Brunt–Väisälä 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.

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:

$\rho_0 \frac{\partial^2 z'}{\partial t^2} = - g (\rho (z)-\rho (z+z'))$

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

$\frac{\partial^2 z'}{\partial t^2} = \frac{g}{\rho_0} \frac{\partial \rho (z)}{\partial z} z'$

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.

See also

• Buoyancy
• Bénard cell