Exact solution to the main turbulence problem for a compressible medium and the universal −8/3 law turbulence spectrum of breaking waves

Abstract

An exact analytical solution to the one-dimensional compressible Euler equations in the form of a nonlinear simple wave is obtained. In contrast to the well-known Riemann solution, the resulting solution and the time of its collapse t0 have an explicit dependence on the initial conditions. For the non-zero dissipation the regularization of the solution over an unlimited time interval is justified. Based on this solution of the Euler equations, an exact explicit and closed description for any single- and multi-point characteristics of turbulence in a compressible medium are obtained, and Onsager's dissipative anomaly is considered. The exact turbulence energy universal spectrum E(k)∝k−8/3, corresponding to the time t→t0 of the shock arising, is stated. That spectrum is more relevant to the strong acoustic turbulence than the well-known spectrum E(k)∝k−2. Installed, spectrum−8/3 is also matched with the observed compressible turbulence spectrum in the magnetosheath and solar wind. The turbulence energy dissipation rate fluctuations universal spectrum ED(k)∝k−2/3 is obtained and corresponds to the known observation data in the atmospheric surface layer.