Reynolds number of transition and self-organized criticality of strong turbulence

A turbulent flow is characterized by velocity fluctuations excited in an extremely broad interval of wave numbers k > Lambda(f), where Lambda(f) is a relatively small set of the wave vectors where energy is pumped into fluid by external forces. Iterative averaging over small-scale velocity fluctuations from the interval Lambda(f) < k <= Lambda(0), where eta = 2 pi/Lambda(0) is the dissipation scale, leads to an infinite number of relevant scale-dependent coupling constants (Reynolds numbers) Re-n (k) = O(1). It is shown that in the infrared limit k -> Lambda(f), the Reynolds numbers Re(k) -> Re-tr, where Retr is the recently numerically and experimentally discovered universal Reynolds number of smooth transition from Gaussian to anomalous statistics of spatial velocity derivatives. The calculated relation Re(Lambda(f)) = Re-tr selects the lowest-order nonlinearity as the only relevant one. This means that in the infrared limit k -> Lambda(f), all high-order nonlinearities generated by the scale elimination sum up to zero.