Negative-diffusion criterion for viscous disk instability (${\partial }_{\Sigma }(\bar{\nu }\Sigma )<0$)

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.