# Exner function

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The **Exner function** is a parameter used in [atmospheric modeling](/source/Atmospheric_model). 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](/source/HPa), but sometimes as the [surface pressure](/source/Atmospheric_pressure#Surface_pressure)); R_d is the [specific gas constant](/source/Specific_gas_constant) for dry air; c_p is the [specific heat capacity of dry air](/source/Table_of_specific_heat_capacities) at constant pressure; T is the [absolute temperature](/source/Absolute_temperature); and \theta is the [potential temperature](/source/Potential_temperature).[1][2][3] The non-dimensional form can be used as a vertical coordinate in some [numerical weather prediction](/source/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](/source/Felix_Maria_von_Exner-Ewarten).[4]

## References

1. Holton, James R. (2004). [*An Introduction to Dynamic Meteorology*](https://www.google.com/books/edition/An_Introduction_to_Dynamic_Meteorology/fhW5oDv3EPsC?hl=en&gbpv=1&dq=exner+function&pg=PA109&printsec=frontcover). Burlington, MA: [Elsevier Academic Press](/source/Elsevier_Academic_Press). p. 109. ISBN 978-0-12-354015-7. Retrieved 2025-11-18.

1. Bluestein, Howard B. (2013). [*Severe Convective Storms and Tornadoes*](https://www.google.com/books/edition/Severe_Convective_Storms_and_Tornadoes/0eU_AAAAQBAJ?hl=en&gbpv=1&dq=exner+function&pg=PA61&printsec=frontcover). London: [Springer-Verlag](/source/Springer-Verlag). pp. 60–61. ISBN 978-3-642-05381-8. Retrieved 2025-11-18.

1. ["CF Standard Name Table"](https://cfconventions.org/Data/cf-standard-names/current/build/cf-standard-name-table.html). *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.

1. Koertge, Noretta (2008). ["EXNER-EWARTEN, FELIX MARIA VON"](https://www.google.com/books/edition/New_Dictionary_of_Scientific_Biography_I/lO_bzgEACAAJ?hl=en). *New Dictionary of Scientific Biography*. Retrieved 2025-11-18.

## See also

- [Barometric formula](/source/Barometric_formula)
- [Climate model](/source/Climate_model)
- [Euler equations](/source/Euler_equations)
- [Fluid dynamics](/source/Fluid_dynamics)
- [General circulation model](/source/General_circulation_model)
- [Numerical weather prediction](/source/Numerical_weather_prediction)
- [Primitive equations](/source/Primitive_equations)

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Adapted from the Wikipedia article [Exner function](https://en.wikipedia.org/wiki/Exner_function) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Exner_function?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.