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Molecular gas disks are generally Toomre stable ($Q_T>$1) and yet clearlygravitationally unstable to structure formation as evidenced by the existenceof molecular clouds and ongoing star formation. This paper adopts a 3Dperspective to obtain a general picture of instabilities in flattened rotatingdisks, using the 3D dispersion relation to describe how disks evolve whenperturbed over their vertical extents. By explicitly adding a verticalperturbation to an unperturbed equilibrium disk, stability is shown to varywith height above the mid-plane. Near to $z$=0 where the equilibrium density isroughly constant, instability takes on a Jeans-like quality, occurring onscales larger than the Jeans length and subject to a threshold$Q_M=\kappa^2/(4\pi G\rho)=1$ or roughly $Q_T\approx 2$. Far from themid-plane, on the other hand, stability is pervasive, and the threshold for thetotal disk (out to $z=\pm\infty$) to be stabilized is lowered to $Q_T=1$ as aconsequence. In this new framework, gas disks are able to fragment throughpartial 3D instability even where total 2D instability is suppressed. Thegrowth rates of the fragments formed via 3D instability are comparable to, orfaster than, Toomre instabilities. The rich structure in molecular disks on thescale of 10s of pc can thus be viewed as a natural consequence of their 3Dnature and their exposure to a variety of vertical perturbations acting onroughly a disk scale height, i.e. due to their situation within the moreextended galaxy potential, participation in the disk-halo flow, and exposure tostar formation feedback.

Molecular Clouds as Gravitational Instabilities in Rotating Disks: A Modified Stability Criterion