Lecture 3 - Access · IFTSIntensiveCourseonAdvancedPlasmaPhysics-Spring2019 Lecture3– 3 That the SAW energy functional is a proper variational principle, thus, logically follows

IFTS Intensive Course on Advanced Plasma Physics-Spring 2019 Lecture 3 – 1

Lecture 3

Variational formulation and variational principle.

Integral form of the dispersion relation: Kinetic theory

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 10.th, 2019

Kinetic theory and global dispersion relationof Alfven waves in tokamaks

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

Fulvio Zonca


The shear Alfven wave variational principle.

✷ So far, we have demonstrated that the following SAW energy functional[Chen and Hasegawa 1991]

F [Φs, δPcomp] = −1

2

∫

∇Φ†s · δJdr =

1

2

∫

Φ†s∇ · δJdr ,

reduces identically to the MHD energy principle in the low frequency limit,when the leading order asymptotic expressions for plasma displacement andmagnetic fluctuation are used for SAW (Lecture 2).

✷ Here, integrating by parts and using homogeneous boundary conditions(ideal conducting wall) at the plasma surface, we note that the SAW energyfunctional is obtained by multiplication of the vorticity equation (∇ · δJ =0) by Φ†

s and integrating over the plasma domain [Chen and Hasegawa 1991].

Fulvio Zonca

http://www.afs.enea.it/zonca/references/seminars/IFTS_spring19/lecture2.pdf


✷ That the SAW energy functional is a proper variational principle, thus,logically follows from its equivalence to the MHD energy principle in theproper limit.

E: Review and discuss the MHD energy principle as a variational formulation ofmagnetohydrodynamics.

✷ Here, we follow the general methodology introduced in Lecture 2 [Gerjuoy1983] (see also Spring 2018 Lectures) and show that the SAW energy func-tional is indeed a proper variational principle [Z & C 2014, C & Z RMP2016].

✷ Introduce the following definitions and variational form (Lecture 2):• SAW vorticity equation (p.9): BA[Φs, δPcomp] = 0

• SSW parallel force balance (p.10): BS[Φs, δPcomp] = 0

Fv[Φs,t, δPcomp,t,Lt, Lt] = F [Φs,t, δPcomp,t] + L†atBA[Φs,t, δPcomp,t] + B†

A[Φs,t, δPcomp,t]Lbt

+ L†atBS[Φs,t, δPcomp,t] + B†

S[Φs,t, δPcomp,t]Lbt .

Fulvio Zonca


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



✷ Here, we follow [Chen and Hasegawa 1991] as well as [Z & C 2014, C & ZRMP 2016] and choose the trial functions [Φs,t, δPcomp,t] such that

BS[Φs,t, δPcomp,t] = 0 .

Thus, the auxiliary fields L are redundant (Lecture 2) and the variationalform reduces to

Fv[Φs,t, δPcomp,t,Lt] = F [Φs,t, δPcomp,t]+L†atBA[Φs,t, δPcomp,t]+B†

A[Φs,t, δPcomp,t]Lbt .

E: Comment about the choice BS[Φs,t, δPcomp,t] = 0 and discuss why the varia-tional principle thereby obtained is nonetheless general and useful for the intendedscope.

✷ The condition that Fv is a genuine variational principle reduces to

δFv = δF [Φs, δPcomp] + L†aδBA[Φs, δPcomp] + δB†

A[Φs, δPcomp]Lb = 0 .

Fulvio Zonca



E: Show that the above condition actually ensures that the considered functionalis a proper variational principle. What can you say about variations of Φs andδPcomp? Are they independent?

✷ Explicitating the various terms, and noting BA[Φs, δPcomp] = ∇ ·δJ , we canrewrite

δFv = δ1

2

∫

Φ†s∇ · δJdr +

∫

L†aδ (∇ · δJ) dr +

∫

δ (∇ · δJ)† Lbdr = 0 .

✷ Imposing independent variations of Φs and Φ†s, the vanishing of the variation

with respect to δΦs imposes

L†a = −

1

2Φ†

s .

E: Demonstrate the previous result. Hint: impose the vanishing of the variationwith respect to δ (∇ · δJ).

Fulvio Zonca


✷ Meanwhile, integrating by parts and imposing the vanishing of the variationwith respect to δ∇Φ†

s, we obtain

−1

2

∫

δ(

∇Φ†s

)

· δJdr −

∫

δ(

∇iΦ†s

)

δ(δJ†k)

δ(

∇iΦ†s

)

(r)∇kLb(r)dr = 0 .

✷ Thus, the problem of determining Lb in the most general case in-volves the definition of variational/functional derivative (Lecture 1 ofSpring 2018 Lectures, p. 20), which may be difficult to solve.

✷ The problem has a simple and general solution, when the current δJ isrelated to Φs as a linear functional through, e.g., a linear integral operatorΣ, defined as (see Lecture 4 of Spring 2018 Lectures for more details)

δJ(r, t) =

∫

Σ(r, t, r − x, t− τ) ·∇Φs(r − x, t− τ)dxdτ ,

δJ †(r, t) =

∫

∇Φ†s(r − x, t− τ) ·Σ†(r − x, t− τ, r, t)dxdτ .

using the definition [Σ†(r, t, r′, t′)]µν = [Σ(r′, t′, r, t)]∗νµ.

Fulvio Zonca






✷ Thus, using the above definitions of integral operator and its adjoint, wehave

∫

δ (∇ · δJ)† Lbdr =

= −

∫

δ(

∇Φ†s

)

·

[∫

Σ†(r, t, r + x, t+ τ) ·∇Lb(r + x, t+ τ)dxdτ

]

dr

✷ Meanwhile, we also have

−1

2

∫

δ(

∇Φ†s

)

· δJdr =

= −1

2

∫

δ(

∇Φ†s

)

·

[∫

Σ(r, t, r − x, t− τ) ·∇Φs(r − x, t− τ)dxdτ

]

dr

✷ Summing the RHS of the last two equations and equating to zero gives theequation for Lb.

Fulvio Zonca


✷ Substituting into the variational principle

Fv = −

∫

δJ † ·∇Lbdr

= −

∫

∇Φ†s ·

[∫

Σ†(r, t, r + x, t+ τ) ·∇Lb(r + x, t+ τ)dxdτ

]

dr

=1

2

∫

∇Φ†s ·

[∫

Σ(r, t, r − x, t− τ) ·∇Φs(r − x, t− τ)dxdτ

]

dr

=1

2

∫

∇Φ†s · δJdr .

✷ This result confirms that the functional on p. 4 is a genuine variationalprinciple (E: Comment about the sign change. Does it have any impact onthe present result?). This was to be expected and coincides with the factthat the MHD energy principle reduces to the SAW energy functional inthe proper limit.

Fulvio Zonca


E: Assume now that δJ is a linear functional depending on ∇Φs and higherorder derivatives. Calculate the functional derivative and derive an equation forLb. Can you show that the SAW energy functional is still a good variationalprinciple? (Hint: see Spring 2018 Lectures).

✷ This result is very general and relies on the fact that δJ is a linear func-tional. However, the present method suggest a practical way to construct avariational principle that reduces to the MHD energy principle even in themore general nonlinear case. Doing so is beyond the scope of these Lectures.

✷ Next, we will show the criterion for separating inertia from potential energybased on spatial scale separation: inertia is due to radial singular structuresconnected with the SAW continuous spectrum, while potential energy isdue to regular structures connected with the usual MHD-like fluctuationbehavior.

✷ We will finally show that the SAW energy functional is an integral formfor the fluctuation dispersion relation and extend it to the general kineticformulation.

Fulvio Zonca



Application to the SAW energy functional

✷ The MSD approach (Lecture 1), with its general validity [Lu 12], shows thatthe most natural way to isolate radial singular structures (|kr| → ∞) is totake the |ϑ| → ∞ limit after transforming from (r, θ, ζ) to (r, ϑ, ζ) space.

✷ This is achieved by the transformation PBn(r, ϑ) : f(r, θ, ζ) 7→ fn(r, ϑ),

f(r, θ, ζ) = einζ∑

m∈Z

e−imθ

∫

ei(m−nq)ϑPBn(r, ϑ) [f ] dϑ .

✷ When applied to a functional, such as the SAW energy functional, the MSDintegral transformation is applied to the product of two fluctuations ⇒ aconvolution integral is expected [Z & C 2014]

PBn(r, ϑ) [fg] = 2π∑

h,j∈Z

e−2iπhjqPBn−h(r, ϑ) [f ]PBh(r, ϑ+ 2πj) [g] .

Fulvio Zonca



✷ This definition follows from the definition of the integral transform definingthe MSD (E: Show that this is indeed the case!).

✷ Now apply the approach to the SAW energy functional, which, from pp. 2,4, is given by

F [Φs, δPcomp] = −1

2

∫

∇Φ†s · δJdr =

1

2

∫

Φ†s∇ · δJdr ,

✷ Applying toroidal flux coordinates

∫

dr(...) =

∫ 2π

0

dζ

∫ 2π

0

dθ

∫ a

0

J (dψ/dr)dr(...) .

✷ Using the MSD rule introduced on p. 10, and noting J (r, θ) = J (r, ϑ) forthe Jacobian,

F [Φs, δPcomp] = (2π)3∑

j∈Z

∫ a

0

drdψ/dr

2ei2πnqj

∫ ∞

−∞

J dϑΦ†sn(r, ϑ+2πj)∇·δJn(r, ϑ) .

Fulvio Zonca


E: Derive the above result step by step, using the MSD transformation rule andproperties summarized at p. 10. Can you repeat this calculation just based onthe general MSD definition?

E: Comment about the structure of the SAW energy functional in the MSDrepresentation. Is it more or less complicated than its original form? How manyvariables/dimensions are involved? Why?

✷ The natural way to separate inertia from potential energy, based on spatialscale separation, is connected with |kr| → ∞ as |ϑ| → ∞.

• introduce ϑ1 ≫ 1 as a large control parameter

• recall that in (r, ϑ) space ∇‖ = (JB0)−1∂ϑ

• use the same integration by parts of Lecture 2 to write

F [Φs, δPcomp] =1

2

∫

(

|∇⊥Ψs|2

4π+ Φ†

s∇ · δJ⊥

)

dr .

Fulvio Zonca



✷ Explicitly writing the MSD representation of the SAW energy functional

F [Φs, δPcomp] = (2π)3∑

j∈Z

∫ a

0

drdψ/dr

2ei2πnqj

∫ ∞

−∞

J dϑ

[

1

4π∇⊥Ψ

†sn(r, ϑ+ 2πj)

·∇⊥Ψsn(r, ϑ) + Φ†sn(r, ϑ+ 2πj)∇ · δJ⊥n(r, ϑ)

]

= δW − δI .

✷ By definition, we now divide the (−∞,∞) interval in ϑ in three segments:(−∞,−ϑ1], [−ϑ1, ϑ1], and [ϑ1,∞). Conventionally, and using ϑ1 ≫ 1 as aparameter, we identify [Z & C 2014]

δW = limϑ1→∞

(2π)3∑

j∈Z

∫ a

0

drdψ/dr

2ei2πnqj

∫ ϑ1

−ϑ1

J dϑ

[

1

4π∇⊥Ψ

†sn(r, ϑ+ 2πj)

·∇⊥Ψsn(r, ϑ) + Φ†sn(r, ϑ+ 2πj)∇ · δJ⊥n(r, ϑ)

]

,

and −δI the remaining contribution due to radial singular structures.

Fulvio Zonca


✷ The integration over (−∞,−ϑ1] and [ϑ1,∞) is carried out as follows, recall-ing that Ψs = −(c/JB0)∂ϑΦs and δJ‖ = −(c/4π)∇2

⊥Ψs.

✷ Let ∇2⊥ ≡ −k2ϑκ

2⊥ and kϑ ≡ (−nq/r), with κ2⊥ ≃ s2ϑ2|∇r|2 in the |ϑ| ≫ 1

region of interest (s being the magnetic shear; cf. Lecture 2).

✷ Integrating by parts (note the sign and that kϑ,n = −kϑ,−n)(∫ −ϑ1

−∞

+

∫ ∞

ϑ1

)[

J dϑ

4π∇⊥Ψ

†sn(r, ϑ+ 2πj) ·∇⊥Ψsn(r, ϑ)

]

≃

(∫ −ϑ1

−∞

+

∫ ∞

ϑ1

)[

k2ϑc2dϑ

4πJB20

s2ϑ2|∇r|2∂ϑΦ†sn(r, ϑ+ 2πj)∂ϑΦsn(r, ϑ)

]

≃ −

[

k2ϑc2

4πJB20

s2ϑ2|∇r|2Φ†sn(r, ϑ+ 2πj)∂ϑΦsn(r, ϑ)

]ϑ1

−ϑ1

+

(∫ −ϑ1

−∞

+

∫ ∞

ϑ1

)

[

JB0dϑΦ†sn(r, ϑ+ 2πj)∇‖

(

δJ‖B0

(r, ϑ)

)]

.

Fulvio Zonca



✷ In the last expression, the second integral term adds up exactly to the∝ ∇ · δJ⊥ term and makes the whole integral in dϑ vanish, due to the SAWvorticity equation.

✷ This result shows that the inertial term is due to the first line contributionin the last expression, which is due to the jump of the solution across theregion [−ϑ1, ϑ1].

✷ For convenience, it is possible to introduce φs ≡ κ⊥Φs. In this way

δI = (2π)3∑

j∈Z

∫ a

0

drdψ/dr

2ei2πnqj

[

k2ϑc2

4πJB20

φ†sn(r, ϑ+ 2πj)∂ϑφsn(r, ϑ)

]ϑ→0+

ϑ→0−.

✷ Note that, here, the value of θ1 has been taken as → 0+. This is consistentwith the present asymptotic matching. In fact, ϑ1 → ∞ in the short lengthscale limit adopted for computing δW ; and ϑ1 → 0+ in the large |ϑ| region,where the radial singular structures are computed [Z & C 2014].

Fulvio Zonca


The integral form of the dispersion relation

✷ From Spring 2017 Lecture 3 (p. 26), we know that the general solutionfor the radial singular structures of the SAW continuous spectrum is in theform

φs = eiνϑP (ϑ) ,

where ν is the Floquet characteristic exponent and P (ϑ) is periodic.

✷ This means that, using the Poisson summation formula in Lecture 1 p. 33,the inertia term picks up radial singular contributions at nq −m ≃ Reν.

✷ Assuming that this occurs at one isolated position r = r0, the expressionfor 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−.

Fulvio Zonca




✷ Using the same (arbitrary) normalization, one can set

δW = 2π2

[

|kϑ|c2(dψ/dr)

JB20 |s|

2

]

r=r0,ϑ=0

∣

∣

∣φsn0+

∣

∣

∣

2

δWf .

✷ This definition was introduced for the analysis of the fishbone instability,using the jump of the MHD radial displacement across the singular layer asnormalization condition [CWR 1984].

✷ In this way, the variational principle obtained for the SAW energy functionalbecomes an integral form of the dispersion relation, which reads

i|s|Λ = δWf .

E: Derive the explicit form of iΛ for the fishbone problem. (Hint: look at pp.9-13 of Spring 2017 Lecture 3).

E: Read the classic paper by [CWR 1984] and discuss the meaning of adding a

Fulvio Zonca



δWk term on the RHS of the above dispersion relation to describe the fishboneexcitation by energetic particles.

✷ The integral (global) form of the dispersion relation derived above is thesimplest form of the general fishbone like dispersion relation (GFLDR) [Z &C 2014], which can be further reduced to a local WKB dispersion relationfor short wavelength modes (cf. Lecture 4).

Fulvio Zonca



Variational principle: kinetic extensions

✷ The linear gyrokinetic analysis (cf. Lecture 4) adopts the common de-scription of plasma fluctuations in terms of perturbed scalar potential δφ,parallel vector potential δA‖, and parallel magnetic field δB‖. Meanwhile,the Coulomb gauge ∇ · δA = 0 is assumed, as usual.

✷ Following the general method [Gerjuoy 83], we choose the trial functions[δφt, δA‖,t, δB‖,t] such that the perpendicular pressure balance (p. 7) andthe quasineutrality conditions (p. 9) are satisfied.

✷ Thus, similar to the case considered in Lecture 2, δφ, δA‖ and δB‖ are notvaried independently. The SAW energy functional then becomes

Fv[δφt, δA‖,t,Lt] = F [δφt, δA‖,t]

+ L†atBA[δφt, δA‖,t] + B†

A[δφt, δA‖,t]Lbt ;

while the condition that Fv[δφt, δA‖,t,Lt] be a proper variational principle

Fulvio Zonca




can be written as

δFv = δF [δφ, δA‖] + L†aδBA[δφ, δA‖] + δB†

A[δφ, δA‖]Lb = 0 .

✷ Note that these equations are in one-to-one correspondence with equationson p. 2 and p. 4, respectively.

✷ Here,BA[δφ, δA‖] = ∇ · δJ = 0

formally represents the linear gyrokinetic vorticity equation (p. 16Lecture 4).

✷ Meanwhile, the functional F [δφ, δA‖] is given by [Z & C 2014; C & Z 2016]

F [δφ, δA‖] =1

2

∫(

∂−1t δE† · δJ +

|δB⊥|2

4π

)

dr

≃1

2

∫

∂−1t δφ†

∇ · δJdr .

Fulvio Zonca



✷ In order to demonstrate this form of the SAW energy functional, we notethat

δE⊥ = −∇⊥δφ

[

1 +O

(

β

k⊥Lp

)]

;

consistent with the linear gyrokinetic orderings introduced in Lecture 4.

✷ Thus, given the leading order expression of the plasma displacement andthe expression of the perpendicular magnetic field fluctuation, we can write

∂tΦs = δφ ,

Ψs = δA‖ ;

as leading order relationship between field variables adopted in the fluiddescription (Lecture 2) and in the present kinetic description.

✷ The kinetic SAW energy functional then corresponds to the fluid limit (p.2) once ∂−1

t is interpreted as formal inverse of ∂t; i.e., we formally solveΦs = ∂−1

t δφ.

Fulvio Zonca




✷ This one-to-one correspondence allows to readily use previous results todemonstrate that the kinetic SAW energy functional is a proper variationalprinciple under the very general assumption that δJ is a linear functionalof ∇δφ and higher order derivatives.

✷ Below, we demonstrate that the form given above is also the proper ex-tension of the SAW energy functional, i.e.it applies when the gyrokineticdescription of Lecture 4 is adopted.

✷ Let us rewrite Faraday’s and Ampere’s law in the form (here, as usual, weneglect the displacement current term, due to the |ω| ≪ |k|c assumption)

δB = −c∇× ∂−1t δE ,

∇× δB =4π

cδJ ,

where we have used the ∂−1t integral operator with its definition introduced

above.

Fulvio Zonca



✷ We can, thus, write the following identity

∂−1t δE† · δJ =

c

4π∂−1t δE† · ∇ × δB

= −∇ ·[ c

4π∂−1t δE† × δB

]

−|δB|2

4π.

✷ Separating parallel from perpendicular current on the l.h.s., and noting theδE⊥ expression on p. 21 as well as

∂−1t δE‖ = −b0 ·∇∂−1

t δφ−1

cδA‖ ,

we can rewrite the identity above as follows

−∂−1t ∇δφ† · δJ −

1

cδA†

‖δJ‖ +|δB|2

4π≃

≃ −∇ ·[ c


]

.

Fulvio Zonca


✷ Here, the notation ≃ is a reminder that terms of order O (β/k⊥Lp) andO (|ω|2/k2c2) are consistently neglected.

✷ Meanwhile, because of the linear gyrokinetic orderings, the term ∝ |δB‖|2

on the r.h.s. is of order O(

β2/k2‖L2p

)

and could also be neglected.

✷ The relevant expression can be finally cast as

−∂−1t ∇δφ† · δJ −

1

cδA†

‖δJ‖ +|δB⊥|

2

4π≃

≃ −∇ ·[ c


]

.

✷ This equation is the energy conservation (Poynting) theorem, rewritten inthe proper approximated form consistent with the gyrokinetic orderingsintroduced in Lecture 4.

Fulvio Zonca



✷ We can further reduce this expression noting

|δB⊥|2 =

(

∇× b0δA†‖

)

· δB⊥ =

= ∇ ·(

b0δA†‖ × δB⊥

)

+ b0δA†‖ · (∇× δB⊥)

= ∇ ·(


)

+4π

cδA†

‖δJ‖ .

✷ Thus, by direct substitution, we obtain

−∂−1t ∇δφ† · δJ ≃ −∇ ·

(


4π

)

−∇ ·[ c


]

.

Fulvio Zonca


✷ Substituting this expression into the kinetic SAW energy functional (p.20) and integrating over the plasma domain taking into account homoge-neous boundary conditions, we obtain the demonstration that it is a propergeneral extension of the SAW energy functional that applies for the lineargyrokinetic description of Lecture 4 and that provides the required kineticvariational principle.

✷ This derivation also demonstrates that the integrand of the SAW energyfunctional represents the energy density, whose evolution balances the di-vergence of the Poynting flux. In MSD variables this reads:

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)

]

.

Fulvio Zonca



E: Derive explicitly the kinetic SAW energy functional using the MSD approachof Lecture 1.

✷ Here, the integrand of the integration in the radial flux coordinate r is anexpression of the magnetic flux surface integrated energy density.

✷ When an eigenmode is formed dynamically, the aforementioned integrandrepresents the magnetic flux surface integrated energy flux, which corre-spond to the time varying mode structure evolving toward the time asymp-totic (fixed) eigenmode structure [Z & C 2014; C & Z 2016].

✷ Even when the time asymptotic eigenmode structure is eventually formed,the magnetic flux surface integrated energy flux does not vanish, sinceenergy sources/sinks generally come from different radial locations (cf.Lecture 4). This is clearly true, a fortiori, when nonlinear interactions areaccounted for [C & Z 2016].

E: Give a physical reason why the above statement is true and provide a practicalexample to support it.

Fulvio Zonca




✷ Time scales of radial envelope evolution due to the magnetic flux surfaceintegrated energy flux are the temporal meso-scales that will be discussedmore in discussed in Lecture 4.

✷ Due to the approximations introduced in the derivations, the present de-scription is expected to break down when the time, measured in unitsof these temporal meso-scales, is of the order of the smallest among

O (k⊥Lp/β), O (k2c2/|ω|2) and O(

k2‖L2p/β

2)

.

E: Give a physical reason why the above statement is true and provide a practicalexample to support it.

✷ Descriptions that are valid on even longer time scales are possible, based onthe general method [Gerjuoy 83] discussed here and in Lecture 2. However,they are beyond the intended scope of these lectures.

Fulvio Zonca




References and reading material

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

I. M. Gelfand and S. V. Fomin, Calculus of Variations, Prentice-Hall, Inc. (En-glewood Cliffs, NJ, 1963).

E. Gerjuoy, A. Rau, and L. Spruch, 1983, Rev. Mod. Phys. 55, 725 (1983).

I. B. Bernstein, E. A. Frieman, M. D. Kruskal and R. M. Kulsrud”, Proc. Roy.Soc. Ser. A 244, 17 (1958).

M. D. Kruskal and C. R. Oberman, Phys. Fluids 1, 275 (1958).

M. N. Rosenbluth and N. Rostoker, Phys. Fluids 2, 23 (1959).

J. P. Freidberg, Ideal Magnetohydrodynamics, Plenum Press (New York and Lon-don, 1987).

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

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

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

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Lecture 3 - Access · IFTSIntensiveCourseonAdvancedPlasmaPhysics-Spring2019 Lecture3– 3 That the SAW energy functional is a proper variational principle, thus, logically follows