Holographic Turbulence and the Fractal Dimension of the Turbulent Horizon

TL;DR

The paper probes holographic turbulence in a (2+1)D boundary fluid by solving the fully nonlinear Einstein–scalar system in $\mathrm{AdS}_4$ and driving the boundary with a random scalar source to achieve a quasi-steady turbulent state. The authors employ the Bondi-Sachs formalism to evolve bulk fields and extract the boundary energy spectrum, finding a scaling $E(k)\sim k^{-1.79}$ with a compressible-dominated decomposition $E_c(k)\sim k^{-1.80}$ and $E_i(k)\sim k^{-1.99}$, as well as a fractal horizon with $D\approx 2.65$. A decaying-turbulence analysis recovers an inverse energy cascade with transient $E(k)\sim k^{-5/3}$ and later $E(k)\sim k^{-5}$ behavior. These results corroborate previous holographic studies while extending the framework to driven turbulence and providing a nonlinear, fully gravitational measurement of horizon fractal structure, with potential implications for understanding turbulence in strongly coupled quantum fluids through holography.

Abstract

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 $\mathrm{AdS}_4$ 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 $D\approx2.65$, which matches the result from simulating the boundary conformal fluid.

Figures (10)

• Figure 1: Schematic foliation of $\mathrm{AdS_{4}}$ spacetime. Initial data are given on the ingoing null hypersurfaces (blue dashed lines) at $v_{0}$ and radial domain is chosen as hypersurfaces between $z=\mathrm{const}$ and the AdS boundary $z=0$. The evolution repeatedly follows along the vector $\partial_{v}$ (red arrows) from one null hypersurfaces into a next one.
• Figure 2: The vorticity field $\omega=\partial_{x}u_{y}-\partial_{y}u_{x}$ of the boundary fluid at $v=1000,1400,2000,3000$. The flow is transformed from an unstable shear flow to a homogenous and isotropic decaying turbulent flow where the inverse cascade is manifestly shown from the first row profiles. The stage at $v=1400$ corresponds to the point where the velocity components $u_{x}$ and $u_{y}$ reach approximately the same magnitude of order. The plots in the second row show the corresponding energy spectrum with fitted scaling power around two subranges $k\in\left(5,10\right)$ and $k\in\left(10,35\right)$.
• Figure 3: Mean kinetic energy $E_{\mathrm{total}}=\frac{1}{2L^{2}}\int d^{2}x\rho\bm{u}^{2}$ of the fluid and its $u_{x}$ and $u_{y}$ contributions from $v=0$ to $v=10000$. It shows that the shear flow transforms into turbulence by two stages: while the total kinetic energy $E$ decreases, nonlinear instability triggers $u_{x}$ grows exponentially until it reaches the same level of $u_{y}$ (at $v\approx 1400$) and then both $u_{x}$ and $u_{y}$ decay at a similar rate.
• Figure 4: Vorticity field $\omega$ , velocity field component $u_{x}$ and the energy spectrum $E\left(k\right)$ of the driven turbulence at $v=100, 1000, 5000, 10000$. The vorticity field (top row) is manifestly homogeneous and isotropic. Vorticies grow from the driving scales $k_{f}$ to the largest scales around $k=10$ which agree with the energy spectrum. The large scale structures observed in the velocity component $u_{x}$ (middle row) are similar to those in $u_{y}$. A narrow peak in the energy spectrum $E\left(k\right)$ (bottom row) appears around the driving scale $k_{f}=100$ where energy is injected there and transferred into large scales.
• Figure 5: Mean kinetic energy \ref{['eq:def_mean_Ek']} of the driven turbulence. It is normalized by the value at $v=1$ since they vanishes initially. It rapidly raises from a small value and then fluctuates around a nearly constant value. Shaded area shows the range used in the time average of the scaling powers.