Bypass to Turbulence in Hydrodynamic Accretion: Lagrangian Analysis of Energy Growth

Despite observational evidence for cold neutral astrophysical accretiondisks, the viscous process which may drive the accretion in such systems is notyet understood. While molecular viscosity is too small to explain the observedaccretion efficiencies by more than ten orders of magnitude, the absence of anylinear instability in Keplerian accretion flows is often used to rule out thepossibility of turbulent viscosity. Recently, the fact that some fine tuneddisturbances of any inviscid shear flow can reach arbitrarily large transientgrowth has been proposed as an alternative route to turbulence in thesesystems. We present an analytic study of this process for 3D plane wavedisturbances of a general rotating shear flow in Lagrangian coordinates, anddemonstrate that large transient growth is the generic feature ofnon-axisymmetric disturbances with near radial leading wave vectors. Themaximum energy growth is slower than quadratic, but faster than linear in time.The fastest growth occurs for two dimensional perturbations, and is onlylimited by viscosity, and ultimately by the disk vertical thickness. After including viscosity and vertical structure, we find that, as a functionof 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 isrequired to sustain turbulence in Keplerian disks. Therefore, assuming thatthere exists a non-linear feedback process to replenish the seeds for transientgrowth, astrophysical accretion disks must be well within the turbulent regime.However, large 3D numerical simulations running for many orbital times, and/orwith fine tuned initial conditions, are required to confirm Keplerianhydrodynamic turbulence on the computer.Comment: 25 preprint pages, 2 figures, some modifications mainly to the Discussions section, Accepted for publication in Ap