PSFC/JA-10-36 Particle interactions with spatially localized wavepackets Y. Koinis, a K. Hizanidis, a and A.K. Ra October 010 Plasa Science and Fusion Center, Massachusetts Institute of Technology Cabridge, MA 0139 U.S.A. a National Technical University of Athens Association EURATOM-Hellenic Republic Zografou, Athens 15773. Greece This work was supported by the U.S. Departent of Energy, This work is supported by DoE grants DE-FG0-99ER-5451 and DE- FG0-91ER-54109, and by Association EURATOM, Hellenic Republic. Reproduction, translation, publication, use and disposal, in whole or in part, by or for the United States governent is peritted. To be published in the Proceedings of the 37 th European Physical Society Conference on Plasa Physics (010).
Particle interactions with spatially localized wavepackets Y. Koinis 1, K. Hizanidis 1, A.K. Ra 1 School of Electrical and Coputer Engineering, NTUA, Athens GR 15773, Greece Plasa Science and Fusion Center, MIT, Cabridge MA 0139, USA Wave-particle interaction is one of the ost well studied subjects in plasa physics. Particle dynaics in the presence of electrostatic or electroagnetic waves has been one of the ain paradigs on which the odern theory of nonlinear Hailtonian dynaics and chaos has been applied [1]. However, alost all previous studies of wave-particle interactions fro the point of view of Hailtonian dynaics have been focused on waves having discrete spectra, naely, periodic waves. In a previous work [] we have studied particle interactions with localized wavepackets propagating in the absence of a agnetic field or along a unifor agnetic field. In this work we study such interactions in the general case where the wavepackets propagate at an angle to a agnetic field. The Hailtonian describing particle otion in a unifor constant agnetic field B=B 0 z is H 0 = p-(e/c)a /(M), where A=-B 0 yx is the vector potential corresponding to the agnetic field and p=mv+(e/c)a is the canonical oentu. Utilizing the generating function F 1 =MΩ[(y-Y) cotφ/-xy] (Ω=eB 0 /Mc, is the gyration frequency) we transfor to guiding center variables with the new Hailtonian being H 0 = P z /M + P φ Ω. The new variables are the guiding center position (X,Y), the z coordinate and oentu (z,p z ) and the gyration angle and angular oentu (φ, P φ ). Under the presence of a localized electrostatic wave field Φ 0 (r-v g t) sin(k r-ωt), with wavenuber k = k z+k y and group velocity V g, the Hailtonian is PZ H = + PΩ φ + eφ0 ( X Vxt+ ρcos φ, Y Vyt ρsin φ, z Vzt) Jsin( k z+ k Y φ ωt) M where ρ(p φ ) = (P φ /ΜΩ) 1/ is the Laror radious and J (k ρ) are Bessel functions. The wave fields range fro ordinary wavepackets to ultra short few-cycle and subcycle transient pulses. Note that for the latter, the assuption of adiabaticity for aplitude odulation, coonly adopted in previous works, does not hold. We consider the presence of the localized wave as a perturbation to the particle otion in the constant unifor agnetic field. The unperturbed
frequencies of particle otion are ω φ = Ω and ω z = P z /M (= v z ). The perturbation results in resonances between the degrees of freedo given by the condition P z k ω Ω M = In order to analyze particle dynaics we utilize the canonical perturbation ethod [1]. According to this ethod we construct a near-identity canonical transforation resulting to a new Hailtonian where the dependence on canonical positions is "pushed" to higher order. According to a standard procedure [1], the first order generating function is calculated by integrating the perturbative part of the Hailtonian along unperturbed particle orbits. Although our approach is general, in the following we focus on a localized wave of Gaussian for (,, ) exp x y z Φ0 xyz = a In order to consider finite Laror effects, i.e. take into account the fact that the Laror radious can be coparable to the spatial with of the wave, we use a first order, with respect to (ρ/α ), Taylor expansion of Φ 0. Therefore we obtain the first order generating function as ee B 4A C Ω BΩ w1 = exp exp exp iψ + i A 4A 4A A where: ρ J + R J e J e K At+ B iθ1 iθ1 ( 1 1 ) ( ) + ρ V + + + a A X Y z ( Pz / M) t R =,, a a a V V x y Vz ( Pz / M) V =,, a iθ iθ ( J 1e J+ 1e ) BK( At B) Λ( At B) A = V, B= R V, C = R X θ = tan, θ = tan Y 1 1 1 R = X + Y a, V = V + V a x y [ ( / ) ] ( ) z V V Ψ = k z P M t + k Y φ Ωt B+ iω Ω = k ( Pz / M) Ω ω, Β = A x y
The nonperiodic tie dependence of the generating function is given through the functions t x π Kt () = e dx= [ 1+ erf(), t] li Kt () = π t + t 1 x t Λ() t = xe dx= e, li Λ() t = 0 t + The agnitude of w 1 depends exponentially on the oentu parallel to the agnetic field P z, the gyration frequency Ω and the wave frequency ω, through the exponential ter ( Ω A exp 4 ). The generating function has a significant agnitude in the phase space area localized around the locations where the resonance conditions Ω = 0 are fulfilled, with the width of these areas being inversely proportional to the transit tie of the localized field through the particle (A ~ V/a). The effect of the cross-section of the wave-particle scattering is taken into account through the exponential ter B 4A C R V exp = exp 4A V First-order finite Laror radius effects are taken into account through ters proportional to (ρ/α ). Having calculated w 1, we can construct first-order approxiate invariants ( z, φ, ) P P Χ of the particle otion as follows: w P = P 1 Q where (P, Q) = (P z, z), (P φ, φ), (ΜΩΧ, Υ) are the respective pairs of canonically conjugate variables. By setting two variable pairs equal to constants the contour plots of each one of these approxiate invariants provides analytically the Poincare surface of section in the plane defined by the third pair. Moreover, we can study the axiu canonical oentu variation after an interaction of the particle with the localized wave through the equation w 1 P = P ax z, φυ, Q where w = li w. 1 1 t + The above results refer to the study of single particle dynaics under interaction with the localized wave field. Based on the generalized Madey s theore [3], [4], we can utilize the results of first-order perturbation theory in order to calculate position averaged quantities, depending on the canonical oenta, with up to second-order accuracy. Therefore the
averaged canonical oentu variation of an enseble of particles having different initial gyration angles, z and Y positions, after a single interaction with the localized wave, is 1 w 1 Δ = z0, φ0, Y0 P 0 Q0 ( P) z0, φ0, Y0 where we have substitute the initial values of the canonical variables in the r.h.s. This equation provides the canonical oentu variation depending on the initial values of P z, P φ and X. Additional averaging over the initial guiding center coordinate X results in spaceaveraged parallel and angular oentu variations that are of interest to calculations on energy transfer through wave-particle interactions with applications to heating and current drive in agnetized plasas. In addition we can study the transient diffusion of particle oentu and X position for an enseble of particles interacting with the localized wave field by utilizing the following evolution equation for the particle distribution function averaged over (z 0, φ 0, Υ 0 ) F( P) 1 F( ) (, t) P = DP t P P where P = (P z, P φ, Χ) and Q = (z, φ, Υ) and DP (, t) w w 1 1 = Q0 Q0 Q is the tie-dependent diffusion tensor [5]. In conclusion, we have studied particle interaction with spatially localized electrostatic waves in a constant unifor agnetic field. The localized fields ay range fro ordinary wavepackets to ultra short few-cycle and subcycle transient pulses, since no adiabaticity assuption is considered. The utilization of the canonical perturbation theory allowed for the construction of approxiate invariants of the otion containing all the essential inforation for the strongly inhoogeneous phase space of the syste, corresponding to the chaotic scattering and transient oentu variation of the particles. Moreover, the collective particle behaviour has been studied by calculating position-averaged oentu variations and transient oentu and position diffusion. 0 References [1] A. J. Lichtenberg and M. A. Lieberan, (Springer-Verlag, New York, 199). [] Y. Koinis, K. Hizanidis and A.K. Ra, Phys. Rev. Lett. 96, 0500 (006). [3] P.E. Latha, S.M. Miller and C.D. Striffler, Phys. Rev. A 45, 1197 (199). [4] Y. Koinis, Phys. Rev. E 77, 016404 (008).
[5] Y. Koinis, A.K. Ra and K. Hizanidis, Phys. Rev. Lett. 104, 35001 (010).