Joint multifractal measures: Theory and applications to turbulence

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Abstract
A high-Reynolds-number turbulent flow subsumes several intermittent fields; some examples are the rates of dissipation of turbulent energy and scalar variance, square of turbulent vorticity and rate of strain, etc. These intermittent fields display different degrees of correlation among them. Motivated by the need for characterizing such coexisting distributions of intermittent fields in fully developed turbulence, the multifractal formalism—which we have already found useful in describing such intermittent distributions singly—is extended to more than one variable. The formalism is first illustrated by studying joint log-normal as well as joint binomial distributions. It is then applied to simultaneous measurements in several classical turbulent flows of the joint distribution of a component of the dissipation of kinetic energy, the dissipation rate of passive scalar variance, as well as the square of a component of turbulent vorticity. This allows simple but realistic models of simultaneous cascades of more than one variable to be developed.