Radial buoyancy frequency ($N_r^2$)

The radial buoyancy frequency describes restoration of a radially displaced parcel in a stratified astrophysical disk. With a dimensionless specific entropy $S$, its square is proportional to $-{\rho }^{-1}(dP/dr)(dS/dr)$. Negative $N_r^2$ means an adverse entropy stratification, but epicyclic motion can still stabilize the parcel. For smooth thin disk profiles, $|N_r^2|$ is usually of order $(H/r{)}^2{\Omega }^2$.

For ideal axisymmetric radial displacements in a Keplerian shearing sheet, the hydrodynamic squared oscillation frequency is $N_r^2+{\kappa }^2$, with radial epicyclic frequency $\kappa =\Omega$. Exponential instability occurs when $N_r^2+{\kappa }^2<0$. This is the radial special case of rotating-fluid convective stability, rather than the complete set of three-dimensional Solberg–Høiland criteria.