Fully Developed Turbulence and Complex Time Singularities

In 1981 Frisch and Morf (1) proposed to relate the intermittent bursts shown by a high pass filtered turbulent flow to complex time singularities of the solution of Navier Stokes equations. Their arguments may be summarized as follows: let v(t) be a velocity signal measured at one point in a turbulent flow. If there is dissipation, v(t) will in general never become singular on the real time axis but it is free to develop singularities as soon as one goes into the complex time domain. These spontaneous singularities are a salient feature of non linear differential equations. Their positions cannot be easily predicted but their exponent as well as their amplitude (to be defined below) are intimately linked to the structure of the equations governing the flow.