Skip to main content Accessibility help

We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.

Close cookie message
×
Home
  • Home
Article
  • Cited by 10

  • Access
  • Open access

From one-dimensional fields to Vlasov equilibria: theory and application of Hermite polynomials

  • O. Allanson (a1), T. Neukirch (a1), S. Troscheit (a1) and F. Wilson (a1)

Abstract

We consider the theory and application of a solution method for the inverse problem in collisionless equilibria, namely that of calculating a Vlasov–Maxwell equilibrium for a given macroscopic (fluid) equilibrium. Using Jeans’ theorem, the equilibrium distribution functions are expressed as functions of the constants of motion, in the form of a Maxwellian multiplied by an unknown function of the canonical momenta. In this case it is possible to reduce the inverse problem to inverting Weierstrass transforms, which we achieve by using expansions over Hermite polynomials. A sufficient condition on the pressure tensor is found which guarantees the convergence and the boundedness of the candidate solution, when satisfied. This condition is obtained by elementary means, and it is clear how to put it into practice. We also argue that for a given pressure tensor for which our method applies, there always exists a positive distribution function solution for a sufficiently magnetised plasma. Illustrative examples of the use of this method with both force-free and non-force-free macroscopic equilibria are presented, including the full verification of a recently derived distribution function for the force-free Harris sheet (Allanson et al., Phys. Plasmas, vol. 22 (10), 2015, 102116). In the effort to model equilibria with lower values of the plasma ${\it\beta}$ , solutions for the same macroscopic equilibrium in a new gauge are calculated, with numerical results presented for ${\it\beta}_{pl}=0.05$ .

  • View HTML

      • Send article to Kindle

      To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

      Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

      Find out more about the Kindle Personal Document Service.

      From one-dimensional fields to Vlasov equilibria: theory and application of Hermite polynomials
      Available formats
      ×

      Send article to Dropbox

      To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

      From one-dimensional fields to Vlasov equilibria: theory and application of Hermite polynomials
      Available formats
      ×

      Send article to Google Drive

      To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

      From one-dimensional fields to Vlasov equilibria: theory and application of Hermite polynomials
      Available formats
      ×

Copyright

© Cambridge University Press 2016
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.

Corresponding author

†Email address for correspondence: oliver.allanson@st-andrews.ac.uk

References

Hide All
Abraham-Shrauner, B. 1968 Exact, stationary wave solutions of the nonlinear Vlasov equation. Phys. Fluids 11, 1162–1167.
Abraham-Shrauner, B. 2013 Force-free Jacobian equilibria for Vlasov–Maxwell plasmas. Phys. Plasmas 20 (10), 102117.
Allanson, O., Neukirch, T., Wilson, F. & Troscheit, S. 2015 An exact collisionless equilibrium for the force-free Harris sheet with low plasma beta. Phys. Plasmas 22 (10), 102116.
Artemyev, A. V., Vasko, I. Y. & Kasahara, S. 2014 Thin current sheets in the Jovian magnetotail. Planet. Space Sci. 96, 133–145.
Bell, E. T. 1934 Exponential polynomials. Ann. of Math. (2) 35 (2), 258–277.
Bilodeau, G. G 1962 The Weierstrass transform and Hermite polynomials. Duke Math. J. 29 (2), 293–308.
Birn, J., Drake, J. F., Shay, M. A., Rogers, B. N., Denton, R. E., Hesse, M., Kuznetsova, M., Ma, Z. W., Bhattacharjee, A., Otto, A. et al. 2001 Geospace environmental modeling (GEM) magnetic reconnection challenge. J. Geophys. Res. 106 (A3), 3715–3719.
Birn, J., Galsgaard, K., Hesse, M., Hoshino, M., Huba, J., Lapenta, G., Pritchett, P. L., Schindler, K., Yin, L., Büchner, J. et al. 2005 Forced magnetic reconnection. Geophys. Res. Lett. 32 (6), l06105.
Birn, J. & Priest, E. 2007 Reconnection of Magnetic Fields: Magnetohydrodynamics and Collisionless Theory and Observations. Cambridge University Press.
Biskamp, D. 2000 Magnetic Reconnection in Plasmas, Cambridge Monographs on Plasma Physics, vol. 3, p. xiv, 387 p. Cambridge University Press, ISBN: 0521582881.
Camporeale, E., Delzanno, G. L., Lapenta, G. & Daughton, W. 2006 New approach for the study of linear Vlasov stability of inhomogeneous systems. Phys. Plasmas 13 (9), 092110.
Channell, P. J. 1976 Exact Vlasov–Maxwell equilibria with sheared magnetic fields. Phys. Fluids 19, 1541–1545.
Comtet, L. 1974 Advanced Combinatorics: The Art of Finite and Infinite Expansions, Enlarged Edition. D. Reidel.
Connon, D. F.2010 Various applications of the (exponential) complete Bell polynomials. ArXiv e-prints.
Eddington, A. S. 1913 On a formula for correcting statistics for the effects of a known error of observation. Mon. Not. R. Astron. Soc. 73, 359–360.
Evans, L. C. 2010 Partial Differential Equations, 2nd edn, Graduate Studies in Mathematics, vol. 19. American Mathematical Society.
Fitzpatrick, R. 2014 Plasma Physics: An Introduction. CRC Press, Taylor & Francis Group.
Fruit, G., Louarn, P., Tur, A. & Le QuéAu, D. 2002 On the propagation of magnetohydrodynamic perturbations in a Harris-type current sheet 1. Propagation on discrete modes and signal reconstruction. J. Geophys. Res. 107, SMP 39-1–18.
Grad, H. 1949a Note on $N$ -dimensional Hermite polynomials. Commun. Pure Appl. Maths 2, 325–330.
Grad, H. 1949b On the kinetic theory of rarefied gases. Commun. Pure Appl. Maths 2, 331–407.
Grad, H. 1961 Boundary layer between a plasma and a magnetic field. Phys. Fluids 4, 1366–1375.
Gradshteyn, I. S. & Ryzhik, I. M. 2007 Table of Integrals, Series, and Products, 7th edn. Elsevier/Academic.
Guo, F., Li, H., Daughton, W. & Liu, Y.-H. 2014 Formation of hard power laws in the energetic particle spectra resulting from relativistic magnetic reconnection. Phys. Rev. Lett. 113, 155005.
Harrison, M. G. & Neukirch, T. 2009a One-dimensional Vlasov–Maxwell equilibrium for the force-free Harris sheet. Phys. Rev. Lett. 102 (13), 135003.
Harrison, M. G. & Neukirch, T. 2009b Some remarks on one-dimensional force-free Vlasov–Maxwell equilibria. Phys. Plasmas 16 (2), 022106.
Hesse, M., Kuznetsova, M., Schindler, K. & Birn, J. 2005 Three-dimensional modeling of electron quasiviscous dissipation in guide-field magnetic reconnection. Phys. Plasmas 12 (10), 100704.
Hewett, D. W., Nielson, C. W. & Winske, D. 1976 Vlasov confinement equilibria in one dimension. Phys. Fluids 19, 443–449.
Howes, G. G., Cowley, S. C., Dorland, W., Hammett, G. W., Quataert, E. & Schekochihin, A. A. 2006 Astrophysical gyrokinetics: basic equations and linear theory. Astrophys. J. 651, 590–614.
Kölbig, K. S. 1994 The complete Bell polynomials for certain arguments in terms of Stirling numbers of the first kind. J. Comput. Appl. Math. 51 (1), 113–116.
Kolotkov, D. Y., Vasko, I. Y. & Nakariakov, V. M. 2015 Kinetic model of force-free current sheets with non-uniform temperature. Phys. Plasmas 22 (11), 112902.
Mynick, H. E., Sharp, W. M. & Kaufman, A. N. 1979 Realistic Vlasov slab equilibria with magnetic shear. Phys. Fluids 22, 1478–1484.
Neukirch, T., Wilson, F. & Harrison, M. G. 2009 A detailed investigation of the properties of a Vlasov–Maxwell equilibrium for the force-free Harris sheet. Phys. Plasmas 16 (12), 122102.
Northrop, T. G. 1961 The guiding center approximation to charged particle motion. Ann. Phys. 15, 79–101.
Panov, E. V., Artemyev, A. V., Nakamura, R. & Baumjohann, W. 2011 Two types of tangential magnetopause current sheets: cluster observations and theory. J. Geophys. Res. 116, A12204.
Petrukovich, A., Artemyev, A., Vasko, I., Nakamura, R. & Zelenyi, L. 2015 Current sheets in the earth magnetotail: plasma and magnetic field structure with cluster project observations. Space Sci. Rev. 188, 311–337.
Priest, E. & Forbes, T. 2000 Magnetic Reconnection. Cambridge University Press.
Riordan, J. 1958 An Introduction to Combinatorial Analysis. Wiley, Chapman & Hall.
Sansone, G. 1959 Orthogonal Functions. Interscience.
Schekochihin, A. A., Parker, J. T., Highcock, E. G., Dellar, P. J., Dorland, W. & Hammett, G. W. 2016 Phase mixing versus nonlinear advection in drift-kinetic plasma turbulence. J. Plasma Phys. 82, 905820212 (47 pages).
Schindler, K. 2007 Physics of Space Plasma Activity. Cambridge University Press.
Suzuki, A. & Shigeyama, T. 2008 A novel method to construct stationary solutions of the Vlasov–Maxwell system. Phys. Plasmas 15 (4), 042107.
Tasso, H. & Throumoulopoulos, G. 2014 Tokamak-like Vlasov equilibria. Eur. Phys. J. D 68, 175–181.
Vasko, I. Y., Artemyev, A. V., Petrukovich, A. A. & Malova, H. V. 2014 Thin current sheets with strong bell-shape guide field: cluster observations and models with beams. Ann. Geophys. 32, 1349–1360.
Watson, G. N. 1933 Notes on generating functions of polynomials: (2) Hermite polynomials. J. Lond. Math. Soc. s1‐8 (3), 194–199.
Widder, D. V. 1951 Necessary and sufficient conditions for the representation of a function by a Weierstrass transform. Trans. Am. Math. Soc. 71, 430–439.
Widder, D. V. 1954 The convolution transform. Bull. Am. Math. Soc. 60 (5), 444–456.
Wilson, F. & Neukirch, T. 2011 A family of one-dimensional Vlasov–Maxwell equilibria for the force-free Harris sheet. Phys. Plasmas 18 (8), 082108.
Wilson, F., Neukirch, T., Hesse, M., Harrison, M. G. & Stark, C. R. 2016 Particle-in-cell simulations of collisionless magnetic reconnection with a non-uniform guide field. Phys. Plasmas 23 (3), 032302.
Wolf, K. B. 1977 On self-reciprocal functions under a class of integral transforms. J. Math. Phys. 18 (5), 1046–1051.
Yamada, M., Kulsrud, R. & Ji, H. 2010 Magnetic reconnection. Rev. Mod. Phys. 82, 603–664.
Zocco, A. 2015 Linear collisionless Landau damping in Hilbert space. J. Plasma Phys. 81 (4), 049002.
MathJax
MathJax is a JavaScript display engine for mathematics. For more information see http://www.mathjax.org.
  • Cited by 10

  • Access
  • Open access

From one-dimensional fields to Vlasov equilibria: theory and application of Hermite polynomials

  • O. Allanson (a1), T. Neukirch (a1), S. Troscheit (a1) and F. Wilson (a1)

Metrics

Altmetric attention score

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