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.
References
See also Barometric formula Climate model Euler equations Fluid dynamics General circulation model Numerical weather prediction Primitive equations