Intermittency and Anomalous Scaling of Passive Scalars in Any Space Dimension

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 3D 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 soluble ``rapid-change'' model with velocity delta-correlated in time. Our results include as a special case an exponent-inequality derived previously by Constantin \& Procaccia [Nonlinearity {\bf 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.