Exner function

The Exner function is a parameter used in atmospheric modeling. Depending on the application, the Exner function may be defined as Π = c p ( p p 0 ) R d / c p = c p T θ {\displaystyle \Pi =c_{p}\left({\frac {p}{p_{0}}}\right)^{R_{d}/c_{p}}=c_{p}{\frac {T}{\theta }}} or as a non-dimensional form Π = ( p p 0 ) R d / c p = T θ {\displaystyle \Pi =\left({\frac {p}{p_{0}}}\right)^{R_{d}/c_{p}}={\frac {T}{\theta }}} where p 0 {\displaystyle p_{0}} is a standard reference surface pressure (usually taken as 1000 hPa, but sometimes as the surface pressure); R d {\displaystyle R_{d}} is the specific gas constant for dry air; c p {\displaystyle c_{p}} is the specific heat capacity of dry air at constant pressure; T {\displaystyle T} is the absolute temperature; and θ {\displaystyle \theta } is the potential temperature. The non-dimensional form can be used as a vertical coordinate in some numerical weather prediction applications, resulting in a simpler mathematical formulation compared to using pressure as the vertical coordinate. It is named after Felix Maria von Exner-Ewarten.

Parameter in atmospheric modeling

The Exner function is a parameter used in atmospheric modeling. Depending on the application, the Exner function may be defined as

Π = c p ( p p 0 ) R d / c p = c p T θ {\displaystyle \Pi =c_{p}\left({\frac {p}{p_{0}}}\right)^{R_{d}/c_{p}}=c_{p}{\frac {T}{\theta }}}

or as a non-dimensional form

Π = ( p p 0 ) R d / c p = T θ {\displaystyle \Pi =\left({\frac {p}{p_{0}}}\right)^{R_{d}/c_{p}}={\frac {T}{\theta }}}

where p 0 {\displaystyle p_{0}} is a standard reference surface pressure (usually taken as 1000 hPa, but sometimes as the surface pressure); R d {\displaystyle R_{d}} is the specific gas constant for dry air; c p {\displaystyle c_{p}} is the specific heat capacity of dry air at constant pressure; T {\displaystyle T} is the absolute temperature; and θ {\displaystyle \theta } is the potential temperature.^[1][2][3] The non-dimensional form can be used as a vertical coordinate in some numerical weather prediction applications, resulting in a simpler mathematical formulation compared to using pressure as the vertical coordinate.^[2] It is named after Felix Maria von Exner-Ewarten.^[4]

References

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1. ^ Holton, James R. (2004). An Introduction to Dynamic Meteorology. Burlington, MA: Elsevier Academic Press. p. 109. ISBN 978-0-12-354015-7. Retrieved 2025-11-18.
2. ^ ^{a b} Bluestein, Howard B. (2013). Severe Convective Storms and Tornadoes. London: Springer-Verlag. pp. 60–61. ISBN 978-3-642-05381-8. Retrieved 2025-11-18.
3. ^ "CF Standard Name Table". CF Metadata Conventions. 2025-07-24. Retrieved 2025-11-18. The term "Exner function" is applied to various quantities in the literature. "Dimensionless Exner function" is the standard name of (p/p0)^(R/Cp), where p is pressure, p0 a reference pressure, R the gas constant and Cp the specific heat at constant pressure. This quantity is also the ratio of in-situ to potential temperature.
4. ^ Koertge, Noretta (2008). "EXNER-EWARTEN, FELIX MARIA VON" (PDF). New Dictionary of Scientific Biography. Retrieved 2025-11-18.

See also

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• Barometric formula
• Climate model
• Euler equations
• Fluid dynamics
• General circulation model
• Numerical weather prediction
• Primitive equations

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