We establish anomalous inertial range scaling of structure functions for a model of advection of a passive scalar by a random velocity field. The velocity statistics is taken gaussian with decorrelation in time and velocity differences scaling as |x| κ/2 in space, with 0 ≤ κ < 2. The scalar is driven by a gaussian forcing acting on spatial scale L and decorrelated in time. The structure functions for the scalar are well defined as the diffusivity is taken to zero and acquire anomalous scaling behavior for large pumping scales L. The anomalous exponent is calculated explicitly for the 4 th structure function and for small κ and it differs from previous predictions. For all but the second structure functions the anomalous exponents are nonvanishing.In 1941 A.N. Kolmogorov argued that in fully developed turbulence a range of spatial scales exists where the velocity structure functions acquire a form independent of the IR and UV cutoffs provided by the scale of energy pumping and dissipation respectively. Ever since a debate has been going on as to whether there are corrections to the scaling exponents predicted by Kolmogorov and whether such corrections depend on the dissipation or the pumping scale or both. This question being still quite open for Navier-Stokes turbulence both experimentally and theoretically, it is useful to consider it in the context of simpler models that are nevertheless expected to display phenomena similar to the Navier-Stokes equations.In this letter we consider one such model that has attracted much attention recently [1,2,3,4,5,6], namely that of passive advection in a random velocity field v(t, x) of a scalar quantity T whose density T (t, x) satisfies the equationwhere ν denotes the molecular diffusivity of the scalar T and f (t, x) is an external source driving the system. We take v(t, x) and f (t, x) to be mutually independent Gaussian random fields with zero mean and covariances

In this paper, we rigorously construct Liouville Quantum Field Theory on the Riemann sphere introduced in the 1981 seminal work by Polyakov. We establish some of its fundamental properties like conformal covariance under PSL2(C)-action, Seiberg bounds, KPZ scaling laws, KPZ formula and the Weyl anomaly formula. We also make precise conjectures about the relationship of the theory to scaling limits of random planar maps conformally embedded onto the sphere.

The anomalous scaling in the Kraichnan model of advection of the passive scalar by a random velocity field with non-smooth spatial behavior is traced down to the presence of slow resonance-type collective modes of the stochastic evolution of fluid trajectories. We show that the slow modes are organized into infinite multiplets of descendants of the primary conserved modes. Their presence is linked to the non-deterministic behavior of the Lagrangian trajectories at high Reynolds numbers caused by the sensitive dependence on initial conditions within the viscous range where the velocity fields are more regular. Revisiting the Kraichnan model with smooth velocities we describe the explicit solution for the stationary state of the scalar. The properties of the probability distribution function of the smeared scalar in this state are related to a quantum mechanical problem involving the Calogero-Sutherland Hamiltonian with a potential.

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