  We offer a new example of conformal invariance far from equilibrium -- the
inverse cascade of Surface Quasi-Geostrophic (SQG) turbulence. We show that
temperature isolines are statistically equivalent to curves that can be mapped
into a one-dimensional Brownian walk (called Schramm-Loewner Evolution or
SLE$_\kappa$). The diffusivity is close to $\kappa=4$, that is iso-temperature
curves belong to the same universality class as domain walls in the O(2) spin
model. Several statistics of temperature clusters and isolines are measured and
shown to be consistent with the theoretical expectations for such a spin system
at criticality. We also show that the direct cascade in two-dimensional
Navier-Stokes turbulence is not conformal invariant. The emerging picture is
that conformal invariance may be expected for inverse turbulent cascades of
strongly interacting systems.
