Holographic Turbulence and the Fractal Dimension of the Turbulent Horizon

We study two-dimensional turbulence driven by a scalar operator within the framework of the AdS/CFT correspondence, where the external driving source is used to sustain a quasi-steady turbulent state. We numerically construct dynamical and spatially inhomogeneous turbulent black holes in the asymptotically ${\mathrm{A}\mathrm{d}\mathrm{S}}_4$ spacetime by solving the full nonlinear equations of motion in the Bondi-Sachs formalism. The inverse energy cascade and the corresponding energy spectrum of both decaying and driven turbulence are analyzed. The scalar driving leads to a compressible energy dominated flow, and the corresponding scaling power laws agree well with previous simulations of two-dimensional turbulence in compressible fluids. Furthermore, we take a direct estimate of the fractal structure of the turbulent black hole, obtaining a fractal dimension $D\approx 2.65$, which matches the result from simulating the boundary conformal fluid.