SCALING EXPONENTS FOR TURBULENCE AND OTHER RANDOM-PROCESSES AND THEIRRELATIONSHIPS WITH MULTIFRACTAL STRUCTURE

In the recent literature on high-Reynolds-number turbulence, several d ifferent types of scaling exponents-such as multifractal exponents for velocity increments, for energy and scalar dissipation, for the squar e of the local vorticity, and so forth-have been introduced. More rece ntly, a new exponent called the cancellation exponent has been introdu ced for characterizing rapidly oscillating quantities. Not all of thes e exponents are independent; some of them are simply related to more f amiliar scaling for velocity and temperature structure functions eithe r exactly or through plausible hypotheses familiar for turbulence. A p rimary purpose of this paper is to establish the interrelationships am ong the various exponents. In doing so, we obtain several additional r elations. Much of the paper is relevant to general stochastic processe s, although the discussion is heavily influenced by the turbulent cont ext. We first examine the case of one-dimensional random processes and subsequently consider two- and three-dimensional processes. Special c onsideration is given to characteristic values appropriate to the geom etry of turbulence, as well as the lifetimes of eddies of various scal es. Finally, we discuss some properties of the tails of the probabilit y density function to which the scaling properties of high-order struc ture functions are related and discuss the implications of multifracta lity on their structure.