Universal spectrum in the infrared range of two-dimensional turbulent flows

Abstract

The low-wavenumber behavior of decaying turbulence governed by the generalized two-dimensional (2D) fluid system, the so-called α-turbulence system, is investigated theoretically and through direct numerical simulation. This system is governed by the nonlinear advection equation for an advected scalar q and is characterized by the relationship between q and the stream function ψ: \documentclass[12pt]{minimal}\begin{document}$q=-(-\bm {\nabla }^2)^{\alpha/2 }\psi$\end{document}q=−(−∇2)α/2ψ. Here, the parameter α is a real number that does not exceed 3. The enstrophy transfer function in the infrared range (k → 0) is theoretically derived to be \documentclass[12pt]{minimal}\begin{document}$T_{\alpha }^{\mathcal {Q}}(k \rightarrow 0) \sim k^5$\end{document}TαQ(k→0)∼k5 using a quasi-normal Markovianized model of the generalized 2D fluid system. This leads to three canonical cases of the infrared enstrophy spectrum, which depend on the initial conditions: Qα(k → 0) ∼ Jk, Qα(k → 0) ∼ Lk3, and Qα(k → 0) ∼ Ik5, where J, L, and I are various integral moments of two-point correlation for q. The prefactors J and L are shown to be invariants of the system, while I is an increasing function of time. The evolution from a narrow initial enstrophy spectrum exhibits a universal infrared enstrophy spectrum of the form Qα(k → 0) ∼ k5, which is independent of α. These results are verified by direct numerical simulations of the generalized 2D fluid system.