Fluid and Kinetic Modelling of Instabilities and Transport in ExB plasma Discharges

Transcription

1 Fluid and Kinetic Modelling of Instabilities and Transport in ExB plasma Discharges IEPC Presented at the 35th International Electric Propulsion Conference Georgia Institute of Technology Atlanta, Georgia USA A. Smolyakov 1, O. Koshkarov 2, I. Romadanov 3, S. Janhunen 4, O. Chapurin 5, University of Saskatchewan, Saskatoon, SK, S7N 5E2, Canada Y. Raitses 6, I. Kaganovich 7 Princeton Plasma Physics Laboratory, Princeton, New Jersey, 08540, USA D. Sydorenko 8 University of Alberta, Centennial Centre for Interdisciplinary Science, Edmonton, AB T6G2E9, Canada Abstract: Partially-magnetized plasmas with magnetized electrons and non-magnetized ions are typically in strongly non-equilibrium state due to presence of crossed electric and magnetic fields, inhomogeneities of plasma density, temperature, magnetic field and beams of accelerated ions. These free energy sources result in a number of instabilities leading to turbulence, anomalous transport, and appearance of coherent structures as found in experiments. This paper provides an overview of the instabilities that exist in such plasmas. A nonlinear fluid model has been developed for description of the Simon-Hoh, lower-hybrid and ion-sound instabilities. This fluid model accounts for the electron inertia and gyroviscosity providing the physics based cut-off of the mode growth at high wavenumber. The results of nonlinear simulations demonstrate turbulence, anomalous current and tendency toward the formation of coherent structures. It is shown that resistive current flow instability is responsible for the appearance of large scale axial structures coexisting with small scale fluctuations. The kinetic electron-cyclotron drift instability that occur at high wavenumbers is also discussed and results of the PIC 1D simulations are presented. P I. Introduction lasmas with moderate values of the magnetic field where ions are not or weakly magnetized such as in Hall thrusters and related systems, have distinctly different properties from strongly magnetized plasmas for fusion 1 Professor, Department of Physics and Engineering Physics, andrei.smolyakov@usask.ca. 2 PhD student, Department of Physics and Engineering Physics, olk548@mail.usask.ca. 3 PhD student, Department of Physics and Engineering Physics, ivr509@mail.usask.ca. 4 Post-Doc, Department of Physics and Engineering Physics, salomon.janhunen@usask.ca. 5 PhD student, Department of Physics and Engineering Physics, alex.chapurin@usask.ca. 6 Principal Research Physicist, Plasma Physics Laboratory, yraitses@pppl.gov. 7 Principal Research Physicist, Plasma Physics Laboratory, ikaganov@pppl.gov. 8 Research Scientist, Department of Physics, sydorenk@ualberta.ca 1

2 applications. The natural scale separation between the ion and electron Larmor radii further exploited by the application of the external electric field is used to create and control low temperature plasmas, localize and control ionization, separate, extract and accelerate ions. These unique possibilities are widely used in various devices for electric propulsion and material processing 1-5. In most applications, plasmas with strong electron currents due to the applied E and B field, involve density, magnetic field and temperature gradients. Often, there are also directed flows of ions which are non-magnetized and thus drifting strongly with respect to electrons. All these represent strongly non-equilibrium conditions with large reservoirs of free energy making plasma prone to a number of instabilities. A resulting turbulent behavior is ubiquitously observed in many devices across a wide range of spatial and temporary scales Despite the long history of experiments and observations in various Hall plasma devices the physical picture of small scale fluctuations and coherent structures is poorly understood. As a result, rescaling of the current designs to new regimes and different power is very difficult and costly in absence of the first principle physics based models. One of the manifestations of turbulence is enhanced (anomalous) electron conductivity in such plasmas that typically exceeds the classical values by one or two orders of magnitude 11. Another often observed phenomena is the excitation of meso-scale structures with a characteristic length scale between the geometric scale length of the device and the characteristic length scale of small scale fluctuations. The nature of fluctuations, anomalous transport and structures have direct impact on plasma temperature, profiles of the electric field, walls erosion and deep space operation of plasma propulsion systems. Development of the physics based models is imperative for predictive design of future space propulsion systems. The problem of turbulence and transport in Hall plasmas is another incarnation of the fundamental yet unsolved problem of plasma and fluid turbulence. In the last decades, there have been significant advances in the physics of high temperature fusion plasmas, in particular in theoretical understanding of crucial instabilities responsible for anomalous transport in tokamaks as well as in the development of numerical tools capable of quantitative modeling of anomalous transport. A number of large scale numerical codes have been developed are at the maturity level that is able to provide quantitative predictions of the stability criteria and resulting level of the anomalous transport for different modes. On the contrary, plasma turbulence of low temperature partially magnetized plasmas with ExB drift is poorly understood; many basic questions still remain unanswered 12, the available numerical tools do not match the maturity of those developed for fusion applications and are not able to provide reliable predictions of the anomalous transport. Full kinetic simulations are direct and could be the most comprehensive approach to model the experimental conditions. However, in many cases, such simulations are as complex as experiments and can be difficult to interpret. In many cases, kinetic simulations with realistic parameters are very expensive and practically out of reach even for modern high-performance computers. On other hand, fluid simulations could be faster and cheaper numerical tools for simulations of nonlinear plasma dynamics. Such simulations are easier to interpret, provide much greater flexibility in separating various physics elements, and are vital in developing of physical intuition and physical models for complex processes in plasmas. Here we describe the fluid models and our results of linear and nonlinear simulations of partially magnetized plasmas of interest for electric propulsion. The plasma discharges are of primary interest; however, these models are also directly applicable to other systems such as FRC, ECR and helicon discharges. We also describe the results of our PIC simulations of the related kinetic drift instability. II. Linear theory A. Advanced fluid model The nonlinear fluid model has been developed for description of the instabilities driven by density and magnetic field gradient and collisions in presence of the electron current due to the drift. The model is based on the reduced electron dynamics for magnetized electrons in the low frequency approximation, and full ion dynamics in neglect of the magnetic field. The model describes the Simon-Hoh, lower-hybrid and ion sound instabilities including the effects of the ion beam velocity The model also includes the transverse electron inertia and finite electron temperature effects (electron gyro-viscosity tensor) and thus incorporates the effects of finite electron Larmor radius. For small but finite values of the perpendicular wavevector 1, the Larmor radius effects in this model are asymptotically exact (as compared to the full kinetic theory), and for large values, 1, our model provides the qualitatively correct Pade type approximation to the exact Bessel functions of the kinetic theory. Therefore, this advanced fluid model provides physically correct picture of electron dynamics in the whole range of the wavelengths, including the short wavelength limit, 1. As a result, our model is able to describe the required transition to the ion sound instability in the limit 1 when the electrons become un-magnetized. Our model predicts the 2

3 excitation of the ion-sound instabilities driven the electron drift current, density gradient and collisions, Fig. 1. Further extension of this mode includes the effect of pressure anisotropy that are important for the inhomogeneous magnetic field 15. Figure 1. Magnetization as a function of applied field Effect of the gyro-viscosity and conversion into the ion sound mode, note the almost linear dependence on the wave-vector corresponding to the ion sound dispersion,. (a) Destabilization of the ion-sound mode by density gradient and collisions; (b) Destabilization by density gradient, E B drift and collisions. B. Local analysis of linear instabilities and dispersion relation numerical tool. The local analysis of linear instabilities is a simple approach which provides a useful assessment of the relevance and importance of various instabilities for typical experimental conditions and facilitate the development of physical intuition. Based on our advanced fluid model that describes a range of instabilities for a wide range of the wavevectors including the small-scale modes 1, we have performed an extensive study of the Simon-Hoh, lowerhybrid and ion-sound instabilities An unexpected role of two dimensional perturbations for Simon-Hoh mode has been revealed. Contrary to the previous belief that for the long wavelengths the Simon-Hoh is the low frequency mode with the eigen-mode frequency well below of the frequency,, it has been established that the growth rate is the non-monotonous function of the transverse wave vector in the radial (axial) direction,, and the growth rate is maximal for highly anisotropic modes with 2 / /, where is the azimuthal wave vectors, and / is the electron drift frequency. For such modes, the growth rate and real part of the mode frequency become equal,. Our models have demonstrated that the Simon-Hoh and lower-hybrid modes are in fact a single continuous mode. The small-scale modes are the most unstable, and for typical experimental plasma parameters the mode growth rate is maximal for the mode with 1, the exact value depends on the values of the and frequencies. It has been also found that the lower-hybrid mode can be destabilized by the density gradient and collisions even in absence of the drift. Another important finding was the discovery of the destabilization mechanism due to the ion beam alone even in the neglect of the electron current. Our analysis of various destabilization mechanism noted above revealed that the parametric dependencies of the mode frequencies and growth rate are quite complex and quite sensitive to absolute and relative values of specific plasma parameters such as magnetic field, density gradient, electric field, electron temperature, ion beam velocity and neutral gas pressure. We have developed the interactive dispersion relation solver 16 which can be conveniently used to assess the particular instabilities for specific plasma parameters in a given experiment. C. Nonlocal analysis of linear instabilities Most of the previous studies of gradient-drift modes were based on the local approximation. However, for plasmas with complex profiles of plasma density, magnetic and electric field, the local approximation breaks down and nonlocal analysis of the global modes has to be invoked taking into account the variations of plasma parameters. We have developed the linear spectral eigen-value code using Chebyshev polynomials to approximate the eigen-modes. Using this code, the properties of the global modes for realistic plasma parameters profiles (electric field, density and magnetic field) and the effects of cylindrical geometry for Penning discharge and Hall thruster were studied 17. Nonlocal theory shows significant differences from the local model. This difference is especially important for the long wavelength modes which are expected to provide the dominant contributions to the anomalous transport. One of 3

4 the important conclusions is the presence of unstable global modes in the regions where the local theory predicts stability, Fig. 2, and existence of localized unstable solutions in sheared flows, Fig.3. Figure 2. (a) Full spectrum of unstable eigenvalues for the constant shear profile; (b) multiple unstable eigen-functions; wavenumber : the eigen-function with the largest growth rate.. 1 (red); the ground state unstable eigen-function with.. 2 (blue); the eigenfunction with largest real frequency.. 3 (black). Figure 3. The localized eigen-functions for the extended domain,. The eigenfunction with the largest growth rate,.. 1 (blue);.. 2 (red). III. Anomalous transport and structures in nonlinear simulations A. Model benchmarking Our nonlinear fluid model has been implemented in the BOUT++ fluid simulations framework which is based on finite difference schemes in magnetic field aligned coordinates and allows the 3D simulations in complex magnetic geometry. The linear differential operators along and perpendicular of the magnetic field as well as nonlinear Poisson bracket employed in BOUT++ have been extensively tested before 18. Our numerical implementation has been tested against the linear eigen-mode solvers which shows good agreement, see Fig 4. 4

5 Figure 4. Linear benchmark of BOUT++ simulations against the theoretical eigen-value solutions. In nonlinear regime, we have tested our simulations for saturation starting from different initial conditions and grid convergence. It has been shown that the final saturated state is independent of the initial state and is well convergent for different grid resolutions and values of the hyperviscosity coefficients. To test the fidelity of the finite difference nonlinear simulations, we have been developing a high resolution pseudo-spectral code for a double periodic region. The identical system of nonlinear equations has been simulated with finite difference BOUT++ and a pseudo-spectral code. The comparison of the result has been encouraging, Fig. 5. Pseudo-spectral methods have been successful in modeling of incompressible fluids. However, straightforward application of this approach to more general hyperbolic problems which tend to develop shock wave type solutions may lead to stability problems due to Gibbs phenomenon and spectral blocking. We are now employing smooth filters which allow the pseudo-spectral method to be stable for almost singular solutions. Figure 5. Comparison of density a) and potential fluctuations b) in nonlinear simulations with BOUT++ and pseudo-spectral code. B. Anomalous electron current and structures in nonlinear fluid simulations In our nonlinear simulations, the saturation of turbulence and formation of the coherent structures have been demonstrated 14. It is shown that the instability reaches the saturation in nonlinear regime at a level which is independent of the initial state. Significant anomalous current due to turbulent fluctuations has been found in the first principle nonlinear simulation model 14. The anomalous (turbulent) current is strongly intermittent and the structures observed in the current density are correlated with density and potential structures, Figs 6 and 7. The density and current structures are reminiscent of experimentally observed structures (spokes) in Hall thrusters and magnetrons and the values of the anomalous Hall parameter are consistent with experiments in Penning discharge and PIC simulations 14. 5

6 Figure 6. The time evolution of the density and potential energy integrals (a); the time dependence of the anomalous current (in units of the classical collisional current) (b). Figure 7a. Large scale structures in density (right) and electric current (left). Figure 7b. Large scale structures in potential (left) and vorticity(right). C. Energy cascade into large length scale modes: modulational instability. Using the approach of the modulational instability, we have examined the condensation of small scale lowerhybrid frequency modes into the long wavelength structures. It is shown that the bath of small scale modes is unstable with respect to the secondary instability of the large scale quasi-mode perturbations 19. The large-scale structures are not linear unstable eigen-modes but the nonlinearly driven modes supported by the nonlinear energy transfer from small scale modes. It is suggested that the large scale slow coherent modes observed in a number of Hall plasma devices may be explained as a result of such secondary instabilities. 6

7 D. Resistive instability, axial modes and structures The resistive axial modes occur as a result of the axial current flow instability due to phase shift and positive feedback between the dissipative electron response and ballistic (inertial) response of ions We have developed a full nonlinear model for such modes taking into account the electron inertia effects 22. The electron inertia is important to limit the modes growth at small scales and thus selects the value of the wavelength for the most unstable modes. The axial flow instability has relatively low growth rate compared to azimuthal modes (of higher frequencies) which are driven by collisions and density gradients. The significance of axial modes however is in the high amplitude of the saturated states. The mode saturation occurs due to ion dynamics resulting in appearance of highly nonlinear quasicoherent structures resembling the cnoidal waves, Fig. 8. We detected this mode also in 2D simulations which show coexistence od large scale and small fluctuations, Fig 9. It has been suggested 20 that that this instability play a crucial role for breathing modes oscillations 23. We have included the ionization effects into the axial mode theory and performed nonlinear simulations. The obtained structures closely resemble the breathing modes, Fig 10. It is worth noting that our model does not include the effects of the electron temperature. Figure 8. Nonlinear development and axial motion of large scale (axial) modes (plasma density), z is the axial direction. Figure 9. The dimensionless amplitude of the vorticity from 2D nonlinear simulations. The small-scale modes exist on the background of large axial mode, Lx is axial direction, and Ly is azimuthal direction of the Hall thruster. 7

8 Figure 10. Axial propagation of the ion density structures in the model with ionization effects. E. The axial modes driven by the ion-beam in a finite length system The ion beam can be another source of the axial modes. Such modes are related to the lower-hybrid instability driven by the ion flow in a finite length system 24. Similar instability mechanism was also identified for ion sound waves in umagnetized plasmas 25. We have studied the linear and nonlinear stages of this instability and found that the instability results in stationary axial structures 26. It is interesting that the profiles of the perturbed potential in stationary state is not monotonous, Fig 11. Figure 11. (a) Eigen-functions of density and ion velocity perturbations. (b) Stationary profiles of density and ion velocity. IV. Role of kinetic effects and electron-cyclotron resonances: Electron-Cyclotron-Drift-Instability. The electron-cyclotron instability has attracted an interest in view of its possible role in driving the instabilities of plasma discharges 6,9,10, It has been suggested that this instability simply becomes the ion sound mode 28,29,32. We have studied this mode with the combination of the analytical theory and kinetic simulations 34. Our results indicated that at moderately short wavelength 1, the eigen-modes of partially magnetized plasma indeed become similar to the ion-sound mode, e.g. as is shown in Fig. 1. However, in case of the kinetic modes strongly driven by electron-cyclotron resonances, the situation is more complex. Our 1D PIC kinetic simulations show that initially there are multiple unstable electron-cyclotron harmonics which are excited according to the linear theory, as shown in Fig 13. 8

9 Figure 13. Development of multiple unstable cyclotron modes of the instability as a function of time. Amplitude of higher harmonics reduces with time. At a later stage, the nonlinear interactions result in robust reorganization with a single dominant mode selected by the resonance condition. This mode remains strongly coherent well into the nonlinear stage. This primary cyclotron resonance is a source of the energy that cascades further down to the ion-sound and lower hybrid modes. Our simulations demonstrate the energy accumulation at longer wavelengths. The inverse cascade is also evident in the appearance of secondary longer wavelength envelope on the background of the small-scale mode with the dominant wavelength 2 /, Fig 14. Figure 14. Coherent density structure generated by instability (left). Nonlinear modulational instability resulting in inverse cascade and energy transfer to long wavelengths (right) in nonlinear kinetic simulations. We have also performed 2D simulations corresponding to the r plane of the Hall thruster. The magnetic field is in the radial (r) direction, while is the periodic azimuthal direction, similar to Ref. 35. For the 2D case, our simulations are: initial temperature is 10 ev, we have 2048 cells in r for cm and 512 in theta for l θ of 1.35 cm, 800 particles per cell. The dielectric walls are at r 0 r 40 mm so the sheath is formed at both walls. We show the evolution of the perturbed ion density in Figure 15. The pictures show the evolution of the ion density fluctuations: first we see the saturation of the linear modes, then a fast non-linear cascade to low-k through strong modulation particularly close to the sheath boundary, and subsequent ballistic release of fluctuations into the sheath region so at the fluctuations penetrate the sheath. The spatial Fourier content is still dominated by the fundamental cyclotron harmonic as in 1D case and the features of the cascade to long wavelength modes are strongly evident. 9

10 Figure 15. Snapshots of ion density fluctuations in the simulation over four distinct regimes of non-linear evolution. First, modes saturate and assume an amplitude-modulated form, then a strong cascade to low-k occurs. After this, the perturbations penetrate the sheath regions. V.Conclusion The advanced fluid model for Hall plasmas with magnetized electrons and un-magnetized fluid ions has been presented. The model extends the collisionless Simon-Hoh instability into the short-wave-length and higher frequency regimes including the lower-hybrid modes. It is shown that the lower-hybrid mode destabilized by the density gradient is a natural extention of the Simon-Hoh instability. The gyro-viscosity effects, which are of the same order as the electron inertia, are important for plasmas with finite electron temperature plasma. The lower hybrid modes in plasmas E B drift can be destabilized by the density gradient as well as electron-neutral collisions which are especially effective at shorter wave-lengths. The effects of the gyro-viscosity describe the transition of the lower-hydrid waves into the ion sound waves for short wavelengths. The reduced nonlinear fluid model has been implemented in the high performance BOUT++ framework. Initial value simulations in the linear regime have been bench-marked against the results obtained by the eigenvalue solvers. In nonlinear simulations the saturation of turbulence and formation of the coherent structures have been demonstrated. It is shown that nonlinear simulations reach saturation at the level which is independent of the initial state. Significant anomalous current due to turbulent fluctuations has been found in the nonlinear state. It is shown that the anomalous (turbulent) current is strongly intermittent and the structures observed in the current density are well correlated with density and potential structures. The density and current structures are reminiscent to experimentally observed 10

11 structures (spokes) in Hall thrusters and magnetrons and the values of the anomalous Hall parameter are consistent with experiments in Penning discharge and PIC simulations. The axial instability driven by the ion and electron flows and formation of the associated nonlinear structures have been investigated. The formation of large scale axial structures has also been observed in nonlinear 2D simulations. Our 1D and 2D PIC simulations demonstrate strongly nonlinear waves driven by the fundamental cyclotron resonance as well as features of the cascade toward longer wavelength. The nonlinear model and simulations presented here provide the first principles calculations of the anomalous electron current from turbulent fluctuations. Further work includes expansion of the model to include ionization effects and self-consistent multi-scale simulations allowing for slow evolution of background plasma parameters. Acknowledgments This work was supported by the Air Force Office of Scientific Research. References 1 S. N. Abolmasov, Physics and engineering of crossed-field discharge devices, Plasma Sources Sci. Technol. 21, (2012). 2 J. B. Parker, Y. Raitses, and N. J. Fisch, Transition in electron transport in a cylindrical Hall thruster, Appl. Phys. Lett. 97, (2010) 3 J. P. Boeuf, G. Fubiani, G. Hagelaar, N. Kohen, L. Pitchford, P. Sarrailh, A. Simonin, Physics and Modeling of the Negative Ion Source for the ITER Neutral Beam Injection, IAEA-CN-180, in Proceedings of the 23rd IAEA Fusion Energy Conference, Daejeon, October A. Anders, P. Ni, and A. Rauch, Drifting localization of ionization runaway: Unraveling the nature of anomalous transport in high power impulse magnetron sputtering, J. Appl. Phys. 111, (2012) 5 D. Lundin, U. Helmersson, S. Kirkpatrick, S. Rohde and N. Brenning, Anomalous electron transport in high power impulse magnetron sputtering, Plasma Sources Sci. Technol. 17, (2008) 6 J. C. Adam, A. Heron, and G. Laval, Study of stationary plasma thrusters using two-dimensional fully kinetic simulations, Phys. Plasmas 11, 295 (2004). 7 S. Tsikata, J. Cavalier, A. Héron, C. Honoré, and N. Lemoine, D. Grésillon, and D. Coulette, An axially propagating two-stream instability in the Hall thruster plasma, Phys. Plasmas 21, (2014). 8 J. P. Boeuf and B. Chaudhury, Rotating Instability in Low-Temperature Magnetized Plasmas, Physical Review Letters 111, (2013). 9 J.-P. Boeuf, Rotating structures in low temperature magnetized plasmas insight from particle simulations, Frontiers in Physics 2 (2014). 10 J. P. Boeuf, Tutorial: Physics and modeling of Hall thrusters, Journal of Applied Physics 121, 24 (2017). 11 N.B. Meezan, W.A. Hargus, Jr., and M.A. Cappelli, Phys. Rev. E 63, (2001) 12 A. Smolyakov, Y Raitses, I. Kaganovich, J.P. Boeuf, K. Matyash, R. Schneider, F. Taccogna, M. Cappelli, Turbulence, anomalous transport and structures in low temperature Hall plasmas with ExB drift, Invited White Paper, for Frontiers of Plasma Science, Panel 2: Plasma Turbulence and Transport, Department of Energy, Office of Fusion Energy Sciences, 13 W. Frias, Plasma instabilities in Hall thrusters, PhD Thesis, University of Saskatchewan, A. I. Smolyakov, O Chapurin, W Frias, O Koshkarov, I Romadanov, T Tang, M Umansky, Y Raitses, I D Kaganovich and V P Lakhin. Fluid theory and simulations of instabilities: Turbulent transport and coherent structures in partially-magnetized plasmas of E B discharges, 2017 Plasma Phys. Control. Fusion , 15 O. Chapurin, A. Smolyakov, On the electron drift velocity in plasma devices with ExB drift. J of Applied Physics 119, (2016). 16 I. Romadanov, A. Smolyakov, W.Frias, O. Chapurin, O.Koshkarov, Dispersion Relation Tool for generalized lower-hybrid mode with, density gradient, equilibrium ExB drift, collisions and finite electron Larmor radius, arxiv: , 17 I. Romadanov, A.I. Smolyakov, Y. Raitses, I. Kaganovich, S. Ryzhkov, Nonlocal gradient-drift instabilities in Hall E B discharges, Phys. Plasmas 23, (2016); 18 B.D. Dudson, J. Madsen, J. Omotani, P. Hill, L. Easy, M. Loiten, Verification of BOUT++ by the method of manufactured solutions, Physics of Plasmas, 2016, / V. P. Lakhin, V. I. Ilgisonis, A. I. Smolyakov and E. A. Sorokina. Nonlinear excitation of long-wavelength modes in Hall plasmas, Phys. Plasmas 23, (2016); 11

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