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. 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 $\rho _{0}$ 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, $N 2 > 0$ and the angular frequency of oscillation is given $N$ . If the acceleration is away from the initial position ( $N 2 < 0$ ), 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.

Brunt–Väisälä frequency

Derivation for a general fluid

In meteorology and oceanography

Context

See also