The area rule for circulation in three-dimensional turbulence

Abstract

Significance Circulation around closed loops is important in classical and quantum fluids, as well as condensed matter in the solid state. This paper deals with the statistical theory of circulation in high–Reynolds number turbulence and has important implications for the structure of turbulent vorticity, which is a quantity of central interest in turbulence. We focus particularly on the so-called area rule. The rule states that the statistical properties of circulation around closed contours depend solely on the area of the minimal surface for the contour, not on its shape. We demonstrate that the area rule works to a good approximation but falls short in some respects; in particular, the minimal area alone is insufficient to specify all the statistical properties.