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Exponential decay of the power spectrum of turbulence. (English) Zbl 1081.35505
Summary: 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 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 $$\to \pm \infty$$. 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.

##### MSC:
 35Q30 Navier-Stokes equations 76F99 Turbulence 76M25 Other numerical methods (fluid mechanics) (MSC2010)
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##### References:
 [1] 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). · doi:10.1103/PhysRevA.44.3622 [2] P. Constantin and C. Foias,Navier-Stokes Equations (University of Chicago Press, Chicago, 1988). · Zbl 0687.35071 [3] N. Dunford and J. T. Schwartz,Linear Operators. Part I: General Theory (Intersiience, New York, 1958). [4] L. Hörmander,The Analysis of Linear Partial Differential Operators (Springer-Verlag, Berlin, 1983). [5] 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). · Zbl 0016.08604 · doi:10.2307/1968511 [6] Y. W. Lee,Statistical Theory of Communication (Wiley, New York, 1960). [7] 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). · Zbl 0006.16702 [8] W. Rudin,Real and Complex Analysis (McGraw-Hill, New York, 1966). · Zbl 0142.01701 [9] R. Temam,Navier-Stokes Equations: Theory and Numerical Analysis (North-Holland, Amsterdam, 1977). · Zbl 0383.35057 [10] N. Wiener,The Fourier Integral and Certain of Its Applications (Dover, New York, 1958). · Zbl 0082.29101 [11] 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). · doi:10.1103/PhysRevLett.64.2140
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