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High wavenumber spectrum of a passive scalar in isotropic turbulence. (English) Zbl 0596.76067
Summary: The time-dependent solution is found for R. H. Kraichnan’s random- straining model [J. Fluid Mech. 64, 737-762 (1974; Zbl 0291.76022)] of the viscous-diffusive subrange. Temporal decay increases very rapidly with increasing spatial dimension. The Batchelor spectrum is recovered as the conditional variance of spectral amplitudes in excess of a threshold value. A by-product of the analysis is a trivial derivation of the inversion theorem for the Kontorovich-Lebedev transform.
##### MSC:
 76F05 Isotropic turbulence; homogeneous turbulence
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##### References:
 [1] H. Tennekes and J. L. Lumley,A First Course in Turbulence(MIT, Cambridge, MA, 1972). · Zbl 0285.76018 [2] Batchelor, J. Fluid Mech. 5 pp 113– (1959) [3] Kraichnan, J. Fluid Mech. 64 pp 737– (1974) [4] N. G. van Kampen,Stochastic Processes in Physics and Chemistry(North-Holland, Amsterdam, 1981). · Zbl 0511.60038 [5] In the Stratonovitch interpretation, the right-hand side of (1) is interpreted as the limit of a centered difference. The Stratonovitch interpretation is equivalent4 to taking the limit of a vanishingly small decorrelation time for the straining field. In the I to interpretation of (1), a forward difference is assumed. Ito theory is obtained if {$$\alpha$$} is replaced with {$$\alpha$$}-1 in (4). [6] Gargett, J. Fluid Mech. 159 pp 379– (1985) [7] G. N. Watson,Theory of Bessel Functions(Cambridge U.P., Cambridge, 1966). · Zbl 0174.36202 [8] F. Oberhettinger and L. Badii,Tables of Laplace Transforms(Springer, Berlin, 1973). · Zbl 0285.65079 [9] Measurements may also be affected by the sensitivity of the tail of the spectrum to large-scale spatial fluctuations in the effective time-integrated strain history. The author is indebted to Dr. R. H. Kraichnan for this remark and other helpful comments. [10] N. N. Lebedev, I. P. Skalskaya, and Y. S. Ufland,Worked Problems in Applied Mathematics(Dover, New York, 1955).
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