Equivalent Potential Temperature (Bolton)

Equivalent potential temperature puts the heat and the moisture of an air parcel into one number. It is the temperature the parcel would have if it were lifted until all its water vapour had condensed and fallen out, then brought down dry to 1000 hPa. Because it is conserved while the parcel rises and sinks, even through cloud, it labels air masses, and it shows where the atmosphere is convectively unstable: that is where $\theta_E$ falls with height.

Reading the terms. The first factor, $(1000/p)$ raised to $0.2854\,(1 - 0.28\times10^{-3} r)$, is the dry part: it brings the temperature $T$ to 1000 hPa. The exponential factor adds the effect of the vapour and grows with the mixing ratio $r$, entered in grams per kilogram inside the formula. The temperature $T_L$ is the one the parcel reaches at its lifting condensation level, from Bolton's equation 15, $T_L = 1/(1/(T_d - 56) + \ln(T/T_d)/800) + 56$, all in kelvins. Pressure may be entered in any unit and is used in hectopascals. The dew point and the mixing ratio go together: for a parcel that is not saturated, take $r$ from the dew point and the pressure, and keep the dew point at or below the temperature.

Worked example. Bolton's Table 3 gives $\theta_E$ by numerical integration for a parcel initially saturated, so $T_d = T$ and $r$ is the saturation mixing ratio. At 1000 hPa and 30 degrees Celsius (303.15 K) with $r$ = 27.58 g/kg, the result is 386.28 K. The page also matches 283.60 K at 1000 hPa and 0 degrees with $r$ = 3.825 g/kg, 394.71 K at 700 hPa and 20 degrees with $r$ = 21.48 g/kg, and 391.82 K at 200 hPa and -30 degrees with $r$ = 1.591 g/kg, within 0.04 K of every row. Run backward, the pressure behind 386.28 K at 30 degrees saturated is 1000 hPa.

Source and limits. The formula is equation 43 of Bolton (1980), The computation of equivalent potential temperature, in Monthly Weather Review. Bolton states a largest error of 0.3 K against numerical integration of the full problem. It assumes the water vapour condenses and is removed along the way (the pseudo-adiabatic process), with no heat from freezing.