Gl. Eyink, INTERMITTENCY AND ANOMALOUS SCALING OF PASSIVE SCALARS IN ANY SPACE DIMENSION, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics, 54(2), 1996, pp. 1497-1503
We establish exact inequalities for the structure-function scaling exp
onents 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 ar
e valid for three-dimensional Navier-Stokes turbulence and satisfied w
ithin errors by present experimental data. The inequalities also hold
for any ''synthetic'' turbulent velocity statistics with a finite corr
elation in time. We show that for time-correlation exponents of the ve
locity smaller than the ''local turnover'' exponent, the scalar spectr
al 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 ''rapidchange'' model with velocity d
elta correlated in time. Our results include as a special case an expo
nent inequality derived previously by Constantin and Procaccia [Nonlin
earity 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 improve
d inequalities are based upon a rigorous version of the refined simila
rity hypothesis for passive scalars. These are compared with the relat
ions implied by ''fusion rules'' hypothesized for scalar gradients.