Viscous instability of an accretion disk

The negative-diffusion criterion for viscous disk instability is ${\partial }_{\Sigma }(\bar{\nu }\Sigma )<0$. A surface density enhancement then transports angular momentum less effectively and grows. Freezing the Keplerian viscous diffusion equation locally gives growth proportional to the square of radial wavenumber, within the thin-disc transport range; wavelengths must still be sufficiently long compared with the disk scale height for that closure. The formal short-wave limit is a backward heat equation.

Freezing coefficients locally in the Keplerian viscous diffusion equation gives ${\Sigma }_{1,t}\simeq 3{\partial }_{\Sigma }(\bar{\nu }\Sigma ){\Sigma }_{1,rr}$. A radial Fourier mode grows at rate $-3{\partial }_{\Sigma }(\bar{\nu }\Sigma )k_r^2$. Negative response reverses ordinary smoothing and gives a backward heat equation, within the wavelength range where the thin disc transport closure applies.

The response to a small surface density perturbation is $\delta (\bar{\nu }\Sigma )=q\bar{\nu }\delta \Sigma$. For an instantaneous law $\bar{\nu }\propto r^a{\Sigma }^b$, $q=b+1$. This exponent is not the orbital shear exponent or a satellite-to-star mass ratio, which also often use the letter $q$.

In a characteristic vertical model of a radiation-pressure-dominated alpha disk with constant opacity, $\mu \sim \alpha p/\Omega \sim c/\kappa$ and $p\sim \Sigma {\Omega }^{2}H$ imply $H\propto {\Sigma }^{-1}$ and $\bar{\nu }\Sigma \propto {\Sigma }^{-1}$. The resulting $d(\bar{\nu }\Sigma )/d\Sigma <0$ gives a viscous instability of an accretion disk, distinct from a thermal instability of a radiation-pressure-dominated alpha disk.

In a local Keplerian sheet, slow viscous drift gives ${\partial }_t\Sigma =3{\partial }_x^2(\nu \Sigma )$. For $\nu =A{\Sigma }^2$, the compact profile $\Sigma =\sigma \sqrt{1-x^2/w^2}$ preserves its shape, conserves $\sigma w$, and spreads with $w\propto t^{1/4}$ at late times.