等离子体激波面上的静电不稳定性
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摘要
等离子体激波面中存在很强的密度和温度梯度,且此极端梯度使激波面中的电子流速和离子流速产生差别,从而造成电荷分离产生强电场。这种激波面中的强电场和参量梯度单独或共同驱动静态激波面上的不同于以往电中性给出的流体不稳定模式(如瑞利-泰勒不稳定性)的新型的静电不稳定模式。
本文从等离子体的双流体方程和泊松方程出发,分别从高频(ω≥ω_(pe))和低频(ω≤ω_(pi))两个不同频段讨论了一维平面激波波前中这种新型的静电不稳定性。得到:1)、在高频情况下,不稳定模由电场或(和)参量梯度驱动,且得到的不稳定波的频率的实部类似于电子静电波,增长率在波矢平行激波方向(k=k_xe_x)要大垂直(k=k_ye_y)方向两个量级。2)、在低频情况下,不稳定模也是由电场或(和)参量梯度驱动,且得到的不稳定波的频率的实部类似于离子静电波,增长率在平行和垂直方向大小在同一量级。另外,我们还得到了一支绝对不稳定模。同时,在两种情况下我们均给出了频率的实部和增长率与平衡量(n_(e0)(x),u_(e0)(x),T_(e0)(x) ,E_0(x))的关系。
In a plasma shock front, there are extremely high density and temperature gradients. So, the particle and heat diffusive fluxes caused by the density and temperature gradients are very large and the corresponding difference between the electron and ion diffusive fluxes will produce large charge separation (strong electrostatic field) in front. The one of the effects of this electrostatic field on the non-neutral shock is that it-self or it together with the parameter gradients can drive several instabilities on the equilibrium (steady state) shock front. These instabilities are a kind of new electrostatic modes, which are qualitatively different from those well known neutral modes (e.g. Rayliegh-Taylor or Richtmyer-Meshkov instability) driven only by the parameters gradient in the neutral shock front or discontinuity.
These new mentioned instabilities are investigated in this text with high frequency (ω≥ω_(pe)) and low frequency (ω≤ω_(pi)) respectively in a plane (one dimensional) plasma shock front by double fluid equations and Poisson equation. It can be concluded that: 1) In the case of high frequency, the instabilities are driven by the electrostatic field or /and parameter gradients, and the real parts of instabilities are similar to the one of electron electrostatic (plasma) wave, the growth rate in paralleling direction is about two magnitudes larger than in the perpendicular direction. 2) In the case of low frequency, the instabilities are also driven by the electrostatic field or/and parameter gradients, and the real parts of instabilities are similar to the one of ion electrostatic (plasma) wave, the growth rate in paralleling direction is in the same magnitude as the growth rate in the perpendicular direction. But more we find that there exit an absolute unstable mode—zero-frequency instability mode. This text we also give the dependencies of real frequencies and growth rates on the steady profiles of n_(e0)(x),u_(e0)(x),T_(e0)(x) and E_0(x) in both cases.
本文从等离子体的双流体方程和泊松方程出发,分别从高频(ω≥ω_(pe))和低频(ω≤ω_(pi))两个不同频段讨论了一维平面激波波前中这种新型的静电不稳定性。得到:1)、在高频情况下,不稳定模由电场或(和)参量梯度驱动,且得到的不稳定波的频率的实部类似于电子静电波,增长率在波矢平行激波方向(k=k_xe_x)要大垂直(k=k_ye_y)方向两个量级。2)、在低频情况下,不稳定模也是由电场或(和)参量梯度驱动,且得到的不稳定波的频率的实部类似于离子静电波,增长率在平行和垂直方向大小在同一量级。另外,我们还得到了一支绝对不稳定模。同时,在两种情况下我们均给出了频率的实部和增长率与平衡量(n_(e0)(x),u_(e0)(x),T_(e0)(x) ,E_0(x))的关系。
In a plasma shock front, there are extremely high density and temperature gradients. So, the particle and heat diffusive fluxes caused by the density and temperature gradients are very large and the corresponding difference between the electron and ion diffusive fluxes will produce large charge separation (strong electrostatic field) in front. The one of the effects of this electrostatic field on the non-neutral shock is that it-self or it together with the parameter gradients can drive several instabilities on the equilibrium (steady state) shock front. These instabilities are a kind of new electrostatic modes, which are qualitatively different from those well known neutral modes (e.g. Rayliegh-Taylor or Richtmyer-Meshkov instability) driven only by the parameters gradient in the neutral shock front or discontinuity.
These new mentioned instabilities are investigated in this text with high frequency (ω≥ω_(pe)) and low frequency (ω≤ω_(pi)) respectively in a plane (one dimensional) plasma shock front by double fluid equations and Poisson equation. It can be concluded that: 1) In the case of high frequency, the instabilities are driven by the electrostatic field or /and parameter gradients, and the real parts of instabilities are similar to the one of electron electrostatic (plasma) wave, the growth rate in paralleling direction is about two magnitudes larger than in the perpendicular direction. 2) In the case of low frequency, the instabilities are also driven by the electrostatic field or/and parameter gradients, and the real parts of instabilities are similar to the one of ion electrostatic (plasma) wave, the growth rate in paralleling direction is in the same magnitude as the growth rate in the perpendicular direction. But more we find that there exit an absolute unstable mode—zero-frequency instability mode. This text we also give the dependencies of real frequencies and growth rates on the steady profiles of n_(e0)(x),u_(e0)(x),T_(e0)(x) and E_0(x) in both cases.
引文
[1] Landau, Lifshitz. Fluid Mechanics. Pergamon Press, 1987.
[2] W. Greenberg, H. K. Sen, Y. M. Treve. Hydrodynamic Model of Diffusion Effects on Shock Structure in a Plasma. Phys. Fluid, 1960, 3(3): 379~386
[3] W. Greenberg, Y. M. Treve. Shock Wave and Solitary Wave Structure in a Plasma Phys. Fluids, 1960, 3(5): 769~785
[4] M. Y. Jaffrin, R. F. Probstein. Structure of a Plasma Shock Wave. Phys. Fluid, 1964, 7(10): 1658~1674
[5] T. J. M.博伊德, J. J.桑德森,戴世强,陆志云译.等离子体动力学.北京:科学出版社, 1977.
[6] Xu N, Wang L, Hu X. Extreme electrostatic phenomena in a single sonoluminescing bubble. Phys. Rev. Lett, 1999, 83(12): 2441~2444
[7]徐宁.单泡声致发光的等离子体过程及数值模拟: [博士学位论文].北京:中国科学院物理研究所,2000.
[8] Rayleigh L. Investigation of the character of the equilibrium of an incompressible heavy fluid of variable density. Proc. London Math. Soc., 1883, 14: 170~177
[9] Taylor G I. The instability of liquid surfaces when accelerated in a direction perpendicular to their planes. I. Proc. R. Soc. A, 1950, 201: 192~196
[10] Richtmyer R D. Taylor instability in shock acceleration of compressible fluids. Comm. Pure Appl. Math., 1960, 8: 297~319
[11] Meshkov E E. Instability of the interface of two gases accelerated by a shock wave. Sov. Fluid Dyn., 1969, 4: 101~108
[12]李维新.一维不定常流与冲击波,北京:国防工业出版社,2003.
[13] R.柯朗, K. O.弗里德里克斯,李维新等译.超声速流与冲击波.北京:科学出版社, 1986.
[14] Zel’dovich Y B, Raizer Y P,张树材译.激波和高温流体动力学现象物理学.北京:科学出版社, 1980.
[15]胡业民.等离子体激波结构与非线性极化率对参量过程的影响: [博士学位论文].合肥:中国科学技术大学, 2003.
[16]胡希伟.等离子体理论基础.北京:北京大学出版社,2006.
[17]何勇.等离子体中自生电场对激波面和界面不稳定性的影响:[博士学位论文].武汉:华中科技大学,2007.
[18] Hu Xi Wei. Jump Conditions of a Non-Neutral Plasma Shock with Current and Potential Difference. Chin. Phys. Lett, 2002, 19(6): 822~824
[19] Hu Y, Hu X. The properties and structures of a plasma non-neutral shock. Phys. Plasmas, 2003, 10(7): 2704~2711
[20] Hu Ye Min, Hu Xi Wei. Properties and structure of a plasma non-neutral shock. Chin. Phys. Lett, 2004, 21(1): 133~136
[21] He Yong, Hu Xi Wei, Hu Ye Min. Jump Conditions of a Shock with Current in Cylindrical Non-Neutral Plasma. Chin. Phys. Lett, 2006, 23(1): 165~168
[22] He Y, Hu X, Hu Y. Rankine-Hugoniot relations of an axial shock in cylindrical non-neutral plasma. Phys. Plasmas, 2006, 13(9): 092116(5)
[23] D’yakov S P. Stability of shock waves. Zh. Eksp Teor. Fiz., 1954, 27: 288
[24] Kontorovich V M. The question on stability of shock waves. Journal of Experimental and Theoretical Physics, 1957, 6: 1179
[25] Sinkevich O A. Stability of a plane, ionizing shock wave in a magnetic field. Fluid dynamics, 1972, 14(1): 122~128
[26] G. R. Fowles, G. W. Swan. Stability of Plane Shock Wave. Phys Rev Lett, 1973, 30(21): 1023~1025
[27] G. W. Swan, G. R. Fowles. Shock Wave Instability. Physics of Fluids, 1975, 18(1): 28~35
[28] G. R. Fowles, A. F. P. Houwing. Instabilities of Shock and Detonation Wave. Physics of Fluids, 1984, 27(8): 1982~1990
[29] L. N. Tsintsadzea P K S, Tsintsadze N L. Corrugation instability of radiative shock waves in a relativistically hot plasma. Phys. Plasmas, 1997, 4(11): 3923~3927
[30] Jason. W. Bates, David C. Montgomery. The D’yakov-Kontorovich instability of shock waves in real gases. Phys Rev Lett, 2000, 84: 1180~1183
[31] Jason. W. Bates. Studies of non-classical shock wave phenomena. Shock Wave, 2002, 12: 31~37
[32]徐复,陈乐山.管道内激波的不稳定性.应用数学和力学, 1993, 14(12): 1093~1104
[33]黄迎雷,崔季平.激波稳定性(1).力学学报, 1989, 21(2): 168~175
[34]黄迎雷,崔季平.激波稳定性(2).力学学报, 1989, 21(3): 280~289
[35]徐跃民,周国成,朱连芳.弓激波前内等离子体动力学横场流静电不稳定性.空间科学学报,1991, 11(2): 91~97
[36] S. L. Braginskii. Transport processes in a plasma, in Reviews of Plasma Physics. New York: Academic, 1965.
[37] Casanova M, Larroche O. Kinetic simulation of a collisional shock wave in a plasma. Phys. Rev. Lett., 1991, 67(16): 2143~2146
[38]哈瑟加瓦著,王水译,方励之校.等离子体不稳定性和非线性效应.北京:科学出版社,1981.
[39]徐家鸾,金尚宪.等离子体物理学.北京:原子能出版社,1981.
[40] Lv Jian Hong, He Yong, Hu Xi Wei. Electrostatic Instabilities at High Frequency in a Plasma Shock Front. Chin. Phys. Lett, 2007, 24(4): 1000~1003
[41]马腾才,胡希伟,陈银华.等离子体物理原理合肥:中国科学技术大学出版社1988.
[42] F. F.陈著,林光海译.等离子体物理学导论.北京:人民教育出版社,1980.
[2] W. Greenberg, H. K. Sen, Y. M. Treve. Hydrodynamic Model of Diffusion Effects on Shock Structure in a Plasma. Phys. Fluid, 1960, 3(3): 379~386
[3] W. Greenberg, Y. M. Treve. Shock Wave and Solitary Wave Structure in a Plasma Phys. Fluids, 1960, 3(5): 769~785
[4] M. Y. Jaffrin, R. F. Probstein. Structure of a Plasma Shock Wave. Phys. Fluid, 1964, 7(10): 1658~1674
[5] T. J. M.博伊德, J. J.桑德森,戴世强,陆志云译.等离子体动力学.北京:科学出版社, 1977.
[6] Xu N, Wang L, Hu X. Extreme electrostatic phenomena in a single sonoluminescing bubble. Phys. Rev. Lett, 1999, 83(12): 2441~2444
[7]徐宁.单泡声致发光的等离子体过程及数值模拟: [博士学位论文].北京:中国科学院物理研究所,2000.
[8] Rayleigh L. Investigation of the character of the equilibrium of an incompressible heavy fluid of variable density. Proc. London Math. Soc., 1883, 14: 170~177
[9] Taylor G I. The instability of liquid surfaces when accelerated in a direction perpendicular to their planes. I. Proc. R. Soc. A, 1950, 201: 192~196
[10] Richtmyer R D. Taylor instability in shock acceleration of compressible fluids. Comm. Pure Appl. Math., 1960, 8: 297~319
[11] Meshkov E E. Instability of the interface of two gases accelerated by a shock wave. Sov. Fluid Dyn., 1969, 4: 101~108
[12]李维新.一维不定常流与冲击波,北京:国防工业出版社,2003.
[13] R.柯朗, K. O.弗里德里克斯,李维新等译.超声速流与冲击波.北京:科学出版社, 1986.
[14] Zel’dovich Y B, Raizer Y P,张树材译.激波和高温流体动力学现象物理学.北京:科学出版社, 1980.
[15]胡业民.等离子体激波结构与非线性极化率对参量过程的影响: [博士学位论文].合肥:中国科学技术大学, 2003.
[16]胡希伟.等离子体理论基础.北京:北京大学出版社,2006.
[17]何勇.等离子体中自生电场对激波面和界面不稳定性的影响:[博士学位论文].武汉:华中科技大学,2007.
[18] Hu Xi Wei. Jump Conditions of a Non-Neutral Plasma Shock with Current and Potential Difference. Chin. Phys. Lett, 2002, 19(6): 822~824
[19] Hu Y, Hu X. The properties and structures of a plasma non-neutral shock. Phys. Plasmas, 2003, 10(7): 2704~2711
[20] Hu Ye Min, Hu Xi Wei. Properties and structure of a plasma non-neutral shock. Chin. Phys. Lett, 2004, 21(1): 133~136
[21] He Yong, Hu Xi Wei, Hu Ye Min. Jump Conditions of a Shock with Current in Cylindrical Non-Neutral Plasma. Chin. Phys. Lett, 2006, 23(1): 165~168
[22] He Y, Hu X, Hu Y. Rankine-Hugoniot relations of an axial shock in cylindrical non-neutral plasma. Phys. Plasmas, 2006, 13(9): 092116(5)
[23] D’yakov S P. Stability of shock waves. Zh. Eksp Teor. Fiz., 1954, 27: 288
[24] Kontorovich V M. The question on stability of shock waves. Journal of Experimental and Theoretical Physics, 1957, 6: 1179
[25] Sinkevich O A. Stability of a plane, ionizing shock wave in a magnetic field. Fluid dynamics, 1972, 14(1): 122~128
[26] G. R. Fowles, G. W. Swan. Stability of Plane Shock Wave. Phys Rev Lett, 1973, 30(21): 1023~1025
[27] G. W. Swan, G. R. Fowles. Shock Wave Instability. Physics of Fluids, 1975, 18(1): 28~35
[28] G. R. Fowles, A. F. P. Houwing. Instabilities of Shock and Detonation Wave. Physics of Fluids, 1984, 27(8): 1982~1990
[29] L. N. Tsintsadzea P K S, Tsintsadze N L. Corrugation instability of radiative shock waves in a relativistically hot plasma. Phys. Plasmas, 1997, 4(11): 3923~3927
[30] Jason. W. Bates, David C. Montgomery. The D’yakov-Kontorovich instability of shock waves in real gases. Phys Rev Lett, 2000, 84: 1180~1183
[31] Jason. W. Bates. Studies of non-classical shock wave phenomena. Shock Wave, 2002, 12: 31~37
[32]徐复,陈乐山.管道内激波的不稳定性.应用数学和力学, 1993, 14(12): 1093~1104
[33]黄迎雷,崔季平.激波稳定性(1).力学学报, 1989, 21(2): 168~175
[34]黄迎雷,崔季平.激波稳定性(2).力学学报, 1989, 21(3): 280~289
[35]徐跃民,周国成,朱连芳.弓激波前内等离子体动力学横场流静电不稳定性.空间科学学报,1991, 11(2): 91~97
[36] S. L. Braginskii. Transport processes in a plasma, in Reviews of Plasma Physics. New York: Academic, 1965.
[37] Casanova M, Larroche O. Kinetic simulation of a collisional shock wave in a plasma. Phys. Rev. Lett., 1991, 67(16): 2143~2146
[38]哈瑟加瓦著,王水译,方励之校.等离子体不稳定性和非线性效应.北京:科学出版社,1981.
[39]徐家鸾,金尚宪.等离子体物理学.北京:原子能出版社,1981.
[40] Lv Jian Hong, He Yong, Hu Xi Wei. Electrostatic Instabilities at High Frequency in a Plasma Shock Front. Chin. Phys. Lett, 2007, 24(4): 1000~1003
[41]马腾才,胡希伟,陈银华.等离子体物理原理合肥:中国科学技术大学出版社1988.
[42] F. F.陈著,林光海译.等离子体物理学导论.北京:人民教育出版社,1980.