The length-scale distribution function of the distance between extremal points in passive scalar turbulence

In order to extract small-scale statistical information from passive scalar fields obtained by direct numerical simulation (DNS) a new method of analysis is introduced. It consists of determining local minimum and maximum points of the fluctuating scalar field via gradient trajectories starting from every grid point in the directions of ascending and descending scalar gradients. The ensemble of grid cells from which the same pair of extremal points is reached determines a spatial region which is called a ‘dissipation element’. This region may be highly convoluted but on average it has an elongated shape with, on average, a nearly constant diameter of a few Kolmogorov scales and a variable length that has the mean of a Taylor scale. We parameterize the geometry of these elements by the linear distance between their extremal points and their scalar structure by the absolute value of the scalar difference at these points. The joint p.d.f. of these two parameters contains most of the information needed to reconstruct the statistics of the scalar field. It is decomposed into a marginal p.d.f. of the linear distance and a conditional p.d.f. of the scalar difference. It is found that the conditional mean of the scalar difference follows the 1/3 inertial-range Kolmogorov scaling over a large range of length-scales even for the relatively small Reynolds number of the present simulations. This surprising result is explained by the additional conditioning on minima and maxima points. A stochastic evolution equation for the marginal p.d.f. of the linear distance is derived and solved numerically. The stochastic problem that we consider consists of a Poisson process for the cutting of linear elements and a reconnection process due to molecular diffusion. The resulting length-scale distribution compares well with those obtained from the DNS.