Exact Area Law for Planar Loops in Turbulence in Two and Three Dimensions

Abstract

We study properties of the minimal surface in the Area Law Solution \cite{M93}, \cite{M19a}, \cite{M19b}. We find out that Area Law holds exactly for 2D turbulence as well as for arbitrary planar loop in higher dimensions. This relies on our previous result $\hat{I}\pm =\frac{1}{2}$ in which case the second moment of circulation can be proven to reduce to the area inside the planar loop. In $d=3$, we demonstrate how the Stokes condition ${\partial }_i\ddot{I}{‰}_i(r)=0$ is exactly satisfied for the minimal surface solution in virtue of vanishing mean curvature at the minimal surface. In order to satisfy Loop Equation beyond planar loops, we introduce self-consistent conformal metric on the surface designed to preserve Stokes condition but to compensate the terms in the loop equation. We derive nonlinear integral equation for this conformal metric as a function of a point on a surface.

9 pages, 0 figures, fixed typos, modified Orientation Symmetry discussion, added comments about the second moment computatiuon vs experiment, removed one equation