Alpha disk ($\alpha$-disk)

The alpha-disk prescription writes turbulent kinematic viscosity as $\nu =\alpha c_{s}H\simeq \alpha c_s^2/\Omega$, with dimensionless $\alpha \lesssim 1$. It reflects eddies no larger than the disk thickness and no faster than the sound speed.

The Shakura--Sunyaev thin disk is a geometrically thin, optically thick, radiatively efficient accretion-disk model whose turbulent stress is parametrized by the alpha prescription.

The magnetorotational instability destabilizes a magnetized differentially rotating flow when angular velocity decreases outward. Magnetic tension couples neighboring fluid elements, allowing the inner one to lose angular momentum and fall inward while the outer one gains angular momentum and moves outward.

For a vertical magnetic field in a Keplerian disk, axisymmetric vertical MRI modes are unstable when $k^2{v}_A^2<3{\Omega }^2$. The shortest unstable wavelength is ${\lambda }_{\mathrm{crit}}=2\pi v_A/(\sqrt{3}\Omega )$.

A local thermal equilibrium is stable when a small temperature increase raises cooling faster than heating at fixed surface density. The standard total-pressure alpha prescription makes a radiation-pressure-dominated, Thomson-opacity disk thermally unstable.