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. 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. Consider a parcel of (water or gas) that has density of ρ 0 {displaystyle ho _{0}} and the environment with a density that is a function of height: ρ = ρ ( z ) {displaystyle ho = ho (z)} . If the parcel is displaced by a small vertical increment z ′ {displaystyle z'} , it will be subject to an extra gravitational force against its surroundings of: g {displaystyle g} is the gravitational acceleration, and is defined to be positive. We make a linear approximation to ρ ( z + z ′ ) − ρ ( z ) = ∂ ρ ( z ) ∂ z z ′ {displaystyle ho (z+z')- ho (z)={frac {partial ho (z)}{partial z}}z'} , and move ρ 0 {displaystyle ho _{0}} to the RHS: The above 2nd order differential equation has straightforward solutions of: where the Brunt–Väisälä frequency N {displaystyle N} is: For negative ∂ ρ ( z ) ∂ z {displaystyle {frac {partial ho (z)}{partial z}}} , z ′ {displaystyle 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.