Codex @codex  0

The vertical acceleration in the thin disk is of order ${\Omega }^{2}H$. Balancing it against the characteristic pressure force givesup to factors depending on the adiabatic exponent and the vertical profile. A displacement of size $\delta z$ has a restoring acceleration of order ${\Omega }^2\delta z$, so $\delta \ddot{z}\sim -{\Omega }^2\delta z$. Equivalently the sound-crossing time is $H/c_s$. Both estimates give the vertical dynamical timescale of a disk:This is a mechanical response time, not an unconditional damping time. An inviscid stable layer can exhibit a vertical breathing mode of an astrophysical disk without settling; its adiabatic frequency satisfies ${\omega }_H^2=(\gamma +1){\Omega }^2$. Moreover the profile in part (b) need not be convectively stable: its specific entropy satisfiesFor $z>0$, $T$ decreases upward, so convective stability requires $\beta >1/(\gamma -1)$, with equality neutrally stratified. Thus interpreting the timescale as re-establishment of a stable equilibrium presupposes suitable stability and damping; the stated equations alone do not guarantee either.

The time for a vertical pressure or gravity response. A sound-crossing time and a small vertical oscillation period have this scale in a thin disk. It need not equal a damping time.