A study of transient dynamics of perturbations in Keplerian discs using a variational approach

Abstract

We study linear transient dynamics in a thin Keplerian disc employing a method based on variational formulation of optimisation problem. It is shown that in a shearing sheet approximation due to a prominent excitation of density waves by vortices the most rapidly growing shearing harmonic has azimuthal wavelength, ${\lambda }_y$, of order of the disc thickness, $H$, and its initial shape is always nearly identical to a vortex having the same potential vorticity. Also, in the limit ${\lambda }_y\gg H$ the optimal growth $G\propto (\Omega /\kappa {)}^4$, where $\Omega$ and $\kappa$ stand for local rotational and epicyclic frequencies, respectively, what suggests that transient growth of large scale vortices can be much stronger in areas with non-Keplerian rotation, e.g. in the inner parts of relativistic discs around the black holes. We estimate that if disc is already in a turbulent state with effective viscosity given by the Shakura parameter $\alpha <1$, the considered large scale vortices with wavelengths $H/\alpha >{\lambda }_y>H$ have the most favourable conditions to be transiently amplified before they are damped. At the same time, turbulence is a natural source of the potential vorticity for this transient activity. We extend our study to a global spatial scale showing that global perturbations with azimuthal wavelengths more than an order of magnitude greater than the disc thickness still are able to attain the growth of dozens of times in a few Keplerian periods at the inner boundary of disc.

arXiv comment21 pages, 10 figures, accepted for publication in MNRAS

References in corpus (12)

• The excitation of spiral density waves through turbulent fluctuations in accretion discs II: Numerical Simulations with MRI driven turbulence
• Emergent Mesoscale Phenomena in Magnetized Accretion Disc Turbulence
• Hydrodynamic stability of rotationally supported flows: Linear and nonlinear 2D shearing box results
• Bypass to Turbulence in Hydrodynamic Accretion: Lagrangian Analysis of Energy Growth
• On hydrodynamic shear turbulence in Keplerian disks: via transient growth to bypass transition
• The excitation of spiral density waves through turbulent fluctuations in accretion discs I: WKBJ theory
• Bypass to Turbulence in Hydrodynamic Accretion Disks: An Eigenvalue Approach
• Turbulence in Global Simulations of Magnetized Thin Accretion Disks
• Spiral density wave generation by vortices in Keplerian flows
• Global transient dynamics of three-dimensional hydrodynamical disturbances in a thin viscous accretion disk
• Non-exponential hydrodynamical growth in density-stratified thin Keplerian discs
• An optimal transient growth of small perturbations in thin gaseous discs

Cited by in corpus (10)

• Dynamics of homogeneous shear turbulence: A key role of the nonlinear transverse cascade in the bypass concept
• Magnetorotational instability: nonmodal growth and the relationship of global modes to the shearing box
• Zero net flux MRI-turbulence in disks $-$ sustenance scheme and magnetic Prandtl number dependence
• Nonlinear transverse cascade and sustenance of MRI-turbulence in Keplerian disks with an azimuthal magnetic field
• Active modes and dynamical balances in MRI-turbulence of Keplerian disks with a net vertical magnetic field
• Transient dynamics of perturbations in astrophysical disks
• A viscous instability in axially symmetric laminar shear flows
• Transient growth of perturbations on scales beyond the accretion disc thickness
• Subcritical transition to turbulence in accretion disc boundary layer
• On dynamics of nonmagnetic accretion disks