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Self-similar solution in the Leith model of turbulence: anomalous power law and asymptotic analysis. (English) Zbl 1304.35546

Summary: We consider a Leith model of turbulence [C. E. Leith, “Diffusion approximation to inertial energy transfer in isotropic turbulence”, Phys. Fluids 10, Article ID 1409 (1967; doi:10.1063/1.1762300)] 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 \(t_{\ast}\). We prove that this solution has a power-law asymptotic with an anomalous exponent \(x^{\ast}\), which is less than the Kolmogorov value, \(x^{\ast} > 5/3\). This is a result that was previously discovered by numerical simulations in [C. Connaughton and the second author, “Warm cascades and anomalous scaling in a diffusion model of turbulence”, Phys. Rev. Lett. 92, Article ID 044501 (2004; doi:10.1103/PhysRevLett.92.044501)]. We also prove the convergence to this self-similar solution of the spectrum evolving from an arbitrary finitely supported initial data as \(t \to t_{\ast}\).

MSC:

35Q35 PDEs in connection with fluid mechanics
35K57 Reaction-diffusion equations
76F99 Turbulence
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