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} {(\frac{p}{p_{0}})}^{R_{d} / c_{p}} = c_{p} \frac{T}{θ}

or as a non-dimensional form

Π = {(\frac{p}{p_{0}})}^{R_{d} / c_{p}} = \frac{T}{θ}

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 $θ$ 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

^ Page Module:Citation/CS1/styles.css has no content.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 Page Module:Citation/CS1/styles.css has no content.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.
^ Page Module:Citation/CS1/styles.css has no content."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.
^ Page Module:Citation/CS1/styles.css has no content.Koertge, Noretta (2008). "EXNER-EWARTEN, FELIX MARIA VON" (PDF). New Dictionary of Scientific Biography. Retrieved 2025-11-18.