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MINI REVIEW article

Front. Astron. Space Sci., 06 October 2022
Sec. Space Physics
This article is part of the Research Topic Quasilinear and Nonlinear Wave-Particle Interactions in Magnetospheric Plasmas View all 10 articles

Hamiltonian formulations of quasilinear theory for magnetized plasmas

  • 1Department of Physics, Saint Michael’s College, Colchester, VT, United States
  • 2Department of Physics and Astronomy, Rice University, Houston, TX, United States

Hamiltonian formulations of quasilinear theory are presented for the cases of uniform and nonuniform magnetized plasmas. First, the standard quasilinear theory of Kennel and Engelmann (Kennel, Phys. Fluids, 1966, 9, 2377) is reviewed and reinterpreted in terms of a general Hamiltonian formulation. Within this Hamiltonian representation, we present the transition from two-dimensional quasilinear diffusion in a spatially uniform magnetized background plasma to three-dimensional quasilinear diffusion in a spatially nonuniform magnetized background plasma based on our previous work (Brizard and Chan, Phys. Plasmas, 2001, 8, 4762–4771; Brizard and Chan, Phys. Plasmas, 2004, 11, 4220–4229). The resulting quasilinear theory for nonuniform magnetized plasmas yields a 3 × 3 diffusion tensor that naturally incorporates quasilinear radial diffusion as well as its synergistic connections to diffusion in two-dimensional invariant velocity space (e.g., energy and pitch angle).

1 Introduction

The complex interaction between charged particles and electromagnetic-field wave fluctuations in a magnetized plasma represents a formidable problem with crucial implications toward our understanding of magnetic confinement in laboratory and space plasmas (Kaufman and Cohen, 2019). These wave-particle interactions can be described either linearly, quasi-linearly, or nonlinearly, depending on how the background plasma is affected by the fluctuating wave fields and the level of plasma turbulence associated with them (Davidson, 1972).

In linear plasma wave theory (Stix, 1992), where the field fluctuations are arbitrarily small, the linearized perturbed Vlasov distribution of each charged-particle species describes the charged-particle response to the presence of small-amplitude electromagnetic waves which, when coupled to the linearized Maxwell wave equations, yields a wave spectrum that is supported by the uniform background magnetized plasma (Stix, 1992).

In weak plasma turbulence theory (Sagdeev and Galeev, 1969; Galeev and Sagdeev, 1983), the background plasma is considered weakly unstable so that a (possibly discrete) spectrum of field perturbations grow to finite but small amplitudes. While these small-amplitude fluctuations interact weakly among themselves, they interact strongly with resonant particles, which satisfy a wave-particle resonance condition in particle phase space (described in terms of unperturbed particle orbits). These resonant wave-particle interactions, in turn, lead to a quasilinear modification of the background Vlasov distribution on a long time scale compared to the fluctuation time scale (Kaufman, 1972a; Dewar, 1973).

Lastly, in strong plasma turbulence theory (Dupree, 1966), nonlinear wave-wave and wave-particle-wave interactions cannot be neglected, and wave-particle resonances include perturbed particle orbits (Galeev and Sagdeev, 1983). The reader is referred to a pedagogical review by Krommes (Krommes, 2002) on the theoretical foundations of plasma turbulence as well as a recent study on the validity of quasilinear theory (Crews and Shumlak, 2022). In addition, the mathematical foundations of quasilinear theory for inhomogeneous plasma can be found in the recent work by Dodin (Dodin, 2022).

1.1 Motivation for this work

The primary purpose of the present paper is to present complementary views of two-dimensional quasilinear diffusion in a uniform magnetized plasma. First, we review the quasilinear theory derived by Kennel and Engelmann (Kennel and Engelmann, 1966), which represents the paradigm formulation upon which many subsequent quasilinear formulations are derived (Stix, 1992). (We mainly focus our attention on non-relativistic quasilinear theory in the text and summarize the extension to relativistic quasilinear theory in Supplementary Appendix A) As an alternative formulation of quasilinear theory, we present a Hamiltonian formulation that relies on the use of guiding-center theory for a uniform magnetic field (Cary and Brizard, 2009). In this Hamiltonian formulation, the quasilinear diffusion equation is described in terms of a diffusion tensor whose structure is naturally generalized to three-dimensional quasilinear diffusion in a nonuniform magnetized plasma, as shown in the works of Brizard and Chan (Brizard and Chan, 2001; Brizard and Chan, 2004).

Next, two formulations of three-dimensional quasilinear theory are be presented. First, we present a generic quasilinear formulation based on the action-angle formalism (Kaufman, 1972b; Mahajan and Chen, 1985), which applies to general magnetic-field geometries. This formulation is useful in highlighting the modular features of the quasilinear diffusion tensor. Our second three-dimensional quasilinear formulation is developed for the case of an axisymmetric magnetic field B0 = ∇ψ ×φ, for which the drift action Jd=qψ/c is expressed simply in terms of the magnetic flux ψ. The presentation of this case is based on a summary of the non-relativistic limit of our previous work (Brizard and Chan, 2004).

1.2 Notation for quasilinear theory in a uniform magnetized plasma

In a homogeneous magnetic field B0=B0ẑ, the unperturbed Vlasov distribution f0(v) (for a charged-particle species with charge q and mass M) is a function of velocity v alone and the perturbed Vlasov-Maxwell fields (δf, δE, δB) can be decomposed in terms of Fourier components: δf=δf̃(v)exp(iϑ)+c.c. and (δE,δB)=(δẼ,δB̃)exp(iϑ)+c.c., where the wave phase is ϑ(x, t) = k xω t and the dependence of the eikonal (Fourier) amplitudes (δf̃,δẼ,δB̃) on (k, ω), which is denoted by a tilde, is hidden. According to Faraday’s law, we find δB̃=(kc/ω)×δẼ, which implies kδB̃=0. For the time being, however, we will keep the perturbed electric and magnetic fields separate, and assume that the uniform background plasma is perturbed by a monochromatic wave with definite wave vector k and wave frequency ω.

Following the notation used by Kennel and Engelmann (Kennel and Engelmann, 1966), the velocity v and wave vector k are decomposed in terms of cylindrical components

v=vẑ+vcosϕx̂+sinϕŷk=kẑ+kcosψx̂+sinψŷ,

so that k v = kv + kv cos(ϕψ), where ϕ is the gyroangle phase and ψ is the wave-vector phase. We note that the unperturbed Vlasov equation ∂f0/∂ϕ = 0 implies that f0(v) is independent of the gyroangle ϕ, i.e., f0(v, v). In what follows, we will use the definition

kk=cosψx̂+sinψŷ=12eiψx̂iŷ+12eiψx̂+iŷ12K̂+K̂,

and the identity

vv̂=cosϕx̂+sinϕŷeiϕψK̂/2+eiϕψK̂/2.

We note that, in the work of Kennel and Engelmann (Kennel and Engelmann, 1966), the right-handed polarized electric field is δẼRδẼK̂eiψ and the left-handed polarized electric field is δẼLδẼK̂eiψ; we will refrain from using these components in the present work.

2 Kennel-Engelmann quasilinear diffusion equation

In this Section, we review the quasilinear theory presented by Kennel and Engelmann (Kennel and Engelmann, 1966) for the case of a uniform magnetized plasma. Here, we make several changes in notation from Kennel and Engelmann’s work in preparation for an alternative formulation presented in Section 3.

2.1 First-order perturbed Vlasov equation

The linearized perturbed Vlasov equation is expressed in terms of the first-order differential equation for the eikonal amplitude δf̃(v):

iωkvδf̃Ωδf̃ϕΩeiΘϕeiΘδf̃=qMδẼ+vc×δB̃f0v

where Ω = qB0/(Mc) denotes the (signed) gyrofrequency and the solution of the integrating factor Θ/∂ϕ ≡Ω−1/dt = (k vω)/Ω yields

Θϕ=kvωΩϕ+kvΩsinϕψφϕ+λsinϕψ,

where λ = kv/Ω. The perturbed Vlasov Eq. 2.1 is easily solved as

δf̃v=qeiΘMΩϕeiΘδẼ+vc×δB̃f0vdϕ,

where a prime denotes a dependence on the integration gyroangle ϕ′. Here, we can write the perturbed evolution operator

qMΩδẼ+vc×δB̃vδṼv+δṼv+δϕ̃ϕ,

which is expressed in terms of the velocity-space eikonal amplitudes.

δṼ=qMΩδẼ+vc×δB̃ẑ,
δṼ=qMΩδẼ+vẑc×δB̃̂,
δϕ̃=qMΩδẼ+vẑc×δB̃ϕ̂vδB̃B0,

where ϕ̂=̂/ϕ=ẑ×̂. Whenever direct comparison with the work of Kennel and Engelmann (Kennel and Engelmann, 1966) is needed, we will use Faraday’s law to express δB̃=(kc/ω)×δẼ. With this substitution (see Supplementary Appendix A for details), for example, we note that Eqs. 2.4.7.–.Eqs. 2.2.7 agree exactly with Eq. 2.12 of Kennel and Engelmann (Kennel and Engelmann, 1966).

We now remark that, since ∂f0(v, v)/∂ϕ vanishes, only the first two terms in Eq. 2.4 are non-vanishing when applied to f0. Hence, Eq. 2.3 contains the integrals.

eiΘϕeiΘdϕ,
eiΘϕeiΘ̂dϕ.

In order to evaluate these integrals, we use the Bessel-Fourier decomposition eiΘ=eiφ=J(λ)ei(ϕψ), so that the scalar integral Eq. 2.8 becomes

eiΘϕeiΘdϕ=m,=iΔJmλJλeimϕψ,

where the resonant denominator is

ΔΩkv+Ωω,

while, using the identity Eq. 1.3, the vector integral Eq. 2.9 becomes

eiΘϕeiΘ̂dϕ=m,=iΔJmλJλeimϕψ,

where we introduced the vector-valued Bessel function

JλK̂2J+1λ+K̂2J1λ,

with the identity

kJ=J+1+J1k/2=Ω/vJ,

which follows from a standard recurrence relation for Bessel functions. The perturbed Vlasov distribution (Eq. 2.3) is thus expressed as

δf̃=m,iΔJmλeimϕψδṼf0v+δṼf0v,

where the Bessel-Fourier components are

δṼ=qMΩδẼJλvẑ×δB̃B0Jλ,
δṼ=qMΩδẼ+vẑc×δB̃Jλ.

Once again, Eqs. 2.15.17.–.Eqs. 2.2.17 agree exactly with Eq. 2.19 of Kennel and Engelmann (Kennel and Engelmann, 1966) when Faraday’s law is inserted in Eqs. 2.16, 2.17; see Supplementary Appendix A for details. The relativistic version of Eqs. 2.15.17.–.Eqs. 2.2.17, which was first derived by Lerche (Lerche, 1968), is also shown in Supplementary Appendix A.

2.2 Quasilinear diffusion in velocity space

We are now ready to calculate the expression for the quasilinear diffusion equation for the slow evolution (τ = ϵ2t) of the background Vlasov distribution

1Ωf0τ=ReqMΩδẼ+vc×δB̃δf̃v=ReδṼv+δṼv+δϕ̃ϕδf̃,

where ϵ denotes the amplitude of the perturbation fields, ⟨ ⟩ denotes a gyroangle average, and (δṼ,δṼ,δϕ̃) are the complex conjugates of Eqs. 2.52.7. In addition, the real part appears on the right side of Eq. 2.18 as a result of averaging with respect to the wave phase ϑ. We note that Kennel and Engelmann (Kennel and Engelmann, 1966) ignore the term ∂f2/∂t on the left side of Eq. 2.18, which is associated with the second-order perturbed Vlasov distribution f2 generated by non-resonant particles (Kaufman, 1972a; Dewar, 1973). While this term was shown by Kaufman (Kaufman, 1972a) to be essential in demonstrating the energy-momentum conservation laws of quasilinear theory, it is also omitted here and the right side of Eq. 2.18 only contains resonant-particle contributions.

First, since Eqs. 2.5, 2.6 are independent of v and v, respectively, we find

δṼδf̃v+δṼδf̃v=vδṼδf̃+vδṼδf̃=vδṼδf̃+1vv×vδṼδf̃δṼvδf̃,

where we took into account the proper Jacobian (v) in cylindrical velocity space (v, v, ϕ). On the other hand, the third term in Eq. 2.18 can be written as

δϕ̃δf̃ϕ=δϕ̃ϕδf̃=qMΩδẼ+vẑc×δB̃̂vδf̃δṼvδf̃,

where the last term in Eq. 2.7 is independent of the gyroangle ϕ. Since this term cancels the last term in Eq. 2.19, the quasilinear diffusion Eq. 2.18 becomes

1Ωf0τ=vReδṼδf̃1vvvReδṼδf̃.

Next, using the identity Eq. 1.3, we find

mJmλ1,̂eimϕψ=Jλ,Jλ,

so that, from Eq. 2.15, we find

mJmλδṼeimϕψ=qMΩδẼJλvẑ×δB̃B0JλδṼ,
mJmλδṼeimϕψ=qMΩδẼ+vẑc×δB̃JλδṼ.

Hence, the quasilinear diffusion Eq. 2.20 can be written as

1Ωf0τ=vRe=iΔδṼδṼf0v+δṼf0v1vvvRe=iΔδṼδṼf0v+δṼf0vvDf0v,

where the diagonal diffusion coefficients are

DẑDẑ==ReiΔ|δṼ|2,
D̂D̂==ReiΔ|δṼ|2,

while the off-diagonal diffusion coefficients are

DẑD̂==ReiΔReδṼδṼ,
D̂Dẑ==ReiΔReδṼδṼ,

which are defined to be explicitly symmetric (i.e., D=D). Here, using the Plemelj formula (Stix, 1992), we find

ReiΔ=ReiΩωkvΩ=πΩδωrkvΩ,

where we assumed ω = ωr + i γ and took the weakly unstable limit γ → 0+. Hence, the quasilinear diffusion coefficients (2.24)–(2.27) are driven by resonant particles, which satisfy the resonance condition kv‖resωΩ. The reader is referred to the early references by Kaufman (Kaufman, 1972a) and Dewar (Dewar, 1973) concerning the role of non-resonant particles in demonstrating the energy-momentum conservation laws of quasilinear theory.

Eq. 2.25 from Kennel and Engelmann (Kennel and Engelmann, 1966) (see Supplementary Appendix A) can be expressed as the dyadic diffusion tensor

DReiΔṽṽ=ReiΔδṼẑ+δṼ̂δṼẑ+δṼ̂,

which is Hermitian since the term − i Δ is replaced with Re( − i Δ). Here, the perturbed velocity

ṽ=δṼẑ+δṼ̂=qδẼMΩJλẑẑ+Jλ̂+ẑ×δB̃B0Jλv̂vẑ

explicitly separates the electric and magnetic contributions to the quasilinear diffusion tensor Eq. 2.29. In particular, the role of the perturbed perpendicular magnetic field is clearly seen in the process of pitch-angle diffusion because of the presence of the terms (v̂vẑ) associated with it. We also note that the parallel component of the perturbed magnetic field, δB̃=ẑδB̃, does not contribute to quasilinear diffusion in a uniform magnetized plasma. The components of the perturbed electric field, on the other hand, involve the parallel component, δẼ=ẑδẼ, as well as the right and left polarized components, δẼR=δẼ(x̂iŷ)/2 and δẼL=δẼ(x̂+iŷ)/2, respectively, appearing through the definition Eq. 2.13.

Lastly, we note that the dyadic form Eq. 2.29 of the quasilinear diffusion tensor in the quasilinear diffusion Eq. 2.23 can be used to easily verify that the unperturbed entropy S0f0lnf0d3v satisfies the H Theorem:

dS0dt=ϵ2f0τlnf0+1d3v=ϵ2ReiΔf0ṽlnf0v2d3v>0.

Once again, the energy-momentum conservation laws in quasilinear theory will not be discussed here. Instead the interested reader can consult earlier references (Kaufman, 1972a; Dewar, 1973), as well as Chapters 16–18 in the standard textbook by Stix (Stix, 1992).

2.3 Quasilinear diffusion in invariant velocity space

In preparation for Section 3, we note that a natural choice of velocity-space coordinates, suggested by guiding-center theory, involves replacing the parallel velocity v with the parallel momentum p = M v and the perpendicular speed v with the magnetic moment μ=Mv2/(2B0). We note that these two coordinates are independent dynamical invariants of the particle motion in a uniform magnetic field.

With this change of coordinates, the quasilinear diffusion Eq. 2.23 becomes

1Ωf0τpDppf0p+Dpμf0μ+μDμpf0p+Dμμf0μ,

where the quasilinear diffusion coefficients are

Dpp=M2D=ReiΔ|δP̃|2Dpμ=M2v/B0D=ReiΔReδP̃δμ̃Dμμ=Mv/B02D=ReiΔ|δμ̃|2,

with the eikonal amplitudes

δP̃=qΩδẼJ+vcJ×δB̃ẑ,
δμ̃=qB0ΩδẼ+vẑc×δB̃vJ,

and the symmetry Dμp = D follows from the assumption of a Hermitian diffusion tensor. Lastly, as expected, we note that the eikonal amplitude for the perturbed kinetic energy

δẼMvṽ=vδP̃+δμ̃B0=qΩδẼvJẑ+vJ,

only involves the perturbed electric field. Hence, another useful representation of quasilinear diffusion in invariant velocity (E,μ) space is given by the quasilinear diffusion equation

1Ωf0τvE1vDEEf0E+DEμf0μ+vμ1vDμEf0E+Dμμf0μ,

where the quasilinear diffusion coefficients are

DEE=ReiΔ|δẼ|2DEμ=ReiΔReδẼδμ̃Dμμ=ReiΔ|δμ̃|2,

and the Jacobian 1/v is a function of (E,μ): |v|=(2/M)(EμB0), while the sign of v is a constant of the motion in a uniform magnetic field.

3 Hamiltonian quasilinear diffusion equation

In Section 2, we reviewed the standard formulation of quasilinear theory in a uniform magnetized plasma (Kennel and Engelmann, 1966). In this Section, we introduce the Hamiltonian formulation of the Vlasov equation from which we will derive the Hamiltonian quasilinear diffusion equation, which will then be compared with the Kennel-Engelmann quasilinear diffusion Eq. 2.23.

In order to proceed with a Hamiltonian formulation, however, we will be required to express the perturbed electric and magnetic fields in terms of perturbed electric and magnetic potentials. We note that, despite the use of these potentials, the gauge invariance of the Hamiltonian quasilinear diffusion equation will be guaranteed in the formulation adopted here.

3.1 Non-adiabatic decomposition of the perturbed Vlasov distribution

The Hamiltonian formulation of quasilinear diffusion begins with the representation of the perturbed electric and magnetic fields in terms of the perturbed electric scalar potential δΦ and the perturbed magnetic vector potential δA, where δE = −∇δΦ − c−1∂δA/∂t and δB = ∇ × δA. Hence, we find the identity

δE+vc×δB=δΦvcδA1cdδAdtδΨ1cdδAdt,

where d/dt denotes the unperturbed time derivative. We note that the gauge transformation

δΦ,δA,δΨδΦ1cδχt,δA+δχ,δΨ1cdδχdt

guarantees the gauge invariance of the right side of Eq. 3.1.

Next, by removing the perturbed magnetic vector potential δA from the canonical momentum

P=mv+qA0+ϵδA/cP0=mv+qA0/c,

the noncanonical Poisson bracket (which can also be expressed in divergence form)

f,g=1Mfgvfvg+qB0M2cfv×gv
=v1Mf+qB0Mc×fvgfvgM

only contains the unperturbed magnetic field B0., where f and g are arbitrary functions of (x, v).

The removal of the perturbed magnetic vector potential δA from the noncanonical Poisson bracket Eq. 3.3, however, implies that the perturbed Vlasov distribution can be written as

δf=qcδAf0v+δgqcδAx,f0+δs,f0,

where the non-adiabatic contribution δg is said to be generated by the perturbation scalar field δs (Brizard, 1994; Brizard, 2018; Brizard and Chandre, 2020), which satisfies the first-order eikonal equation

ikvωδs̃Ωδs̃ϕ=qδΦ̃vcδÃqδΨ̃.

Hence, the eikonal solution for δs̃ is expressed with the same integrating factor used in Eq. 2.3:

δs̃v=qΩeiΘϕδΨ̃eiΘdϕ=qΩm,iΔJmλeimϕψδΨ̃,

where the gyroangle Fourier component of the effective perturbed potential is

δΨ̃δΦ̃vcδÃJλvcδÃJλ,

and the eikonal amplitude of the non-adiabatic perturbed Vlasov distribution is

δg̃=eiϑδs̃eiϑ,f0=1Mikδs̃+Ωẑ×δs̃vf0v=ikMf0vδs̃ΩB0δs̃ϕf0μ,

where μM|v|2/(2B0) denotes the magnetic moment. We note that, under the gauge transformations Eq. 3.2, the scalar field δs transforms as δsδs − (q/c) δχ (Brizard, 1994; Brizard, 2018; Brizard and Chandre, 2020), and the expression Eq. 3.5 for the perturbed Vlasov distribution is gauge-invariant. Moreover, under the gauge transformation Eq. 3.2, the eikonal Fourier amplitude Eq. 3.8 transforms as

δΨ̃δΨ̃+icωkvΩJδχ̃,

which is consistent with Eq. 3.2.

Next, since the components of the Poisson bracket Eq. 3.3 are constant, the unperturbed time derivative of δf yields the linearized perturbed Vlasov equation

dδfdt=qcdδAdtx,f0+qcδAv,f0+dδsdt,f0=qcdδAdtx,f0+qcδAv,f0+qδΨ,f0=qcdδAdt+qδΨx,f0qMδE+vc×δBf0v,

which implies that the non-adiabatic decomposition Eq. 3.5 is a valid representation of the perturbed Vlasov distribution.

3.2 Second-order perturbed Vlasov equation

In order to derive an alternate formulation of quasilinear theory for uniform magnetized plasmas, we begin with second-order evolution of the background Vlasov distribution

f0τ=qMδE+vc×δBδfv=qδΨ+qcdδAdtx,δf,

where, once again, τ = ϵ2t denotes the slow quasilinear diffusion time scale, we have ignored the second-order perturbed Vlasov distribution f2, and we have inserted Eqs. 3.1, 3.3. The first term on the right side of Eq. 3.12 can be written as

qδΨx,δf=qδΨ,δfqδΨvv,δf=qδΨ,δf+qcδAv,δf=qδΨ,qcδAx,f0+δg+qcδAv,δf,

where we have inserted the non-adiabatic decomposition Eq. 3.5, so that the first term can be written as

qδΨ,qcδAx,f0=q2cδΨ,δAx,f0+δAδΨ,x,f0=q2Mc2δAδAx,f0+qcδAqδΨ,x,f0,

where we used {δΨ, δA} = (δA/Mc) δA. The second term on the right side of Eq. 3.12, on the other hand, can be written as

qcdδAdtx,δf=q2c2dδAdtx,x,f0δA+ddtqcδAx,δgqcδAv,δg+qcδAx,f0,qδΨ.

Next, by using the Jacobi identity for the Poisson bracket (3.3):

f,g,h+g,h,f+h,f,g=0,

which holds for arbitrary functions (f, g, h), we obtain

q2cδAδΨ,x,f0+x,f0,δΨ=q2cδAf0,x,δΨf0,δH2,

where δH2 = q2|δA|2/(2Mc2) is the second-order perturbed Hamiltonian. We now look at the first term on the right side of Eq. 3.15, which we write as

q2c2dδAdtx,x,f0δA=ddtq2c2δAx,x,f0δAq2c2δAv,x,f0+x,v,f0δAq2c2δAx,x,f0dδAdt.

Because of the symmetry of the tensor x,{x,f0}, the last term on the right side (omitting the minus sign) is equal to the left side, so that we obtain

q2c2dδAdtx,x,f0δA=ddtq22c2δAx,x,f0δAq2c2δAv,x,f0δA,

where we used the Jacobi identity Eq. 3.16 to find {x, {v, f0}} = {v, {x, f0}}, since {f0, {x, v}} = 0.

When these equations are combined into Eq. 3.12, we obtain the final Hamiltonian form of the second-order perturbed Vlasov equation

f0τ=δH,δg+δH2,f0+ddtqcδAx,δg+q22c2δAx,x,f0δA,

where δH = q δΨ = q δΦ − q δA v/c and δg = {δs, f0}.

3.3 Hamiltonian quasilinear diffusion equation

We now perform two separate averages of the second-order perturbed Vlasov Eq. 3.19: we first perform an average with respect to the wave phase ϑ, which will be denoted by an overbar, and, second, we perform an average with respect to the gyroangle ϕ. We begin by noting that the averaged second-order perturbed Hamiltonian δH̄2=q2|δÃ|2/(2Mc2) is a constant and, therefore, its contribution in Eq. 3.19 vanishes upon eikonal-phase averaging. Likewise, the total time derivative in Eq. 3.19 vanishes upon eikonal-phase averaging.

The Hamiltonian quasilinear diffusion equation is, therefore, defined as

f0τ12δH,δḡ=12qMcδAδḡ+12vδH+ΩqcδA×ẑδgM̄=12vδH+ΩqcδA×ẑδgM̄,

where we used the divergence form Eq. 3.4 of the Poisson bracket and the eikonal average of the spatial divergence vanishes. Next, the eikonal average of the first term on the last line of the right side of Eq. 3.20 yields

δHδḡ=ikδH̃δg̃δH̃δg̃,

so that

vδHδgM̄=pikδH̃δg̃δH̃δg̃+1B0μikvδH̃δg̃δH̃δg̃,

where p = M v and μ = M|v|2/2B0. The eikonal average of the second term on the last line of the right side of Eq. 3.20, on the other hand, yields

Ω2vvvMqcδÃϕ̂δg̃+qcδÃϕ̂δg̃ΩB0μReqcδÃvϕδg̃,

so that by combining Eqs. 3.21, 3.22 into Eq. 3.20, we find

1Ωf0τ=pkΩReiδH̃δg̃+1B0μReqcδÃvϕ+ikvΩδH̃δg̃,

In order to evaluate the gyroangle averages in Eq. 3.23, we need to proceed with a transformation from particle phase space to guiding-center phase space, which is presented in the next Section.

4 Guiding-center Hamiltonian quasilinear diffusion equation

In this Section, we use the guiding-center transformation (Northrop, 1963) in order to simplify the calculations involved in obtaining an explicit expression for the Hamiltonian quasilinear diffusion Eq. 3.23 that can compared with the standard quasilinear diffusion Eq. 2.32 obtained from Kennel-Engelmann’s work (Kennel and Engelmann, 1966).

4.1 Guiding-center transformation

In a uniform background magnetic field, the transformation from particle phase space to guiding-center phase space is simply given as x = X + ρ, where the particle position x is expressed as the sum of the guiding-center position X and the gyroradius vector ρẑ×v/Ω, while the velocity-space coordinates (p, μ, ϕ) remain unchanged (Cary and Brizard, 2009). Hence, the eikonal wave phase ϑ = k xω t becomes

ϑ=kX+ρωt=θ+kρθ+Λ,

where θ denotes the guiding-center eikonal wave phase and Λ ≡ λ sin(ϕψ). Next, the particle Poisson bracket (Eq. 3.3) is transformed into the guiding-center Poisson bracket (Cary and Brizard, 2009)

F,Ggc=ẑFGpFpGΩB0FϕGμFμGϕcẑqB0F×G,

where the last term vanishes in the case of a uniform background plasma since the guiding-center functions F and G depend on the guiding-center position only through the guiding-center wave phase θ (with ∇θ = k).

4.2 First-order perturbed guiding-center Vlasov equation

The guiding-center transformation induces a transformation on particle phase-space functions f to a guiding-center phase-space function F through the guiding-center push-forward Tgc1:FTgc1f. For a perturbed particle phase-space function δgδg̃exp(iϑ)+c.c., we find the perturbed guiding-center phase-space function δGδG̃exp(iθ)+c.c., where the eikonal amplitude δG̃ is given by the push-forward expression as

δG̃=δg̃eiΛ=eiθδS̃eiθ,f0gc=ikf0pδS̃ΩB0f0μδS̃ϕ.

The eikonal amplitude of the guiding-center generating function δS̃=δs̃exp(iΛ) satisfies an equation derived from the first-order eikonal Eq. 3.6:

ikvωδS̃ΩδS̃ϕ=δH̃eiΛδH̃gc.

The solution of the first-order guiding-center eikonal Eq. 4.4 makes use of the gyroangle expansion δS̃==δS̃exp[i(ϕψ)], which yields the Fourier component

δS̃=iΔΩδH̃eiΛ+iϕψ=iΔΩqδΨ̃.

Inserting this solution into Eq. 4.3, with the gyroangle expansion δG̃==δG̃exp[i(ϕψ)], yields

δG̃=ikf0p+ΩB0f0μδS̃=qΩδΨ̃Δkf0p+ΩB0f0μ.

Hence, the solution for the eikonal amplitude δg̃ appearing in Eq. 3.23 can be obtained from the guiding-center pull-back expression δg̃=δG̃exp(iΛ).

4.3 Guiding-center Hamiltonian quasilinear diffusion equation

Using the solution Eq. 4.6 for δG̃, we are now ready to calculate the quasilinear diffusion Eq. 3.23 and obtaina simple dyadic form for the quasilinear diffusion tensor.

4.3.1 Quasilinear diffusion in guiding-center (p, μ)-space

Now that the solution for the eikonal amplitude δg is obtained in terms of the guiding-center phase-space function δg̃=δG̃exp(iΛ), we are now able to evaluate the gyroangle-averaged expressions in Eq. 3.23. We begin with the gyroangle-averaged quadratic term

δH̃δg̃=δH̃δG̃eiΛ=δH̃eiΛδG̃==δG̃δH̃eiΛ+iϕψ==qδΨ̃δG̃==q2Ω|δΨ̃|2Δkf0p+ΩB0f0μ,

so that

kΩReiδH̃δg̃==kDkf0p+ΩB0f0μ,

where we introduced the quasilinear perturbation potential

D=ReiΔ|q/ΩδΨ̃|2ReiΔ|δJ̃|2,

and

pkΩReiδH̃δg̃pDHppf0p+DHpμf0μ,

where DHpp=k2D and DHpμ=k(Ω/B0)D.

Next, we find

qcδÃvϕ+ikvΩδH̃δg̃=qcδÃvϕ+ikvΩδH̃δG̃eiΛ=qcδÃvϕeiΛiqΛϕδΦ̃vcδÃvcδÃ×eiΛδG̃=ϕδH̃eiΛδG̃=δH̃eiΛδG̃ϕ==iδG̃δH̃eiΛ+iϕψ==iqδΨ̃δG̃==iΔΩq/Ω2|δΨ̃|2kf0p+ΩB0f0μ,

so that

1B0μReqcδÃvϕ+ikvΩδH̃δg̃=μ=ΩB0Dkf0p+ΩB0f0μμDHμpf0p+DHμμf0μ,

where DHμp=(Ω/B0)kD and DHμμ=(Ω/B0)2D. We can now write the Hamiltonian quasilinear diffusion Eq. 3.23 as

1Ωf0τ=pDHppf0p+DHpμf0μ+μDHμpf0p+DHμμf0μ.

This quasilinear diffusion equation will later be compared with the standard quasilinear diffusion Eq. 2.32 derived by Kennel and Engelmann (Kennel and Engelmann, 1966).

4.3.2 Quasilinear diffusion in guiding-center (Jg,E)-space

Before proceeding with this comparison, however, we consider an alternate representation for the Hamiltonian quasilinear diffusion Eq. 4.13, which will be useful in the derivation of a quasilinear diffusion equation for nonuniform magnetized plasmas. If we replace the guiding-center parallel momentum p with the guiding-center kinetic energy E=p2/2m+μB0, and the guiding-center magnetic moment μ with the gyroaction Jg=μB0/Ω, the Fourier eikonal solution Eq. 4.6 becomes

δG̃=qδΨ̃f0E+qΩδΨ̃Δωf0E+f0Jg,

where the first term on the right side is interpreted as a guiding-center adiabatic contribution to the perturbed Vlasov distribution (Brizard, 1994), while the remaining terms (proportional to the resonant denominator Δ) are non-adiabatic contributions.

By substituting this new solution in Eq. 4.8, we find

kΩReiδH̃δg̃==kDωf0E+f0Jg,

while

ReqcδÃvϕ+ikvΩδH̃δg̃==ΩDωf0E+f0Jg,

where the guiding-center adiabatic contribution has cancelled out. The guiding-center quasilinear diffusion Eq. 4.13 becomes

1Ωf0τ=vE1vDHEEf0E+DHEJf0Jg+vJg1vDHJEf0E+DHJJf0Jg,

where the guiding-center quasilinear diffusion tensor is represented in 2 × 2 matrix form as

DH=2ωωω2D.

We note that, because of the simple dyadic form of Eq. 4.18, other representations for the guiding-center quasilinear diffusion tensor DH can be easily obtained, e.g., by replacing the guiding-center gyroaction Jg with the pitch-angle coordinate ξ=1μB0/E. We also note that the dyadic quasilinear tensor (4.18) has a simple modular form compared to the dyadic form Eq. 2.29.

4.4 Comparison with Kennel-Engelmann quasilinear theory

We can now compare the Kennel-Engelmann quasilinear diffusion Eq. 2.32 with the guiding-center Hamiltonian quasilinear diffusion Eq. 4.13. First, we express the perturbed fields Eqs. 2.34, 2.35 in terms of the perturbed potentials (δΦ, δA):

δP̃=MδṼ=qΩJikδΦ̃+iωcδÃvcikδÃikδÃJ=ikδJ̃+iωkvΩqδÃΩcJ,

and

δμ̃=MvB0δṼ=qvJB0ΩikδΦ̃+iωcδÃ+vcikδÃikδÃ=iΩB0δJ̃+iωkvΩqδÃcB0ΩvJ,

which are both gauge invariant according to the transformation (Eq. 3.10). Hence, these perturbed fields are expressed in terms of a contribution from the perturbed action δJ̃ and a contribution that vanishes for resonant particles (i.e., k v‖res = ωΩ). We note that, in the resonant-particle limit (Δ), the difference between the Kennel-Engelmann formulation and the Hamiltonian formulation vanishes. For example, the Kennel-Engelmann quasilinear diffusion coefficient Dpp=Re(iΔ)|δP̃|2 is expressed as

Dpp=ReiΔk2|δJ̃|2+2kJReδJ̃ΔqδÃΩc+qΩc2|δÃ|2J2|Δ|2DHpp,

which yields DHpp in the resonant-particle limit (Δ).

In summary, we have shown that, in the resonant-particle limit (Δ), the Hamiltonian quasilinear diffusion Eq. 4.13 is identical to the standard quasilinear diffusion Eq. 2.32 derived by Kennel and Engelmann (Kennel and Engelmann, 1966) for the case of a uniform magnetized plasma. In the next Section, we will see how the Hamiltonian quasilinear formalism can be extended to the case of a nonuniform magnetized plasma.

5 Hamiltonian quasilinear formulations for nonuniform magnetized plasma

In this Section, we briefly review the Hamiltonian formulation for quasilinear diffusion in a nonuniform magnetized background plasma. In an axisymmetric magnetic-field geometry, the 2 × 2 quasilinear diffusion tensor in velocity space is generalized to a 3 × 3 quasilinear diffusion tensor that includes radial quasilinear diffusion. In a spatially magnetically-confined plasma, the process of radial diffusion is a crucial element in determining whether charged particles leave the plasma. A prime example is provided by the case of radial diffusion in Earth’s radiation belt, which was recently reviewed by Lejosne and Kollmann (Lejosne and Kollmann, 2020).

We present two non-relativistic Hamiltonian formulations of quasilinear diffusion in a nonuniform magnetized plasmas. The first one based on the canonical action-angle formalism (Kaufman, 1972b; Mahajan and Chen, 1985; Mynick and Duvall, 1989; Schulz, 1996) and the second one based on a summary of our previous work (Brizard and Chan, 2004).

5.1 Canonical action-angle formalism

When a plasma is confined by a nonuniform magnetic field, the charged-particle orbits can be described in terms of 3 orbital angle coordinates θ (generically referred to as the gyration, bounce, and precession-drift angles) and their canonically-conjugate 3 action coordinates J (generically referred to as the gyromotion, bounce-motion, and drift-motion actions). In principle, these action coordinates are adiabatic invariants of the particle motion and they are calculated according to standard methods of guiding-center theory (Tao et al., 2007; Cary and Brizard, 2009), which are expressed in terms of asymptotic expansions in powers of a small dimensionless parameter ϵB = ρ/LB ≪ 1 defined as the ratio of a characteristic gyroradius (for a given particle species) and the gradient length scale LB associated with the background magnetic field B0. When an asymptotic expansion for an adiabatic invariant J=J0+ϵBJ1 is truncated at first order, for example, we find dJ/dtϵB2 and the orbital angular average dJ/dt=0 is the necessary condition for the adiabatic invariance of J. The reader is referred to Refs. (Cary and Brizard, 2009) and (Tao et al., 2007) where explicit expansions for all three guiding-center adiabatic invariants are derived in the non-relativistic and relativistic limits, respectively, for arbitrary background magnetic geometry.

The canonical action-angle formulation of quasilinear theory assumes that, in the absence of wave-field perturbations, the action coordinates J are constants of motion dJ/dt = − ∂H0/θ = 0, which follows from the invariance of the unperturbed Hamiltonian H0(J) on the canonical orbital angles θ. In this case, the unperturbed Vlasov distribution F0(J) is a function of action coordinates only. We note that the action coordinates considered here are either exact invariants or adiabatic invariants (Kaufman, 1972b; Mynick and Duvall, 1989) of the particle motion, and it is implicitly assumed that any adiabatic action invariant used in this canonical action-angle formulation of quasilinear theory can be calculated to sufficiently high order in ϵB within a region of particle phase space that excludes non-adiabatic diffusion in action space (Bernstein and Rowlands, 1976). For example, see Ref. (Brizard and Markowski, 2022) for a brief discussion of the breakdown of the adiabatic invariance of the magnetic moment (on the bounce time scale) for charged particles trapped by an axisymmetric dipole magnetic field.

In the presence of wave-field perturbations, the perturbed Hamiltonian can be represented in terms of a Fourier decomposition in terms of a discrete wave spectrum ωk and orbital angles (with Fourier-index vector m):

δHJ,θ,t=m,kδH̃Jexpimθiωkt+c.c.,

where the parametric dependence of δH̃ on the Fourier indices (m, k) is hidden. The perturbed Vlasov distribution δF is obtained from the perturbed Vlasov equation

δFt+δFθH0J=δHθF0J,

from which we obtain the solution for the Fourier component δf̃:

δF̃=δH̃ωkmΩmF0J,

where Ω(J) ≡ ∂H0/J denotes the unperturbed orbital-frequency vector.

The quasilinear wave-particle interactions cause the Vlasov distribution F0(J, τ) to evolve on a slow time scale τ = ϵ2t, represented by the quasilinear diffusion equation

F0J,ττ=12δH,δF=12JδHθδF=Jm,kmImδH̃δF̃=JImm,kmm|δH̃|2ωkmΩF0JJDQLF0J,

where ⟨ ⟩ includes orbital-angle averaging and wave time-scale averaging, and the canonical quasilinear diffusion tensor

DQLm,kmmπδωkmΩ|δH̃|2

is expressed in terms of a dyadic Fourier tensor mm, a wave-particle resonance condition obtained from the Plemelj formula

Im1ωkmΩ=ReiωkmΩ=πδωkmΩ,

and the magnitude squared of the perturbed Hamiltonian Fourier component δH̃(J), which is an explicit function of the action coordinates J and the perturbation fields [see of Ref. (Brizard and Chan, 2004), for example]. We note that the perturbed Hamiltonian δH̃(J) will, therefore, include terms that contain a product of an adiabatic action coordinate (such as the gyro action Jg) and a wave perturbation factor (such as δB/B0ϵδ). This means that an expansion of an adiabatic action coordinate (e.g., Jg=Jg(0)+ϵBJg(1)+) in the factor |δH̃|2 in Eq. 5.5 results in a leading term of order ϵδ2, followed by negligible terms of order ϵBϵδ2ϵδ2. Hence, only a low-order expansion (in ϵB) of the adiabatic action coordinates JJ0 is needed in an explicit evaluation of Eq. 5.5. In addition, we note that the form Eq. 5.4, with Eq. 5.5, guarantees that the Vlasov entropy S0 = − ∫F0 ln F0 d3J

dS0dt=ϵ2F0τlnF0+1d3J=ϵ2m,kF0mlnF0J2πδωkmΩ|δH̃|2d3J>0

satisfies the H Theorem. Lastly, we note that collisional transport in a magnetized plasma can also be described in terms of drag and diffusion in action space (Bernstein and Molvig, 1983).

5.2 Local and bounce-averaged wave-particle resonances in quasilinear theory

The canonical action-angle formalism presented in Section 5.1 unfortunately makes use of the bounce action Jb=p(s)ds, which is a nonlocal quantity (Northrop, 1963), while the drift action Jd(q/2πc)ψdφ=qψ/c is a local coordinate in an axisymmetric magnetic field B = ∇ψ ×φ, where the drift action is canonically conjugate to the toroidal angle φ. In our previous work (Brizard and Chan, 2001; Brizard and Chan, 2004), we replaced the bounce action with the guiding-center kinetic energy E in order to obtain a local quasilinear diffusion equation in three-dimensional Ji=(Jb,E,Jd) guiding-center invariant space:

F0τ=JDQLF0J=1τbJiτbDQLijF0Jj,

where the bounce period τb ≡∮ ds/v is the Jacobian. In addition, the 3 × 3 quasilinear diffusion tensor

DQL=,k,m2ωkmωkωk2ωkmmmωkm2Γkm

is defined in terms of the Fourier indices (associated with the gyroangle ζ) and m (associated with the toroidal angle φ) and the wave frequency ωk, while the scalar Γℓkm was shown in Ref. (Brizard and Chan, 2004) to include the bounce-averaged wave-particle resonance condition

ωk=ωcb+nωb+mωdb,

where ωcb=(q/Mc)Bb and ωdb are the bounce-averaged cyclotron and drift frequencies, respectively, and ωb = 2π/τb is the bounce frequency. Here, the bounce-average operation is defined as

b1τbσsLsUds|v|,

where σv/|v| denotes the sign of the parallel guiding-center velocity, and the points sL,U(J) along a magnetic field line are the bounce (turning) points where v changes sign (for simplicity, we assume all particles are magnetically trapped). In this Section, we present a brief derivation of the quasilinear diffusion Eq. 5.7, with the 3 × 3 quasilinear diffusion tensor Eq. 5.8 and the wave-particle resonance condition Eq. 5.9, based on our previous work (Brizard and Chan, 2004), which is presented here in the non-relativistic limit.

We begin with the linear guiding-center Vlasov equation in guiding-center phase space (s,φ,ζ;J):

d0δFdt=δFt+δF,Egc=F0,δHgc,

where the perturbed Hamiltonian is a function of the guiding-center invariants (Jb,E,Jd) as well as the angle-like coordinates (s, φ, ζ). The unperturbed guiding-center Poisson bracket, on the other hand, is

F,Ggc=FζGJgFJgGζ+FφGJdFJdGφ+d0FdtFtGEFEd0GdtGt,

and d0/dt = /∂t + v /∂s + ωd /∂φ + ωc /∂ζ denotes the unperturbed Vlasov operator (s denotes the local spatial coordinate along an unperturbed magnetic-field line). Since the right side of Eq. 5.11 is

F0,δHgc=F0JgδHζ+F0JdδHφ+F0Ed0δHdtδHt,

we can introduce the non-adiabatic decomposition (Chen and Tsai, 1983)

δFδHF0E+δG,

where the non-adiabatic contribution δG satisfies the perturbed non-adiabatic Vlasov equation

d0δGdt=F0Jgζ+F0JdφF0EtδHF̂δH.

Next, since the background plasma is time independent and axisymmetric, and the unperturbed guiding-center Vlasov distribution is independent of the gyroangle, we perform Fourier transforms in (φ, ζ, t) so that Eq. 5.15 becomes

vsiωkωcmωdδG̃s,σL̂δG̃s,σ=iFδH̃s,σ,

where the amplitudes (δG̃,δH̃) depend on the spatial parallel coordinate s and the sign σ = v/|v| = ±1, as well as the invariants J, while the operator F̂ becomes iF, with

FωkF0E+F0Jg+mF0Jd.

In order to remove the dependence of the perturbed Hamiltonian δH̃ on σ (which appears through the combination vδÃ), we follow our previous work (Brizard and Chan, 2004) and introduce the gauge δÃδα̃/s and the transformation (δG̃,δH̃)(δG̃,δK̃), where δG̃=δG̃+i(q/c)Fδα̃ and δK̃=δH̃+(q/c)L̂δα̃, so that Eq. 5.16 becomes L̃δG̃(s,σ)=iFδK̃(s).

In order to obtain an integral solution for δG̃, we now introduce the integrating factor

vsiωkωcmωdδG̃s,σeiσθvseiσθδG̃s,σ=iFδK̃s,

where

θssLsωkωcsmωdsds|v|

is defined in terms of the lower (L) turning point sL(J). The solution of Eq. 5.18 is, therefore, expressed as

δG̃s,σ=δḠeiσθ+iσeiσθsLsδK̃seiσθsds|v|F

where the constant amplitude δḠ is determined from the matching conditions δG̃(sL,+1)=δG̃(sL,1) and δG̃(sU,+1)=δG̃(sU,1) at the two turning points. At the lower turning point, the matching condition implies that δḠ is independent of σ. The matching condition at the upper turning point, on the other hand, is expressed as

eiΘδḠ+iτb2δK̃eiθbeiΘF=eiΘδḠiτb2δK̃eiθb×eiΘF,

which yields

δḠ=τb2cotΘδK̃cosθb+δK̃sinθbF,

where

ΘθsU=τb2ωkωcbmωdb.

We note that cot Θ in Eq. 5.21 has singularities at , which immediately leads to the resonance condition Eq. 5.9.

Now that the solution δG̃ has been determined, we can proceed with the derivation of the quasilinear diffusion equation, which has been shown by Brizard and Chan (Brizard and Chan, 2004) to be expressed as

F0τ=1τbEτb,k,mωkΓkmF+1τbJgτb,k,mΓkmF+1τbJdτb,k,mmΓkmF1τbJiτbDQLijF0Jj,

which requires us to evaluate ΓkmF1ImδH̃δG̃b=F1ImδK̃δG̃b, which is found to be expressed as

Γkm=τb2ImcotΘδK̃cosθb2,

where, using the Plemelj formula with the identity cotz=n=(znπ)1, we finally obtain

Γkm=δK̃cosθb2n=πδωkωcbnωbmωdb.

This expression completes the derivation of the quasilinear diffusion tensor Eq. 5.8 and the perturbed Hamiltonian δK̃ is fully defined in Ref. (Brizard and Chan, 2004). We note that, in the limit of low-frequencies electromagnetic fluctuations, we also recover our previous work (Brizard and Chan, 2001) from Eq. 5.8.

We now make a few remarks concerning the bounce-averaged wave-particle resonance condition Eq. 5.9. First, in the case of a uniform magnetized plasma (with the drift frequency ωd ≡ 0), we substitute the eikonal representations δG̃=δḠexp(iks) and δH̃=δH̄exp(iks) in Eq. 5.16 and we recover the uniform quasilinear diffusion Eq. 4.17. Second, the bounce-averaged wave-particle resonance condition Eq. 5.9 assumes that the waves are coherent on the bounce-time scale, which is not realistic for high-frequency (VLF), short-wavelength whistler waves (Stenzel, 1999; Allanson et al., 2021). We recover a local wave-particle resonance condition by introducing the bounce-angle coordinate ξ(s) (Brizard, 2000), which is defined by the equation /ds = ωb/v, so that v /∂s in Eq. 5.16 is replaced with ωb /∂ξ. Next, by introducing the bounce-angle Fourier series δG̃=n=δḠexp(inξ) and δH̃=n=δH̄exp(inξ) in Eq. 5.16, the integral phase Eq. 5.19 is replaced by the new integral phase

σχs=σθsnξs=σsLsωkωcsmωdsnωbds|v|.

If we now evaluate this integral by stationary-phase methods (Stix, 1992), the dominant contribution comes from points s0 along a magnetic-field line where

0=χs0=|vs0|1ωkωcs0mωds0nωb,

which yields the local wave-particle resonance condition, provided v(s0) ≠ 0 (i.e., the local resonance does not occur at a turning point).

6 Summary

In the present paper, we have established a direct connection between the standard reference of quasilinear theory for a uniform magnetized plasma by Kennel and Engelmann (Kennel and Engelmann, 1966) and its Hamiltonian formulation in guiding-center phase space. We have also shown that the transition to a quasilinear theory for a nonuniform magnetized plasma is greatly facilitated within a Hamiltonian formulation. The main features of a Hamiltonian formulation of quasilinear theory is that the quasilinear diffusion tensor has a simple modular dyadic form in which a matrix of Fourier indices is multiplied by a single quasilinear scalar potential, which includes the resonant wave-particle delta function. This simple modular is observed in the case of a uniform magnetized plasma, as seen in Eq. 4.18, as well as in the case of a nonuniform magnetized plasma, as seen in Eq. 5.8. In particular, we note that the quasilinear diffusion tensor Eq. 5.8 naturally incorporates quasilinear radial diffusion as well as its synergistic connections to diffusion in two-dimensional invariant velocity space. These features are easily extended to the quasilinear diffusion of relativistic charged particles that are magnetically confined by nonuniform magnetic fields.

Author contributions

AB has written 90% of the manuscript. AC has added technical references as well as historical context.

Funding

This work was partially funded by grants from (AB) NSF-PHY 2206302 and (AC) NASA NNX17AI15G and 80NSSC21K1323.

Conflict of interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Publisher’s note

All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.

Supplementary material

The Supplementary Material for this article can be found online at: https://www.frontiersin.org/articles/10.3389/fspas.2022.1010133/full#supplementary-material

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Keywords: quasilinear theory, guiding-center approximation, wave-particle resonance, Hamiltonian formulation, action-angle coordinates

Citation: Brizard AJ and Chan AA (2022) Hamiltonian formulations of quasilinear theory for magnetized plasmas. Front. Astron. Space Sci. 9:1010133. doi: 10.3389/fspas.2022.1010133

Received: 02 August 2022; Accepted: 22 August 2022;
Published: 06 October 2022.

Edited by:

Oliver Allanson, University of Exeter, United Kingdom

Reviewed by:

Anton Artemyev, University of California, Los Angeles, United States

Copyright © 2022 Brizard and Chan. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

*Correspondence: Alain J. Brizard, abrizard@smcvt.edu

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