Bypass to turbulence in hydrodynamic accretion: Lagrangian analysis of energy growth

Despite observational evidence for cold neutral astrophysical accretion disks, the viscous process that 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 10 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 three-dimensional plane wave disturbances of a general rotating shear flow in Lagrangian coordinates and demonstrate that large transient growth is a generic feature of nonaxisymmetric disturbances with near radial leading wavevectors. The maximum energy growth is slower than quadratic but faster than linear in time. The fastest growth occurs 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 greater than or similar to 10(4) is required to sustain turbulence in Keplerian disks. Therefore, assuming that there exists a nonlinear feedback process to replenish the seeds for transient growth, astrophysical accretion disks must be well within the turbulent regime. However, large three-dimensional numerical simulations running for many orbital times, and/or with fine-tuned initial conditions, are required to confirm Keplerian hydrodynamic turbulence on the computer.