Plasma and Fusion Research

Volume 4, 030 (2009)

Regular Articles


Statistical Theory of Plasmas Turbulence
Eun-jin KIM and Johan ANDERSON
Department of Applied Mathematics, University of Sheffield, Sheffield, S3 7RH, U.K.
(Received 31 August 2008 / Accepted 20 April 2009 / Published 23 June 2009)

Abstract

We present a statistical theory of intermittency in plasma turbulence based on short-lived coherent structures (instantons). In general, the probability density functions (PDFs) of the flux R are shown to have an exponential scaling P(R) ∝ exp (-cRs ) in the tails. In ion-temperature-gradient turbulence, the exponent takes the value s = 3/2 for momentum flux and s = 3 for zonal flow formation. The value of s follows from the order of the highest nonlinear interaction term and the moments for which the PDFs are computed. The constant c depends on the spatial profile of the coherent structure and other physical parameters in the model. Our theory provides a powerful mechanism for ubiquitous exponential scalings of PDFs, often observed in various tokamaks. Implications of the results, in particular, on structure formation are further discussed.


Keywords

turbulence, structure, shear flow, confinement, probability density function (PDF)

DOI: 10.1585/pfr.4.030


References

  • [1] S. Zweben, Phys. Fluids 28, 974 (1985).
  • [2] M. Endler et al. and ASDEX team, Nucl. Fusion 35, 1307 (1995).
  • [3] R.A. Moyer et al., Plasma Phys. Control. Fusion 38, 1273 (1996).
  • [4] D.A. Russell, J.R. Myra and D.A. D'Ippolito, Phys. Plasmas 14, 102307 (2007).
  • [5] O.E. Garcia et al., Nucl. Fusion 47, 667 (2007).
  • [6] D.A. Russell et al., Phys. Rev. Lett. 93, 265001 (2004).
  • [7] B.D. Scott, Plasma Phys. Control. Fusion 49, S25 (2007).
  • [8] X.Q. Xu et al., Phys. Plasmas 10, 1773 (2003).
  • [9] S.I. Krasheninnikov, D.A. D'Ippolito and J.R. Myra, J. Plasma Phys. 74, 679 (2008).
  • [10] J.R. Myra, D.A. Russell and D.A. D'Ippolito, Phys. Plasmas 15, 032304 (2008).
  • [11] Z. Yan, G.R. Tynan, J.H. Yu et al., On the statistical properties of turbulent Reynolds stress, 49th APSDPP November 12-16, 2007, Orlando, Fl. USA.
  • [12] S.J. Zweben et al., Plasma Phys. Control. Fusion, 49, S1 (2007).
  • [13] F. Sattin et al., Plasma Phys. Control. Fusion, 48, 1033 (2006).
  • [14] E. Kim and P.H. Diamond, Phys. Rev. Lett. 90, 185006 (2003).
  • [15] K. Itoh et al., J. Plasma Fusion Res. 79, 608 (2003).
  • [16] S. Coleman, Aspects of Symmetry (Cambridge University Press, Cambridge, 1985).
  • [17] G. 't Hooft, Phys. Rev. Lett. 37, 8 (1976).
  • [18] V. Gurarie and A. Migdal, Phys. Rev. E 54, 4908 (1996).
  • [19] G. Falkovich et al., Phys. Rev. E 54, 4896 (1996).
  • [20] E. Balkovsky et al., Phys. Rev. Lett. 78, 1452 (1997).
  • [21] J. Fleischer and P.H. Diamond, Phys. Lett. A 283, 237 (2001).
  • [22] H.W. Wyld, Ann. Phys. 14, 143 (1961).
  • [23] J. Zinn-Justin, Quantum Field Theory and Critical Phenomena (Clarendon Press, Oxford, 1989).
  • [24] P.C. Martin, E.D. Sigga and H.A. Rose, Phys. Rev. E 8, 423 (1973).
  • [25] A. Hasegawa and K. Mima, Phys. Rev. Lett. 39, 205 (1977).
  • [26] W. Horton, Rev. Mod. Phys. 71, 735 (1999).
  • [27] E. Kim and P.H. Diamond, Phys. Plasmas 9, 71 (2002); Phys. Rev. Lett. 88, 225002 (2002).
  • [28] E. Kim et al., Nucl. Fusion 43, 961 (2003).
  • [29] J. Anderson and E. Kim, Phys. Plasmas 15, 052306 (2008).
  • [30] J. Anderson and E. Kim, Phys. Plasmas 15, 082312 (2008).
  • [31] J. Anderson, H. Nordman, R. Singh et al., Phys. Plasmas 9, 4500 (2002).
  • [32] B.G. Hong, F. Romanelli and M. Ottaviani, Phys. Fluids B 3, 615 (1991).
  • [33] E. Kim and J. Anderson, Phys. Plasmas 15, 114506 (2008).
  • [34] H.-L. Liu, J. Atmospheric Sci. 64, 580 (2007).
  • [35] E. Kim, H.-L. Liu and J. Anderson, Phys. Plasmas 16, 052304 (2009).
  • [36] C. Hildago et al., New J. Phys. 4, 51 (2002).

This paper may be cited as follows:

Eun-jin KIM and Johan ANDERSON, Plasma Fusion Res. 4, 030 (2009).