Hydrodynamic response of rotationally supported flows in the small shearing box model

¹ Department of Physics, Technion-Israel Institute of Technology, 32000 Haifa, Israel e-mail: phassaf@techunix.technion.ac.il
² Department of Geophysics and Space Sciences, Tel-Aviv University, Tel-Aviv, Israel
³ Department of Astronomy, City College of San Francisco, San Francisco, CA 94112, USA
⁴ Department of Astronomy, Columbia University, New York, NY 10025, USA

Received: 15 March 2007
Accepted: 1 April 2008

Abstract

We examine the hydrodynamic response of the inviscid small shearing box model of a midplane section of a rotationally supported astrophysical disk. We formulate an energy functional  for the general nonlinear problem. We find that the fate of disturbances is related to the conservation of this quantity which, in turn, depends on the boundary conditions utilized: is conserved for channel boundary conditions, while it is not conserved, in general, for shearing box conditions. Linearized disturbances subject to channel boundary conditions have normal-modes described by Bessel Functions and are qualitatively governed by a quantity Σ, which is a measure of the ratio between the azimuthal and vertical wavelengths. Inertial oscillations ensue if Σ > 1 – otherwise disturbances must be treated generally as an initial value problem. We reflect upon these results and offer a speculation.