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  • Open Access

Generalized Scaling of Focal Temperature in Converging Shock Waves

Sourabh Bhardwaj*, Nicholas Apazidis, and Michael Liverts

  • *Contact author: sourabhb@kth.se
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Phys. Rev. Lett. 136, 215101 – Published 28 May, 2026

DOI: https://doi.org/10.1103/bl3z-8krg

Abstract

A scaling framework unifying the markedly different and independently studied cylindrical and spherical shock convergence is presented. For ionizing argon, we show that the focal temperature becomes invariant to shock symmetry and initial shock conditions when scaled by the prefocus shock Mach number, after accounting for pressure effects. The resulting collapsed focal temperature–Mach number relation is governed by the thermodynamics of argon, up to an equilibrium temperature of 35 000 K. Such a scaling enables both predictive estimation of focal temperatures over a wide range of initial conditions and, conversely, determination of the parameters required to achieve a target temperature in a given medium.

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References (44)

  1. A. Rososhek, S. Efimov, A. Virozub, D. Maler, and Y. E. Krasik, Particularities of shocks generated by underwater electrical explosions of a single wire and wire arrays, Appl. Phys. Lett. 115, 074101 (2019).
  2. M. Liverts and N. Apazidis, Limiting temperatures of spherical shock wave implosion, Phys. Rev. Lett. 116, 014501 (2016).
  3. O. Antonov, S. Efimov, D. Yanuka, M. Kozlov, V. T. Gurovich, and Y. E. Krasik, Generation of converging strong shock wave formed by microsecond timescale underwater electrical explosion of spherical wire array, Appl. Phys. Lett. 102, 124104 (2013).
  4. J. Yan, J. Li, X.-T. He, L. Wang, Y. Chen, F. Wang, X. Han, K. Pan, J. Liang, Y. Li et al., Experimental confirmation of driving pressure boosting and smoothing for hybrid-drive inertial fusion at the 100-kJ laser facility, Nat. Commun. 14, 5782 (2023).
  5. R. Nora, W. Theobald, R. Betti, F. J. Marshall, D. T. Michel, W. Seka, B. Yaakobi, M. Lafon, C. Stoeckl, J. Delettrez et al., Gigabar spherical shock generation on the OMEGA laser, Phys. Rev. Lett. 114, 045001 (2015).
  6. K. Takayama and T. Saito, Shock wave/geophysical and medical applications, Annu. Rev. Fluid Mech. 36, 347 (2004).
  7. A. Laso Garcia, L. Yang, V. Bouffetier, K. Appel, C. Baehtz, J. Hagemann, H. Höppner, O. Humphries, T. Kluge, M. Mishchenko et al., Cylindrical compression of thin wires by irradiation with a Joule-class short-pulse laser, Nat. Commun. 15, 7896 (2024).
  8. S. Woosley and T. Janka, The physics of core-collapse supernovae, Nat. Phys. 1, 147 (2005).
  9. D. J. Flannigan and K. S. Suslick, Inertially confined plasma in an imploding bubble, Nat. Phys. 6, 598 (2010).
  10. S. J. Putterman and K. R. Weninger, Sonoluminescence: How bubbles turn sound into light, Annu. Rev. Fluid Mech. 32, 445 (2000).
  11. D. Veysset, A. A. Maznev, T. Pezeril, S. Kooi, and K. A. Nelson, Interferometric analysis of laser-driven cylindrically focusing shock waves in a thin liquid layer, Sci. Rep. 6, 24 (2016).
  12. L. Dresselhaus-Cooper, J. E. Gorfain, C. T. Key, B. K. Ofori-Okai, S. J. Ali, D. J. Martynowych, A. Gleason, S. Kooi, and K. A. Nelson, Single-shot multi-frame imaging of cylindrical shock waves in a multi-layered assembly, Sci. Rep. 9, 3689 (2019).
  13. T. Pezeril, G. Saini, D. Veysset, S. Kooi, P. Fidkowski, R. Radovitzky, and K. A. Nelson, Direct visualization of laser-driven focusing shock waves, Phys. Rev. Lett. 106, 214503 (2011).
  14. N. Apazidis and V. Eliasson, Shock Focusing Phenomena (Springer, New York, 2018).
  15. G. C. Burdiak, S. V. Lebedev, R. P. Drake, A. J. Harvey-Thompson, G. F. Swadling, F. Suzuki-Vidal, J. Skidmore, L. Suttle, E. Khoory, L. Pickworth et al., The production and evolution of multiple converging radiative shock waves in gas-filled cylindrical liner Z-pinch experiments, High Energy Density Phys. 9, 52 (2013).
  16. S. Bhardwaj, S. Rojas Mata, N. Apazidis, and M. Liverts, Influence of flow nonuniformities and real gas effects on cylindrical shock wave convergence, Phys. Fluids 36, 124101 (2024).
  17. G. B. Whitham, Linear and Nonlinear Waves (Wiley, New York, 1974).
  18. See Supplemental Material at http://link.aps.org/supplemental/10.1103/bl3z-8krg for shadowgraph video showing transformation of planar shock to cylindrical shock.
  19. Y. Saillard, H. Barbry, and C. Mounier, Transformation of a plane uniform shock into cylindrical or spherical uniform shock by wall shaping, in Shock Waves and Shock Tubes: Proceedings of the Fifteenth International Symposium on Shock Waves and Shock Tubes, Berkeley, California, edited by D. Bershader and R. K. Hanson (Stanford University Press, Stanford, CA, 1986), pp. 147–154.
  20. H. Barbry, C. Mounier, and Y. Saillard, Transformation of a uniform plane shock into a uniform cylindrical or spherical shock, Technical Report No. CEA-N-2516, CEA Centre d’Études de Limeil, 1986.
  21. L. Z. Dumitrescu, On efficient shock-focusing configurations, in Shock Tubes and Waves, edited by R. D. Archer and B. E. Milton (Sydney Shock Tube Symposium Publishers, Sydney, 1984), pp. 196–204.
  22. M. Kjellander, N. Tillmark, and N. Apazidis, Energy concentration by spherical converging shocks generated in a shock tube, Phys. Fluids 24, 126103 (2012).
  23. Z. Zhai, C. Liu, F. Qin, J. Yang, and X. Luo, Generation of cylindrical converging shock waves based on shock dynamics theory, Phys. Fluids 22, 041701 (2010).
  24. G. Guderley, Starke kugelige und zylindrische Verdichtungsstöße in der Nähe des Kugelmittelpunktes BZW. Der Zylinderachse, Luftfahrtforschung 19, 302 (1942).
  25. N. F. Ponchaut, H. G. Hornung, D. I. Pullin, and C. A. Mouton, On imploding cylindrical and spherical shock waves in a perfect gas, J. Fluid Mech. 560, 103 (2006).
  26. H. R. Griem and F. R. Scott, Plasma spectroscopy, Am. J. Phys. 33, 864 (1965).
  27. J. B. Shumaker and C. H. Popenoe, A study of equilibrium in argon arcs, J. Res. Natl. Bur. Stand., Sect. A 76, 71 (1972).
  28. K. C. Lapworth, Spectroscopic temperature measurements in high-temperature gases and plasmas, J. Phys. E 7, 413 (1974).
  29. T. Fujimoto and R. W. P. McWhirter, Validity criteria for local thermodynamic equilibrium in plasma spectroscopy, Phys. Rev. A 42, 6588 (1990).
  30. G. Cristoforetti, E. Tognoni, and L. A. Gizzi, Thermodynamic equilibrium states in laser-induced plasmas: From the general case to laser-induced breakdown spectroscopy plasmas, Spectrochim. Acta B Atom. Spectros. 90, 1 (2013).
  31. J. Colgan, D. P. Kilcrease, N. H. Magee, Jr., M. E. Sherrill, J. Abdallah, Jr., P. Hakel, C. J. Fontes, J. A. Guzik, and K. A. Mussack, A new generation of Los Alamos opacity tables, Astrophys. J. 817, 116 (2016).
  32. G. Tchuen, F. Fogang, Y. Burtschell, and P. Woafo, A hybrid numerical method and its application to inviscid compressible flow problems, Comput. Phys. Commun. 185, 479 (2014).
  33. J. H. Carpenter, S. Root, K. R. Cochrane, D. G. Flicker, and T. K. R. Mattsson, Equation of state of argon: Experiments on Z, density functional theory simulations, and wide-range model, Report No. SAND2012-7991, Sandia National Laboratories, 2012.
  34. M. Kjellander, N. Tillmark, and N. Apazidis, Shock dynamics of strong imploding cylindrical and spherical shock waves with real gas effects, Phys. Fluids 22, 116102 (2010).
  35. J. L. Stollery, Real gas effects on shock-tube performance at high shock strengths, Report No. CP-403, Aeronautical Research Council, 1958.
  36. Y. B. Zel’dovich and Y. P. Raizer, Physics of Shock Waves and High-Temperature Hydrodynamic Phenomena (Courier Corporation, Mineola, NY, 2002).
  37. H. Petschek and S. Byron, Approach to equilibrium ionization behind strong shock waves in argon, Ann. Phys. (N.Y.) 1, 270 (1957).
  38. L. M. Biberman, A. K. Mnatsakanyan, and I. T. Yakubov, Ionization relaxation behind strong shock waves in gases, Sov. Phys. Usp. 13, 728 (1971).
  39. G. B. Whitham, A new approach to problems of shock dynamics. Part I. Two-dimensional problems, J. Fluid Mech. 2, 145 (1957).
  40. C. J. Davie and R. G. Evans, Symmetry of spherically converging shock waves through reflection, relating to the shock ignition fusion energy scheme, Phys. Rev. Lett. 110, 185002 (2013).
  41. S. H. R. Hosseini and K. Takayama, Implosion from a spherical shock wave reflected from a spherical wall, J. Fluid Mech. 530, 223 (2005).
  42. W. Mostert, D. I. Pullin, R. Samtaney, and V. Wheatley, Spontaneous singularity formation in converging cylindrical shock waves, Phys. Rev. Fluids 3, 071401(R) (2018).
  43. M. Kjellander, N. Tillmark, and N. Apazidis, Thermal radiation from a converging shock implosion, Phys. Fluids 22, 046102 (2010).
  44. S. Bhardwaj, N. Apazidis, and M. Liverts, Dataset for cylindrical and spherical shock convergence in argon, 10.5281/zenodo.20036814 (2026).

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Phys. Rev. Lett. 136, 215101– Published 28 May, 2026
  • Received 6 November 2025
  • Accepted 1 May 2026
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Funded by Bibsam.

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