Gravito-turbulent self-regulation

In a cooling self-gravitating disk, $Q<1$ excites gravitational turbulence whose dissipation raises the sound speed, while $Q>1$ suppresses that heating and lets cooling lower the sound speed. A sustained state can therefore regulate itself near $Q\simeq 1$.

A mixing-length estimate for gravitational turbulence takes the largest unstable wavelength $\ell$ and orbital growth rate $\Omega$, giving turbulent velocity $u_{\mathrm{turb}}\sim \Omega \ell$ and kinematic viscosity $\nu \sim u_{\mathrm{turb}}\ell \sim \Omega {\ell }^2$.