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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 and the environment with a density that is a function of height: . If the parcel is displaced by a small vertical increment , it will subject to an extra gravitational force against its surroundings of:
g is the gravitational acceleration, and is defined to be positive. We make a Newton's method approximation to , and move to the RHS:
The above 2nd order differential equation has straightforward solutions of:
where the Brunt–Väisälä frequency N is:
For negative , 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,
In the ocean where salinity is important, or in fresh water lakes near freezing, where density is not a linear function of temperature,
- , where , the potential density, depends on both temperature and salinity.
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.