Topology and the Conformal Invariance of Nodal Lines in Two-Dimensional Active Scalar Turbulence

The inverse cascade in two-dimensional hydrodynamic turbulence exhibits a mysterious phenomenon. Numerical simulations have shown that the nodal isolines of certain scalars actively transported in the flow (eg, the vorticity in Navier-Stokes theory) obey Schramm-Loewner evolution (SLE), which indicates the presence of conformal invariance. Therefore, these turbulent isolines are somehow in the same class as cluster boundaries in equilibrium statistical mechanical models at criticality, such as critical percolation. In this paper, we propose that the inverse cascade is characterized by a local energy (or in some cases, enstrophy) flux field that spontaneously breaks time reversal invariance. The turbulent state consists of random constant flux domains, with the nodal isolines acting as domain walls where the local flux vanishes. The generalized circulation of the domains is proportional to a topological winding number. We argue that these turbulent states are gapped states, in analogy with quantum Hall systems. The turbulent flow consists of many strongly coupled vortices that are analogous to quasi-particles. The nodal isolines are associated with the gapless topological degrees of freedom in the flow, where scale invariance is enhanced to conformal invariance. We introduce a concrete model of this behavior using a two-dimensional effective theory involving the canonical Clebsch scalars. This theory has patch solutions that exhibit power law scaling. The fractional winding number associated with the patches can be related to the Kolmogorov-Kraichnan scaling dimension of the corresponding fluid theory. We argue that the fully developed inverse cascade is a scale invariant gas of these patches. This theory has a conformally invariant sector described by a Liouville conformal field theory whose central charge is fixed by the fractional winding number.