The stability of accretion tori - I. Long-wavelength modes of slender tori

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, a, to orbital radius, r₀. In equilibrium it rotates on cylinders wilh angular velocity profile $$\Omega\propto r^{-q}$$. Linear modes are proportional to $$\text {exp} \enspace i(m\phi-\omega t)$$. For small $$\beta\equiv ma/r_0$$. we justify the use of height-averaged equations by appealing to approximate vertical hydrostatic equilibrium. The effective polytropic index for the resulting two-dimensional problem is $$N\equiv n+ \frac12$$; 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 three-dimensional system is small everywhere along the principal branch even for β∼0.5, and is less than 1 per cent for the fastest-growing mode. We analytically solve the artificial case $$n=-\frac12$$, which is two-dimensionally incompressible, and show that it has all the qualitative features of the general case, except that it does not have a resonance at coroiation. In the general case, with $$n\gt-\frac12$$ and $$q\lt 2$$, the corotation resonance absorbs energy and angular momentum, so the growing and decaying modes do not occur in complex–conjugate pairs. We solve a second special case, namely n=2−q=0, almost analytically in three dimensions, without height-averaging. 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 Kelvin–Helmholtz 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.