Physics of modes in a differentially rotating system - analysis of the shearing sheet

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Abstract
We analyse the linear non-vortical modes of the shearing sheet, a model compressible two-dimensional fluid system with constant density, constant shear, and Coriolis force. This model has several features found in differentially rotating systems of interest in astrophysics, such as disc galaxies, accretion tori, planetary rings, protostellar nebulae, and possibly even rotating stars. The linear modal analysis of the shearing sheet leads to an eigenvalue problem based on the Parabolic Cylinder differential equation. A detailed analysis of the solutions is presented. From this we extract the following physical principles: (i) Each mode has an associated ‘corotation radius’, where the modal pattern is stationary as viewed from a frame moving with the fluid. A mode consists of wave-like disturbances in supersonic ‘permitted regions’ on either side of corotation, and exponential variation in a sonic ‘barrier region’ around corotation. (ii) We identify a particular conserved action that is positive for fluid on one side of corotation, and negative on the other side. Because of the change of sign of the action, a wave incident on the corotation barrier is reflected with increased amplitude. The strength of the resulting ‘corotation amplifier’ depends critically on the amplitude for wave penetration through the barrier, (iii) No instability is possible unless there is feedback introduced into the amplifier. In the shearing sheet, such feedback can take place only at the boundaries, (iv) Unstable modes invariably have equal amounts of positive and negative action, which requires that corotation must occur within the fluid. There are no such restrictions on neutral modes, (v) A semi-infinite shearing sheet has no neutral modes, only unstable modes that are characterized by a resonant ‘cavity’ between corotation and the edge of the sheet. Growth occurs because the action in the cavity has the opposite sign to that which leaks out to infinity through the corotation barrier, (vi) For a finite shearing sheet with two walls there are two cavities, two amplifiers, and two feedback loops. When tunnelling is small, most of the modes are neutral. Rare growing modes are produced whenever two stringent phase conditions are simultaneously met, i.e. when both cavities are resonant, (vii) When tunnelling is large, both neutral and unstable modes are common. Most often, one of the cavities takes charge of an unstable mode, and so the modes behave similarly to the semi-infinite system, (viii) When the equilibrium shearing sheet is perturbed slightly by introducing density and/or velocity perturbations, a new qualitative effect occurs in the form of a ‘corotation resonance’. Action is absorbed at corotation, and neutral modes are converted into growing or decaying modes. Numerical results are presented to illustrate several of the above features.

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