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Deflection effect of electromagnetic field generated byWeibel instability on proton probe

Du Bao Cai Hong-Bo Zhang Wen-Shuai Chen Jing Zou Shi-Yang Zhu Shao-Ping

Deflection effect of electromagnetic field generated byWeibel instability on proton probe

Du Bao, Cai Hong-Bo, Zhang Wen-Shuai, Chen Jing, Zou Shi-Yang, Zhu Shao-Ping
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  • The electric and magnetic fields generated by the Weibel instability, most of which have a tube-like structure, are of importance for many relevant physical processes in the astrophysics and the inertial confinement fusion. Experimentally, proton radiography is a commonly used method to diagnose the Weibel instability, where the proton deflection introduced from the self-generated electric field is usually ignored. This assumption, however, is in conflict with the experimental observations by Quinn, Fox and Huntington, et al. because the magnetic field with a tube-like structure cannot introduce parallel flux striations on the deflection plane in the proton radiography. In this paper, we re-examine the nature of the proton radiography of the Weibel instability numerically. Two symmetric counterstreaming plasma flows are used to generate the electron Weibel instability with the three-dimensional particle-in-cell simulations. The proton radiography of the Weibel instability generated electric and magnetic fields are calculated with the ray tracing method. Three cases are considered andcompared: only the self-generated electric field E is included, only the self-generated magnetic field B is included, both the electric field E and magnetic field B are included. It is shown that when only E is included, the probe proton flux density perturbation on the detection plane, i.e., δn/n0, is much larger than that when only B is included. Also, when both E and B are included, δn/n0 is almost the same as that when only E is included. This suggests that in the proton radiography of the Weibel instability generated electric and magnetic fields, the deflection from the electric field dominates the radiography, whereas the magnetic field has an ignorable influence. Our conclusion is quite different from that obtained on the traditional assumption that the electric field is ignorable in the radiography. This mainly comes from the spatial structure of the Weibel instability generated magnetic field, which is tube-like and points to the azimuthal direction around the current filaments. When the probe protons pass through the field region, the deflection from the azimuthal magnetic field can be compensated for completely by itself along the passing trajectories especially if the deflection distance inside the field region is small. Whereas for the electric field, which is in the radial direction, the deflection to the probe protons will not be totally compensated for and will finally introduce an evident flux density perturbation into the detection plane. This understanding can beconducive to the comprehension of the experimental results about the proton radiography of the Weibel instability.
      Corresponding author: Cai Hong-Bo, Cai_hongbo@iapcm.ac.cn ; Zhu Shao-Ping, zhu_shaoping@iapcm.ac.cn
    [1]

    Weibel E S 1959 Phys. Rev. Lett. 2 83

    [2]

    Fried B D 1959 Phys. Fluids 2 337

    [3]

    Honda M, Meyer-ter-Vehn J, Pukhov A 2000 Phys. Rev. Lett. 85 2128

    [4]

    Ross J S, Park H S, Berger R, Divol L, Kugland N L, Rozmus W, Ryutov D, Glenzer S H 2013 Phys. Rev. Lett. 110 145005

    [5]

    Fiuza F, Fonseca R A, Tonge J, Mori W B, Silva L O1 2012 Phys. Rev. Lett. 108 235004

    [6]

    Ardaneh K, Cai D S, Nishikawa K I, Lembége B 2015 Astrophys. J. 811 57

    [7]

    Quinn K, Romagnani L, Ramakrishna B, Sarri G, Dieckmann M E, Wilson P A, Fuchs J, Lancia L, Pipahl A, Toncian T, Willi O, Clarke R J, Notley M, Macchi A, Borghesi M 2012 Phys. Rev. Lett. 108 135001

    [8]

    Kugland N L, Ryutov D D, Chang P Y, Drake R P, Fiksel G, Froula D H, Glenzer S H, Gregori G, Grosskopf M, Koenig M, Kuramitsu Y, Kuranz C, Levy M C, Liang E, Meinecke J, Miniati F, Morita T, Pelka A, Plechaty C, Presura R, Ravasio A, Remington B A, Reville B, Ross J S, Sakawa Y, Spitkovsky A, Takabe H, Park H S 2012 Nat. Phys. 8 809

    [9]

    Fox W, Fiksel G, Bhattacharjee A, Chang P Y, Germaschewski K, Hu S X, Nilson P M 2013 Phys. Rev. Lett. 111 225002

    [10]

    Huntington C M, Fiuza F, Ross J S, Zylstra A B, Drake R P, Froula D H, Gregori G, Kugland N L, Kuranz C C, Levy M C, Li C K, Meinecke J, Morita T, Petrasso R, Plechaty C, Remington B A, Ryutov D D, SakawaY, Spitkovsky A, Takabe H, Park S H 2015 Nat. Phys. 11 173

    [11]

    Tzoufras M, Ren C, Tsung F S, Tonge J W, Mori W B, Fiore M, Fonseca R A, Silva L O 2006 Phys. Rev. Lett. 96 105002

    [12]

    Dieckmann M E 2009 Plasma Phys. Control. Fusion 51 124042

    [13]

    Kugland N L, Ryutov D D, Plechaty C, Ross J S, Park H S 2012 Rev. Sci. Instruments 83 101301

    [14]

    Wang W W, Cai H B, Teng J, Chen J, He S K, Shan L Q, Lu F, Wu Y C, Zhang B, Hong W, Bi B, Zhang F, Liu D X, Xue F B, Li B Y, Liu H J, He W, Jiao J L, Dong K G, Zhang F Q, He Y L, Cui B, Xie N, Yuan Z Q, Tian C, Wang X D, Zhou K N, Deng Z G, Zhang Z M, Zhou W M, Cao L F, Zhang B H, Zhu S P, He X T, Gu Y Q 2018 Phys. Plasmas 25 083111

    [15]

    Bret A, Gremillet L, Dieckmann M E 2010 Phys. Plasmas 17 120501

    [16]

    Gao L, Nilson M P, Igumenshchev I V, Haines M G, Froula D H, Betti R, Meyerhofer D D 2015 Phys. Rev. Lett. 114 215003

    [17]

    Du B, Wang X F 2018 AIP Adv. 8 125328

    [18]

    Cai H B, Mima K, Zhou W M, Jozaki T, Nagatomo H, Sunahara A, Mason R J 2009 Phys. Rev. Lett. 102 245001

    [19]

    Cagas P, Hakim A, Scales W, Srinivasan B 2017 Phys. Plasmas 24 112116

    [20]

    Alves E P, Zrake J, Fiuza F 2018 Phys. Rev. Lett. 121 245101

    [21]

    Sentoku Y, Mima K, Sheng Z M, Kaw P, Nishihara K, Nishikawa K 2002 Phys. Rev. E 65 046408

    [22]

    Shukla C, Kumar A, Das A, Patel B G 2018 Phys. Plasmas 25 022123

    [23]

    Li C K, Séguin F H, Frenje J A, Rygg J R, Petrasso R D, Town R P J, Amendt P A, Hatchett S P, Landen O L, Mackinnon A J, Patel P K, Smalyuk V A, Sangster T C, Knauer J P 2006 Phys. Rev. Lett. 97 135003

    [24]

    Cecchetti C A, Borghesi M, Fuchs J, Schurtz G, Kar S, Macchi A, Romagnani, Wilson P A, Antici P, Jung R, Osterholtz J, Pipahl C, Willi O, Schiavi A, Notley M, Neely D 2009 Phys. Plasmas 16 043102

  • 图 1  Weibel不稳定性的质子照相示意图

    Figure 1.  Schematic diagram of the proton radiography of the Weibel instability

    图 2  Weibel不稳定性 (a)自生磁场By和(b)自生电场Ext = 1.06 ps时的三维空间分布

    Figure 2.  Three demensional distributions of the Weibel instability generated (a) magnetic field By and (b) electric field Ex at t = 1.06 ps

    图 3  Weibel不稳定性自生磁场和电场能量随着时间的演化

    Figure 3.  Evolution of the energy of the Weibel instability generated magnetic and electric fields.

    图 4  t = 1.06 ps时, z = 0平面上(a)磁场强度|B|、(b)电场强度|E|、(c)磁场方向和(d)电场方向的分布情况以及y = 0平面上(e) y向磁场By和(f) y向电场Ey的分布情况

    Figure 4.  Spatial distributions of (a) the magnetic field strength |B|, (b) the electric field strength |E|, (c) the direction of B and (d) the direction of E on the z = 0 plane, (e) the y component of the magnetic field By and (f) the y component of the electric field Ey on the y = 0 plane at t = 1.06 ps.

    图 5  t = 4.78 ps时, z = 0平面上(a)磁场强度|B|、(b)电场强度|E|、(c)磁场方向和(d)电场方向的分布情况以及y = 0平面上(e) y向磁场By和(f) y向电场Ey的分布情况

    Figure 5.  Spatial distributions of (a) the magnetic field strength |B|, (b) the electric field strength |E|, (c) the direction of B and (d) the direction of E on the z = 0 plane, (e) the y component of the magnetic field By and (f) the y component of the electric field Ey on the y = 0 plane at t = 4.78 ps.

    图 6  t = 1.06 ps时, (a)只考虑电场E、(b)只考虑磁场B以及(c)同时考虑电场E和磁场B三种情况下探测面上的质子通量密度扰动分布信息

    Figure 6.  Proton flux density perturbations on the detection plane when (a) only the electric field is included, (b) only the magnetic field is included and (c) both the electric and magnetic fields are included at t = 1.06 ps.

    图 7  t = 4.78 ps时(a)只考虑电场E、(b)只考虑磁场B以及(c)同时考虑电场E和磁场B三种情况下探测面上的质子通量密度扰动分布信息

    Figure 7.  Proton flux density perturbations on the detection plane when (a) only the electric field is included, (b) only the magnetic field is included and (c) both the electric and magnetic fields are included at t = 4.78 ps.

  • [1]

    Weibel E S 1959 Phys. Rev. Lett. 2 83

    [2]

    Fried B D 1959 Phys. Fluids 2 337

    [3]

    Honda M, Meyer-ter-Vehn J, Pukhov A 2000 Phys. Rev. Lett. 85 2128

    [4]

    Ross J S, Park H S, Berger R, Divol L, Kugland N L, Rozmus W, Ryutov D, Glenzer S H 2013 Phys. Rev. Lett. 110 145005

    [5]

    Fiuza F, Fonseca R A, Tonge J, Mori W B, Silva L O1 2012 Phys. Rev. Lett. 108 235004

    [6]

    Ardaneh K, Cai D S, Nishikawa K I, Lembége B 2015 Astrophys. J. 811 57

    [7]

    Quinn K, Romagnani L, Ramakrishna B, Sarri G, Dieckmann M E, Wilson P A, Fuchs J, Lancia L, Pipahl A, Toncian T, Willi O, Clarke R J, Notley M, Macchi A, Borghesi M 2012 Phys. Rev. Lett. 108 135001

    [8]

    Kugland N L, Ryutov D D, Chang P Y, Drake R P, Fiksel G, Froula D H, Glenzer S H, Gregori G, Grosskopf M, Koenig M, Kuramitsu Y, Kuranz C, Levy M C, Liang E, Meinecke J, Miniati F, Morita T, Pelka A, Plechaty C, Presura R, Ravasio A, Remington B A, Reville B, Ross J S, Sakawa Y, Spitkovsky A, Takabe H, Park H S 2012 Nat. Phys. 8 809

    [9]

    Fox W, Fiksel G, Bhattacharjee A, Chang P Y, Germaschewski K, Hu S X, Nilson P M 2013 Phys. Rev. Lett. 111 225002

    [10]

    Huntington C M, Fiuza F, Ross J S, Zylstra A B, Drake R P, Froula D H, Gregori G, Kugland N L, Kuranz C C, Levy M C, Li C K, Meinecke J, Morita T, Petrasso R, Plechaty C, Remington B A, Ryutov D D, SakawaY, Spitkovsky A, Takabe H, Park S H 2015 Nat. Phys. 11 173

    [11]

    Tzoufras M, Ren C, Tsung F S, Tonge J W, Mori W B, Fiore M, Fonseca R A, Silva L O 2006 Phys. Rev. Lett. 96 105002

    [12]

    Dieckmann M E 2009 Plasma Phys. Control. Fusion 51 124042

    [13]

    Kugland N L, Ryutov D D, Plechaty C, Ross J S, Park H S 2012 Rev. Sci. Instruments 83 101301

    [14]

    Wang W W, Cai H B, Teng J, Chen J, He S K, Shan L Q, Lu F, Wu Y C, Zhang B, Hong W, Bi B, Zhang F, Liu D X, Xue F B, Li B Y, Liu H J, He W, Jiao J L, Dong K G, Zhang F Q, He Y L, Cui B, Xie N, Yuan Z Q, Tian C, Wang X D, Zhou K N, Deng Z G, Zhang Z M, Zhou W M, Cao L F, Zhang B H, Zhu S P, He X T, Gu Y Q 2018 Phys. Plasmas 25 083111

    [15]

    Bret A, Gremillet L, Dieckmann M E 2010 Phys. Plasmas 17 120501

    [16]

    Gao L, Nilson M P, Igumenshchev I V, Haines M G, Froula D H, Betti R, Meyerhofer D D 2015 Phys. Rev. Lett. 114 215003

    [17]

    Du B, Wang X F 2018 AIP Adv. 8 125328

    [18]

    Cai H B, Mima K, Zhou W M, Jozaki T, Nagatomo H, Sunahara A, Mason R J 2009 Phys. Rev. Lett. 102 245001

    [19]

    Cagas P, Hakim A, Scales W, Srinivasan B 2017 Phys. Plasmas 24 112116

    [20]

    Alves E P, Zrake J, Fiuza F 2018 Phys. Rev. Lett. 121 245101

    [21]

    Sentoku Y, Mima K, Sheng Z M, Kaw P, Nishihara K, Nishikawa K 2002 Phys. Rev. E 65 046408

    [22]

    Shukla C, Kumar A, Das A, Patel B G 2018 Phys. Plasmas 25 022123

    [23]

    Li C K, Séguin F H, Frenje J A, Rygg J R, Petrasso R D, Town R P J, Amendt P A, Hatchett S P, Landen O L, Mackinnon A J, Patel P K, Smalyuk V A, Sangster T C, Knauer J P 2006 Phys. Rev. Lett. 97 135003

    [24]

    Cecchetti C A, Borghesi M, Fuchs J, Schurtz G, Kar S, Macchi A, Romagnani, Wilson P A, Antici P, Jung R, Osterholtz J, Pipahl C, Willi O, Schiavi A, Notley M, Neely D 2009 Phys. Plasmas 16 043102

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  • Received Date:  21 May 2019
  • Accepted Date:  03 July 2019
  • Available Online:  01 September 2019
  • Published Online:  20 September 2019

Deflection effect of electromagnetic field generated byWeibel instability on proton probe

    Corresponding author: Cai Hong-Bo, Cai_hongbo@iapcm.ac.cn
    Corresponding author: Zhu Shao-Ping, zhu_shaoping@iapcm.ac.cn
  • 1. Institute of Applied Physics and Computational Mathematics, Beijing 100094, China
  • 2. HEDPS, Center for Applied Physics and Technology, Peking University, Beijing 100871, China
  • 3. IFSA Collaborative Innovation Center, Shanghai Jiao Tong University, Shanghai 200240, China
  • 4. Graduate School, China Academy of Engineering Physics, Beijing 100088, China
  • 5. STPPL, Research Center of Laser Fusion, China Academy of Engineering Physics, Mianyang 621900, China

Abstract:  The electric and magnetic fields generated by the Weibel instability, most of which have a tube-like structure, are of importance for many relevant physical processes in the astrophysics and the inertial confinement fusion. Experimentally, proton radiography is a commonly used method to diagnose the Weibel instability, where the proton deflection introduced from the self-generated electric field is usually ignored. This assumption, however, is in conflict with the experimental observations by Quinn, Fox and Huntington, et al. because the magnetic field with a tube-like structure cannot introduce parallel flux striations on the deflection plane in the proton radiography. In this paper, we re-examine the nature of the proton radiography of the Weibel instability numerically. Two symmetric counterstreaming plasma flows are used to generate the electron Weibel instability with the three-dimensional particle-in-cell simulations. The proton radiography of the Weibel instability generated electric and magnetic fields are calculated with the ray tracing method. Three cases are considered andcompared: only the self-generated electric field E is included, only the self-generated magnetic field B is included, both the electric field E and magnetic field B are included. It is shown that when only E is included, the probe proton flux density perturbation on the detection plane, i.e., δn/n0, is much larger than that when only B is included. Also, when both E and B are included, δn/n0 is almost the same as that when only E is included. This suggests that in the proton radiography of the Weibel instability generated electric and magnetic fields, the deflection from the electric field dominates the radiography, whereas the magnetic field has an ignorable influence. Our conclusion is quite different from that obtained on the traditional assumption that the electric field is ignorable in the radiography. This mainly comes from the spatial structure of the Weibel instability generated magnetic field, which is tube-like and points to the azimuthal direction around the current filaments. When the probe protons pass through the field region, the deflection from the azimuthal magnetic field can be compensated for completely by itself along the passing trajectories especially if the deflection distance inside the field region is small. Whereas for the electric field, which is in the radial direction, the deflection to the probe protons will not be totally compensated for and will finally introduce an evident flux density perturbation into the detection plane. This understanding can beconducive to the comprehension of the experimental results about the proton radiography of the Weibel instability.

    • Weibel不稳定性是一种基于等离子体温度各向异性的微观不稳定性[1,2]. 在惯性约束聚变以及天体物理领域, Weibel不稳定性对于等离子体能量输运、无碰撞冲击波的形成以及伽马射线暴等物理过程具有关键的影响[36]. 伴随着近二十年来高功率激光技术的不断发展, 以膨胀等离子体为基础的Weibel不稳定性实验研究受到了广泛关注并取得了大量突破性进展[710].

      通常, 可以通过两束对称等离子体的对穿来产生Weibel不稳定性[9,10]. 对穿方向的等离子体在不稳定性反馈机制下发生箍缩而形成丝状电流结构, 产生环绕着电流丝的管状磁场. 粒子模拟(particle-in-cell, PIC)结果表明, 这些管状磁场在平行于电流的方向上基本均匀分布, 而在垂直于电流方向上则具有随机分布的特征[11]. 可以在非磁化等离子体中产生磁场是Weibel不稳定性的一个重要特性, 但在Weibel不稳定性的发展过程中, 电流的箍缩作用同时也会使等离子体在垂直于电流方向上形成静电场[12].

      实验上, 质子束照相技术是诊断电磁场的重要方法[13]. 该技术的基本原理是: 探针质子束在穿越电磁场区域时, 电磁场会引起质子束的偏转, 并最终对探测面上的质子通量密度分布造成调制. 利用探测面上的质子束通量密度分布并借助一定的反演方法, 可以定量或者定性地诊断被探测电磁场的分布信息. 得益于国内外高能质子源品质的不断优化, 质子束照相技术已经成为一种诊断Weibel不稳定性的常用方法[14]. 图1为常见的Weibel不稳定性质子照相示意图, 其中等离子体沿着z方向进行对穿, 探针质子束沿着x方向对其进行侧向照相.

      Figure 1.  Schematic diagram of the proton radiography of the Weibel instability

      在对Weibel不稳定性进行质子束照相时, 横向电场和磁场均可对探针质子束造成偏转. 准确地判断被诊断电磁场的属性是进行深入物理分析的基本前提. 传统上, 在对Weibel不稳定性的质子照相结果进行定性和定量分析时, 一般情况下忽略来自电场的影响, 而认为引起探针质子束偏转的主要是磁场[7,10]. 但是, 该假设的正确性却一直没有得到针对性的定量检验.

      Quinn等[7]、Fox等[9]以及Huntington等[10]关于Weibel不稳定性的质子束照相实验结果显示, 探测面上的通量密度大部分呈现出沿z方向平行的丝状调制结构. 这表明, 探针质子束受到的偏转作用主要在y方向. 但当探针质子束沿着x方向穿透Weibel不稳定性的发生区域时, 如果造成质子束偏转的作用力来自于横向磁场Bx + By, 那么磁场对质子束的偏转作用力将沿着z方向. 这与实验观察不符合. 而且, PIC模拟结果显示, 一些情况下横向静电场的能量只比磁场能量小约一个数量级[15], 即$\left| {{E}} \right| \sim c\left| {{B}} \right|/\sqrt {10} $. 而实验上质子源多基于靶后鞘层加速(target normal sheath acceleration)产生, 其动能的最大值一般约为20 MeV[16]. 此时, 来自电场的偏转作用力$q\left| {{E}} \right|\sim q\dfrac{c}{{\sqrt {10} }}\left| {{B}} \right|$甚至比磁场的洛伦兹作用力$q{u_0}\left| {{B}} \right|$更大. 这里c为真空光速, q为质子电荷, u0为质子运动速度. 基于以上分析, 在对Weibel不稳定性进行质子束照相时, 忽略电场的偏转作用可能并不是一个很好的假设.

      为了检验Weibel不稳定性质子束照相中电场可被忽略这一假设的合理性, 本文首先使用三维PIC程序模拟了对称等离子体在对穿时电子Weibel不稳定性的自生电磁场. 其次, 利用径迹追踪法(ray tracing method)[17]计算了当只考虑Weibel不稳定性自生电场或磁场时的质子束照相过程, 对比分析了探测面上质子通量密度分布与同时存在自生电磁场时的差别. 与电场影响可被忽略的传统认识相反, 本文中模拟结果表明, 磁场对探针质子束的偏转作用远小于电场. 这主要是因为在探针质子束的穿越路径上, Weibel不稳定性自生环向磁场的偏转作用力总是被自身中和并抵消, 而自生电场却并没有这一限制.

    2.   PIC模拟
    • 为甄别Weibel不稳定性质子束照相的电磁场属性, 本文采用径迹追踪法分别模拟三种情况下的质子束照相过程: 只考虑电场, 只考虑磁场, 以及同时考虑电场和磁场. 其中, Weibel不稳定性的自生电场和磁场可以通过三维PIC模拟程序获得.

      本文使用三维PIC程序Ascent模拟了两束对称的氢等离子体在对穿过程中的自生电磁场[18]. 模拟中, 氢等离子体沿着z方向对穿, 电子和离子的漂移速度均为u0 = ± 0.5c. 等离子体的密度为1.1 × 1023 m–3, 电子温度和离子温度均为5 keV. 等离子体的空间尺寸为Lx × Ly × Lz = 318 μm × 318 μm × 159 μm, 模拟网格大小为Δx = Δy = Δz = Lx/256, 每个网格中放置54个粒子, 时间步长为1.92 fs. 模拟使用了周期边界条件.

      图2为模拟区域内y向自生磁场Byx向自生电场Ext = 1.06 ps时的三维空间分布情况. 从图2可以看出, 磁场和电场的主要特征是沿电流方向呈管状分布. 但电场具有更小的横向周期长度.

      Figure 2.  Three demensional distributions of the Weibel instability generated (a) magnetic field By and (b) electric field Ex at t = 1.06 ps

      图3为模拟区域内自生磁场总能量εB和电场总能量εE随着时间的演化情况. 图中曲线显示出典型的电子Weibel不稳定性磁场和电场能量的变化规律[15]. 即在线性增长阶段, 因为电流的箍缩, 等离子体的部分动能被转化为磁场能量. 同时, 随着箍缩作用的加剧, 空间电荷效应引起的静电场能量也开始上升, 直至t = 1.22 ps时达到饱和. 此时, 电场能量与磁场能量相当, 这主要是因为当电子束温度较低而对穿速度较大时, 强烈的箍缩作用可产生明显的空间电荷分离[19]. 而在Weibel不稳定性达到饱和后的非线性发展阶段, 电场和磁场能量通过离子的静电响应和磁重联机制被缓慢地转移给等离子体并最终导致了等离子体的热化[5,11,2022]. 此外, 电场能量和磁场能量具有明显的相关性, 这主要是因为当横向位移电流较小时, 径向电场和磁场压力达到平衡状态, 即$E = - \dfrac{{\nabla {B^2}}}{{e{n_{\rm{e}}}{\mu _0}}}$, 其中ne为等离子体密度, μ0为真空磁导率[12].

      Figure 3.  Evolution of the energy of the Weibel instability generated magnetic and electric fields.

      图4t = 1.06 ps时的磁场和电场在z = 0和y = 0平面上的空间分布情况. 根据图3显示, 此时磁场能量和电场能量接近线性增长的峰值时刻, Weibel不稳定性即将达到饱和, 而且磁场能量略大于电场能量. 其中, z = 0平面上的电场和磁场强度分布, 即图4(a)图4(b), 均显示出随机分布的特征. 图4(c)图4(d)中磁场和电场的矢量分布显示, 磁场的方向为环向, 电场方向为径向. 电磁场的这种指向符合Weibel不稳定性的典型图像, 即磁场围绕着z向丝状电流产生, 而电场则是由x-y平面内箍缩作用引起的电荷累积所导致[15]. y = 0平面上的电场和磁场结构, 即图4(e)图4(f), 则显示出丝状特征. 这种二维各向同性随机的分布同样符合Weibel不稳定性的典型特征[15]. 此时, 磁场的峰值强度约51 T, 电场的峰值强度约为1.1 × 1010 V/m.

      Figure 4.  Spatial distributions of (a) the magnetic field strength |B|, (b) the electric field strength |E|, (c) the direction of B and (d) the direction of E on the z = 0 plane, (e) the y component of the magnetic field By and (f) the y component of the electric field Ey on the y = 0 plane at t = 1.06 ps.

      图5t = 4.78 ps时的磁场和电场分布情况. 根据图3显示, 此时Weibel不稳定性已进入饱和后的非线性演化阶段, 磁场和电场的能量均有所下降, 但电场能量因为下降得更快而只约为磁场能量的0.03倍. 与t = 1.06 ps时对比, 除了磁场方向仍然为环向, 电场方向仍然为径向外, t = 4.78 ps时的电场和磁场空间结构同样显示出二维各向同性随机分布的特征, 但在z方向更加均匀平滑. 此时, 磁场强度峰值约33 T, 电场的峰值强度约为2.8 × 109 V/m. 另外, 横向空间周期明显变长, 说明发生了磁重联[22].

      Figure 5.  Spatial distributions of (a) the magnetic field strength |B|, (b) the electric field strength |E|, (c) the direction of B and (d) the direction of E on the z = 0 plane, (e) the y component of the magnetic field By and (f) the y component of the electric field Ey on the y = 0 plane at t = 4.78 ps.

      根据以上三维PIC模拟结果, 我们发现在对穿等离子体的Weibel不稳定性演化过程中, 无论是饱和前的线性增长阶段还是饱和后的非线性发展阶段, 自生电场和磁场均具有二维各向同性随机分布的特征.

    3.   质子束照相的数值模拟
    • 基于PIC模拟给出的电磁场三维分布数据, 本文采用径迹追踪法分别模拟了三种情况下的质子束照相过程: 即只考虑电场, 只考虑磁场, 以及同时考虑电场和磁场. 在径迹追踪法模拟中, 本文使用了动能为20 MeV平行质子束作为探针.

      t = 1.06 ps时, 三种情况下探测器上的质子束通量密度扰动的分布δn/n0图6所示, 其中δn = nn0, n为电场或者磁场不为零时的质子通量密度分布, n0为电场或者磁场为零时的质子通量密度分布. 此时, 探测器到场区域的距离LD = 0.5 mm. 从图6可见, 当只有电场时, 引起的密度扰动的最大值达到(δn/n0)max = 28.1, 但当只有磁场时, 引起的密度扰动的最大值(δn/n0)max只有约0.4, 远小于电场偏转引起的密度扰动. 而且, 无论有没有磁场, 探测面上的密度扰动信息几乎没有发生改变. 模拟还发现, 即使改变探测器距离LD为0.1 mm, 只考虑电场时的最大密度扰动(δn/n0)max = 1.2, 仍然远大于只考虑磁场时的(δn/n0)max = 0.2. 这说明, 在对线性演化阶段的Weibel不稳定性进行质子束照相时, 相比于电场而言, 磁场对探针质子束的偏转作用可以忽略不计. 此外, 因为线性发展阶段电磁场变化较快, 在实验上对此时的Weibel不稳定性进行质子照相时, 探针质子束穿越等离子体区域时感受到的其实是电场和磁场偏转作用的时间累加效果, 这种运动模糊效应将引起图6(a)(c)中δn/n0空间分布的模糊化和(δn/n0)max的下降. 为了规避运动模糊效应带来的影响, 我们同样观察了t = 4.78 ps时的质子照相情况. 如图3所示, 此时Weibel不稳定性进入非线性区, 电场和磁场变化非常缓慢, 运动模糊效应可被忽略不计.

      Figure 6.  Proton flux density perturbations on the detection plane when (a) only the electric field is included, (b) only the magnetic field is included and (c) both the electric and magnetic fields are included at t = 1.06 ps.

      t = 4.78 ps时, 三种情况下探测器上的质子束通量密度扰动的分布δn/n0图7所示. 因为此时电磁场相比于t = 1.06 ps时均较弱, 所以为了清晰地观察此时电磁场引起的密度扰动, 已将探测器到场区域的距离增大到LD = 2 mm. 由图7可见, 与t = 1.06 ps时一样, 电场引起的质子束通量密度扰动最大值((δn/n0)max = 8.8)远大于磁场引起的质子束通量密度扰动最大值((δn/n0)max = 1.5), 而且磁场存在与否同样不会对探测面上的质子通量密度扰动产生明显影响. 这说明, 在对非线性阶段的Weibel不稳定性进行质子束照相时, 拍摄到的仍然只是电场的信息, 磁场的分布信息并不会被反映在探测器上.

      Figure 7.  Proton flux density perturbations on the detection plane when (a) only the electric field is included, (b) only the magnetic field is included and (c) both the electric and magnetic fields are included at t = 4.78 ps.

      此外, 图6中质子通量密度条纹具有局部平行的特征, 这主要是由于电场自身在z方向的非均匀分布引起的. 而图7中的质子通量密度条纹显示出平行状特征. 这些特征均与Quinn等[7]、Fox等[9]以及Huntington等[10]的实验室观察是一致的.

      总之, 无论在Weibel不稳定性饱和前的线性增长阶段还是饱和后的非线性演化阶段, 对其进行质子束照相时, 拍摄到的都只是自生电场的分布信息, 自生磁场的影响可以忽略不计. 这种结果与传统上电场偏转作用可被忽略的认识是刚好相反.

    4.   讨 论
    • 当按照图1所示的布局安排对Weibel不稳定性进行质子束照相时, 可以引起沿着x方向运动的质子束偏转的主要有EyBy. 其中Ey对质子束造成的偏转在y方向上, 而By对质子束造成的偏转在z方向上. 探针质子束离开场区域时的偏转速度由其穿越轨迹上EyBy的路径积分决定, 即$\int {{{{E}}_y}} {\rm{d}}x$$\int {{{{B}}_y}} {\rm{d}}x$. 因为EyBy都具有随机分布的特征, 电场和磁场对质子束的贡献都需要经过矢量求和. 不过, 如图4(c)图5(c)所示, Weibel不稳定性产生的磁场是环向的, 当质子束在沿着x方向穿越时, 每一根电流丝产生的y方向磁场都将被自生完全中和掉, 即$\int {{{{B}}_y}} {\rm{d}}x = 0$. 质子束侧向照相时的这种环向磁场自我中和现象在Li等[23]和Cecchetti等[24]的实验里也同样被观测到. 但是对于Weibel不稳定产生的静电场而言, 因为指向径向(如图4(d)图5(d)所示), 在探针质子束的穿越路径上尽管会部分消除电场的作用却并没有$\int {{{{E}}_y}} {\rm{d}}x = 0$这一限制. 这也是本文模拟中磁场的偏转作用可被忽略不计的根本原因.

      另外, 在t = 4.78 ps时, 电场的能量只约为磁场能量的0.03倍, 其对探针质子束的静电作用力远小于磁场的洛伦兹力, 但仍然在探针质子束离开场区域时产生了远强于磁场的偏转效果. 这可佐证上述解释的唯一性.

    5.   结 论
    • 为甄别Weibel不稳定性质子束照相中的电磁场属性, 本文利用三维PIC程序模拟了对穿等离子体Weibel不稳定性的自生电磁场, 并使用径迹追踪法分别模拟了只考虑电场、只考虑磁场以及同时考虑电磁场时的质子束照相过程. 对比分析发现, 自生电场是形成探测器上质子通量密度条纹的主要原因, 而自生磁场对照相结果不会产生明显的影响. 这主要是因为在探针质子束的穿越路径上, 围绕在丝状电流周围的环形磁场总是被自身中和并抵消, 但电场并没有这一限制. 因此, 在对Weibel不稳定性进行质子束照相时, 过去的研究中忽略电场影响的假设并不合理. 与之相反的是, 磁场的影响可以被忽略. 本研究可帮助理解Weibel不稳定性的质子束照相实验结果, 对于使用质子束照相定量诊断Weibel不稳定性有一定的促进作用.

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