Brunt–Väisälä Frequency

In atmospheric dynamics, oceanography, asteroseismology and geophysics, the Brunt–Väisälä frequency, or buoyancy frequency, is a measure of the stability of a fluid to vertical displacements such as those caused by convection. More precisely it is the frequency at which a vertically displaced parcel will oscillate within a statically stable environment. It is named after David Brunt and Vilho Väisälä. It can be used as a measure of atmospheric stratification.

Derivation for a general fluid

Consider a parcel of water or gas that has density $\rho_0$. This parcel is in an environment of other water or gas particles where the density of the environment is a function of height: $\rho = \rho (z)$. If the parcel is displaced by a small vertical increment $z'$, ''and it maintains its original density so that its volume does not change,'' it will be subject to an extra gravitational force against its surroundings of:

\rho_0 \frac = - g \left rho (z)-\rho (z+z')\right/math>

where $g$ is the gravitational acceleration, and is defined to be positive. We make a linear approximation to $\rho (z+z') - \rho (z) = \frac z'$, and move $\rho_0$ to the RHS:

$\frac = \frac \frac z'$

The above second-order differential equation has the following solution:

$z' = z'_0 e^$

where the Brunt–Väisälä frequency $N$ is:

$N = \sqrt$

For negative $\frac$, the displacement $z'$ has oscillating solutions (and N gives our angular frequency). If it is positive, then there is run away growth – i.e. the fluid is statically unstable.

In meteorology and astrophysics

For a gas parcel, the density will only remain fixed as assumed in the previous derivation if the pressure, $P$, is constant with height, which is not true in an atmosphere confined by gravity. Instead, the parcel will expand adiabatically as the pressure declines. Therefore, a more general formulation used in meteorology is:

$N \equiv \sqrt,$ where $\theta$ is potential temperature, $g$ is the local acceleration of gravity, and $z$ is geometric height.

Since $\theta = T (P_0/P)^$, where $P_0$ is a constant reference pressure, for a perfect gas this expression is equivalent to:
$N^2 \equiv g\left(\frac\frac - \frac\frac\frac\right) = g \left(\frac\frac - \frac\frac\frac\right),$
where in the last form $\gamma = c_P/c_V$, the adiabatic index. Using the ideal gas law, we can eliminate the temperature to express $N^2$ in terms of pressure and density:
$N^2 \equiv g\left(\frac\frac\frac - \frac\frac\right) = g\left(\frac\frac - \frac\right).$

This version is in fact more general than the first, as it applies when the chemical composition of the gas varies with height, and also for imperfect gases with variable adiabatic index, in which case $\gamma \equiv \gamma_= \left(\frac\right)_$, i.e. the derivative is taken at constant entropy, $S$.

If a gas parcel is pushed up and $N^2>0$, the air parcel will move up and down around the height where the density of the parcel matches the density of the surrounding air. If the air parcel is pushed up and $N^2=0$, the air parcel will not move any further. If the air parcel is pushed up and $N^2<0$, (i.e. the Brunt–Väisälä frequency is imaginary), then the air parcel will rise and rise unless $N^2$ becomes positive or zero again further up in the atmosphere. In practice this leads to convection, and hence the Schwarzschild criterion