Potential Temperature Calculator

Potential temperature theta is the temperature an air parcel would reach if compressed or expanded adiabatically to a reference pressure, conventionally 1000 hPa. It follows from the first law of thermodynamics applied to an ideal gas undergoing an adiabatic change, which yields the Poisson relation theta = T (P0/P)^(R/Cp). For dry air R is 287.05 J per kg per K and Cp is about 1004 J per kg per K, so the exponent kappa is close to 0.286, often written as the fraction 2 over 7 that kinetic theory predicts for a diatomic gas. Because theta is conserved whenever no heat is added and no condensation occurs, it labels an air parcel independently of the level at which it is observed. That conservation is what makes theta the working stability variable. If theta increases with height the layer is statically stable; a constant theta layer is neutral and typically well mixed by daytime convection; theta decreasing with height is absolutely unstable and overturns quickly. Tropospheric values typically run from about 290 K near the surface in mid latitudes to 340 K or more at the tropopause, where theta rises very steeply across the stable boundary. Virtual potential temperature, theta times (1 + 0.608 w), corrects for the buoyancy of water vapor and differs by roughly 1 to 3 K in the moist lower troposphere. The main limitation is that theta stops being conserved as soon as condensation, radiative cooling or surface heat flux acts on the parcel, which is why equivalent potential temperature is preferred for moist convection.

The adiabatic pressure-temperature relation underlying the formula traces to Simeon Denis Poisson, who published it in the 1820s as part of the developing theory of gases, and it was refined once Rudolf Clausius and William Thomson placed thermodynamics on a firm footing in the 1850s. The specific application to the atmosphere came in 1888, when Wilhelm von Bezold introduced the term potential temperature in his papers on the thermodynamics of the atmosphere, while Hermann von Helmholtz was independently treating the same reduction of temperature to a standard pressure in his work on atmospheric motions. Potential temperature moved from theory into daily analysis in the middle of the twentieth century. Carl-Gustaf Rossby and his collaborators promoted isentropic analysis during the 1930s, and Jerome Namias with the United States Weather Bureau published practical guidance around 1940 for plotting charts on constant theta surfaces, a technique that made moisture transport visible in a way that constant pressure charts did not. The radiosonde networks that expanded after the Second World War supplied the paired temperature and pressure measurements theta requires at every level. Modern numerical weather prediction models carry potential temperature, or a closely related variable together with the Exner function, as a prognostic thermodynamic variable rather than carrying temperature itself.