Goldreich, Peter and Goodman, Jeremy and Narayan, Ramesh (1986) The stability of accretion tori. I. Longwavelength modes of slender tori. Monthly Notices of the Royal Astronomical Society, 221 (2). pp. 339364. ISSN 00358711. http://resolver.caltech.edu/CaltechAUTHORS:20130313071925944

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Abstract
We elucidate the inviscid instabilities of an isentropic torus found previously by Papaloizou & Pringle. The torus is a polytrope of index, n, and has a small ratio of minor radius, ɑ, to orbital radius, r_0. In equilibrium it rotates on cylinders with angular velocity profile Ω,∝r^(q). Linear modes are proportional to exp i(mØwt). For small β≡mα/r_0 , we justify the use of heightaveraged equations by appealing to approximate vertical hydrostatic equilibrium. The effective polytropic index for the resulting twodimensional problem is N≡n+½; thus an incompressible torus in three dimensions behaves compressibly in two. Height averaged modes obey an ordinary differential equation, which we solve numerically to obtain the growth rate as a function of q, n, and β. The error made in predicting the growth rate of the actual threedimensional system is small everywhere along the principal branch even for β~0.5, and is less than 1 per cent for the fastestgrowing mode. We analytically solve the artificial case n=½, which is twodimensionally incompressible, and show that it has all the qualitative features of the general case, except that it does not have a resonance at corotation. In the general case, with n>½ and q<2, the corotation resonance absorbs energy and angular momentum, so the growing and decaying modes do not occur in complexconjugate pairs. We solve a second special case, namely n=2q=0, almost analytically in three dimensions, without heightaveraging. Papaloizou & Pringle asserted that this system is stable but we show that there is an unstable mode for small β just as in the other systems. In fact this principal unstable branch, with corotation at the pressure maximum, is qualitatively the same for all n and is essentially independent both of compressibility and of the gradient in vorticity per unit surface density. Thus the modes are not sonic, nor are they similar to those of the KelvinHelmholtz instability. Instead they are composed of two edge waves, akin to surface waves in water although modified by shear and rotation, coupled across a forbidden region around corotation.
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Additional Information:  © 1986 Royal Astronomical Society. Accepted 1986 February 5. Received 1986 February 5; in original form 1985 November 11. We thank Roger Blandford for encouragement and advice and John Papaloizou and Jim Pringle for sending us their papers far in advance of publication. We note that Omar Blaes and Wolfgang Glatzel have independently derived many of the results reported in this paper. Our research was supported in part by the NSF through grants AST8313725 and AST8213001.  
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Record Number:  CaltechAUTHORS:20130313071925944  
Persistent URL:  http://resolver.caltech.edu/CaltechAUTHORS:20130313071925944  
Official Citation:  The stability of accretion tori. I  Longwavelength modes of slender tori Goldreich, P.; Goodman, J.; Narayan, R. Monthly Notices of the Royal Astronomical Society vol. 221, July 15, 1986, p. 339364  
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ID Code:  37482  
Collection:  CaltechAUTHORS  
Deposited By:  Ruth Sustaita  
Deposited On:  13 Mar 2013 14:55  
Last Modified:  30 Aug 2013 18:45 
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