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The last stage of evolution toward the stationary Kolmogorov spectrum ofhydrodynamic turbulence is studied using the Leith model. This evolution isshown to manifest itself as a reflection wave in the wavenumber spacepropagating from the largest toward the smallest wavenumbers, and is describedby a self-similar solution of a new (third) kind. This stage follows thepreviously studied stage of an initial explosive propagation of the spectralfront from the smallest to the largest wavenumbers reaching arbitrarily largewavenumbers in a finite time, and which was described by a self-similarsolution of the second kind. Nonstationary solutions corresponding to"warmcascades" characterised by a thermalised spectrum at large wavenumbers are alsoobtained.

Self-similar formation of the Kolmogorov spectrum in the Leith model of turbulence