# Brunt–Väisälä frequency

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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.

## Derivation for a general fluid

Consider a parcel of (water or gas) that has density of $\rho _{0}$ and the environment with a density that is a function of height: $\rho =\rho (z)$ . If the parcel is displaced by a small vertical increment $z'$ , it will be subject to an extra gravitational force against its surroundings of:

$\rho _{0}{\frac {\partial ^{2}z'}{\partial t^{2}}}=-g(\rho (z)-\rho (z+z'))$ ${\frac {\partial ^{2}z'}{\partial t^{2}}}={\frac {g}{\rho _{0}}}{\frac {\partial \rho (z)}{\partial z}}z'$ The above 2nd order differential equation has straightforward solutions of:

$z'=z'_{0}e^{{\sqrt {-N^{2}}}t}\!$ where the Brunt–Väisälä frequency N is:

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

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

$N\equiv {\sqrt {-{\frac {g}{\rho }}{\frac {d\rho }{dz}}}}$ , where $\rho$ , 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.