Different dynamic regimes of stimulated electron-cyclotron emission from mirror-confined non-equilibrium plasma

This is a self-archived version of an original article. This version may differ from the original in pagination and typographic details. Author(s): Title: Year: Version: Copyright: Rights: Rights url: Please cite the original version: CC BY 3.0 https://creativecommons.org/licenses/by/3.0/ Different dynamic regimes of stimulated electron-cyclotron emission from mirrorconfined non-equilibrium plasma © The Authors 2019 Published version Gospodchikov, E. D.; Shalashov, A. G.; Izotov, I. V.; Skalyga, V. A.; Tarvainen, O. Gospodchikov, E. D., Shalashov, A. G., Izotov, I. V., Skalyga, V. A., & Tarvainen, O. (2019). Different dynamic regimes of stimulated electron-cyclotron emission from mirror-confined nonequilibrium plasma. In EPS 2019 : Proceedings of the 46th European Physical Society Conference on Plasma Physics (Article P1.4013). European Physical Society. Europhysics Conference Abstracts, 43C. http://ocs.ciemat.es/EPS2019PAP/pdf/P1.4013.pdf 2019 Different dynamic regimes of stimulated electron-cyclotron emission from mirror-confined non-equilibrium plasma E. D. Gospodchikov1, A. G. Shalashov1, I. V. Izotov1, V. A. Skalyga1 and O. Tarvainen2 1 Institute of Applied Physics RAS, Nizhniy Novgorod, Russia 2 Department of Physics, University of Jyvaskyla, Jyvaskyla, Finland Introduction Studies of cyclotron instabilities have led to the plasma cyclotron maser paradigm, which explains a rich class of phenomena of coherent plasma emission [1]. In particular, electron cyclotron instabilities caused by resonant interaction between energetic electrons and electromagnetic waves are typical for open magnetic traps with electron cyclotron resonance (ECR) plasma heating [2]. One of the present applications is related to a development of ECR ion sources. Particle ejections, which are inherent to the burst regime of the cyclotron instability, cause oscillations of the plasma potential and the beam current accompanied with a significant decrease of the average ion charge [3]. Recently we demonstrated experimentally that tuning of the ECR position in a MHD-stable minimum-B open magnetic trap allows switching from the generation of periodic bursts of electromagnetic radiation to a continuous-wave (cw) low-power regime of emission [4]. In this way we eventually avoid non-desirable effects of bursts and improve the ion source performance. Similar systems have been previously studied in the context of space cyclotron masers in planet magnetospheres and other astrophysical objects [5]. However, a laboratory experiment is characterized by a very different source of fast electrons, thus the existing theories need to be reconsidered [6]. Maser equations Assuming that the cyclotron instability evolves slowly compared to the bounceoscillations of resonant electrons in a trap and a frequency spectrum of wave turbulence is narrow compared to the electron-cyclotron frequency, a self-consistent evolution of particles and waves may be described by the bounce-averaged quasilinear equations [6]: ( ) Edd F K t E J F DED t F c      − ∂ ∂ = ∂ ∂ +       ∂ ∂ + ∂ ∂ = ∂ ∂∫ ∫ ∞ νυκ κκκ κ 0 1 0, , (1) where ),,( υ κ tF is the electron distribution function over electron velocity υ and pitch-angle BB υυκ / min⊥ = in the trap center, maxmin/BB c∝ κ is loss-cone boundary, J is a source of non-equilibrium electrons. The quasi-linear diffusion coefficient ED takes into account the electron scattering into the loss-cone by unstable waves with average energy E; 0 D account 46th EPS Conference on Plasma Physics P1.4013 for other mechanism of electron losses such as Coulomb collisions, for simplicity we assume DED 00 =. The wave energy )(tE is determined from the balance equation, in which the instability grow rate is proportional to κ ∂ ∂ /F, and ν stands for wave dissipation rate. One can seek a solution of (1) as series over eigenmodes ( ) κφ n of the quasilinear operator. To be definite, we consider the strongest interaction case with the extraordinary wave at the fundamental ECR harmonic, then constD ≈ κ / and constK ≈ 24 / κυ . Introducing ,),()(),()( 0 4 0 4 nn n n n nn n nDtjdJtfdF φµ κ φ κ κφυυκφυυ −=       ∂ ∂ ∂ ∂ == ∑ ∫ ∑ ∫∞∞ (2) one can rewrite equations (1) as an infinite set of ordinary differential equations [4]: Efk dt dE fEEj dt df n nnnnn n     −=+−= ∑ νµ ,)( 0, (3) To determine source J of non-equilibrium particles let us consider resonant electrons heated by an external monochromatic radiation under ECR condition. Due to interaction with the heating waves, such electrons redistribute along the curves of quasilinear diffusion, const/2 22 =− ⊥ c γυγ , where γ is the relativistic factor and ⊥ υ is calculated at the position of “cold” ECR. Assuming that the electrons accelerated from very low energy, they all belong to the same curve 2/2 22 ≈− ⊥c γυγ : we obtain some universal relation ECR BB )1/(2 min 2 γκ += between the electron pitch angle and energy. At some * κ this curve crosses another line corresponding to quasilinear diffusion induced by excited unstable waves. Apparently such crossing may be treated as δ -like distribution over pitch-angles, i.e. *)( 0 κφ nn Jj =. Thus, laboratory plasma masers are characterized by a source with a broad spectrum over n φ allowing simultaneous excitation and interaction of many eigenmodes in (3). Dynamic regimes: theory and experiment In spite of high dimension, equations (3) have only two steady-state solutions: one corresponding zero wave energy, i.e. a regime with no maser generation, and one corresponding to non-zero wave energy, i.e. a regime of cw generation. Switching between these two modes can be demonstrated even if only one mode 1 = n is taken into account in (2). One can easily find a bifurcation criterion for such system: if 0111 Ekj νµ > then the zero-wave-solution becomes unstable while the second solution corresponds to a stable regime of stationary generation. The situation becomes less trivial when we take into account other modes. It is possible that both stationary solutions are linearly unstable, then, from topological considera46th EPS Conference on Plasma Physics P1.4013 tions, our system must have a stable limit cycle corresponding to a regime of burst generation. The simplest case of such behavior may be studied within the adiabatic approximation, 0/ =dtdfn for all 2 ≥ n, which is justified by a fast growth of n µ with the mode number (12 10 µµ ≈, 2 n n∝ µ ). We again may consider the only differential equation for 1 f assuming nnn Ejf µ /≈ for 2 ≥ n. Then, the condition for the existence of a stable limit cycle can be found as 0 11 >jk and 0/ 2< ∑ ≥nnnn jk µ . Physically this condition implies that the burst regime is realized when the lowest (non-adiabatic) mode is destabilizing for waves while all higher (adiabatic) modes act as a non-linear absorber [4]. Next, we may consider first *n modes as non-adiabatic, assuming nnn Ejf µ /≈ for *nn > . This generalization is needed to describe the case of “weak quasilinear diffusion” which corresponds to limited strength of source J typical for a laboratory experiment. The boundary *n between adiabatic and non-adiabatic modes may be determined selfconsistently using the following condition: 42 * 2 111 *)(// − ∝≈ nkj n µµν [6]. For 1* > n one more bifurcation is possible, namely, a pair birth of stable and unstable limit cycles while the stability of the stationary point of the system does not change. Physically this may be understood as follows. During the developed burst regime, there is a deep modulation of the wave power E(t), the system spends much of the time in a state where E is close to its minimum value and the quasi-linear diffusion is weakened. In this case, the effective boundary between the adiabatic and nonadiabatic modes is defined from 2 *111 *)(// − ∝≈ nkj n µµν , i.e. it shifts towards larger mode numbers and, as a consequence, Lyapunov’s exponent for (3) can change sign. The range of parameters where it happens corresponds to the unstable limit cycle. The described set of bifurcations is illustrated in figure 1 which shows the results of numerical solution of system (3) for different values of the control parameter * κ . One can see a hysteresis typical for “hard birth” of two limit cycles. The stable limit cycle corresponding to the burst regime merges with the unstable one and disappears at 0.88* = κ , as a result, the system abruptly switches from burst to cw regime for increasing * κ . The region of attraction of a stationary point collapses to zero and the stationary point becomes unstable at 0.835* = κ , as a result, the system abruptly switches from cw to burst regime at this point for decreasing * κ . More details may be found in [6]. Recently the predicted hysteresis dynamics was found experimentally at the JYFL Ion Source (University of Jyvaskyla). In experiment we control the dynamic regimes by varying the source strength J determined by the ECRH power supporting the discharge and parameter 46th EPS Conference on Plasma Physics P1.4013 ECRmin /* BB∝ κ determined by the external magnetic field. Figure 2 shows the results of observations of dynamic modes of generation of stimulated ECR radiation for different values of ECRH power for the increasing and decreasing magnetic field. Observed coincidence of the theoretical predictions with the experiment shows the adequacy of the cyclotron maser model, and allows one to search for more efficient modes of operation of the ECR ion source. The work is supported by RFBR (project no. 19-02-00767). Fig. 1. Characteristic phase trajectories and the peak wave energy in a settled oscillatory regime as a function of the ECR position for fixed particle source strength and dissipation. Numerical solution of (3) for the first sixteen modes being taken into account. Andronov-Hopf bifurcations are indicated by the large points. Fig. 2. Different regimes (stable, cw, pulsed) of plasma emission measured at JYFL Ion Source for different experimental parameters for increasing and decreasing confining magnetic field. [1] Trakhtengerts V. Yu. and Rycroft M. J., Whistler and Alfven mode cyclotron masers in space. Cambridge University Press, NewYork, 2008 [2] Shalashov A. G. et al., Phys. Plasmas, 24 (2017) 032111 [3] Skalyga V. et al., Phys. Plasmas, 22 (2015) 083509 [4] Shalashov A. G. et al., Phys. Rev. Lett., 114 (2018) 205001 [5] Bespalov P. A., Phys. Scripta, T2/2 (1982) 576 [6] Shalashov A. G. et al., Eur. Phys. Lett., 124 (3) (2018) 35001 46th EPS Conference on Plasma Physics P1.4013