Revised Universality Concept in the Theory of Turbulence

  • G. E. Falkovich
Part of the Progress in Nonlinear Differential Equations and Their Applications book series (PNLDE, volume 11)


The initial concept of spectrum universality was formulated withinin the theory of hydrodynamic turbulence and was later extended to the theory of wave turbulence. It purports that the turbulence spectrum in the inertial range of scales to be a power function of the wavenumber. This spectrum is determined by a single external parameter, namely, the flux of energy or any other conserved integral. Such oneflux isotropic spectra have been obtained as exact solutions of respective equations. However, some of those solutions were proved to be structurally unstable with respect to the variation of boundary conditions. It is possible, for instance, that a slightly nonisotropic pump produces the strongly nonisotropic spectrum of fully developed wave turbulence. Such instabilities, as shown here, are necessarily connected with the presence of other conserved integrals and they describe a process of rearrangement from the one-flux spectrum to the multi-flux one. This paper demonstrates, however, that the universality concept still survives, although in a more sophisticated form: A steady spectrum is defined by the fluxes of all conserved integrals flowing at the same direction. A criterion which permits obtaining directions of fluxes is also presented. Such a multi-flux spectrum is also universal since it is defined by the fluxes only, being independent of the fine structure of the source. The multi-flux spectrum is structurally stable, since any variation of the pump changes only those values of fluxes which are solution parameters.


Energy Flux Incompressible Fluid Momentum Flux Steady Solution Structural Instability 
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© Springer Science+Business Media New York 1993

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  • G. E. Falkovich

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