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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.