Abstract
The analyticity on a strip of the solutions of Navier-Stokes equations in 2D is shown to explain the observed fast decay of the frequency power spectrum of the turbulent velocity field. Some subtleties in the application of the Wiener-Khinchine method to turbulence are resolved by showing that the frequency power spectrum of turbulent velocities is in fact a measure exponentially decaying for frequency →±∞. Our approach also shows that the conventional procedures used in analyzing data in turbulence experiments are valid even in the absence of the ergodic property in the flow.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
E. Ching, P. Constantin, L. P. Kadanoff, A. Libchaber, I. Procaccia, and X.-Z. Wu, Transitions in convective turbulence: The role of thermal plumes,Phys. Rev. A 44:8091–8102 (1991).
P. Constantin and C. Foias,Navier-Stokes Equations (University of Chicago Press, Chicago, 1988).
N. Dunford and J. T. Schwartz,Linear Operators. Part I: General Theory (Intersiience, New York, 1958).
L. Hörmander,The Analysis of Linear Partial Differential Operators (Springer-Verlag, Berlin, 1983).
N. Krylov and N. N. Bogoliubov, La théorie générale de la mesure dans son application à l'étude des systèmes dinamiques de la mécanique non linèaire,Ann. Math. 38:65–113 (1937).
Y. W. Lee,Statistical Theory of Communication (Wiley, New York, 1960).
J. Leray, Etude de divers équations intégrales non linéaires et de quelques problèmes que posent l'hydrodinamique,J. Math. Pures Appl. 9e Ser. 12:1–82 (1933).
W. Rudin,Real and Complex Analysis (McGraw-Hill, New York, 1966).
R. Temam,Navier-Stokes Equations: Theory and Numerical Analysis (North-Holland, Amsterdam, 1977).
N. Wiener,The Fourier Integral and Certain of Its Applications (Dover, New York, 1958).
X.-Z. Wu, L. P. Kadanoff, A. Libchaber, and M. Sano, Frequency power spectrum of temperature fluctuations in free convection,Phys. Rev. Lett. 64:2140–2143 (1990).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Bercovici, H., Constantin, P., Foias, C. et al. Exponential decay of the power spectrum of turbulence. J Stat Phys 80, 579–602 (1995). https://doi.org/10.1007/BF02178549
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/BF02178549