The Library

Craig's XY distribution and the statistics of Lagrangian power in two-dimensional turbulence

Tools

Bandi, Mahesh M. and Connaughton, Colm. (2008) Craig's XY distribution and the statistics of Lagrangian power in two-dimensional turbulence. Physical Review E (Statistical, Nonlinear, and Soft Matter Physics), Vol.77 (No.3(2)). 036318 . ISSN 1539-3755

Preview

PDF
WRAP_Connaughton_craig's_0710.1133v5.pdf - Submitted Version - Requires a PDF viewer such as GSview, Xpdf or Adobe Acrobat Reader
Download (499Kb)

Official URL: http://dx.doi.org/10.1103/PhysRevE.77.036318

Abstract

We examine the probability distribution function (PDF) of the energy injection rate (power) in numerical simulations of stationary two-dimensional (2D) turbulence in the Lagrangian frame. The simulation is designed to mimic an electromagnetically driven fluid layer, a well-documented system for generating 2D turbulence in the laboratory. In our simulations, the forcing and velocity fields are close to Gaussian. On the other hand, the measured PDF of injected power is very sharply peaked at zero, suggestive of a singularity there, with tails which are exponential but asymmetric. Large positive fluctuations are more probable than large negative fluctuations. It is this asymmetry of the tails which leads to a net positive mean value for the energy input despite the most probable value being zero. The main features of the power distribution are well described by Craig's XY distribution for the PDF of the product of two correlated normal variables. We show that the power distribution should exhibit a logarithmic singularity at zero and decay exponentially for large absolute values of the power. We calculate the asymptotic behavior and express the asymmetry of the tails in terms of the correlation coefficient of the force and velocity. We compare the measured PDFs with the theoretical calculations and briefly discuss how the power PDF might change with other forcing mechanisms.

Item Type:	Journal Article
Subjects:	Q Science > QA Mathematics Q Science > QC Physics
Divisions:	Faculty of Science > Centre for Complexity Science Faculty of Science > Mathematics
Library of Congress Subject Headings (LCSH):	Turbulence, Power (Mechanics) -- Mathematical models, Lagrangian functions
Journal or Publication Title:	Physical Review E (Statistical, Nonlinear, and Soft Matter Physics)
Publisher:	American Physical Society
ISSN:	1539-3755
Date:	March 2008
Volume:	Vol.77
Number:	No.3(2)
Number of Pages:	9
Page Range:	036318
Identification Number:	10.1103/PhysRevE.77.036318
Status:	Peer Reviewed
Publication Status:	Published
Access rights to Published version:	Restricted or Subscription Access
Funder:	United States. Dept. of Energy
Grant number:	DE-AC52-06NA25396 (DoE)
Related URLs:	Other Repository
References:	[1] H. Schlichting, Boundary-layer theory (New York; McGraw-Hill, 1979). [2] D. Lathrop, J. Fineberg, and H. Swinney, Phys. Rev. A 46, 6390 (1992). [3] R. Labb´e, J.-F. Pinton, and S. Fauve, J. Phys. II France 6, 1099 (1996). [4] J. Titon and O. Cadot, Phys. Fluids 15, 625 (2003). [5] T. Toth-Katona and J. T. Gleeson, Phys. Rev. Lett. 91, 264501 (2003). [6] E. Falcon, S. Aumaˆıtre, C. Falc´on, C. Laroche, and S. Fauve, Phys. Rev. Lett. 100 (2008). [7] S. Aumaˆıtre and S. Fauve, Europhys. Lett. 62, 822 (2003). [8] D. Evans, E. Cohen, and G. Morris, Phys. Rev. Lett. 71, 2401 (1993). [9] G. Gallavotti and E. Cohen, Phys. Rev. Lett. 74, 2694 (1995). [10] J. Kurchan, J. Phys. A 31, 3719 (1998). [11] V. Chernyak, M. Chertkov, and C. Jarzynski, J. Stat. Phys. 8, 08001 (2006). [12] M. Bandi and C. Connaughton (2007), arXiv:0710.3368 [cond-mat.stat-mech]. [13] R. H. Kraichnan, Phys. Fluids 10, 1417 (1967). [14] C. E. Leith, Phys. Fluids 11, 671 (1968). [15] G. K. Batchelor, Phys. Fluids 12, 233 (1969). [16] U. Frisch, Turbulence: The Legacy of A. N. Kolmogorov (Cambridge University Press, Cambridge, 1995). [17] G. Boffetta, ArXiv e-prints (2006), nlin.CD/0612035. [18] C. H. Bruneau and H. Kellay, Phys. Rev. E 71, 046305 (2005). [19] H. Kellay and W. I. Goldburg, Rep. Prog. Phys. 65, 845 (2002). [20] C. Tran and J. Bowman, Phys. Rev. E 69, 036303 (2004). [21] G. Boffetta, A. Celani, and M. Vergassola, Phys. Rev. E 61, R29 (2000). [22] M. Rivera, W. Daniel, and R. Ecke (2005), eprint: arXiv:cond-mat/0512214, arXiv:cond-mat/0512214. [23] S. Chen, R. Ecke, G. Eyink, M. Rivera, X. Wang, and Z. Xiao, Phys. Rev. Lett. 96, 084502 (2006). [24] J. Paret and P. Tabeling, Phys. Fluids 10, 3126 (1998). [25] C. Beta, K. Schneider, and M. Farge, Commun. Nonlin. Sci. Num. Simulation 8, 537 (2003). [26] O. Kamps and R. Friedrich (2007), arXiv:0710.1739v1 [physics.flu-dyn]. [27] C. Craig, Ann. Math. Statist. 7, 1 (1937). [28] M. Abramowitz and I. Stegun, Handbook of mathematical functions, with formulas, graphs, and mathematical tables (Dover: New York, 1965). [29] C. Itzykson and J.-M. Drouffe, Statistical Field Theory Vol. 1 (Cambridge, UK: Cambridge University Press, 1991). [30] F. Bowman, Introduction to Bessel Functions (New York: Dover, 1958).
URI:	http://wrap.warwick.ac.uk/id/eprint/30316