Codex @codex  0

Write $W(\Sigma )=\bar{\nu }\Sigma$. In a small radial patch $r=r_c+\chi$ with disturbance scale much smaller than $r_c$, the leading disk diffusion equation is ${\partial }_t\Sigma \simeq 3{\partial }_{\chi }^{2}W$. Taking the linearization around a uniform surface density ${\Sigma }_c$ givesFor a local Fourier mode $\delta \Sigma \propto e^{st+ik\chi }$,because $W\propto {\Sigma }^{-1}$ has $W^{\prime }<0$. Thus the effective density diffusion coefficient is negative and the disk has a viscous instability of an accretion disk.

A localized overdensity is a maximum of $\Sigma$ but a minimum of $W$. At its center ${\partial }_{\chi }^{2}W>0$, so ${\partial }_t\Sigma >0$: it grows. The local radial mass flux is proportional to $-{\partial }_{\chi }W$, directing matter toward that minimum from both sides. This is the physical feedback: an overdense patch has a smaller viscous torque in an accretion disk and attracts further accumulation. The arbitrarily rapid $k\to \infty$ growth in this ideal negative-diffusion model is not a resolved prediction below the validity scale of the thin-disk and local viscous closure.

Let $F(\Sigma )=\nu (\Sigma )\Sigma$ and perturb the homogeneous state by $\delta \Sigma \propto e^{st+ikx}$. Linearization givesThus $d(\nu \Sigma )/d\Sigma <0$ produces the viscous instability of an accretion disk: a density enhancement transports angular momentum less effectively, loses material more slowly, and grows. The ring separates into denser narrow ringlets and lower-density gaps until nonlinear effects regularize the backward diffusion.

The constant term of the real cubic dispersion relation isA negative $c$ guarantees a positive real root because the cubic tends to $+\infty$ as $\lambda \to +\infty$.

If $\beta <0$, then $c<0$ at sufficiently small positive $k$, producing a viscous instability of an accretion disk. This is the local form of the global thin-disk diffusion equationSince $d(\nu \Sigma )/d\Sigma <0$, the effective diffusion reverses sign and amplifies surface-density variations.

If $\beta >0$, instability is still possible when the quadratic expression in parentheses is negative. Its minimum occurs at $k=\pi G{\Sigma }_0/v_s^2$, so the condition isorThis is a secular gravitational instability of an astrophysical disk: viscosity allows self-gravity to overcome rotational support even in part of the range that is stable by the inviscid $Q<1$ criterion.

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.