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. Instead of the well-known Riemann solution, the explicit dependence on the arbitrary initial fields for that solution and for its timeis matched with the observed turbulence spectrum in the magnetosheath and solar wind. The observed plasma mirror modes nonlinear mechanism arising out corresponding to this spectrum, is established. This mechanism, in turn, is associated with the formation of localized clusters of charged particles whose motion is described by a slow simple wave.

An Inhomogeneous Steady-State Convection of a Vertical Vortex Fluid

“…It is important to note that the investigation of the Navier -Stokes equations by analytical and numerical methods is necessary for many applications. Therefore, the particular results published in [2,[17][18][19][20][21][22][23][24][25][26][27][29][30][31][32][33][34][35] are of interest. In [17][18][19][20][21][22][23][24][25][26][27][28], the theoretical results have been developed which will prepare the scientific community for solving the Sixth Millennium Prize Problem.…”

“…In [17][18][19][20][21][22][23][24][25][26][27][28], the theoretical results have been developed which will prepare the scientific community for solving the Sixth Millennium Prize Problem. The scientific works [29][30][31][32][33][34][35] present interesting results for applied problems, in particular, for working with the equations describing turbulence and nonclassical strongly stratified media. We note that, for the Navier -Stokes equations and their numerous modifications, no general reliable algorithm has been developed so far which guarantees the existence of a unique "mathematical" and "physical" exact or approximate solution.…”

Cosmic Rays Self-arising Turbulence with Universal Spectrum −8/3

“…Thus, the question of whether there is a correlation between the observed differential energy spectrum of CRs and the turbulent spectra characterizing the shock waves themselves has not been widely considered. For example, in Chefranov & Chefranov (2021), for the inertial subrange, a universal spectrum of strong turbulence with an exponent −8/3 is obtained, corresponding to the time of occurrence of the shock wave. In particular, this made it possible to explain the mechanism of the formation of the −8/3 turbulent spectrum observed in the cosmic plasma of the solar wind and the magnetosphere for the density and magnetic field fluctuations (Passot & Sulem 2015;David & Galtier 2019) and adequacy of the −8/3 energy spectrum for the Kadomtsev-Petviashvili theory of strong acoustic turbulence (Kadomtsev & Petviashvili 1973;L'vov et al 1997;Kuznetsov 2004) is also established in Chefranov & Chefranov (2021).…”

“…In our paper, we obtain the two-point correlation function, and the corresponding universal turbulence spectrum law −8/3 is obtained for the CR number density ( ) n x t ;  fluctuations. It is done on the basis of an exact analytical solution to the problem of compressible turbulence (Chefranov & Chefranov 2021) by using a specific solution of the Boltzmann-Vlasov equations with ( ) ( ) f t x p n x t p ;…”

“…The next Section 2 provides a description of the conditions for the implementation of the theory (Skilling 1975). In Section 3, based on the theory (Chefranov & Chefranov 2021), a turbulent spectrum of fluctuations in the density of CR particles with a universal exponent −8/3 is obtained. Section 4 discusses the comparison of the conclusions of the theory and observational data, as well as the results of the similarity theory (Golitsyn 1997(Golitsyn , 2005(Golitsyn , 2012(Golitsyn , 2021.…”