Holographic Turbulence

We construct turbulent black holes in asymptotically ${\mathrm{AdS}}_4$ spacetime by numerically solving Einstein’s equations. Using the AdS/CFT correspondence we find that 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 whose area growth has a fractal-like structure with fractal dimension $D=d+4/3$.

Figure 1

Left: The boundary vorticity $\omega$ at 3 times. Right: the horizon area element $\sqrt{\gamma }$ at the same 3 times.

Figure 2

Left: The velocity power spectrum $\mathcal{P}$ at time $t=1008$. Right: The normalized horizon curvature power spectrum $A/P$ at 4 different times.

Figure 3

Time evolution of the maximum difference between the exact metric and $0\text{th}$ and 1st order gradient expansion.