼ Ω2

i /ω2pi ≪ 1, with λD the

Debye length and ωpi the ion plasma frequency. Thus, Poisson’s equationbecomes approximately the quasineutrality condition

∑

e 〈δf〉v = 0 ,

where∑

implicitly indicates summation on all particle species and 〈. . .〉vdenotes integration in velocity space.

This equation can be readily rewritten as

∑

⟨

e2

m

∂F0

∂E

⟩

v

δφ+∑

〈eJ0(λ)δg〉v

+∇ ·∑

⟨

e2

m

2µ

Ω2

∂F0

∂µ

(

J20 (λ)− 1

λ2

)⟩

v

∇⊥δφ = 0 .

Fulvio Zonca


Here, we have assumed that most of the equilibrium current is carried byelectrons and that relevant wavelengths are much longer than the electronLarmor radius.

Furthermore,⟨

e−ρ·∇⟩

= J0(λ), J0(λ) is the Bessel function and λ2 =2µB0k

2⊥/Ω

2.

E: Demonstrate that the approximations needed for deriving the quasi-neutralityconditions are justified.

Here, we have followed [Qin 98] to properly account for the correct recoveryof the long wavelength limit λ2 = ǫ2⊥ → 0.

For the finite λ case, spatial scale separation between fluctuations and equi-librium profiles applies and, therefore, J0(λ) can be made commute withequilibrium quantities at the leading order. This greatly simplifies formalanalyses, although, in general, equilibrium nonuniformities can be rigor-ously accounted for at all orders [Brizard 07].

Fulvio Zonca


As usual, the gyrophase average involves the introduction of Bessel functionsas integral operators, e.g.

〈δLg〉 = J0(λ)δL = J0(λ)(

δφ−v‖cδA‖

)

.

The definition of J0(λ) acting on a generic functiong(r) =

∫

g(k) exp(ik · r)dk is

J0(λ)g(r) ≡

∫

eik·rJ0(λ)g(k)dk .

From this definition and from the presence of velocity space integrals in-volving δg, one notes that governing equations are integro-differential in na-ture, reflecting the nonlocal plasma response due to finite orbit excursionson scale lengths that may be of the same order of the mode wavelengthand, in some cases, of the characteristic equilibrium scales, especially whensupra-thermal particles are involved.

Fulvio Zonca


Gyrokinetic vorticity equation

The gyrokinetic vorticity equation (formally ∇ · δJ = 0) is derived from avelocity space integral of the GKE, acted upon by e−ρ·∇ and

∑

e.

Looking at various contributions separately, for the inertia term one finds

∑

⟨

eJ0∂

∂tδg

⟩

v

+∑

⟨

e2

mJ0

[

∂F0

∂EJ0∂

∂tδφ

]⟩

v

=

−∇ ·∑

⟨

e2

m

2µ

Ω2

(

B0∂F0

∂E+∂F0

∂µ

)(

J20 − 1

λ2

)⟩

v

∇⊥∂

∂tδφ ,

where we have omitted the λ dependence of J0 for brevity of notation andnoted the quasi-neutrality condition.

E: Fill in the missing details to derive the inertia response.

Fulvio Zonca


Meanwhile, for the gyrokinetic diamagnetic response, we have

∑

e

⟨

J0

[

c

B0

b×∇ (J0δφ) ·∇F0

]⟩

v

=

−∑

ecb×∇

⟨

2µ

Ω2F0

(

J20 − 1

λ2

)⟩

v

·∇∇2⊥δφ .

Note that here, as for the quasi-neutrality condition, we have followed [Qin98] to properly account for the correct recovery of the long wavelength limitλ2 = ǫ2⊥ → 0.

E: Fill in the missing details to derive the diamagnetic response. Discuss whereyou have to invoke equilibrium charge neutrality condition.

Fulvio Zonca


Considering now the gyrokinetic kink term, we have

∑

e

⟨

J0

[

v‖B0

〈δB⊥g〉 ·∇F0

]⟩

v

= δB⊥ ·∇

(

j‖0B0

)

,

so that the gyrokinetic kink term reduces to the ideal MHD expression.

E: Fill in the missing details to derive the gyrokinetic kink response. Whatwould you need to change in this expression, if energetic particles are contributingimportantly to the equilibrium current? Why?

Here, again, the total parallel current is assumed to be carried by electrons.This means that this expression must be modified when, e.g., a significantfraction of the equilibrium current is carried by energetic ions.

Fulvio Zonca


The gyrokinetic line bending term can be written as [Chen 91]

∑

e⟨

J0[

v‖∇‖δg]⟩

v= B0∇‖

(

σkδj‖B0

)

≃ B0∇‖

(

δj‖B0

)

,

where ∇‖ ≡ b ·∇, the gyrokinetic firehose stability term

σk = 1 +4π

c2

∑

⟨

e2

m

v2‖B0

∂F0

∂µ

(

1− J20

k2⊥

)

⟩

v

≃ 1 ,

for low-β plasmas and we have consistently dropped terms ∝ b ·∇J0.

E: Fill in the missing details to derive the gyrokinetic line bending term. What arethe conditions that you need to assume to approximate the gyrokinetic firehosestability term?

Fulvio Zonca


Finally, the ballooning-interchange contribution

∑

e 〈J0 [vd ·∇δg]〉v ≃c

B0

b× κ ·∇∑

⟨

m(

µB0 + v2‖)

J0δg⟩

v,

is the natural gyrokinetic extension of the ideal MHD response obtainedfrom the Chew–Goldberger–Low pressure tensor [Chew 56].

Combining terms, the linear GK vorticity equation can be cast as

B0∇‖

(

δj‖B0

)

−∇ ·∑

⟨

e2

m

2µ

Ω2

(

B0∂F0

∂E+∂F0

∂µ

)(

J20 − 1

λ2

)⟩

v

∇⊥∂

∂tδφ

−∑

ecb×∇

⟨

2µ

Ω2F0

(

J20 − 1

λ2

)⟩

v

·∇∇2⊥δφ + δB⊥ ·∇

(

j‖0B0

)

+c

B0

b× κ ·∇∑

⟨

m(

µB0 + v2‖)

J0δg⟩

v= 0 .

Fulvio Zonca


Here, δj‖ is related to δA via the parallel component of the low-frequencyAmpere’s law; i.e., adopting the ∇ · δA = 0 Coulomb gauge,

δj‖ =c

4πb ·∇× (∇× δA)

=c

4π

[

−∇2 + κ2 + (∇b) : (∇b)]

δA‖

+(∇× b)‖ δB‖ + (∇b) : (∇δA⊥) +∇ · [(∇b) · δA⊥]

+(κ ·∇b) · δA⊥ + (b ·∇δA⊥) · κ .

Applying the linear gyrokinetic ordering and neglecting δB‖ and δA⊥ in theβ ≪ 1 limit, δj‖ reduces to (with ∇⊥ = ∇− b∇‖)

δj‖ ≃ −c

4π∇2

⊥δA‖ .

E: Demonstrate the identities for δj‖.

Fulvio Zonca


The general fishbone like dispersion relation

From Lecture 3, recall that we constructed the kinetic SAW energy func-tional assuming the trial functions [δφt, δA‖,t, δB‖,t] are chosen such that theperpendicular pressure balance (p. 7) and the quasineutrality conditions (p.9) are satisfied.

Using the MSD representation (Lecture 1)

F [δφ, δA‖] = (2π)3∑

j∈Z

∫ a

0

drdψ/dr

2ei2πnqj

∫ ∞

−∞

J dϑ

[

−∇×(

b0c∂−1t ∇‖δφ

†(r, ϑ+ 2πj))

·δB⊥(r, ϑ)

4π

+∂−1t δφ†(r, ϑ+ 2πj)∇ · δJ⊥(r, ϑ)

]

= δW − δI .

This expression easily allows us to separate inertia from potential energy,based on spatial scale separation, is connected with |kr| → ∞ as |ϑ| → ∞.

Fulvio Zonca

http://www.afs.enea.it/zonca/references/seminars/IFTS_spring19/lecture3.pdf



From Lecture 3, assuming that singular structures occur at one isolatedposition r = r0, the expression for the inertia becomes

δI = 2π2

[

|kϑ|c2(dψ/dr)

JB20 |s|

2

]

r=r0,ϑ=0

∣

∣

∣φsn0+

∣

∣

∣

2

iΛ|s|

iΛ =

(

2∣

∣

∣φsn0+

∣

∣

∣

2)−1

[

φ†sn(r, ϑ)∂ϑφsn(r, ϑ)

]ϑ→0+

ϑ→0−.

Noting that, in the regular regions, δE‖ = 0, that is δA‖ = −c∂−1t ∇‖δφ,

δW = 2π2

[

|kϑ|c2(dψ/dr)

JB20 |s|

2

]

r=r0,ϑ=0

∣

∣

∣φsn0+

∣

∣

∣

2 (

δWf + δWk

)

.

δW = limϑ1→∞

(2π)3∑

j∈Z

∫ a

0

drdψ/dr

2ei2πnqj

∫ ϑ1

−ϑ1

J dϑ

[

δB†⊥(r, ϑ+ 2πj) · δB⊥(r, ϑ)

4π

+∂−1t δφ†(r, ϑ+ 2πj)∇ · δJ⊥(r, ϑ)

]

≡ δWf + δWk

Fulvio Zonca



E: Comment these results in comparison with those of Lecture 3. In particular,discuss where you may expect kinetic effects to be important.

RP: Elaborating further on the previous E, comment on how kinetic effects mayenter changing the generalized inertia. What is the role of finite Larmor radii?And of wave particle resonances? Solve the large-|ϑ| vorticity equation (shortradial scale) and calculate explicitly the generalized inertia. Use these results asboundary condition to solve for the parallel mode structure in the ideal regionand compute δWf + δWk. Can you distinguish between fluid vs. kinetic effects?

Combining these results, the general fishbone like dispersion relation(GFLDR) [Z & C 2014; C & Z 2016] can be written as

i|s|Λ = δWf + δWk

This global dispersion relation is very general and was introduced for thefirst time in the classic work by [CWR 1984] for the investigation of fishboneoscillations. Its general implications are discussed in detail by [Z & C 2014;C & Z 2016]. Kinetic effects are included in Λ and δWk via ∇ · δJ⊥.

Fulvio Zonca



The GFLDR for short wavelength modes

Reconsider the general inertia expression

δI = 2π2∑

r0

[

k2ϑc2(dψ/dr)

JB20 |skϑ|

]

r=r0,ϑ=0

∣

∣

∣φsn0+(r0)

∣

∣

∣

2

iΛ(r0) .

and specialize it to short wavelength modes.

Recall that the∑

r0, extended to all resonances (non only a single isolated

one), is due to the presence of singular contributions at nq − ℓ − Reν, dueto the Poisson summation formula

∑

j∈Z

ei2πj(nq−ν∗) =∑

ℓ∈Z

δ(nq − ν∗ − ℓ) =∑

ℓ∈Z

|skϑ|−1δ(r − r0 − ℓ/nq′)

As the fine radial structures are accounted for by integration in ϑ (theparallel mode structure) the remaining radial variations must occur on themesoscales. We can thus replace

∑

r0

|skϑ|−1 ... (r0) =

∫ a

0

... (r)dr .

Fulvio Zonca


Using this method, the general inertia expression specialized to short wave-length modes becomes

δI = 2π2

∫ a

0

dr

[

k2ϑc2(dψ/dr)

JB20

]

ϑ=0

∣

∣

∣φsn0+(r)

∣

∣

∣

2

iΛ(r) ,

with the expression for Λ remaining the same and being calculated at agiven flux surface r.

E: Comment the previous equation as intuitive form for the general inertia ex-pression and compare it with the previous one. What are the differences?

The same procedure can be adopted for the potential energy expression.

δW = (2π)3∑

j∈Z

∫ a

0

drdψ/dr

2ei2πnqj

∫ ∞

−∞

J dϑ

[

1

4πδB†

⊥n(r, ϑ+ 2πj)

·δB⊥n(r, ϑ) + ∂−1t φ†

n(r, ϑ+ 2πj)∇ · δJ⊥n(r, ϑ)]

.

Fulvio Zonca


Because of the phase factor ∝ ei2πnqj , the integral will rapidly vanish for|nq| ≫ 1 unless j = 0. Thus, the leading order expression is

δW = 2π2

∫ a

0

drdψ/dr

2

∫ ∞

−∞

J dϑ

[

∣

∣

∣δB⊥n(r, ϑ)

∣

∣

∣

2

+ 4π∂−1t φ†

n(r, ϑ)∇ · δJ⊥n(r, ϑ)

]

= 2π2

∫ a

0

dr

[

k2ϑc2(dψ/dr)

JB20

]

ϑ=0

∣

∣

∣φsn0+(r)

∣

∣

∣

2(

δWf (r) + δWk(r))

.

The equation above is the definition of the local contribution to the fluctu-ation potential energy δWf + δWk.

The final expression for δW (r) = δWf(r) + δWk(r) is

δW (r) =[JB2

0 ]ϑ=0/(k2ϑc

2)

2∣

∣

∣φsn0+(r)

∣

∣

∣

2

∫ ∞

−∞

J dϑ

[

∣

∣

∣δB⊥n(r, ϑ)

∣

∣

∣

2

+ 4π∂−1t φ†

n(r, ϑ)∇ · δJ⊥n(r, ϑ)

]

.

Fulvio Zonca


Compare this expression with the result given in Spring 2017 Lecture 3. Can yourecover the result using the general expression?

Collecting results, we can rewrite the SAW energy functional as

F [δφ, δA‖] = δW−δI = 2π2

∫ a

0

dr

[

k2ϑc2(dψ/dr)

JB20

]

ϑ=0

∣

∣

∣φsn0+(r)

∣

∣

∣

2[

δW (r)− iΛ(r)]

.

This equation suggests that global SAW at short wavelength can be viewedas superposition of radial wave packets, satisfying the WKB dispersion re-lation (GFLDR) [Z & C 2014, Chen RMP 2016]

D(r, kr, ω) = δWf (r, kr, ω) + δWk(r, kr, ω)− iΛ(r, ω) = 0 .

Note that here, explicit dependences on mode frequency and radial wavenumber have been shown explicitly.

E: In general Λ(r, ω) does not depend on kr. Why? Why can δWf depend on ω?What about the limiting case of low-frequency ideal MHD?

Fulvio Zonca



E: Reconsider the expression for δW as a weighted integral over δW and the ex-pression for δW introducing δW . Show that δW = |s|δW for a radially localizedmode, where equilibrium is slowly varying and the mode is considered constantover the mode width |nq′|−1 = |skϑ|

−1.

The general result obtained so far (GFLDR) [Z & C 2014, Chen RMP 2016],allows us to study the propagation and stability properties of long as wellas short wavelength SAW in toroidal geometry.

The remaining part of this series of lectures will be focused on discussingthe properties of short wavelength SAW, using the general theory of wavepropagation in slowly varying, weakly non-uniform media; with applicationto the description of Alfven waves in tokamaks.

Fulvio Zonca


Eikonal formulation of wave equations

The WKB eigenvalue problem, generated by the field equations above withoutgoing wave boundary conditions for |ϑ| → ∞ (no energy source at |kr| →∞), with A(r, t) ≡ e(r, t)A(r, t), can be written as

D(r, t, kr, ω) ·A(r, t)eiS(r,t) = 0 ,

where A(r, t) and the polarization vector e(r, t) are two-component columnvectors corresponding to (δφ, δA‖) (kinetic theory) or (Φs, δP comp) in theideal MHD fluid limit (cf. Spring 2018 Course).

The D(r, t, kr, ω) operator in is generally a linear integral operator, and thecorresponding wave equation can be cast as

∫

DDD(r, t, r − s, t− τ) ·A(r − s, t− τ)eiS(r−s,t−τ)dsdτ = F (r, t) .

Fulvio Zonca

http://www.afs.enea.it/zonca/references/seminars/IFTS_spring18/


For the sake of generality, we have added a F (r, t) term on the r.h.s., whichcan describe external forcing as well as nonlinear interactions [Chen RMP16].

As D(r, t, kr, ω) is generally complex, we separate real and imaginary parts

D(r, t, kr, ω) = DR(r, t, kr, ω) + iDI(r, t, kr, ω) ,

and assume that

DR(r, t, kr, ω) = D0R(r, t, kr, ω) + εD1

R(r, t, kr, ω) + ... ,

DI(r, t, kr, ω) = εD1I (r, t, kr, ω) + ... .

At the lowest order, O(1), the wave equation yields

D0R(r, t, kr, ω) ·A(r, t) = 0 .

Fulvio Zonca


In the example considered here, D0R(r, t, kr, ω) is a 2×2 matrix that admits

two independent orthonormal eigenvectors, ei(r, t, kr, ω) [i ∈ (1, 2)], andcan, thus, be reduced to diagonal form.

In particular,

D0R(r, t, kr, ω) =

∑

i=1,2

D0Ri(r, t, kr, ω)eie

+i ,

with

D0Ri(r, t, kr, ω) ≡ e+

i (r, t, kr, ω) ·D0R(r, t, kr, ω) · ei(r, t, kr, ω) .

In the present analysis, D0Ri(r, t, kr, ω) is the lowest order GFLDR for SAW

in tokamaks

D(r, kr, ω) = δWf (r, kr, ω) + δWk(r, kr, ω)− iΛ(r, ω) = 0

Fulvio Zonca


The two independent solutions represent SAW and SSW polarizations, aswell as their kinetic extensions (cf. Lecture 5 and Lecture 6).

Mode dispersion relation (GFLDR) as well as polarization and parallel modestructures are computed from vorticity and quasineutrality (and/or pressurebalance; cf. Lecture 2) equations.

With this information, one can calculate the global mode dispersion relationby means of solution of the radial envelope equation (Spring 2018 Course)∂

∂t

(

∂D0R

∂ωA2

)

−∂

∂r

(

∂D0R

∂krA2

)

+ 2D1AA

2 − 2iD1RA

2

+iA

(

∂2D0R

∂k2r+ 2

∂e+

∂kr·D0

R ·∂e

∂kr

)

∂2A

∂r2= −2ie−iSAe+ · F

−

(

e+ ·d

dte−

d

dte+ · e

)

∂D0R

∂ωA2 +

(

∂e+

∂ω·D0

R ·∂e

∂t−∂e+

∂t·D0

R ·∂e

∂ω

)

A2

−

(

∂e+

∂kr·D0

R ·∂e

∂r−∂e+

∂r·D0

R ·∂e

∂kr

)

A2 .

Fulvio Zonca






Properties of solutions of this equation, given in a Schrodinger like form,including the formation of spatiotemporal meso-scales, global mode struc-tures and dispersion relation, are subject of Spring 2018 Course and willnot be considered in this lecture series.

Fulvio Zonca



References and reading material

L. Chen and F. Zonca, Rev. Mod. Phys. 88, 015008 (2016).

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

F. Zonca, L. Chen, S. Briguglio, G. Fogaccia, G. Vlad and X. Wang, New J. Phys.17, 013052 (2015).

T. M. Antonsen and B. Lane, Phys. Fluids 23, 1205 (1980).

P. J. Catto and W. M. Tang and D. E. Baldwin, Plasma Phys. 23, 639 (1981).

E. A. Frieman and L. Chen, Phys. Fluids 25, 502 (1982).

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

L. Chen and A. Hasegawa, J. Geophys. Res. 96, 1503 (1991).

H. Qin, W. M. Tang and G. Rewoldt, Phys. Plasmas 5, 1035 (1998).

G. Chew, F. Goldberger and F. Low, Proc. Roy. Soc. Ser. A 236, 112 (1956).

Fulvio Zonca


F. Zonca and L. Chen, Phys. Plasmas 21, 072120 (2014).

F. Zonca and L. Chen, Phys. Plasmas 21, 072121 (2014).

L. Chen, R. B. White, and M. N. Rosenbluth, Phys. Rev. Lett. 52, 1122 (1984).

Fulvio Zonca

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