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For a dust layer of extent $L$, its smallest admissible wavenumber is of order $1/L$. The secular gravitational instability band ends at $k_{\mathrm{crit}}=2\pi G{\sigma }_0/c^2=2/(QH)$, with Toomre parameter $Q=c\Omega /(\pi G{\sigma }_0)$ and $H=c/\Omega$. Requiring an admissible mode in the band gives $Q\lesssim L/H$. Boundary conditions fix the order-unity prefactor; a finite disk lifetime also limits practical growth.

The secular gravitational instability needs $|k|<k_{\mathrm{crit}}=2\pi G{\sigma }_0/c^2$. A finite radial extent $L$ removes arbitrarily small wavenumbers; the smallest available one is $k_{\min }\sim \alpha /L$, where $\alpha$ is a boundary-dependent constant of order unity. With $H=c/\Omega$ and the dust Toomre parameter,Thus the finite-size secular gravitational criterion is $\boxed{Q\lesssim L/H}$ at the rough accuracy requested. For example, a periodic radial interval has $k_{\min }=2\pi /L$ and gives $Q<L/(\pi H)$ in this idealized model. The numerical prefactor is not universal, but the scaling is. Finite size therefore restores a practical threshold for the otherwise long-wavelength instability. Growth must additionally occur within the disk lifetime, and a global wavelength comparable to disk radius lies beyond the strictly local shearing sheet approximation.

Turbulence usually makes the onset of dust gravitational instability of an astrophysical disk harder in three complementary ways. It increases the dust random velocity dispersion, raising $c$ and the Toomre parameter; it stirs dust vertically, reducing the self-gravity enhancement of a razor-thin disk; and it mixes density enhancements through turbulent diffusion. The last effect is particularly important for a slowly growing secular gravitational instability, which can be erased before it amplifies appreciably.

A simple diffusion model adds $D{\nabla }^2\sigma$ to the dust continuity equation, giving a damping scale of order $D{k}^2$. Growth then has to compete with mixing as well as pressure and rotational support. In the long-wavelength weak-drag regime where ${\omega }^2\simeq {\Omega }^2$, the indicative competition iswith further pressure and finite-thickness corrections. This is a model-dependent long-wave estimate, not a substitution into the original cubic without changing its continuity equation. In an infinite domain, diffusion proportional to $k^2$ need not remove growth proportional to $|k|$ at every arbitrarily long wavelength; in a finite disk, the remaining wavelengths and their growth times may be inadequate. The conclusion is therefore a higher practical collapse threshold and slower growth, rather than guaranteed stability for all turbulent flows.

Turbulence can also concentrate dust through coherent structures or pressure maxima, increasing local surface density and encouraging collapse. Its net quantitative effect requires a model for stirring, diffusion, thickness and concentration. The deterministic fixed gas flow used in the preceding parts does not specify those statistics. This competition is turbulent mixing of a secularly unstable dust layer.

The mode originating at the neutral root grows for $0<{\omega }^2<{\Omega }^2$. At small nonzero $|k|$ this gives $s_{\sec }\sim 2\pi ϵG{\sigma }_0|k|/\Omega$. Gas drag transfers angular momentum to the fixed gas flow and permits secular gravitational instability even for a dust Toomre parameter above unity. If ${\omega }^2<0$, the small-root continuation is damped, while a separate dynamical root grows. The formula is singular at ${\omega }^2=0$.