Competition between Diffusion and Fragmentation: An Important Evolutionary Process of Nature

We investigate systems of nature where the common physical processes diffusion and fragmentation compete. We derive a rate equation for the size distribution of fragments. The equation leads to a third order differential equation which we solve exactly in terms of Bessel functions. The stationary state is a universal Bessel distribution described by one parameter, which fits perfectly experimental data from two very different system of nature, namely, the distribution of ice crystal sizes from the Greenland ice sheet and the length distribution of alpha-helices in proteins.Comment: 4 pages, 3 figures, (minor changes

Exact Periodic Solutions of Shells Models of Turbulence

We derive exact analytical solutions of the GOY shell model of turbulence. In the absence of forcing and viscosity we obtain closed form solutions in terms of Jacobi elliptic functions. With three shells the model is integrable. In the case of many shells, we derive exact recursion relations for the amplitudes of the Jacobi functions relating the different shells and we obtain a Kolmogorov solution in the limit of infinitely many shells. For the special case of six and nine shells, these recursions relations are solved giving specific analytic solutions. Some of these solutions are stable whereas others are unstable. All our predictions are substantiated by numerical simulations of the GOY shell model. From these simulations we also identify cases where the models exhibits transitions to chaotic states lying on strange attractors or ergodic energy surfaces.Comment: 25 pages, 7 figure

Turbulent Binary Fluids: A Shell Model Study

We introduce a shell (``GOY'') model for turbulent binary fluids. The variation in the concentration between the two fluids acts as an active scalar leading to a redefined conservation law for the energy, which is incorporated into the model together with a conservation law for the scalar. The model is studied numerically at very high values of the Prandtl and Reynolds numbers and we investigate the properties close to the critical point of the miscibility gap where the diffusivity vanishes. A peak develops in the spectrum of the scalar, showing that a strongly turbulent flow leads to an increase in the mixing time. The peak is, however, not very pronounced. The mixing time diverges with the Prandtl number as a power law with an exponent of approximately 0.9. The continuum limit of the shell equations leads to a set of equations which can be solved by a scaling ansatz, consistent with an exact scaling of the Navier-Stokes equations in the inertial range. In this case a weak peak also persists for a certain time in the spectrum of the scalar. Exact analytic solutions of the continuous shell equations are derived in the inertial range. Starting with fluids at rest, from an initial variation of the concentration difference, one can provoke a ``spontaneous'' generation of a velocity field, analogous to MHD in the early universe.Comment: 25 pages including 9 figures in epsf. Some figures have been revised, and a few comments have been inserted in the text. To be published in Physic