7th Asia-Pacific Conference on Plasma Physics, 12-17 Nov, 2023 at Port Messe Nagoya

Hamiltonian formulations of quasilinear theory for magnetized plasmas
1

Alain J Brizard1
Department of physics, Saint Michael’s College (USA)e-mail (speaker): abrizard@smcvt.edu

The complex interaction between charged particles andelectromagnetic-field wave fluctuations in a magnetizedplasma represents a formidable problem with crucial
implications toward our understanding of magnetic
confinement in laboratory and space plasmas [1]. Thesewave-particle interactions can be described either linearly,
quasi-linearly, or nonlinearly, depending on how thebackground plasma is affected by the fluctuating wave
fields and the level of plasma turbulence associated withthem [2].
In linear plasma wave theory [3], 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 ofsmall-amplitude electromagnetic waves which, whencoupled to the linearized Maxwell wave equations, yields
a wave spectrum that is supported by the uniformbackground magnetized plasma [3].
In weak plasma turbulence theory [4], the backgroundplasma is considered weakly unstable so that a (possiblydiscrete) spectrum of field perturbations grows to finite
but small amplitudes. While these small-amplitude
fluctuations interact weakly among themselves, theyinteract strongly with resonant particles, which satisfy awave-particle resonance condition in particle phase space(described in terms of unperturbed particle orbits). These
resonant wave-particle interactions, in turn, lead to aquasilinear modification of the background Vlasov
distribution on a long-time scale compared to thefluctuation time scale [5,6].
Lastly, in strong plasma turbulence theory [7], nonlinear
wave-wave and wave-particle-wave interactions cannot
be neglected, and wave-particle resonances include
perturbed particle orbits [4].

many subsequent quasilinear formulations are derived [3].
As an alternative formulation of quasilinear theory, wepresent a Hamiltonian formulation that relies on the use ofguiding-center theory for a uniform magnetic field. In thisHamiltonian 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.
We conclude our presentation by discussing the works ofBrizard and Chan [9-10], in which the Hamiltonian
quasilinear diffusion of relativistic particles in thepresence of electromagnetic fluctuations in an
axisymmetric dipole geometry is presented.
This work was partially supported by a grant NSFPHY2206302,References[1] A.N. Kaufman & B.I. Cohen, J. Plasma Phys. 85,205850601 (2019).
[2] R.C. Davidson, Methods of nonlinear plasma theory(Academic Press, 1972).
[3] T.H. Stix, Waves in plasmas (AIP, 1992).
[4] A. Galeev & R. Sagdeev, Basic plasma physics I(North Holland, 1983).
[5] A.N. Kaufman, J. Plasma Phys. 8, 1 (1972).
[6] R. Dewar, Phys. Fluids 16, 1102 (1973).
[7] T. Dupree, Phys. Fluids 9, 1773 (1966).
[8] C.Kennel & F.Engelmann, Phys. Fluids 9, 2377 (1966).

Our primary purpose is to present complementary viewsof two-dimensional quasilinear diffusion in a uniform
magnetized plasma. First, we review the (Newtonian)quasilinear theory derived by Kennel and Engelmann [8]which represents the paradigm formulation upon which[9] A. Brizard & A. Chan, Phys. Plasmas 8, 4762 (2001).
[10] A. Brizard & A. Chan, Phys. Plasmas 11, 4220 (2004).