Growing pseudo-eigenmodes and positive logarithmic norms in rotating shear flows

TL;DR

This paper demonstrates that rotating shear flows, including Keplerian flows, can exhibit growing pseudo-eigenmodes and positive logarithmic norms, indicating inherent instability that supports the possibility of hydrodynamic turbulence in astrophysical accretion disks.

Contribution

It introduces the concept of growing pseudo-eigenmodes and positive logarithmic norms in rotating shear flows, providing a new explanation for turbulence in Keplerian accretion disks.

Findings

Pseudo-eigenmodes exhibit large transient energy growth.

Logarithmic norms bound the growth rate, indicating feedback of instability.

Supports hydrodynamic turbulence in astrophysical disks.

Abstract

Rotating shear flows, when angular momentum increases and angular velocity decreases as functions of radiation coordinate, are hydrodynamically stable under linear perturbation. The Keplerian flow is an example of such systems which appears in astrophysical context. Although decaying eigenmodes exhibit large transient energy growth of perturbation which could govern nonlinearity into the system, the feedback of inherent instability to generate turbulence seems questionable. We show that such systems exhibiting growing pseudo-eigenmodes easily reach an upper bound of growth rate in terms of the logarithmic norm of the involved nonnormal operators, thus exhibiting feedback of inherent instability. This supports the existence of turbulence of hydrodynamic origin in the Keplerian accretion disc in astrophysics. Hence, this enlightens the mismatch between the linear theory and experimental/observed data and helps in resolving the outstanding question of origin of turbulence therein.