Intermittency and anomalous scaling of passive scalars in any space dimension

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Abstract
We establish exact inequalities for the structure-function scaling exponents of a passively advected scalar in both the inertial-convective and viscous-convective ranges. These inequalities involve the scaling exponents of the velocity structure functions and, in a refined form, an intermittency exponent of the convective-range scalar flux. They are valid for three-dimensional Navier-Stokes turbulence and satisfied within errors by present experimental data. The inequalities also hold for any ‘‘synthetic’’ turbulent velocity statistics with a finite correlation in time. We show that for time-correlation exponents of the velocity smaller than the ‘‘local turnover’’ exponent, the scalar spectral exponent is strictly less than that in Kraichnan’s [Phys. Rev. Lett. 72, 1016 (1994); Phys. Fluids 11, 945 (1968); J. Fluid Mech. 64, 737 (1974); 62, 305 (1974)] soluble ‘‘rapid-change’’ model with velocity δ correlated in time. Our results include as a special case an exponent inequality derived previously by Constantin and Procaccia [Nonlinearity 7, 1045 (1994)], but with a more direct proof. The inequalities in their simplest form follow from a Kolmogorov-type relation for the turbulent passive scalar valid in each space dimension d. Our improved inequalities are based upon a rigorous version of the refined similarity hypothesis for passive scalars. These are compared with the relations implied by ‘‘fusion rules’’ hypothesized for scalar gradients. © 1996 The American Physical Society.
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