%0 Book Section %1 statphys23_0655 %A Kree, M. %A Kalda, J. %B Abstract Book of the XXIII IUPAP International Conference on Statistical Physics %C Genova, Italy %D 2007 %E Pietronero, Luciano %E Loreto, Vittorio %E Zapperi, Stefano %K diffusion passive scalar statistical statphys23 topic-5 topography turbulent %T On the fractal structure of the passive scalar fields in a fully developed turbulence. %U http://st23.statphys23.org/webservices/abstract/preview_pop.php?ID_PAPER=655 %X The features of turbulent diffusion has been a great challenge for the statistical physics during decades. This is explained by high geometrical complexity of the emerging patterns, and by a large number of qualitatively different mixing regimes. Our aim is to study the statistical topography of a passive scalar (tracer) field in a fully developed two-dimensional turbulence, assuming the lower cut-off scale of the turbulent energy spectrum to be vanishingly small. In this case, the tracer density field is known to be characterized by a cascade of fractal discontinuity fronts A. Celani et al, Phys. Fluids 13, 1768 (2001). Let us suppose that initially, the tracer density has a constant gradient, i.e. the iso-density lines are straight. If the molecular diffusion could be ignored, the value of the tracer density would be ``glued'' to the fluid particles. So, the fractal geometry of separate iso-density lines would correspond to the fractal geometry of a simple fluid line (virtual chain of fluid particles) evolving in the turbulent velocity field. However, the situation is somewhat more complicated. Large gradients of the tracer density field emerge due to the stretching and folding by the velocity field. If the molecular diffusion could be ignored, these large gradients would evolve into discontinuity fronts within a finite time. Then, any non-zero diffusivity can no longer be ignored. This gives rise to a drift and reconnection of the iso-density lines. Still, one can argue that the most significant drift and reconnection will take place only on a sparse, fractal set of the highest local stretching rate. Therefore, the iso-density lines (and the discontinuity fronts) might belong to the same universality class as the fluid lines. In order to test this hypothesis, we have calculated the fractal dimension of the liquid line numerically, and compared the results with earlier experimental data J. Kondev and G. Huber, Phys. Rev. Lett. 86, 5890 (2001). The fluid line is defined as a set of points that co-move with the local velocity field; new points are added, when two neighbouring points depart farther than the lattice resolution $\delta$, and the line is reconnected, when two loops of the line approach closer than $\delta$. We generate a random velocity field, which is delta-correlated in time (Kraichnan regime), and follows a normal scaling power law. The simulations are performed on polygons of variable size (up to the radius $r=4096\delta$, as limited by the capabilities of the computing cluster); the exponents are obtained by extrapolating the finite-size scaling. The result $d1.3,1.33$ is in agreement with the fractal dimension of the iso-density lines ($d1.3$) obtained by Kondev and Huber. We calculate also the size-distribution exponent of the loops breaking apart from the main fluid line as a result of reconnections. The result $1.95$ is in agreement with our theoretical expectation $\alpha=2$.

@incollection{statphys23_0655, abstract = {The features of turbulent diffusion has been a great challenge for the statistical physics during decades. This is explained by high geometrical complexity of the emerging patterns, and by a large number of qualitatively different mixing regimes. Our aim is to study the statistical topography of a passive scalar (tracer) field in a fully developed two-dimensional turbulence, assuming the lower cut-off scale of the turbulent energy spectrum to be vanishingly small. In this case, the tracer density field is known to be characterized by a cascade of fractal discontinuity fronts [A. Celani et al, Phys. Fluids 13, 1768 (2001)]. Let us suppose that initially, the tracer density has a constant gradient, i.e. the iso-density lines are straight. If the molecular diffusion could be ignored, the value of the tracer density would be ``glued'' to the fluid particles. So, the fractal geometry of separate iso-density lines would correspond to the fractal geometry of a simple fluid line (virtual chain of fluid particles) evolving in the turbulent velocity field. However, the situation is somewhat more complicated. Large gradients of the tracer density field emerge due to the stretching and folding by the velocity field. If the molecular diffusion could be ignored, these large gradients would evolve into discontinuity fronts within a finite time. Then, any non-zero diffusivity can no longer be ignored. This gives rise to a drift and reconnection of the iso-density lines. Still, one can argue that the most significant drift and reconnection will take place only on a sparse, fractal set of the highest local stretching rate. Therefore, the iso-density lines (and the discontinuity fronts) might belong to the same universality class as the fluid lines. In order to test this hypothesis, we have calculated the fractal dimension of the liquid line numerically, and compared the results with earlier experimental data [J. Kondev and G. Huber, Phys. Rev. Lett. 86, 5890 (2001)]. The fluid line is defined as a set of points that co-move with the local velocity field; new points are added, when two neighbouring points depart farther than the lattice resolution $\delta$, and the line is reconnected, when two loops of the line approach closer than $\delta$. We generate a random velocity field, which is delta-correlated in time (Kraichnan regime), and follows a normal scaling power law. The simulations are performed on polygons of variable size (up to the radius $r=4096\delta$, as limited by the capabilities of the computing cluster); the exponents are obtained by extrapolating the finite-size scaling. The result $d\in [1.3,1.33]$ is in agreement with the fractal dimension of the iso-density lines ($d\approx 1.3$) obtained by Kondev and Huber. We calculate also the size-distribution exponent of the loops breaking apart from the main fluid line as a result of reconnections. The result $\alpha \approx 1.95$ is in agreement with our theoretical expectation $\alpha=2$.}, added-at = {2007-06-20T10:16:09.000+0200}, address = {Genova, Italy}, author = {Kree, M. and Kalda, J.}, biburl = {https://www.bibsonomy.org/bibtex/2647d60055f2f166c2ac2951664dd66ee/statphys23}, booktitle = {Abstract Book of the XXIII IUPAP International Conference on Statistical Physics}, editor = {Pietronero, Luciano and Loreto, Vittorio and Zapperi, Stefano}, interhash = {2397c2b09413b46eb4bf37b15a8309fa}, intrahash = {647d60055f2f166c2ac2951664dd66ee}, keywords = {diffusion passive scalar statistical statphys23 topic-5 topography turbulent}, month = {9-13 July}, timestamp = {2007-06-20T10:16:26.000+0200}, title = {On the fractal structure of the passive scalar fields in a fully developed turbulence.}, url = {http://st23.statphys23.org/webservices/abstract/preview_pop.php?ID_PAPER=655}, year = 2007 }