Homologous vertical motion of an astrophysical disk

In homologous vertical motion, every height scales by one time-dependent thickness $H(t)$, so $z=H(t)\zeta$ and the vertical velocity is $u_z=\dot{H}\zeta$. A fixed dimensionless density profile then expands and contracts without changing shape.

A laminar column can contract through nearly hydrostatic equilibrium when its cooling time greatly exceeds its vertical dynamical timescale of a disk. For cooling $A{T}^{1+\beta }$ and fixed column mass, a similarity solution has $\rho \propto {\eta }^{-1}$, $T\propto {\eta }^2$, $P\propto \eta$, and vertical velocity $u_z=\dot{\eta }z/\eta$. The coefficient $C$ also depends on the initial mass density normalization.

The vertical breathing mode is a coherent oscillation of a disk's thickness. For an adiabatic gas with pressure proportional to $H^{1-\gamma }$ in a harmonic vertical potential, ${\omega }_H=\sqrt{\gamma +1}{\Omega }_z$.