Nonaxisymmetric stability in the shearing sheet approximation

¹ NORDITA, Blegdamsvej 17, 2100 Copenhagen Ø, Denmark e-mail: brandenb@nordita.dk
² Observatoire Midi-Pyrénées, CNRS et Université Toulouse 3, 14 avenue Edouard Belin, 31400 Toulouse, France

Received: 4 December 2005
Accepted: 6 January 2006

Abstract

Aims.To quantify the transient growth of nonaxisymmetric perturbations in unstratified magnetized and stratified non-magnetized rotating linear shear flows in the shearing sheet approximation of accretion disc flows.

Methods.The Rayleigh quotient in modal approaches for the linearized equations (with time-dependent wavenumber) and the amplitudes from direct shearing sheet simulations using a finite difference code are compared.

Results.Both approaches agree in their predicted growth behavior. The magneto-rotational instability for axisymmetric and non-axisymmetric perturbations is shown to have the same dependence of the (instantaneous) growth rate on the wavenumber along the magnetic field, but in the nonaxisymmetric case the growth is only transient. However, a meaningful dependence of the Rayleigh quotient on the radial wavenumber is obtained. While in the magnetized case the total amplification factor can be several orders of magnitude, it is only of order ten or less in the nonmagnetic case. Stratification is shown to have a stabilizing effect. In the present case of shearing-periodic boundaries the (local) strato-rotational instability seems to be absent.