Bypass to Turbulence in Hydrodynamic Accretion Disks: An Eigenvalue Approach

Cold accretion disks such as those in star-forming systems, quiescentcataclysmic variables, and some active galactic nuclei, are expected to haveneutral gas which does not couple well to magnetic fields. The turbulentviscosity in such disks must be hydrodynamic in origin, notmagnetohydrodynamic. We investigate the growth of hydrodynamic perturbations ina linear shear flow sandwiched between two parallel walls. The unperturbed flowis similar to plane Couette flow but with a Coriolis force included. Althoughthere are no exponentially growing eigenmodes in this system, nevertheless,because of the non-normal nature of the eigenmodes, it is possible to have alarge transient growth in the energy of perturbations. For a constant angularmomentum disk, we find that the perturbation with maximum growth has awave-vector in the vertical direction. The energy grows by more than a factorof 100 for a Reynolds number R=300 and more than a factor of 1000 for R=1000.Turbulence can be easily excited in such a disk, as found in previous numericalsimulations. For a Keplerian disk, on the other hand, similar verticalperturbations grow by no more than a factor of 4, explaining why the samesimulations did not find turbulence in this system. However, certain othertwo-dimensional perturbations with no vertical structure do exhibit modestgrowth. For the optimum two-dimensional perturbation, the energy grows by afactor of ~100 for R~10^4.5 and by a factor of 1000 for R~10^6. It isconceivable that these two-dimensional disturbances might lead toself-sustained turbulence. The Reynolds numbers of cold astrophysical disks aremuch larger even than 10^6, therefore, hydrodynamic turbulence may be possiblein disks.Comment: 39 pages including 9 figures; Final version to appear in The Astrophysical Journa