Brunt–Väisälä frequency

Remarks

A parcel moved from height $z$ to $z + \zeta$ without mixing keeps its density $\bar\rho(z)$, while its new surroundings have density $\bar\rho(z + \zeta)$. It feels the buoyancy force per unit mass $-g \big( \bar\rho(z) - \bar\rho(z + \zeta) \big) / \rho_0 \approx (g / \rho_0) \bar\rho'(z) \zeta = -N^2 \zeta$. Hence $\ddot\zeta = -N^2 \zeta$, the equation of a harmonic oscillator of angular frequency $N$ when $N^2 > 0$. If $N^2 < 0$, the displacement grows exponentially and the fluid is statically unstable.

Example: a linear profile $\bar\rho(z) = \rho_0 (1 - \gamma z)$ with $\gamma > 0$ gives the constant $N^2 = g \gamma$; for $\gamma = 10^{-5}$ m$^{-1}$, $N \approx 10^{-2}$ s$^{-1}$, a period $2 \pi / N$ of about $10$ minutes.

Non-example: a fluid of uniform density has $N = 0$; a displaced parcel feels no restoring force and does not oscillate.