Holographic turbulence

TL;DR

It is argued that statistically steady-state black holes dual to d dimensional turbulent flows have horizons whose area growth has a fractal-like structure with fractal dimension D=d+4/3.

Abstract

We construct turbulent black holes in asymptotically AdS_4 spacetime by numerically solving Einstein equations. Both the dual holographic fluid and bulk geometry display signatures of an inverse cascade with the bulk geometry being well approximated by the fluid/gravity gradient expansion. We argue that statistically steady-state black holes dual to d dimensional turbulent flows have horizons which are approximately fractal with fractal dimension D=d+4/3.

Figures (3)

• Figure 1: Left: The boundary vorticity $\omega$ at three times. Right: the horizon area element $\sqrt{\gamma}$ at the same three times.
• Figure 2: Left: The velocity power spectrum $\mathcal{P}$ at time $t = 1008$. Right: The normalized horizon curvature power spectrum $\mathcal{A}/\mathcal{P}$ at four different times.
• Figure 3: Left: The velocity power spectrum on a semi-log scale at 4 different times. Right: Time evolution of the maximum difference between the exact metric and zeroth and first order gradient expansion.