M.Sc Student | Sabag Evyatar |
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Subject | Holographic Turbulence in a Large Number of Dimensions |

Department | Department of Physics |

Supervisor | Professor Amos Yarom |

Full Thesis text |

`We consider relativistic hydrodynamics in the limit where the number of spatial dimensions is very large. We first show that under certain restrictions, the resulting equations of motion simplify significantly. Holographic theories in a large number of dimensions satisfy the aforementioned restrictions and their dynamics are captured by hydrodynamics with a naturally truncated derivative expansion. The resulting hydrodynamic equations of motion in the aforementioned limit bare a striking resemblance to the compressible Navier-Stokes equations. This resemblance allows for a comparison of the former equations to known results from classical hydrodynamics, especially in the subsonic regime, where we expect to find the Kolmogorov and `__Kraichnan__ scaling laws, in addition to the direct and inverse cascades.

The resulting equations
in a large number of dimensions are non-linear as the Navier-Stokes equations,
and are hard to tackle analytically. Thus, we turn to numerical techniques to
analyze two and three-dimensional turbulent flow of such fluids in various
regimes. The Kolmogorov and __Kraichnan__ scaling laws are not found in the
simulations of two dimensional flows, probably due to insufficient lattice
size. Inverse cascade-like behavior, on the other hand is apparent, and remains
visible even when the Mach number is not small. The three-dimensional turbulent
flow shows a transient effect of the Kolmogorov scaling law as expected for
decaying turbulence, in addition to a direct-cascade like behavior. Remarkably,
these two observations remain true even for high Mach numbers. Finally, using
our improved analytic control in the limit of large number of dimensions, we
studied a proposed relation between the horizon curvature power spectrum and
the hydrodynamic energy power spectrum. We found these quantities to be
linearly related by analytic and numeric analysis for low Mach numbers.
However, the relation becomes unclear as we increase the Mach number.