Covariant Form of the Ponderomotive Potentials in a Magnetized Plasma

Abstract

With use of the Lie transform technique, a simple and compact but quite general expression is derived for the ponderomotive scalar and vector potentials cp av and A av in the covariant form (A av , i(p av ) = A av(1 = -h^F^yty).The four-vector X> v is the particle excursion in four-dimensional space due to the rf electric and magnetic fields, F^v is the high-frequency electromagnetic tensor, and the brackets denote an average over the highfrequency phase k • x -cot. PACS numbers: 52.35.MwThe ponderomotive effect is a primary and fundamental nonlinear wave process both in fusion and space plasmas. Expressions for the ponderomotive force have been obtained mainly by averaging the nonlinear terms in the fluid equations and/or by thermodynamical considerations. Only recently has the force been discussed systematically from a particle aspect. A useful tool for the kinetic approach is the Lie transform technique which originated from Hori 1 and was first applied to a plasma problem by Kawakami. 2 Dewar 3 has recently developed another form of the Lie operator formalism and introduced an oscillation-center coordinate to treat the dynamics in a high-frequency field. Cary and Kaufman 4 and Cary 5 applied this technique to derive the K-\ theorem which relates the ponderomotive Hamiltonian K to the linear susceptibility x of the plasma. The work of Johnston, Kaufman, and Johnston 6 along a similar line also pointed to the existence of a ponderomotive vector potential. To our knowledge, there has not yet emerged a uni-, fied treatment of both scalar and vector ponderomotive potentials which could lead to an easily tractable expression for both.In this Letter we present a quite general but compact expression for the ponderomotive potentials in the oscillation-center coordinates for a single particle. It is demonstrated that the scalar potential cp av and the vector potentials A av can be expressed as (1) (2) An important point in our analysis is the introduction of the quantity |, an excursion of the particle due to rf electric and magnetic fields, E x and B 19 respectively. The brackets denote an average over the rf phase, fiT-iT-w£, attached to the fields E x and B x .To derive the new expression, we start by considering the motion of a test particle in the Vlasov system governed by the relativistic Hamiltonian H, in the laboratory frame, (X, P),Here cp 0 and A 0 {<p l and A x ) are quasistatic (highfrequency) potentials, which are assumed to take the following forms:E x = -V<p x -(l/c)8A x /8* = E x exp(tlT-X-twf),where E 0 , B 0 , E x , and B l are slowly varying in space and time. The parameter € represents the smallness of the high-frequency fields.Dewar's formalism 3 will be followed to transform from the laboratory frame into the oscillation-center one. According to his theory, the generating function W(x, p", t) relates the original Hamiltonian H(X, P, t) to the new Hamiltonian K(x, p, t) through the equation F-K] w=c w dH ' de (5) 240

Variational principles for reduced plasma physics

“…where the first-order spatial displacement ξ is defined in terms of the lowest-order eikonal amplitude [12] ξ ≡ ∂ S 10…”

“…The problem of the low-frequency oscillation-center Hamiltonian dynamics of charged particles in high-frequency, short-wavelength electromagnetic waves is the paradigm for applications of Lie-transform perturbation methods in plasma physics [10,11,12,13]. Here, the fast wave space-time scales are removed asymptotically from the Hamiltonian dynamics by a time-dependent phase-space noncanonical transformation z a = (ct, x; w/c, p) → z a = (ct, x; w/c, p), where x denotes the oscillation-center position, and (w, p ≡ m v) denote the oscillation-center's kinetic energymomentum.…”

“…Over the past 30 years, Lie-transform perturbation methods [1,2,3] have played an important role in the process of dynamical reduction in plasma physics. Standard examples of reduced plasma dynamics include the elimination of fast space-time scales associated with the gyromotion of a charged particle in a strong magnetic field (i.e., the guiding-center reduction [4,5,6] and the gyrocenter reduction [7,8]) or the elimination of the fast eikonal space-time scales associated with charged-particle motion perturbed by a high-frequency, short-wavelength electromagnetic wave (the oscillation-center reduction [9,10,11,12,13,14,15,16,17,18]).…”

An energy and momentum conserving collisional bracket for the guiding-centre Vlasov–Maxwell–Landau model

“…The complexity of the particles' trajectories in relation to the wide gap of time scales spanning from the electron cyclotron motion to the macroscopic phenomena has urged the development of perturbative time scale reduction techniques (see e.g. Grebogi, Kaufman & Littlejohn 1979;Cary & Kaufman 1981;Hatori & Washimi 1981;Kaufman & Holm 1984) to allow us to step over the computational limitations set by the gyromotion of particles (Littlejohn 1981(Littlejohn , 1982(Littlejohn , 1983(Littlejohn , 1984.…”

Dressed-particle approach in the nonrelativistic classical limit

“…Hence the average forces are embedded into the properties of the OC, yielding a quasiparticle with a variable effective mass m eff [4,5]. In each given case, m eff can be Taylor-expanded at nonrelativistic energies so as to appear as an effective potential Ψ [4], e.g., ponderomotive [6,7,8,9] or diamagnetic [10]. Yet the nonrelativistic limit must permit also an independent calculation of Ψ.…”