Symmetries in physics underpin fundamental laws and reveal profound connections between seemingly unrelated phenomena. We present new findings showcasing the presence of conformal invariance within a water-wave turbulence system. This fundamental symmetry, extending scale invariance to conformal transformations, is of great importance for comprehending critical phenomena, scaling behaviors, and emergent properties across classical and quantum field theory domains.

Through experiments conducted in gravity-capillary wave turbulence, we apply techniques derived from the theory of stochastic partial differential equations. Our investigations show that the zero-height isolines of our system align with the universality class of domain walls in the critical 2D Ising model. These results establish a link between water-wave turbulence and the physics ofcritical systems. Furthermore, we found that this symmetry appears to break down with increasing nonlinearity.

Theoretical implications of our work place the study of gravity wave turbulence within the realm of critical phenomena. This approach opens avenues for exploring nonlinearly interacting systems described by wave turbulence, such as quantum fluids and gravitational waves.