On hydrodynamic shear turbulence in stratified Keplerian disks: Transient growth of small-scale 3D vortex mode perturbations

Abstract

Abstract. This is a sequel to Paper I (Chagelishvili et al. 2003), where we presented the so-called bypass concept for the onset of turbulence in shearing flows. According to this concept, which was worked out during the last decade by the hydrodynamic community for spectrally stable flows, vortical perturbations undergo transient growth by extracting energy from the shear (a linear process), thereby reaching an amplitude which is sufficient to allow for non-linear interactions which, by positive feedback, sustain turbulence. In Paper I we described this transient growth for 2D perturbations in a Keplerian disk; we showed that their kinematics was the same as in plane-parallel flow, and thus that they were not modified by the presence of the Coriolis force. In the present paper, we pursue our goal of applying the bypass scenario to astrophysical disks: we investigate the linear dynamics of 3D small-scale vortical perturbations for single spatial harmonics, in stably stratified, differentially rotating disks, again in the framework of a nonmodal analysis. We find that these 3D perturbations also undergo substantial transient growth, and that they reach a peak amplitude that is comparable to that of 2D perturbations, as long as their vertical scale remains of the order of the azimuthal scale. When the vertical wave-number exceeds the azimuthal one, the amplification rate is reduced, but this may be more than compensated to by the huge Reynolds number and the high shear rate characterizing astrophysical Keplerian disks. Whereas in 2D the Coriolis force had no impact on the transient growth, in 3D this force somewhat constricts the characteristics of the perturbation dynamics in disk flows, and the initial transient growth is followed by some reduction in amplitude. These differences are quantitative, rather than of fundamental character. But the 3D case presents two interesting novelties. In plane parallel flow, the perturbations do not decay after their transient amplification, but their energy stays on a plateau before being dissipated through viscous friction. More importantly, especially for the astrophysicist, in disk flow the 3D vortex mode perturbations excite density-spiral waves, whose energy also settles on a plateau before viscous dissipation. These local vortex mode perturbations fit naturally into the bypass concept of hydrodynamic shear turbulence, which was first developed for plane-parallel flows. We submit that these perturbations will also play an important role in the onset and in the maintenance of turbulence in Keplerian disks.

Development of the theory of instabilities of differentially rotating plasma with astrophysical applications

“…In fact, overall energetic dynamics is similar to that occurred in the plane parallel constant shear flow (see Fig. 2 of Tevzadze et al (2003)). So, these investigations strongly suggest that the linear dynamics in vertically stratified 3D hydrodynamic Keplerian disks matches requirements of the bypass concept developed for the plane-parallel flows.…”

“…Described above linear transient growth is also at work in rotating hydrodynamic disk flows; however, the Coriolis force causes a quantitative reduction of the growth rate there which delays the onset of turbulence. Keplerian flows are therefore expected to become turbulent for Reynolds numbers a few orders of magnitude higher than for plane subcritical flows (see: Longaretti (2002), Tevzadze et al (2003)). The possibility of an alternate route to turbulence gave new impetus to the research on the dynamics of astrophysical disks (Lominadze et al (1988), Richard & Zahn 1999, Richard (2001, Ioannou & Kakouris (2001), Tagger (2001), Longaretti (2002), Chagelishvili et al (2003), Tevzadze et al (2003), Klahr & Bodenheimer (2003), Yecko (2004, Afshordi et al (2004, Umurhan & Regev (2004), Umurhan & Shaviv (2005), Klahr (2004Klahr ( ) 2004Bodo et al (2005), Mukhopadhyay et al 322 J. G. Lominadze (2005), Barraco & Marcus (2005), Johnson & Gammie (2005), Umurhan (2006)).…”

“…However, it appears that the combined action of differential rotation and stratification introduces a new degree of freedom that may influence the flow stability and lead to turbulence at a high enough Reynolds number. The study of the linear perturbations in strato-rotational flow in the local limit can be found in Tevzadze et al (2003) and Tevzadze et al (2008) where it is shown the following:…”

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).…”

A note on transition, turbulent length scales and transport in differentially rotating flows

“…Chagelishvili et al (2003) (and sequel Tevzadze et al 2003) proposed the mechanism of transient growth and bypass mechanism (used for many years in the aerodynamics community) to provide the required amplitudes. Even in a linearly stable flow, the linear operator can provide transient amplification of perturbations.…”