The decay of multiscale signals — a deterministic model of Burgers turbulence

This work is devoted to the study of the decay of multiscale deterministic solutions of the unforced Burgers’ equation in the limit of vanishing viscosity. It is well known that Burgers turbulence with a power law energy spectrum E0(k)∼∣k∣n has a self-similar regime of evolution. For n<1, this regime is characterized by an integral scale L(t)∼t2/(3+n), which increases with time due to the multiple mergings of the shocks, and therefore the energy of a random wave decays more slowly than the energy of a periodic signal. In this paper a deterministic model of turbulence-like evolution is considered. We construct the initial perturbation as a piecewise linear analog of the Weierstrass function. The wave numbers of this function form a “Weierstrass spectrum”, which accumulates at the origin in geometric progression. “Reverse” sawtooth functions with negative initial slope are used in this series as basic functions, while their amplitudes are chosen by the condition that the distribution of energy over exponential intervals of wave numbers is the same as for the continuous spectrum in Burgers turbulence. By combining these two ideas, we can obtain an exact analytical solution for the velocity field. We also notice that such multiscale waves may be constructed for multidimensional Burgers’ equation. This solution has scaling exponent h=−12(1+n) and its evolution in time is self-similar with logarithmic periodicity and with the same average law L(t) as for Burgers turbulence. Shocklines form self-similar regular tree-like structures. This model also describes important properties of the Burgers turbulence such as the self-preservation of the evolution of large-scale structures in the presence of small-scale perturbations.