Holographic Turbulence and Numerical Estimate of the Fractal Dimension of the Turbulent Horizon

Abstract

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 ${\mathrm{AdS}}_4$ 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, $E(k)\propto k^{-1.79}$, 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 $D\approx 2.65$, which suggests an interesting universality in the fractal dimension.

version to appear in JHEP