Exner function

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

\Pi =c_{p}*\left({\frac {p}{p_{0}}}\right)^{R_{d}/c_{p}}=c_{p}*{\frac {T}{\theta }}

or as a non-dimensional form

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

where $p_{0}$ is a standard reference surface pressure (usually taken as 1000 hPa, but sometimes as the surface pressure); $R_{d}$ is the specific gas constant for dry air; $c_{p}$ is the specific heat capacity of dry air at constant pressure; $T$ is the absolute temperature; and $\theta$ 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

^ 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.
^ ^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.
^ "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.
^ Koertge, Noretta (2008). "EXNER-EWARTEN, FELIX MARIA VON" (PDF). New Dictionary of Scientific Biography. Retrieved 2025-11-18.