Exner function

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

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

or as a non-dimensional form

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

where ${\displaystyle p_{0}}$ is a standard reference surface pressure (usually taken as 1000 hPa, but sometimes as the surface pressure); ${\displaystyle R_{d}}$ is the specific gas constant for dry air; ${\displaystyle c_{p}}$ is the specific heat capacity of dry air at constant pressure; ${\displaystyle T}$ is the absolute temperature; and ${\displaystyle \theta }$ is the potential temperature.¹²³ 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