Skip to main content Accessibility help
×
Home

Proton imaging of stochastic magnetic fields

  • A. F. A. Bott (a1), C. Graziani (a2), P. Tzeferacos (a1) (a2), T. G. White (a1) (a3), D. Q. Lamb (a2), G. Gregori (a1) and A. A. Schekochihin (a4) (a5)...

Abstract

Recent laser-plasma experiments (Fox et al., Phys. Rev. Lett., vol. 111, 2013, 225002; Huntington et al., Nat. Phys., vol. 11(2), 2015, 173–176; Tzeferacos et al., Phys. Plasmas, vol. 24(4), 2017a, 041404; Tzeferacos et al., 2017b, arXiv:1702.03016 [physics.plasm-ph]) report the existence of dynamically significant magnetic fields, whose statistical characterisation is essential for a complete understanding of the physical processes these experiments are attempting to investigate. In this paper, we show how a proton-imaging diagnostic can be used to determine a range of relevant magnetic-field statistics, including the magnetic-energy spectrum. To achieve this goal, we explore the properties of an analytic relation between a stochastic magnetic field and the image-flux distribution created upon imaging that field. This ‘Kugland image-flux relation’ was previously derived (Kugland et al.Rev. Sci. Instrum. vol. 83(10), 2012, 101301) under simplifying assumptions typically valid in actual proton-imaging set-ups. We conclude that, as with regular electromagnetic fields, features of the beam’s final image-flux distribution often display a universal character determined by a single, field-scale dependent parameter – the contrast parameter $\unicode[STIX]{x1D707}\equiv d_{s}/{\mathcal{M}}l_{B}$ – which quantifies the relative size of the correlation length $l_{B}$ of the stochastic field, proton displacements $d_{s}$ due to magnetic deflections and the image magnification ${\mathcal{M}}$ . For stochastic magnetic fields, we establish the existence of four contrast regimes, under which proton-flux images relate to their parent fields in a qualitatively distinct manner. These are linear, nonlinear injective, caustic and diffusive. The diffusive regime is newly identified and characterised. The nonlinear injective regime is distinguished from the caustic regime in manifesting nonlinear behaviour, but as in the linear regime, the path-integrated magnetic field experienced by the beam can be extracted uniquely. Thus, in the linear and nonlinear injective regimes we show that the magnetic-energy spectrum can be obtained under a further statistical assumption of isotropy. This is not the case in the caustic or diffusive regimes. We discuss complications to the contrast-regime characterisation arising for inhomogeneous, multi-scale stochastic fields, which can encompass many contrast regimes, as well as limitations currently placed by experimental capabilities on one’s ability to extract magnetic-field statistics. The results presented in this paper are of consequence in providing a comprehensive description of proton images of stochastic magnetic fields, with applications for improved analysis of proton-flux images.

Copyright

Corresponding author

Email address for correspondence: archie.bott@physics.ox.ac.uk

References

Hide All
Adler, R. 1981 The Geometry of Random Fields. Society for Industrial and Applied Mathematics (SIAM).
Arévalo, P., Churazov, E., Zhuravleva, I., Hernández-Monteagudo, C. & Revnivtsev, M. 2012 A Mexican hat with holes: calculating low-resolution power spectra from data with gaps. Mon. Not. R. Astron. Soc. 426 (3), 17931807.
Berry, M. V. & Upstill, C. 1980 Catastrophe optics: morphologies of caustics and their diffraction patterns. Prog. Optics. 18, 257346.
Birdsall, C. K. & Langdon, A. B. 1991 Plasma Physics via Computer Simulation. Institute of Physics Publishing.
Blandford, R. & Eichler, D. 1987 Particle acceleration at astrophysical shocks: a theory of cosmic ray origin. Phys. Rep. 154 (1), 175.
Borghesi, M., Campbell, D. H., Schiavi, A., Haines, M. G., Willi, O., MacKinnon, A. J., Patel, P., Gizzi, L. A., Galimberti, M., Clarke, R. J. et al. 2002 Electric field detection in laser-plasma interaction experiments via the proton imaging technique. Phys. Plasmas 9 (5), 22142220.
Borghesi, M., Fuchs, J., Bulanov, S., MacKinnon, A., Patel, P. & Roth, M. 2006 Fast ion generation by high-intensity laser irradiation of solid targets and applications. Fusion Sci. Technol. 49 (3), 412439.
Borghesi, M., Kar, S., Romagnani, L., Toncian, T., Antici, P., Audebert, P., Brambrink, E., Ceccherini, F., Cecchetti, C. A., Fuchs, J. et al. 2007 Impulsive electric fields driven by high-intensity laser matter interactions. Laser Part. Beams 25 (1), 161167.
Brenier, Y. 1991 Polar factorization and monotone rearrangement of vector-valued functions. Commun. Pure Appl. Maths 44 (4), 375417.
Churazov, E., Vikhlinin, A., Zhuravleva, I., Schekochihin, A., Parrish, I., Sunyaev, R., Forman, W., Böhringer, H. & Randall, S. 2012 X-ray surface brightness and gas density fluctuations in the coma cluster. Mon. Not. R. Astron. Soc. 421 (2), 11231135.
Davidson, P. A. 2004 Turbulence: An Introduction for Scientists and Engineers. Oxford University Press.
Dean, E. J. & Glowinski, R. 2006 Numerical methods for fully nonlinear elliptic equations of the Monge–Ampère type. Comput. Meth. Appl. Mech. Engng. 195 (13), 13441386.
Dolginov, A. Z. & Toptygin, I. 1967 Multiple scattering of particles in a magnetic field with random inhomogeneities. Sov. Phys. JETP 24, 1195.
Ensslin, T. & Vogt, C. 2003 The magnetic power spectrum in Faraday rotation screens. Astron. Astrophys. 401 (3), 835848.
Fox, W., Fiksel, G., Bhattacharjee, A., Chang, P.-Y., Germaschewski, K., Hu, S. X. & Nilson, P. M. 2013 Filamentation instability of counterstreaming laser-driven plasmas. Phys. Rev. Lett. 111, 225002.
Gangbo, W. & McCann, R. J. 1996 The geometry of optimal transportation. Acta Math. 177 (2), 113161.
Golitsyn, G. S. 1960 Fluctuations of the magnetic field and current density in a turbulent flow of a weakly conducting fluid. Sov. Phys. Dokl. 5, 536.
Graziani, C., Tzeferacos, P., Lamb, D. Q. & Li, C.2016 Inferring morphology and strength of magnetic fields from proton radiographs. arXiv:1603.08617 [physics.plasm-ph].
Gregori, G., Reville, B. & Miniati, F. 2015 The generation and amplification of intergalactic magnetic fields in analogue laboratory experiments with high power lasers. Phys. Rep. 601, 134.
Hall, D. E. & Sturrock, P. A. 1967 Diffusion, scattering, and acceleration of particles by stochastic electromagnetic fields. Phys. Fluids 10 (12), 26202628.
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. et al. 2015 Observation of magnetic field generation via the Weibel instability in interpenetrating plasma flows. Nat. Phys. 11 (2), 173176.
Jokipii, J. R. 1972 Fokker–Planck equations for charged-particle transport in random fields. Astrophys. J. 172, 319.
Kasim, M. F., Ceurvorst, L., Ratan, N., Sadler, J., Chen, N., Sävert, A., Trines, R., Bingham, R., Burrows, P. N., Kaluza, M. C. et al. 2017 Quantitative shadowgraphy and proton radiography for large intensity modulations. Phys. Rev. E 95, 023306.
Krall, N. A. & Trivelpiece, A. W. 1973 Principles of Plasma Physics. McGraw-Hill.
Kugland, N. L., Ryutov, D. D., Plechaty, C., Ross, J. S. & Park, H.-S. 2012 Invited article: relation between electric and magnetic field structures and their proton-beam images. Rev. Sci. Instrum. 83 (10), 101301.
Levy, M. C., Ryutov, D. D., Wilks, S. C., Ross, J. S., Huntington, C. M., Fiuza, F., Martinez, D. A., Kugland, N. L., Baring, M. G. & Park, H.-S. 2015 Development of an interpretive simulation tool for the proton radiography technique. Rev. Sci. Instrum. 86 (3), 033302.
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. et al. 2006 Measuring E and B fields in laser-produced plasmas with monoenergetic proton radiography. Phys. Rev. Lett. 97, 135003.
Lucy, L. B. 1974 An iterative technique for the rectification of observed distributions. Astron. J. 79, 745.
MacKinnon, A. J., Patel, P. K., Borghesi, M., Clarke, R. C., Freeman, R. R., Habara, H., Hatchett, S. P., Hey, D., Hicks, D. G., Kar, S. et al. 2006 Proton radiography of a laser-driven implosion. Phys. Rev. Lett. 97, 045001.
MacKinnon, A. J., Patel, P. K., Town, R. P., Edwards, M. J., Phillips, T., Lerner, S. C., Price, D. W., Hicks, D., Key, M. H., Hatchett, S. et al. 2004 Proton radiography as an electromagnetic field and density perturbation diagnostic (invited). Rev. Sci. Instrum. 75 (10), 35313536.
Manuel, M. J. E., Zylstra, A. B., Rinderknecht, H. G., Casey, D. T., Rosenberg, M. J., Sinenian, N., Li, C. K., Frenje, J. A., Séguin, F. H. & Petrasso, R. D. 2012 Source characterization and modeling development for monoenergetic-proton radiography experiments on OMEGA. Rev. Sci. Instrum. 83 (6), 063506.
Moffatt, K. 1961 The amplification of a weak applied magnetic field by turbulence in fluids of moderate conductivity. J. Fluid Mech. 11 (4), 625635.
Mondal, S., Narayanan, V., Ding, W. J., Lad, A. D., Hao, B., Ahmad, S., Wang, W. M., Sheng, Z. M., Sengupta, S., Kaw, P. et al. 2012 Direct observation of turbulent magnetic fields in hot, dense laser produced plasmas. Proc. Natl Acad. Sci. USA 109 (21), 80118015.
Nürnberg, F., Schollmeier, M., Brambrink, E., Blažević, A., Carroll, D. C., Flippo, K., Gautier, D. C., Geißel, M., Harres, K., Hegelich, B. M. et al. 2009 Radiochromic film imaging spectroscopy of laser-accelerated proton beams. Rev. Sci. Instrum. 80 (3), 033301.
Parker, E. N. 1965 The passage of energetic charged particles through interplanetary space. Planet. Space Sci. 13 (1), 949.
Richardson, W. H. 1972 Bayesian-based iterative method of image restoration. J. Opt. Soc. Am. 62 (1), 5559.
Romagnani, L., Borghesi, M., Cecchetti, C. A., Kar, S., Antici, P., Audebert, P., Bandhoupadjay, S., Ceccherini, F., Cowan, T., Fuchs, J. et al. 2008 Proton probing measurement of electric and magnetic fields generated by ns and ps laser-matter interactions. Laser Part. Beams 26 (2), 241248.
Romagnani, L., Fuchs, J., Borghesi, M., Antici, P., Audebert, P., Ceccherini, F., Cowan, T., Grismayer, T., Kar, S., Macchi, A. et al. 2005 Dynamics of electric fields driving the laser acceleration of multi-MeV protons. Phys. Rev. Lett. 95, 195001.
Sarri, G., Macchi, A., Cecchetti, C. A., Kar, S., Liseykina, T. V., Yang, X. H., Dieckmann, M. E., Fuchs, J., Galimberti, M., Gizzi, L. A. et al. 2012 Dynamics of self-generated, large amplitude magnetic fields following high-intensity laser matter interaction. Phys. Rev. Lett. 109, 205002.
Schekochihin, A. A., Cowley, S. C., Taylor, S. F., Maron, J. L. & McWilliams, J. C. 2004 Simulations of the small-scale turbulent dynamo. Astrophys. J. 612 (1), 276.
Schekochihin, A. A., Iskakov, A. B., Cowley, S. C., McWilliams, J. C., Proctor, M. R. E. & Yousef, T. A. 2007 Fluctuation dynamo and turbulent induction at low magnetic Prandtl numbers. New J. Phys. 9 (8), 300.
Séguin, F. H., DeCiantis, J. L., Frenje, J. A., Kurebayashi, S., Li, C. K., Rygg, J. R., Chen, C., Berube, V., Schwartz, B. E., Petrasso, R. D. et al. 2004 D3He-proton emission imaging for inertial-confinement-fusion experiments (invited). Rev. Sci. Instrum. 75 (10), 35203525.
Shinozuka, M. & Deodatis, G. 1996 Simulation of multi-dimensional Gaussian stochastic fields by spectral representation. Appl. Mech. Rev. 49 (1), 2953.
Sulman, M. M., Williams, J. F. & Russell, R. D. 2011 An efficient approach for the numerical solution of the Monge–Ampère equation. Appl. Numer. Maths 61 (3), 298307.
Tzeferacos, P., Rigby, A., Bott, A., Bell, A. R., Bingham, R., Casner, A., Cattaneo, F., Churazov, E. M., Emig, J., Fiuza, F. et al. 2017a Numerical modeling of laser-driven experiments aiming to demonstrate magnetic field amplification via turbulent dynamo. Phys. Plasmas 24 (4), 041404.
Tzeferacos, P., Rigby, A., Bott, A., Bell, A. R., Bingham, R., Casner, A., Cattaneo, F., Churazov, E. M., Emig, J., Fiuza, F. et al. 2017b Laboratory evidence of dynamo amplification of magnetic fields in a turbulent plasma. arXiv:1702.03016 [physics.plasm-ph].
Villani, C. 2008 Optimal Transport: Old and New, vol. 338. Springer.
Welch, D. R., Rose, D. V., Clark, R. E., Genoni, T. C. & Hughes, T. P. 2004 Implementation of an non-iterative implicit electromagnetic field solver for dense plasma simulation. Comput. Phys. Commun. 164 (1), 183188.
Wilks, S. C., Langdon, A. B., Cowan, T. E., Roth, M., Singh, M., Hatchett, S., Key, M. H., Pennington, D., MacKinnon, A. & Snavely, R. A. 2001 Energetic proton generation in ultra-intense laser–solid interactions. Phys. Plasmas 8 (2), 542549.
Yamazaki, F. & Shinozuka, M. 1988 Digital generation of non-Gaussian stochastic fields. J. Engng Mech. ASCE 114 (7), 11831197.
Ziegler, J. F. 1999 Stopping of energetic light ions in elemental matter. J. Appl. Phys./Rev. Appl. Phys. 85 (3), 12491272.
MathJax
MathJax is a JavaScript display engine for mathematics. For more information see http://www.mathjax.org.

Keywords

Proton imaging of stochastic magnetic fields

  • A. F. A. Bott (a1), C. Graziani (a2), P. Tzeferacos (a1) (a2), T. G. White (a1) (a3), D. Q. Lamb (a2), G. Gregori (a1) and A. A. Schekochihin (a4) (a5)...

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed.