2764 MONTHLY WEATHER REVIEW VOLUME 136 An Efficient and Accurate Method for Computing the Wet-Bulb Temperature along Pseudoadiabats ROBERT DAVIES-JONES NOAA/National Severe Storms Laboratory, Norman, Oklahoma (Manuscript received 17 April 2007, in final form 7 September 2007) ABSTRACT A new technique for computing the wet-bulb potential temperature of a parcel and its temperature after pseudoadiabatic ascent or descent to a new pressure level is presented. It is based on inverting Bolton’s most ␪ accurate formula for equivalent potential temperature E to obtain the adiabatic wet-bulb temperature Tw on a given pseudoadiabat at a given pressure by an iterative technique. It is found that Tw is a linear func- Ϫ ␬ Ϫ ␬ tion of equivalent temperature raised to the 1/ d (i.e., 3.504) power, where d is the Poisson constant for dry air, in a significant region of a thermodynamic diagram. Consequently, Bolton’s formula is raised to Ϫ ␬ the 1/ d power prior to the solving. A good “initial-guess” formula for Tw is devised. In the pressure range 100 Յ p Յ 1050 mb, this guess is within 0.34 K of the converged solution for wet-bulb potential temperatures ␪ Յ Ϫ Յ ␪ Յ w 40°C. Just one iteration reduces this relative error to less than 0.002 K for 20° w 40°C. The upper bound on the overall error in the computed Tw after one iteration is 0.2 K owing to an inherent uncertainty in Bolton’s formula. With a few changes, the method also works for finding the temperature on water- or ice-saturation reversible adiabats. The new technique is far more accurate and efficient than the Wobus method, which, although little known, is widely used in a software package. It is shown that, although the Wobus function, on which the Wobus method is based, is supposedly only a function of temperature, it has in fact a slight pressure dependence, which results in errors of up to 1.2 K in the temperature of a lifted parcel. This intrinsic inaccuracy makes the Wobus method far inferior to a new algorithm presented herein. 1. Introduction probabilistic models (e.g., Hamill and Church 2000). These parameters all require the computation of adia- Numerical analyses of actual and model-output up- batic wet-bulb temperature, Tw, along water-saturation per-air soundings (e.g., Prosser and Foster 1966; Stack- pseudoadiabats. They should be calculated as accurate- pole 1967; Doswell et al. 1982) are used to determine ly as possible because errors affect statistical measures several weather forecast parameters [e.g., convective of their forecast skill and also conditional tornado prob- available potential energy (CAPE), CAPE in the low- abilities. est 3 km of the sounding, convective inhibition, level of Given the initial state of a parcel, there is no simple free convection, height of the wet-bulb zero, bulk Ri- way to compute its temperature during undiluted chardson number, energy–helicity index, the height to pseudoadiabatic ascent. In contrast, there are precise which penetrative convection can reach, etc.] that iden- explicit formulas for equivalent potential temperature ␪ tify environments that support various types of severe (EPT) E (K) so we can easily calculate the parcel’s weather (e.g., Rasmussen 2003; Thompson et al. 2003) equivalent temperature TE during its ascent. Inconve- and that may factor in the forecast likelihood that a niently, the equivalent temperature of a saturated par- thunderstorm will produce a significant tornado in cel is a complicated function of Tw both explicitly and implicitly through the dependence of the parcel’s satu- ration mixing ratio on its temperature. This has discour- aged meteorologists from trying to invert a formula for Corresponding author address: Dr. Robert Davies-Jones, Na- tional Severe Storms Laboratory, National Weather Center, 120 TE to get an explicit expression for Tw. The general David L. Boren Blvd., Norman, OK 73072. view has been that the problem is mathematically in- E-mail: [email protected] tractable, and that solutions for Tw can be obtained DOI: 10.1175/2007MWR2224.1 Unauthenticated | Downloaded 09/25/21 02:54 PM UTC MWR2224 JULY 2008 DAVIES-JONES 2765 only through numerical integration, using small verti- Package (N-AWIPS/GEMPAK; J. Hart 2007, personal cal steps, of the differential equation governing the communication). Since it is in wide use, its errors pseudoadiabat or through iterative numerical tech- should be evaluated. Despite the advent of more recent niques (e.g., Doswell et al. 1982). This paper demon- empirical data and Bolton’s creation in 1980 of a highly strates that there is in fact an explicit solution if errors accurate formula for EPT, the Wobus method has up to 0.34 K relative to a converged solution are per- never been upgraded since its invention. mitted. If greater accuracy is desired, this solution is an Bolton (1980) first obtained new empirical formulas excellent first guess for an iterative method. for saturation vapor pressure and condensation tem- A variety of numerical techniques have been used to perature. With the use of these formulas, he accurately derive the temperature of a parcel lifted adiabatically determined EPT as a function of condensation tem- (if initially unsaturated), then pseudoadiabatically (i.e., perature and pressure by numerically integrating the with all condensate instantly falling out) to some lower differential equation for the pseudoadiabatic process pressure, p, (e.g., Prosser and Foster 1966; Stackpole from the saturation point to a great height. He then 1967; Doswell et al. 1982). In these procedures for the used the numerical results to obtain accurate formulas automated analyses of soundings, condensation tem- for EPT, of which his Eq. (39) is the most precise. Apart ␪ perature, TL, which is needed for computation of E if from the more accurate, but more complicated, formula the parcel is unsaturated initially, was determined by for condensation temperature developed by Davies- either a search technique (Prosser and Foster), by it- Jones (1983), Bolton’s formulas are the most exact of eration (Stackpole), or by curve fitting (Doswell et al.). their type. For initially saturated air, Bolton’s Eq. (39) ␪ To compute the temperature along pseudoadiabats, is accurate to within 0.2 K in E with this error mostly Prosser and Foster used a computationally fast, but er- owing to variation of cpd, the specific heat of dry air at ror-prone, scheme. First, they approximated the tem- constant pressure, with temperature and pressure (List peratures along three specific pseudoadiabats (the ones 1971, his Table 88). Note that, although cpd is treated as with wet-bulb potential temperatures of 10°,20°, and a constant, the variation of the specific heat of moist air 30°C) by third-order polynomials. Then they obtained at constant pressure, cp, with mixing ratio is parameter- the temperature of the lifted parcel by linear interpo- ized. lation, after computing its wet-bulb potential tempera- This paper devises a new accurate method for com- ␪ ture (WBPT) w from a crude empirical formula. Stack- puting temperature along pseudoadiabats, and hence pole computed the difference between the EPT (via the for reducing the errors involved in evaluating the above imprecise Rossby formula) of the pseudoadiabat and forecast parameters. First an efficient algorithm for in- that of a parcel at pressure p with temperature given by verting Bolton’s Eq. (39) to obtain the wet-bulb tem- the latest iterative solution. He also used the inefficient perature along a given pseudoadiabat at a given pres- interval-halving numerical procedure (Gerald and sure p is formulated (section 2). The output at 1000 mb Wheatley 1984). Doswell et al. reported a similar tech- from this algorithm is then used to determine empirical ␪ ␪ nique, due to Hermann Wobus, with some important formulas for w as a function of E (section 3). Over ␪ Ϫ Ͻ ␪ Ͻ differences. The Wobus method employs the much most of the atmospheric range of W ( 19° w ␪ faster secant method (Gerald and Wheatley 1984), 29°C), a linear relationship is discovered between w Ϫ␭ ␪ ␭ ϵ ␬ ϭ which converges within a few iterations. It also uses and the power of E, where 1/ d ( 3.504) and ␬ ϭ ϭ the Wobus function, which was devised by Wobus in d Rd /cpd ( 0.2854) is the Poisson constant for dry 1968. At the time of its invention, the Wobus method air. In section 4, highly accurate initial guesses for the was much more efficient and faster than other methods. computation of wet-bulb temperature along pseudoa- ␪ ␪ In lieu of E, it uses w, which is computed from the diabats are derived. A new linear relationship between Ϫ␭ Wobus function, WF(TK), of absolute temperature TK Tw and the power of equivalent temperature TE (K) only. is found in a significant region of a thermodynamic dia- The Wobus method is little known because it has gram. One iteration of the algorithm then gives a highly never been documented previously in the formal litera- accurate solution for Tw. Next the algorithm is modified ture. However, it is widely used because it is utilized slightly for computation of temperature along reversible unseen in the National Centers Skew–T/Hodograph adiabats (section 5). The Wobus method is described in Analysis and Research Program (NSHARP; Hart et al. section 6 and its intrinsic errors are evaluated and 1999), which is the interactive software for upper-air found to be quite large. Although the Wobus function profiles in the National Centers’ Advanced Weather is supposedly only a function of temperature, it in fact Interactive Processing System/General Meteorological has a slight dependence on pressure. The linear Unauthenticated | Downloaded 09/25/21 02:54 PM UTC 2766 MONTHLY WEATHER REVIEW VOLUME 136 TABLE 1.