Anomalous scalings in dierential models of turbulence

Dierential models for hydrodynamic, passive-scalar and wave turbulence given by nonlinear rst- and second-order evolution equations for the energy spectrum in the k-space were analysed. Both types of models predict formation an anomalous transient power-law spectra. The second-order models were analysed in terms of self-similar solutions of the second kind, and a phenomenological formula for the anomalous spectrum exponent was constructed using numerics for a broad range of parameters covering all known physical examples. The rst-order models were examined analytically, including nding an analytical prediction for the anomalous exponent of the transient spectrum and description of formation of the Kolmogorov- type spectrum as a reection wave from the dissipative scale back into the inertial range. The latter behaviour was linked to pre-shock/shock singularities similar to the ones arising in the Burgers equation. Existence of the transient anomalous scaling and the reection-wave scenario are argued to be a robust feature common to the nite- capacity turbulence systems. The anomalous exponent is independent of the initial conditions but varies for for dierent models of the same physical system.