Conformal invariance of the 1-point statistics of the zero-isolines of 2 d scalar fields in inverse turbulent cascades

For three decades there have been speculations about the existence of conformal invariance in two-dimensional turbulence and possible implications thereof. These speculations have been confirmed by numerical experiments. However, there is a scarcity of relevant analytical studies on this topic. In our work we analyze the underlying equation for the one-point probability density function of a scalar in two-dimensional turbulence. We derive conditions under which the probability measure is conformally invariant and show that with this transformation certain statistics of non-homogeneous fields can be derived based on solutions of the homogeneous one.