Degrees of freedom of turbulence

Abstract

We compute in the framework of a multifractal model for three-dimensional fully developed turbulence the number of degrees of freedom N as function of the Reynolds number R. $N--- depends on the whole spectrum of singularities h related to the anomalous scaling of the velocity differences. On the other hand, we have also considered what is the total number ${N}_{T}^{\mathrm{*}}$ of equations needed in a computer simulation since N just has theoretical relevance. We stress, however, that the main features of intermittency can be described by an effective number N${\ifmmode \tilde{}\else \~{}\fi{}}^{\mathrm{*}}$ which is much smaller than ${N}_{T}^{\mathrm{*}}$ because N${\ifmmode \tilde{}\else \~{}\fi{}}^{\mathrm{*}}$ neglects very improbable events. We show that ${N}_{T}^{\mathrm{*}}$\ensuremath{\propto}${R}^{3}$ while we get from a fit of experimental data that N${\ifmmode \tilde{}\else \~{}\fi{}}^{\mathrm{*}}$\ensuremath{\propto}${R}^{2.3}$ is of the same order of N\ensuremath{\propto}${R}^{2.2}$. .AE