Structure function of passive scalars in two-dimensional turbulence

Abstract

The structure function of a scalar $\theta (\mathbf{x},t)$, passively advected in a two-dimensional turbulent flow $\mathbf{u}(\mathbf{x},t)$, is discussed by means of the fractal dimension ${\delta }_g^{(1)}$ of the passive scalar graph. A relation between ${\delta }_g^{(1)}$, the scaling exponent ${\zeta }_1^{(\theta )}$ of the scalar structure function $D_1^{(\theta )}(r)$, and the structure function D_2(r) of the underlying flow field is derived. Different from the 3-d case, the 2-d structure function also depends on an additional parameter, characteristic of the driving of the passive scalar. In the enstrophy inertial subrange a mean field approximation for the velocity structure function gives a scaling of the passive scalar graph with ${\delta }_g^{(1)}<2$ for intermediate and large values of the Prandtl number Pr. In the energy inertial subrange a model for the energy spectrum and thus D_2(r) gives a passive scalar graph scaling with exponent ${\delta }_g^{(1)}=5/3$. Finally, we discuss an application to recent observations of scalar dispersion in non-universal 2-d flows.

9 pages, 8 figures