Bypass to Turbulence in Hydrodynamic Accretion: Lagrangian Analysis of Energy Growth
Abstract
Despite observational evidence for cold neutral astrophysical accretion disks, the viscous process which may drive the accretion in such systems is not yet understood. While molecular viscosity is too small to explain the observed accretion efficiencies by more than ten orders of magnitude, the absence of any linear instability in Keplerian accretion flows is often used to rule out the possibility of turbulent viscosity. Recently, the fact that some fine tuned disturbances of any inviscid shear flow can reach arbitrarily large transient growth has been proposed as an alternative route to turbulence in these systems. We present an analytic study of this process for 3D plane wave disturbances of a general rotating shear flow in Lagrangian coordinates, and demonstrate that large transient growth is the generic feature of non- axisymmetric disturbances with near radial wave vectors. The scaling of the maximum energy growth is between linear and quadratic in time, with fastest growth occurring for two dimensional perturbations, and is only limited by viscosity, and ultimately by the disk vertical thickness. After including viscosity and vertical structure, we find that, as a function of the Reynolds number, R, the maximum energy growth is approximately 0.4 (R/log R)^{2/3}, and put forth a heuristic argument for why R > 10^4 is required to sustain turbulence in Keplerian disks. Therefore, while the astrophysical accretion disks must be well within the turbulent regime, large 3D numerical simulations running for many orbital times, and/or with fine tuned initial conditions are required to see Keplerian hydrodynamic turbulence.Keywords
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