Bypass to Turbulence in Hydrodynamic Accretion Disks: An Eigenvalue Approach

Abstract

Cold accretion disks with temperatures below ∼ 3000K are likely to be composed of highly neutral gas. The magnetorotational instability may cease to operate in such disks, so it is of interest to consider purely hydrodynamic mechanisms of generating turbulence and angular momentum transport. With this motivation, we investigate the growth of hydrodynamic perturbations in a linear shear flow sandwiched between two parallel walls. The unperturbed flow is similar to plane Couette flow but with a Coriolis force included. Although there are no exponentially growing eigenmodes in this system, nevertheless, because of the non-normal nature of the eigenmodes, it is possible to have a large transient growth in the energy of perturbations. For a constant angular momentum disk, we find that the perturbation with maximum growth is axisymmetric with vertical structure. The energy grows by more than a factor of 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 numerical simulations. For a Keplerian disk, on the other hand, similar perturbations with vertical structure grow by no more than a factor of 4, explaining why the same simulations did not find turbulence in this system. However, certain other two-dimensional perturbations with no vertical structure do exhibit modest growth. For the optimum two-dimensional perturbation, the energy grows by a factor of ∼ 100 for R ∼ 10 4.5 and by a factor of 1000 for R ∼ 10 6 . Such large Reynolds numbers are hard to achieve in numerical simulations and so the nonlinear development of these kinds of perturbations are only beginning to be investigated. It is conceivable that these nearly two-dimensional disturbances might lead to self-sustained three-dimensional turbulence, though this remains 1 bmukhopa@cfa.harvard.edu 2 nafshord@cfa.harvard.edu 3 rnarayan@cfa.harvard.edu -2to be demonstrated. The Reynolds numbers of cold astrophysical disks are much larger even than 10 6 ; therefore, hydrodynamic turbulence may be possible in disks through transient growth.

Angular momentum transport and turbulence in laboratory models of Keplerian flows

“…We now turn to our arguments as to why finite-size effects may not dominate the flows that we have examined, with the caveat that we cannot rule out the possibility that secondary flows produced by end-wall effects are one of the different mechanisms that may perturb our quasi-Keplerian flows into a turbulent state (see e.g. Mukhopadhyay et al 2005;Mukhopadhyay & Saha 2011, and references therein).…”

A Pure Hydrodynamic Instability in Shear Flows and Its Application to Astrophysical Accretion Disks

“…Here, any length is expressed in units of the size L of the system in the x-direction, the time in units of W -1 , the velocity in W q L (  < q 1 2 ), and other variables are expressed accordingly (see, e.g., Mukhopadhyay et al 2005Mukhopadhyay et al , 2011, for detailed description of the choice of coordinates in a small section). Hence, in dimensionless units, the linearized Navier-Stokes equation and continuity equation (for an incompressible flow) can be recast into the well-known Orr-Sommerfeld and Squire equations, but in the presence of stochastic noise and Coriolis force , given by…”

“…It is a long-standing controversy (see, e.g., Dauchot & Daviaud 1995;Richard & Zahn 1999;Gu et al 2000;Kim & Ostriker 2000;Rudiger & Zhang 2001;Klahr & Bodenheimer 2003;Yecko 2004;Afshordi et al 2005;Dubrulle et al 2005aDubrulle et al , 2005bMukhopadhyay et al 2005Mukhopadhyay et al , 2011Mahajan & Krishan 2008; whether the matter in Rayleigh-stable astrophysical disks is stable or unstable. The answer has profound significance for our understanding of how stars and planets form.…”

“…Nevertheless, the recent experimental results by Paoletti et al (2012) clearly argued for a significant level of transport from hydrodynamics alone. Moreover, the results from direct numerical simulations (Avila 2012) and exploration of transient amplification, in otherwise linearly stable flows, with and without noise (e.g., Trefethen et al 1993;Mukhopadhyay et al 2005;Cantwell et al 2010), also argued for (plausible) hydrodynamic instability and turbulence at low Re. Interestingly, accretion disks have huge Re (10 15 ) (Mukhopadhyay 2013), prompting the belief that they are hydrodynamically unstable.…”

Stability of quasi-Keplerian shear flow in a laboratory experiment

“…The transition has been assumed to be subcritical due to similarities with pipe flow transition, though Schultz-Grunow (1959) stated that the transition appeared to be similar to a Kelvin-Helmholtz instability. No first-principles theory exists for subcritical transition in a rotating shear flow, though recent efforts to identify such a transition mechanism can be found in Chagelishvili et al (2003); Tevzadze et al (2003); Yecko (2004); Umurhan & Regev (2004); Mukhopadhyay et al (2005); Afshordi et al (2005); Lithwick (2007); Rincon et al (2007); Lithwick (2009) ;; Mukhopadhyay & Saha (2011). Phenomenological models have been developed based on the assumption that the observed transition is subcritical in nature (Zeldovich 1981;Richard & Zahn 1999;Longaretti 2002;Dubrulle et al 2005).…”