# Brunt–Väisälä frequency

(Redirected from Brunt-Vaisala frequency)

In atmospheric dynamics, oceanography, asteroseismology and geophysics, the Brunt–Väisälä frequency, or buoyancy frequency, is the angular 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 of ${\displaystyle \rho _{0}}$ and the environment with a density that is a function of height: ${\displaystyle \rho =\rho (z)}$. If the parcel is displaced by a small vertical increment ${\displaystyle z'}$, it will be subject to an extra gravitational force against its surroundings of:

${\displaystyle \rho _{0}{\frac {\partial ^{2}z'}{\partial t^{2}}}=-g(\rho (z)-\rho (z+z'))}$

${\displaystyle g}$ is the gravitational acceleration, and is defined to be positive. We make a linear approximation to ${\displaystyle \rho (z+z')-\rho (z)={\frac {\partial \rho (z)}{\partial z}}z'}$, and move ${\displaystyle \rho _{0}}$ to the RHS:

${\displaystyle {\frac {\partial ^{2}z'}{\partial t^{2}}}={\frac {g}{\rho _{0}}}{\frac {\partial \rho (z)}{\partial z}}z'}$

The above 2nd order differential equation has straightforward solutions of:

${\displaystyle z'=z'_{0}e^{{\sqrt {-N^{2}}}t}\!}$

where the Brunt–Väisälä frequency ${\displaystyle N}$ is:

${\displaystyle N={\sqrt {-{\frac {g}{\rho _{0}}}{\frac {\partial \rho (z)}{\partial z}}}}}$

For negative ${\displaystyle {\frac {\partial \rho (z)}{\partial z}}}$, ${\displaystyle z'}$ 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 oceanography

In the atmosphere,

${\displaystyle N\equiv {\sqrt {{\frac {g}{\theta }}{\frac {d\theta }{dz}}}}}$, where ${\displaystyle \theta }$ is potential temperature, ${\displaystyle g}$ is the local acceleration of gravity, and ${\displaystyle z}$ is geometric height.

In the ocean where salinity is important, or in fresh water lakes near freezing, where density is not a linear function of temperature,

${\displaystyle N\equiv {\sqrt {-{\frac {g}{\rho }}{\frac {d\rho }{dz}}}}}$, where ${\displaystyle \rho }$, the potential density, depends on both temperature and salinity.

## Context

The concept derives from Newton's Second Law when applied to a fluid parcel in the presence of a background stratification (in which the density changes in the vertical). The parcel, perturbed vertically from its starting position, experiences a vertical acceleration. If the acceleration is back towards the initial position, the stratification is said to be stable and the parcel oscillates vertically. In this case, N2 > 0 and the angular frequency of oscillation is given N. If the acceleration is away from the initial position (N2 < 0), the stratification is unstable. In this case, overturning or convection generally ensues.

The Brunt–Väisälä frequency relates to internal gravity waves and provides a useful description of atmospheric and oceanic stability.