Turbulent flows for relativistic conformal fluids in 2+1 dimensions

Abstract

We demonstrate that relativistic conformal hydrodynamics in 2+1 dimensions displays a turbulent behaviour which cascades energy to longer wavelengths on both flat and spherical manifolds. Our motivation for this study is to understand the implications for gravitational solutions through the AdS/CFT correspondence. The observed behaviour implies gravitational perturbations of the corresponding black brane/black hole spacetimes (for sufficiently large scales/temperatures) will display a similar cascade towards longer wavelengths.

Figures (13)

• Figure 1: Cubed Sphere Coordinates. A total of six Cartesian patches are employed to cover the sphere. Only patch boundaries coincide at common points.
• Figure 2: Evolution of the vorticity field for a perturbation $\omega_{p}(\theta, \phi) = Y_{10}^{0} (\theta, \phi)$ and $\delta = 0.2$, on Schwarzschild ($\omega_{0}$). (a) Initial config., (b) beginning of turbulence, (c) fully developed turbulent stage, (d) final state.
• Figure 3: Late times configuration for distinct relevant fields in the non-rotating case ($\omega_{0}=0$ and $T\sim 100$, at $t=1200$). Notice that vortices correspond to minima in the energy. This behaviour is expected as stable vortex structure require a larger surrounding pressure that prevents its dispersion.
• Figure 4: Relevant modes in the power spectrum of the vorticity field for a $\omega_{p}(\theta, \phi) = Y_{10}^{0} (\theta, \phi)$ and $\delta = 0.2$ perturbation, in the non-rotating case (at temperature $T\sim 100$).
• Figure 5: Dependence with temperature: logarithm of the $L_2$ norm of $u^{\theta}$ for different temperatures, in the non-rotating case.