Buoyancy Frequency

Forces on a fluid parcel raised by $\zeta$ from it's rest state at $z_{0}$.

Here we demonstrate that a fluid parcel oscillates around it's rest state at the buoyancy frequency. Credit to Vallis (2006) which has similar derivation to the following.

Consider a stable, motionless stratification, $\rho (z)$. A fluid parcel of volume $V$ at depth $z_{0}$ will therefore have a mass of $\rho (z_{0})V$. If that parcel of fluid is adiabatically lifted by a infinitesimal amount to a depth of $z_{0}+\zeta$ then the fluid will experience a net downward force due to it being heavier than the surrounding fluid.

By Archimedes' principle, the magnitude of the buoyancy force is equivalent to the weight of the fluid displaced by the fluid parcel. Ignoring all drag forces, the net force on the parcel at a depth of $z_{0}+\zeta$ is

${\begin{aligned}F&=B-F_{g}\\&=g\rho (z_{0}+\zeta )V-g\rho (z_{0})V\\&=gV\left(\rho (z_{0}+\zeta )-\rho (z_{0})\right)\\&=gV\left(\rho (z_{0})+\left.{\frac {d\rho (z)}{dz}}\right\vert _{z=z_{0}}\zeta +{\mathcal {O}}\left(\zeta ^{2}\right)-\rho (z_{0})\right)\\&=gV\left.{\frac {d\rho (z)}{dz}}\right\vert _{z=z_{0}}\zeta +{\mathcal {O}}\left(\zeta ^{2}\right)\end{aligned}}$

By assumption, $\zeta \ll 1$, so the higher order terms are negligible compared to the first. Therefore, Newton's second law is,

${\begin{aligned}\rho (z_{0})V{\frac {d^{2}\zeta }{dt^{2}}}&=gV\left.{\frac {d\rho (z)}{dz}}\right\vert _{z=z_{0}}\zeta \\{\frac {d^{2}\zeta }{dt^{2}}}&={\frac {g}{\rho (z_{0})}}\left.{\frac {d\rho (z)}{dz}}\right\vert _{z=z_{0}}\zeta \\{\frac {d^{2}\zeta }{dt^{2}}}+N^{2}\zeta &=0\\\end{aligned}}$

where $N^{2}(z)=-{\frac {g}{\rho (z)}}{\frac {d\rho (z)}{dz}}$. This is simply an undamped harmonic oscillator which vibrates at a frequency of $N$. The negative sign in the definition of $N^{2}$ is necessary to ensure that the buoyancy frequency is real for stable stratification. The buoyancy frequency is $N^{2}(z)=-{\frac {g}{\rho _{0}}}{\frac {d\rho (z)}{dz}}$ if the Boussinesq approximation is made.