Self-similar solution in the Leith model of turbulence: anomalous power law and asymptotic analysis

Abstract

We consider a Leith model of turbulence (Leith C 1967 Phys. Fluids 10 1409) in which the energy spectrum obeys a nonlinear diffusion equation. We analytically prove the existence of a self-similar solution with a power-law asymptotic on the low-wavenumber end and a sharp boundary on the high-wavenumber end, which propagates to infinite wavenumbers in a finite-time t*. We prove that this solution has a power-law asymptotic with an anomalous exponent x*, which is less than the Kolmogorov value, x* > 5/3. This is a result that was previously discovered by numerical simulations in Connaughton and Nazarenko (2004 Phys. Rev. Lett. 92 044501). We also prove the convergence to this self-similar solution of the spectrum evolving from an arbitrary finitely supported initial data as t → t*.