关于双束流Weibel不稳定性的研究
摘要
在空间和实验室等离子体中,粒子速度空间不均匀是一种普遍存在的现象,这种不均匀性能驱动不稳定性。Weibel不稳定性就是由于电子各向温度或者动量异性而导致的一种电磁不稳定性。伴随着该不稳定性,等离子体能激发准静态的磁场。
这些年来,Weibel不稳定性已经引起人们很大的关注,不仅因为其本身的物理机制,还在于它产生的磁场。天体等离子中,Weibel不稳定性常用来作为一种产生磁场的机制;实验室等离子体中,Weibel不稳定性对快点火中超热电子能量的传输有重要的作用。快点火中,对于圆偏振激光,在超热电子传播方向上存在一个轴向磁场,本文的目的就是研究该磁场对Weibel不稳定性的影响。
本文我们先讨论了双束流Weibel不稳定性。这部分首先归纳了电子—离子等离子体的线性色散关系;接着对双束流Weibel不稳定性进行PIC粒子模拟,给出不稳定性的一些特征,并比较了对称流和不对称流的差异;然后从流体方程出发,考虑冷双束流,得到描述非线性发展阶段的方程,在弱非线性近似下,求解该方程,解释了Weibel不稳定性的一些物理图像。
我们重点讨论了导向磁场对Weibel不稳定性的影响。考虑冷的双电子束流,从线性色散关系得出增长率和导向场的关系。理论分析给出,导向场会有效地减少不稳定性的增长率。PIC粒子模拟也得出了这样的结果,并给出在导向场的影响下,非线性阶段的一些复杂的物理现象,自生磁场的饱和水平大于无外场情况等等。在所产生的偶极场和静电场的作用下,一束电子被捕捉在自生磁场的两个峰值之间(梯度为正),另一束被分散到两个峰值之外的区域(梯度为负)。导向磁场的存在使得电子的汇聚更加集中。从饱和之后的电流和磁场分析,饱和之后,电子温度并没有达到各向同性,而是保持一定的异性。有了导向场之后,另一个明显的特征就是导向磁场会被调制为具有k=2的特征(我们初始所加扰动对应k=1),电子的大部分动能相应也会偏转到Weibel场所在的方向。
The Weibel instability is a transverse electromagnetic instability driven by temperature or electron's moment anisotropy (Weibel) or two counter streams (Fried). Apart from its basic theoretical interest, the Weibel instability has attracted much attention in the recent years in the plasma community, due to its capability of generating a (dipolar) magnetic field from essentially any kind of initial, infinitesimal, random noise. The resulting magnetic field could be then the seed for more robust mechanisms, such as the magnetic dynamo, able to produce the large scale fields observed in many celestial bodies. It also plays a crucial role in the transport of fast electrons' energies to the target in fast ignitor scenarios.Weibel's work has stimulated a series of further investigation of the transverse electromagnetic instability in unmagnetized plasma. These papers dealt with the linear, quasilinear and fully nonlinear theories as well as the computer simulation experiments of the instability. At the same time several other authors investigated the electromagnetic instabilities in magnetized plasma for a wide different orientations of the propagation vector. Most recently, Califano et al. have carried out further investigations of Weibel-type instability, where the role of temperature anisotropy is taken by two counterstreaming electron populations. All of previous analyses, including that of Weibel, have been based on the Vlasov-Maxwell formalism. As we know, If a circularly polarized super-intense laser is used to generate relativistic electron beam in the fast ignition, there exists a guiding magnetic field along the beam propagation direction due to the electrons circumgyrating with laser electric field. The motivation of our work is to investigate the effect of the guiding magnetic field on the Weibel instability. We focus our interest on one-dimensional non-relativistic case for simplicity. The main conclusion is easily suitable for the relativistic case.First, we study some work on the magnetic-free electron-ion plasmas. Linear dispersion relation is obtained on the basement of Vlasov-Maxwell formalism for both non-relativistic and relativistic cases. Then PIC simulation results show some physical
phenomena in the linear and in the non-linear regimes. Comparison between two symmetry streams and two non-symmetry ones is also done. In order to interpret the simulation results, we give some fluid description of Weibel instability. Considering two cold electron streams, we derive the equation describing the Weibel instability in the non-linear regime. On the weakly nonlinear regime approximation, we solve the equation and obtain its solution.Next, we pay our interests to the effect of a guiding magnetic field along beam propagation direction on the Weibel instability. Linear dispersion relation is obtained in the presence of such an external magnetic field, which shows the field can suppress or even stabilize the Weibel instability. Comparisons of our PIC simulation results with the analytical ones show very good agreement. The growth rate of Weibel instability decreases draftly as the guiding magnetic field increases and then comes to zero, which means the Weibel instability can't grow. Also observed in the simulation are the suppression of the electrostatic field, a higher level of self-generated magnetic field than that in the absence of guiding magnetic field, mode competition. As we know, on the action of self-generated magnetic field and the guiding magnetic field, one population of electrons will concentrated between two peaks of the field and be trapped there, while other population will diverge besides two peaks and also be trapped there. Then the self-generated magnetic field comes to saturation with a constant value and space configuration. Most of the electrons' kinetic energies will be deviated to the direction of Weibel magnetic field , and the guiding magnetic field will be devised by a self-generated field which has a characteristic of k=2. What's more, the guiding magnetic field also makes each electron population bunch further.
这些年来,Weibel不稳定性已经引起人们很大的关注,不仅因为其本身的物理机制,还在于它产生的磁场。天体等离子中,Weibel不稳定性常用来作为一种产生磁场的机制;实验室等离子体中,Weibel不稳定性对快点火中超热电子能量的传输有重要的作用。快点火中,对于圆偏振激光,在超热电子传播方向上存在一个轴向磁场,本文的目的就是研究该磁场对Weibel不稳定性的影响。
本文我们先讨论了双束流Weibel不稳定性。这部分首先归纳了电子—离子等离子体的线性色散关系;接着对双束流Weibel不稳定性进行PIC粒子模拟,给出不稳定性的一些特征,并比较了对称流和不对称流的差异;然后从流体方程出发,考虑冷双束流,得到描述非线性发展阶段的方程,在弱非线性近似下,求解该方程,解释了Weibel不稳定性的一些物理图像。
我们重点讨论了导向磁场对Weibel不稳定性的影响。考虑冷的双电子束流,从线性色散关系得出增长率和导向场的关系。理论分析给出,导向场会有效地减少不稳定性的增长率。PIC粒子模拟也得出了这样的结果,并给出在导向场的影响下,非线性阶段的一些复杂的物理现象,自生磁场的饱和水平大于无外场情况等等。在所产生的偶极场和静电场的作用下,一束电子被捕捉在自生磁场的两个峰值之间(梯度为正),另一束被分散到两个峰值之外的区域(梯度为负)。导向磁场的存在使得电子的汇聚更加集中。从饱和之后的电流和磁场分析,饱和之后,电子温度并没有达到各向同性,而是保持一定的异性。有了导向场之后,另一个明显的特征就是导向磁场会被调制为具有k=2的特征(我们初始所加扰动对应k=1),电子的大部分动能相应也会偏转到Weibel场所在的方向。
The Weibel instability is a transverse electromagnetic instability driven by temperature or electron's moment anisotropy (Weibel) or two counter streams (Fried). Apart from its basic theoretical interest, the Weibel instability has attracted much attention in the recent years in the plasma community, due to its capability of generating a (dipolar) magnetic field from essentially any kind of initial, infinitesimal, random noise. The resulting magnetic field could be then the seed for more robust mechanisms, such as the magnetic dynamo, able to produce the large scale fields observed in many celestial bodies. It also plays a crucial role in the transport of fast electrons' energies to the target in fast ignitor scenarios.Weibel's work has stimulated a series of further investigation of the transverse electromagnetic instability in unmagnetized plasma. These papers dealt with the linear, quasilinear and fully nonlinear theories as well as the computer simulation experiments of the instability. At the same time several other authors investigated the electromagnetic instabilities in magnetized plasma for a wide different orientations of the propagation vector. Most recently, Califano et al. have carried out further investigations of Weibel-type instability, where the role of temperature anisotropy is taken by two counterstreaming electron populations. All of previous analyses, including that of Weibel, have been based on the Vlasov-Maxwell formalism. As we know, If a circularly polarized super-intense laser is used to generate relativistic electron beam in the fast ignition, there exists a guiding magnetic field along the beam propagation direction due to the electrons circumgyrating with laser electric field. The motivation of our work is to investigate the effect of the guiding magnetic field on the Weibel instability. We focus our interest on one-dimensional non-relativistic case for simplicity. The main conclusion is easily suitable for the relativistic case.First, we study some work on the magnetic-free electron-ion plasmas. Linear dispersion relation is obtained on the basement of Vlasov-Maxwell formalism for both non-relativistic and relativistic cases. Then PIC simulation results show some physical
phenomena in the linear and in the non-linear regimes. Comparison between two symmetry streams and two non-symmetry ones is also done. In order to interpret the simulation results, we give some fluid description of Weibel instability. Considering two cold electron streams, we derive the equation describing the Weibel instability in the non-linear regime. On the weakly nonlinear regime approximation, we solve the equation and obtain its solution.Next, we pay our interests to the effect of a guiding magnetic field along beam propagation direction on the Weibel instability. Linear dispersion relation is obtained in the presence of such an external magnetic field, which shows the field can suppress or even stabilize the Weibel instability. Comparisons of our PIC simulation results with the analytical ones show very good agreement. The growth rate of Weibel instability decreases draftly as the guiding magnetic field increases and then comes to zero, which means the Weibel instability can't grow. Also observed in the simulation are the suppression of the electrostatic field, a higher level of self-generated magnetic field than that in the absence of guiding magnetic field, mode competition. As we know, on the action of self-generated magnetic field and the guiding magnetic field, one population of electrons will concentrated between two peaks of the field and be trapped there, while other population will diverge besides two peaks and also be trapped there. Then the self-generated magnetic field comes to saturation with a constant value and space configuration. Most of the electrons' kinetic energies will be deviated to the direction of Weibel magnetic field , and the guiding magnetic field will be devised by a self-generated field which has a characteristic of k=2. What's more, the guiding magnetic field also makes each electron population bunch further.
引文
1. Weibel E.S. Spontaneously Growing Transverse Waves in a Plasma Due to an Anisotropic Velocity Distribution Physics Review Letter 2, 83 1959
2. Krainow V. P. Weibel Instability in Plasmas Produced by aSuper-intense Femtosecond Laser Pulse Annual Moscow Workshop(Moscow, 3-4 December 2002)
3. Fried B.D. Mechanism for Instability of Transverse Plasmas Waves The Physics of Fluids 2, 337 1959
4. Basu B. Moment Equation Description of Weibel Instability Physics of Plasmas 9, 5131 2002
5.朱少平 等离子体物理讲义 (中国工程物理研究院北京研究生部讲义)2002
6. Medvedev M.V. and Loeb A. Generation of Magnetic Fields in the Relativistic Shock of Gammar-ray Burst Sources The Astrophysical Journal 526, 697 1999
7. Wiersma J. and Achterberg B. Magnetic field Generation in Relativistic Shocks Astron Soc 116, 138 2004
8.常铁强 快点火物理与进展情况(惯性约束聚变暑期班讲义) 2003
9. Silva L.O., Fonseca R. A., Tonge J.W., Mori W.B. and Dawson J.M. On the Role of the Purely Transverse Weibel Instability in Fast Ignitor Scenorios Physics of Plasmas 9, 2458 2002
10. Wilks S.C., Kruer W.L., Tabak M., and Langdon A.B. Absorption of Ultra-Intense Laser Pulses, Physics Review Letter 69, 1383 1992
11. Davidson R.C. , Hammer D.A. , Haber I. , and Wanger C.E. Nonlinear Development of Electromagnetic Instabilities in inisotropic Plasmas The Physisc of Fluids 15, 317 1972
12. DavidsonR. C. and Wu C.S. Ordinary-Mode Electromagnetic Instability in High-beta Plasmas The Physics of Fluids 13, 1407 1970
13.Hoeng J. T. Dispersion Relation in a Bi-Maxwellian Magnetoplasma Chinese Journal of Physics 14, 8 1976
14. Ozaki M. , Sato T. , Horiuchi R. , and Complexity Simulation Group Electromagnetic Instability and Anomalous Resistivity in a Magnetic Neutral Sheet Physics of Plasmas 3, 2265 1996
15. Califano F. , Pegoraro F. , Bulanov S. V. , and Mangeney A. Kinetic Saturation of the Weibel Instability in a Collisionless Plasma Physics Review E 57, 7048 1998
16. Kazimura Y. , Califano F. , Sakai J. , Neubert T. , Pegoraro F., and Bulanov S. V. Magnetic field Generation during the Collision of Electron-ion Plasma Clouds Journal of Phys. Soc. of Japan, 67, 1079 1998
17. Pegoraro F., Bulanov S., Califano F., and Lontano M., Non-linear development of the Weibel instability and magnetic field generation in collisionless plasmas, Phys. Scr.T63, 262 1996
18. Califano F., Prandi R., Pegoraro E., and Lontano M. Non-linear Filamentation Instability Driven by an Inhomogeneous Current in a Collisionless Plasma, Phys. Rev. E 58, 7837 1998
19.Califano F. , Cecchi T., and Chiuderi C. Nonlinear Kinetic Regime of the Weibel Instability in an Electron-ion Plasma Physics of Plasmas 9, 451 2002
20.Bernstein I.B. Waves in a Plasma in a Magnetic Field Physics Review 109, 10 1958
21. Fonseca R. A. , Silva L. O. , Tonge J. , Hemker R. G. , Dawson J. M. and Mori W. B. Three-Dimensional Particle-in-Cell Simulations of the Weibel Instability in Electron-Positron Plasmas IEEE TRANSACTIONS ON PLASMA SCIENCE, 30, 28 2002
22.Fonseca R.A., and Silva L. 0. Three-dimensional Weibel Instability in Astrophysical Scenarios Physics of Plasmas 10, 1979 2002
23. Yang T.-Y.B. and Arons J. Evolution of the Weibel Instability in Relativistically Hot Electron-positron Plasmas Physcis of Plasmas 1, 3059 1994
24. Lutomirski R.F. and Sudan R.N. Exact Nonlinear Electromagnetic Whistler Modes Physical Review 147, 156 1967
25.徐家鸾,金尚宪 等离子体物理学(原子能出版社)1981
26.陈银华 非线性等离子体物理导论(中国科学技术大学近代物理系讲义) 2000
27. Ichimaru S. Statistical Plasma Physics (Addison-Wesley Publishing company, Inc) 1992
28. Yang T.-Y.B. , Gallant Y. , and Arons J. Weibel Instability in Relativistically Hot Magnetized Electron-positron Plasmas Physics Fluids B5, 3369 1993
29. Zheng C.Y., He X.T. , and Zhu S.P. Magnetic Field Generation and Relativistic Electron Dynamics in Circularly Polarized Intense Laser Interaction with Dense Plasma Physics of Plasmas 12 2005
30. Qiao B., Zhu S. P., Zheng C. Y., and He X. T. Plasma Electron Acceleration for Fast Ignition in Intense Laser Plasma Interaction (Accepted by Physics of Plasmas)
31. Kruer W.L. The Physics of Laser Plasma Interactions (Addison-Wesley Publishing company, Inc) 1988
32. Birdsall C.K., and Langdon A.B. Plasma Physics via Computer Simulation (McGraw-Hill Book Company, New York) 1985
33. Zheng C. Y., Zhu S. P., and He X. T. Quasistatic Magnetic Field Generation by an Intense Ultrashort Laser Pulse in Underdense Plasma Chinese Physics Letter 17, 746 2000
2. Krainow V. P. Weibel Instability in Plasmas Produced by aSuper-intense Femtosecond Laser Pulse Annual Moscow Workshop
3. Fried B.D. Mechanism for Instability of Transverse Plasmas Waves The Physics of Fluids 2, 337 1959
4. Basu B. Moment Equation Description of Weibel Instability Physics of Plasmas 9, 5131 2002
5.朱少平 等离子体物理讲义 (中国工程物理研究院北京研究生部讲义)2002
6. Medvedev M.V. and Loeb A. Generation of Magnetic Fields in the Relativistic Shock of Gammar-ray Burst Sources The Astrophysical Journal 526, 697 1999
7. Wiersma J. and Achterberg B. Magnetic field Generation in Relativistic Shocks Astron Soc 116, 138 2004
8.常铁强 快点火物理与进展情况(惯性约束聚变暑期班讲义) 2003
9. Silva L.O., Fonseca R. A., Tonge J.W., Mori W.B. and Dawson J.M. On the Role of the Purely Transverse Weibel Instability in Fast Ignitor Scenorios Physics of Plasmas 9, 2458 2002
10. Wilks S.C., Kruer W.L., Tabak M., and Langdon A.B. Absorption of Ultra-Intense Laser Pulses, Physics Review Letter 69, 1383 1992
11. Davidson R.C. , Hammer D.A. , Haber I. , and Wanger C.E. Nonlinear Development of Electromagnetic Instabilities in inisotropic Plasmas The Physisc of Fluids 15, 317 1972
12. DavidsonR. C. and Wu C.S. Ordinary-Mode Electromagnetic Instability in High-beta Plasmas The Physics of Fluids 13, 1407 1970
13.Hoeng J. T. Dispersion Relation in a Bi-Maxwellian Magnetoplasma Chinese Journal of Physics 14, 8 1976
14. Ozaki M. , Sato T. , Horiuchi R. , and Complexity Simulation Group Electromagnetic Instability and Anomalous Resistivity in a Magnetic Neutral Sheet Physics of Plasmas 3, 2265 1996
15. Califano F. , Pegoraro F. , Bulanov S. V. , and Mangeney A. Kinetic Saturation of the Weibel Instability in a Collisionless Plasma Physics Review E 57, 7048 1998
16. Kazimura Y. , Califano F. , Sakai J. , Neubert T. , Pegoraro F., and Bulanov S. V. Magnetic field Generation during the Collision of Electron-ion Plasma Clouds Journal of Phys. Soc. of Japan, 67, 1079 1998
17. Pegoraro F., Bulanov S., Califano F., and Lontano M., Non-linear development of the Weibel instability and magnetic field generation in collisionless plasmas, Phys. Scr.T63, 262 1996
18. Califano F., Prandi R., Pegoraro E., and Lontano M. Non-linear Filamentation Instability Driven by an Inhomogeneous Current in a Collisionless Plasma, Phys. Rev. E 58, 7837 1998
19.Califano F. , Cecchi T., and Chiuderi C. Nonlinear Kinetic Regime of the Weibel Instability in an Electron-ion Plasma Physics of Plasmas 9, 451 2002
20.Bernstein I.B. Waves in a Plasma in a Magnetic Field Physics Review 109, 10 1958
21. Fonseca R. A. , Silva L. O. , Tonge J. , Hemker R. G. , Dawson J. M. and Mori W. B. Three-Dimensional Particle-in-Cell Simulations of the Weibel Instability in Electron-Positron Plasmas IEEE TRANSACTIONS ON PLASMA SCIENCE, 30, 28 2002
22.Fonseca R.A., and Silva L. 0. Three-dimensional Weibel Instability in Astrophysical Scenarios Physics of Plasmas 10, 1979 2002
23. Yang T.-Y.B. and Arons J. Evolution of the Weibel Instability in Relativistically Hot Electron-positron Plasmas Physcis of Plasmas 1, 3059 1994
24. Lutomirski R.F. and Sudan R.N. Exact Nonlinear Electromagnetic Whistler Modes Physical Review 147, 156 1967
25.徐家鸾,金尚宪 等离子体物理学(原子能出版社)1981
26.陈银华 非线性等离子体物理导论(中国科学技术大学近代物理系讲义) 2000
27. Ichimaru S. Statistical Plasma Physics (Addison-Wesley Publishing company, Inc) 1992
28. Yang T.-Y.B. , Gallant Y. , and Arons J. Weibel Instability in Relativistically Hot Magnetized Electron-positron Plasmas Physics Fluids B5, 3369 1993
29. Zheng C.Y., He X.T. , and Zhu S.P. Magnetic Field Generation and Relativistic Electron Dynamics in Circularly Polarized Intense Laser Interaction with Dense Plasma Physics of Plasmas 12 2005
30. Qiao B., Zhu S. P., Zheng C. Y., and He X. T. Plasma Electron Acceleration for Fast Ignition in Intense Laser Plasma Interaction (Accepted by Physics of Plasmas)
31. Kruer W.L. The Physics of Laser Plasma Interactions (Addison-Wesley Publishing company, Inc) 1988
32. Birdsall C.K., and Langdon A.B. Plasma Physics via Computer Simulation (McGraw-Hill Book Company, New York) 1985
33. Zheng C. Y., Zhu S. P., and He X. T. Quasistatic Magnetic Field Generation by an Intense Ultrashort Laser Pulse in Underdense Plasma Chinese Physics Letter 17, 746 2000