Brunt–Väisälä Frequency
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In
atmospheric dynamics Meteorology is the scientific study of the Earth's atmosphere and short-term atmospheric phenomena (i.e. weather), with a focus on weather forecasting. It has applications in the military, aviation, energy production, transport, agriculture, ...
,
oceanography Oceanography (), also known as oceanology, sea science, ocean science, and marine science, is the scientific study of the ocean, including its physics, chemistry, biology, and geology. It is an Earth science, which covers a wide range of to ...
,
asteroseismology Asteroseismology is the study of oscillations in stars. Stars have many Resonance, resonant modes and frequencies, and the path of sound waves passing through a star depends on the local speed of sound, which in turn depends on local temperature a ...
and
geophysics Geophysics () is a subject of natural science concerned with the physical processes and Physical property, properties of Earth and its surrounding space environment, and the use of quantitative methods for their analysis. Geophysicists conduct i ...
, the Brunt–Väisälä frequency, or buoyancy frequency, is a measure of the stability of a fluid to vertical displacements such as those caused by
convection Convection is single or Multiphase flow, multiphase fluid flow that occurs Spontaneous process, spontaneously through the combined effects of material property heterogeneity and body forces on a fluid, most commonly density and gravity (see buoy ...
. More precisely it is the frequency at which a vertically displaced parcel will oscillate within a statically stable environment. It is named after David Brunt and
Vilho Väisälä Vilho Väisälä (; September 28, 1889 – August 12, 1969) was a Finnish meteorologist and physicist, and founder of Vaisala Oyj. After graduation in mathematics in 1912, Väisälä worked for the Finnish Meteorological Institute in ''ae ...
. 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 \rho_0. This parcel is in an environment of other water or gas particles where the density of the environment is a function of height: \rho = \rho (z). If the parcel is displaced by a small vertical increment z', ''and it maintains its original density so that its volume does not change,'' it will be subject to an extra gravitational force against its surroundings of: \rho_0 \frac = - g \left rho (z)-\rho (z+z')\right/math> where g is the gravitational acceleration, and is defined to be positive. We make a
linear approximation In mathematics, a linear approximation is an approximation of a general function (mathematics), function using a linear function (more precisely, an affine function). They are widely used in the method of finite differences to produce first order ...
to \rho (z+z') - \rho (z) = \frac z', and move \rho_0 to the RHS: \frac = \frac \frac z' The above second-order differential equation has the following solution: z' = z'_0 e^ where the Brunt–Väisälä frequency N is: N = \sqrt For negative \frac, the displacement 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 astrophysics

For a gas parcel, the density will only remain fixed as assumed in the previous derivation if the pressure, P, is constant with height, which is not true in an atmosphere confined by gravity. Instead, the parcel will expand adiabatically as the pressure declines. Therefore, a more general formulation used in meteorology is: N \equiv \sqrt, where \theta is potential temperature, g is the local acceleration of
gravity In physics, gravity (), also known as gravitation or a gravitational interaction, is a fundamental interaction, a mutual attraction between all massive particles. On Earth, gravity takes a slightly different meaning: the observed force b ...
, and z is geometric height. Since \theta = T (P_0/P)^, where P_0 is a constant reference pressure, for a perfect gas this expression is equivalent to: N^2 \equiv g\left(\frac\frac - \frac\frac\frac\right) = g \left(\frac\frac - \frac\frac\frac\right), where in the last form \gamma = c_P/c_V, the
adiabatic index In thermal physics and thermodynamics, the heat capacity ratio, also known as the adiabatic index, the ratio of specific heats, or Laplace's coefficient, is the ratio of the heat capacity at constant pressure () to heat capacity at constant volu ...
. Using the
ideal gas law The ideal gas law, also called the general gas equation, is the equation of state of a hypothetical ideal gas. It is a good approximation of the behavior of many gases under many conditions, although it has several limitations. It was first stat ...
, we can eliminate the temperature to express N^2 in terms of pressure and density: N^2 \equiv g\left(\frac\frac\frac - \frac\frac\right) = g\left(\frac\frac - \frac\right). This version is in fact more general than the first, as it applies when the chemical composition of the gas varies with height, and also for imperfect gases with variable adiabatic index, in which case \gamma \equiv \gamma_= \left(\frac\right)_, i.e. the derivative is taken at constant
entropy Entropy is a scientific concept, most commonly associated with states of disorder, randomness, or uncertainty. The term and the concept are used in diverse fields, from classical thermodynamics, where it was first recognized, to the micros ...
, S. If a gas parcel is pushed up and N^2>0, the air parcel will move up and down around the height where the density of the parcel matches the density of the surrounding air. If the air parcel is pushed up and N^2=0, the air parcel will not move any further. If the air parcel is pushed up and N^2<0, (i.e. the Brunt–Väisälä frequency is imaginary), then the air parcel will rise and rise unless N^2 becomes positive or zero again further up in the atmosphere. In practice this leads to convection, and hence the
Schwarzschild criterion In astrophysics, the Schwarzschild criterion indicates when a stellar medium is stable against convection when the rate of change in temperature'','' T, by altitude'','' z, satisfies : -\frac < \frac where g is
Ledoux criterion if there is compositional stratification) is equivalent to the statement that N^2 should be positive. The Brunt–Väisälä frequency commonly appears in the thermodynamic equations for the atmosphere and in the structure of stars.


In oceanography

In the ocean where
salinity Salinity () is the saltiness or amount of salt (chemistry), salt dissolved in a body of water, called saline water (see also soil salinity). It is usually measured in g/L or g/kg (grams of salt per liter/kilogram of water; the latter is dimensio ...
is important, or in fresh water lakes near freezing, where density is not a linear function of temperature:N\equiv \sqrtwhere \rho, the potential density, depends on both temperature and salinity. An example of Brunt–Väisälä oscillation in a density stratified liquid can be observed in the 'Magic Cork' movi
here
.


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 - i.e. the density can be said to have multiple vertical layers). 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, and the
angular frequency In physics, angular frequency (symbol ''ω''), also called angular speed and angular rate, is a scalar measure of the angle rate (the angle per unit time) or the temporal rate of change of the phase argument of a sinusoidal waveform or sine ...
of oscillation is given . If the acceleration is away from the initial position (), the stratification is unstable. In this case, overturning or convection generally ensues. The Brunt–Väisälä frequency relates to
internal gravity waves Internal waves are gravity waves that oscillate within a fluid medium, rather than on its surface. To exist, the fluid must be stratified: the density must change (continuously or discontinuously) with depth/height due to changes, for example, in ...
: it is the frequency when the waves propagate horizontally; and it provides a useful description of atmospheric and oceanic stability.


See also

*
Buoyancy Buoyancy (), or upthrust, is the force exerted by a fluid opposing the weight of a partially or fully immersed object (which may be also be a parcel of fluid). In a column of fluid, pressure increases with depth as a result of the weight of t ...
*
Bénard cell Benard or Bénard is a surname or given name. Notable people with the name include: Surname * Abraham-Joseph Bénard (1750–1822), French actor of the Comédie-Française * Aimé Bénard (1873–1938), Canadian politician * André Bénard (192 ...


References

{{DEFAULTSORT:Brunt-Vaisala Frequency Atmospheric thermodynamics Atmospheric dynamics Fluid dynamics Oceanography Buoyancy