Holographic turbulence and numerical estimate of the fractal dimension of the turbulent horizon

A bstract We numerically 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 propose a simple and efficient evolution scheme within the Bondi-Sachs formalism. Applying this scheme to numerically solve the full nonlinear equations of motion, we obtain a turbulent black hole in asymptotically AdS 4 spacetime. The inverse energy cascade and the corresponding energy spectrum of both decaying and driven dual turbulence are analyzed. The scalar driving leads to a compressible-energy-dominated flow, and the corresponding power law scaling, E ( k ) ∝ k −1.79 , agrees well with previous simulations of two-dimensional turbulence in weakly coupled compressible fluids in fluid dynamics. This differs from the Kolmogorov –5/3 scaling law. Furthermore, we perform a direct numerical estimate of the fractal structure of the turbulent black hole, obtaining a fractal dimension D ≈ 2.65, which suggests an interesting universality in the fractal dimension.