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

The low-wavenumber behavior of decaying turbulence governed by the generalized two-dimensional (2D) fluid system, the so-called alpha-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 psi: q = -(-del(2))(alpha/2)psi. Here, the parameter a is a real number that does not exceed 3. The enstrophy transfer function in the infrared range (k -> 0) is theoretically derived to be T-alpha(Q)(k -> 0) similar to k(5) 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: Q(alpha)(k -> 0) similar to Jk, Q(alpha)(k -> 0) similar to Lk(3), and Q(alpha)(k -> 0) similar to Ik(5), 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 Q(alpha)(k -> 0) similar to k(5), which is independent of alpha. These results are verified by direct numerical simulations of the generalized 2D fluid system. (C) 2014 AIP Publishing LLC.