Brunt–Väisälä frequency

In atmospheric dynamics, oceanography, 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 subject to an extra gravitational force against its surroundings of:

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

g is the gravitational acceleration, and is defined to be positive. We make a Newton's method approximation to \(\rho (z) - \rho_0 = \frac{\partial \rho (z)}{\partial z} z'\), 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 frequency of oscillation is given by 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 cellde:Brunt-Väisälä-Frequenz