Anomalous scaling in fluid mechanics: The case of the passive scalar

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Abstract
A mechanism for anomalous scaling in turbulent advection of passive scalars is identified as being similar to a recently discovered mechanism in Navier-Stokes dynamics [V. V. Lebedev and V. S. L’vov, JETP Lett. 59, 577 (1994)]. This mechanism is demonstrated in the context of a passive scalar field that is driven by a rapidly varying velocity field. The mechanism is not perturbative, and its demonstration within renormalized perturbation theory calls for a resummation of infinite sets of diagrams. For the example studied here we make use of a small parameter, the ratio of the typical time scales of the passive scalar vs that of the velocity field, to classify the diagrams of renormalized perturbation theory such that the relevant ones can be resummed exactly. The main observation here, as in the Navier-Stokes counterpart, is that the dissipative terms lead to logarithmic divergences in the diagrammatic expansion, and these are resummed to an anomalous exponent. The anomalous exponent can be measured directly in the scaling behavior of the dissipation two-point correlation function, and it also affects the scaling laws of the structure functions. It is shown that when the structure functions exhibit anomalous scaling, the dissipation correlation function does not decay on length scales that are in the scaling range. The implication of our findings is that the concept of an ‘‘inertial range’’ in which the dissipative terms can be ignored is untenable. The consequences of this mechanism for other cases of possible anomalous scaling in turbulence are discussed.