Buoyancy Frequency

Forces on a fluid parcel raised by $z$ 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. Vallis (2006) 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 $z_{0}+z$ then the fluid will experience a net downward force due to it being heavier than the surrounding fluid.

By Archimedes' principle the buoyancy force is equal to the weight of the fluid displaced by the fluid parcel. Therefore, the net force at a depth of $z_{0}+z$ is

${\begin{aligned}F&=B-F_{g}\\&=g\rho (z_{0}+z)V-g\rho (z_{0})V\\&=gV\left(\rho (z_{0}+z)-\rho (z_{0})\right)\\&=gV\left(\rho (z_{0})+{\frac {d\rho (z_{0})}{dz_{0}}}z-\rho (z_{0})\right)\\&=gV{\frac {d\rho (z_{0})}{dz_{0}}}z\end{aligned}}$

By Newton's second law,

${\begin{aligned}\rho (z_{0})V{\frac {d^{2}z}{dt^{2}}}&=gV{\frac {d\rho (z_{0})}{dz_{0}}}z\\{\frac {d^{2}z}{dt^{2}}}&={\frac {g}{\rho (z_{0})}}{\frac {d\rho (z_{0})}{dz_{0}}}z\\{\frac {d^{2}z}{dt^{2}}}+N^{2}z&=0\\\end{aligned}}$

where $N^{2}(z)=-{\frac {g}{\rho (z)}}{\frac {d\rho (z)}{dz}}$. The negative sign is necessary to ensure that $N(z)$ is real for stable stratification.