2764 MONTHLY WEATHER REVIEW VOLUME 136

An Efficient and Accurate Method for Computing the Wet-Bulb 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 of a parcel and its temperature after pseudoadiabatic ascent or descent to a new 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 ␪ 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, , 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

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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 and 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

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TABLE 1. Parameters in (2.1) and (2.2) as they apply to Bolton’s Eqs. (28), (35), (38), and (39) for water-saturation pseudoadiabats ϭ Ϫ and to Saunders’s Eq. (3) for water-saturation reversible adiabats. The of vaporization is given by L L0 L1T, where ϭ ϫ 6 Ϫ1 ϭ ϫ 3 Ϫ1 Ϫ1 the constants L0 2.501 10 Jkg and L1 2.37 10 Jkg K . In the last column, cW is the specific heat of water and Q is the mixing ratio of total water to dry air.

Bolton’s Eq. (28) Bolton’s Eq. (35) Bolton’s Eq. (38) Bolton’s Eq. (39) Saunders’s Eq. (3) ϩ ϩ ϩ k0 (L0 L1C)/cpd 2675 K 3376 K 3036 K (L0 L1C)/(cpd cWQ) ϩ k1 L1/cpd 0 2.54 1.78 L1/(cpd cWQ) k2 0 0 0.810 0.448 0 k3 0 0.28 0.28 0 0 ␯␬ ␬ ϩ d 00 d Rd /(cpd cWQ)

relationship discovered in section 3 is used in section 7 mula, his adaptation of the Betts and Dugan (1973) to formulate a new more accurate Wobus function of formula, and two new formulas. The formulas are writ- ␲ both temperature and pressure. The modified Wobus ten for an unsaturated parcel at the point (TK, ) and method thus obtained is shown to be simply a convo- involve its mixing ratio r, its vapor pressure e and TL. luted version of the new method. The reason why the Bolton’s Eq. (39) is the most accurate. Simpson’s for- original Wobus method works fairly well is addressed in mula does not fit the same mathematical mould as the

section 8 where it is shown that the error in Tw caused other formulas and so is not considered further here. by assuming that the Wobus function is independent of We can apply the formulas to any saturated parcel at ␲ pressure is less than 1 K. (TW, ) simply by replacing TL by TW, r by saturation ␲ Before proceeding, we explain our terminology of mixing ratio rs(TW, ), and e by saturation vapor pres- errors. “Relative error” denotes the error relative to sure es(TW). When this is done, the remaining formulas the converged solution of Bolton’s Eq. (39). “Absolute all have the following form: error” also includes the error inherent in Bolton’s Eq. ͑ ͒ Ϫ␯ es TW ␲͒ (39) itself. The “intrinsic error” of the Wobus method T ϭ T ͫ1 Ϫ ͬ ␲ k3 rs͑TW, exp͓G͑T , ␲͔͒, E W ␲ ␭ W refers to the error resulting from the assumption that p0 the Wobus function is a function of just temperature. ͑2.1͒

where 2. Mathematical formulation of the new method k ͑ ␲͒ ϭ ͩ 0 Ϫ ͓ͪ ͑ ␲͒ ϩ 2͑ ␲͔͒ G TW, k1 rs TW, k2r s TW, , We start by developing the new method. We use Bol- TW ton’s nomenclature here, including his convention that ͑2.2͒ a temperature with a capital subscript is in kelvins and one with a small subscript is in degrees Celsius. Tem- and the constants are listed in Table 1. Note that we peratures with the same letter subscript but different have used Bolton’s Eqs. (24) or (7) and the relationship ϭ ␪ ␲ case are the same variable in different units. The only TE E to write the formulas in the form in (2.1). We departures from these rules are T, the temperature in can find the temperature at pressure p along a given ϭ ϩ ␪ degrees Celsius, TK, the absolute temperature ( T pseudoadiabat with EPT E by solving for TW. How- C, where C ϭ 273.15 K), and ␪, the potential tempera- ever, for the reason given below, it is generally more ture in kelvins. The one exception to Bolton’s nomen- advantageous to solve (2.1) raised to the Ϫ␭ power (i.e., ␲ ␪ clature in this paper is the unit of mixing ratio, which is to solve for given and E): grams per gram instead of grams per kilogram. Any ␭ variable that can be determined uniquely from a ther- C ͩ ͪ ϭ f͑T , ␲͒ W modynamic diagram is a function of just two inde- TE pendent variables, chosen in the following analyses to C ␭ e ͑T ͒ ␭␯ ␲ ϵ s W Ϫ␭k r ͑T , ␲͒ be temperature and the nondimensional pressure ϵ ͩ ͪ ͫ1 Ϫ ͬ ␲ 3 s W ␭ ␭ 1/ TW p ␲ (p/p0) . 0 In his Eqs. (28), (33), (35), (38), and (39), Bolton ϫ ͓Ϫ␭ ͑ ␲͔͒ ͑ ͒ exp G TW, , 2.3 (1980) gives five formulas for the EPT. These are, re- ␭ ϭ ␬ ϭ ϭ spectively, the traditional but inaccurate Rossby for- where 1/ d cpd /Rd 3.504. In (2.3) TW and TE mula, Bolton’s adjustment of the Simpson (1978) for- have been scaled by C simply to avoid large numbers.

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Љ ␲ FIG. 1. The second derivative f of f(TW; ) w.r.t. TW vs Tw at of 1000, 700, 500, and 300 mb. (At 1000 ␪ ␲ mb, Tw is equal to w.) The Taylor series expansion to first order of f(TW; ) at constant pressure, evaluated at the temperature ␶* ϭϪ50(1 Ϫ ␲) (marked by the asterisk), is written at the upper left of each grid window. Throughout the figure | f Љ| is less than 0.0002 within 20°Cof␶. At each pressure therefore, the upper bound for the remainder term in (2.4) within the temperature interval (␶* Ϫ 20°C, ␶* ϩ 20°C) is an order of magnitude ␲ smaller than the first-order term in the written expansion, proving that f(TW; ) is almost a linear function of Tw in this interval.

By Taylor series expansion about a temperature ␶ (K) other words, there is the almost linear relationship be- ␲ ␭ at constant , tween (C/TE) and TW in this interval, ͑ ␲͒ ϭ ͑␶ ␲͒ ϩ Ј͑␶ ␲͒͑ Ϫ ␶͒ ␭ f TW; f ; f ; TW f͑␶*; ␲͒ Ϫ ͑CրT ͒ ϭ ␶ Ϫ E ͑ ͒ TW * , 2.5 ϩ Љ͓␶ ϩ ␤͑ Ϫ ␶͒ ␲͔͑ Ϫ ␶͒2ր fЈ͑␶*; ␲͒ f TW ; TW 2!, Ј ␶ ␲ 0 Ͻ ␤ Ͻ 1 ͑2.4͒ where an expression for f ( ; )is provided in the ap- pendix for the reader’s convenience. This anticipates [The notation f(␶; ␲) indicates that f is a function of ␶ ␶ our later finding that, with a good initial guess 0, one or with ␲ fixed.] At each pressure Ն300 mb, we can two iterations of the algorithm, choose ␶ ϭ ␶* such that the remainder (the last term) is ∈ ␶ Ϫ f͑␶ ; ␲͒ Ϫ ͑CրT ͒␭ much smaller than the first-order term for TW [ * ␶ ϭ ␶ Ϫ n E ͑ ͒ ␶ ϩ nϩ1 n Ј͑␶ ␲͒ , 2.6 20°C, * 20°C] (this can be deduced from Fig. 1). In f n;

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in Newton’s method always provide a precise numerical gence by retaining the second-order term in the series

solution, TW, of (2.3). We can accelerate the conver- expansion and solving the resulting quadratic equation:

Љ͑␶ ␲͒ f n; ͑T Ϫ ␶ ͒2 ϩ fЈ͑␶ ; ␲͒͑T Ϫ ␶ ͒ ϩ f͑␶ ; ␲͒ Ϫ ͑Cր␪ ͒␭ ϭ 0, ͑2.7͒ 2 W n n W n n E

in a form that is accurate for a small second-order term (see Henrici 1964, p. 199). This gives

͓ ͑␶ ␲͒ Ϫ ͑ ր␪ ͒␭͔ 2 f n; C E ␶ ϩ ϭ ␶ Ϫ , ͑2.8͒ n 1 n Ј͑␶ ␲͒ Ϯ Ј͑␶ ␲͒ Ϫ Љ͑␶ ␲͓͒ ͑␶ ␲͒ Ϫ ͑ ր␪ ͒␭͔ 0.5 f n; ͕ f n; 2f n; f n; C E ͖

where f Љ(␶;␲) is given in the appendix. We choose the ␪ ϭ Ϫ ͑ ր␪ ͒␭ ͑ ͒ w 45.674 52.091 C E , 3.3 root that is closest to the linear ␶ ϩ provided by (2.6). n 1 which is valid in some interval around C. This interval In this paper, we use Bolton’s Eq. (39) as the basis for turns out to be fairly large owing to the small second the computation of TW. However, we can compute TW ␪ derivative of f( w;1). A plot (Fig. 2) of the actual (i.e., from Bolton’s Eqs. (28), (35), or (38), or even compute ␪ converged iterative) solution for w as a function of temperature along water- or ice-saturation reversible ␭ (C/␪ ) shows that the linear solution in (3.3) is ap- adiabats from Saunders’s (1957) Eqs. (3) or (4), simply E proximately valid in the interval Ϫ19° Ͻ ␪ Ͻ 29°C. by changing a few parameters in the computer code as w The minimax-polynomial approximation method (Scheid dictated by Table 1. 1989) was used to obtain the minimax line in this in- A first guess ␶ that is accurate to within 10 K is given 0 terval, by ␪ ϭ 45.114 Ϫ 51.489͑␪ րC͒Ϫ␭, ͑3.4͒ ␶ ϭ ͓ ␪ Ϫ ͑ Ϫ ␲͔͒ ͑ ͒ w E 0 min TE, W 150 1 , 2.9 which fits the solution to 0.1°C. Note that (3.3) and ␪ ␪ where W is obtained from E via a formula obtained in (3.4) are very similar. section 3. This is sufficient for convergence of the algo- Is there a better linear relationship than (3.4) be- ␪ ␪ ␪ Ϫ␮ rithm, but not optimal. A far more accurate initial es- tween w and another power of ( E /C), say ( E /C) ? ␪ timate is based on results presented below and so is To answer this question, the standard error of the w supplied later in Eqs. (4.8)–(4.11). predicted by linear regression over the interval Ϫ19°C Ͻ ␪ Ͻ ␮ w 29°C was computed for different values of . ␮ Ϸ 3. Computing ␪ from ␪ The minimum standard error (0.06 K) occurred for W E 3.5, thus confirming that ␮ ϭ ␭ produces the most lin- We first look at the problem of computing WBPT earity. from EPT to gain insight into the more general problem The linear fit naturally breaks down at large values of ␪ ␪ ␪ of computing temperature on a given pseudoadiabat at E because w cannot remain finite as E tends to in- ␪ ␪ ␪ ␪ a given pressure. Using Bolton’s Eq. (39) for E, we find finity and at cold values of E because W tends to E as ␪ Յ from (2.3) applied at 1000 mb that W is the solution of the saturation mixing ratio becomes small. For 377 ␪ Ͻ Յ ␪ Ͻ E 674 K (28.2° w 50°C), a minimax polynomial C ␭ f͑␪ ;1͒ ϵ ͩ ͪ ͓1 Ϫ e ͑␪ ͒րp ͔ exp͓Ϫ␭G͑␪ ,1͔͒ was fitted to the difference between the actual and the W ␪ s W 0 W ␪ Յ ␪ Յ Ϫ W linear solutions. For E 257K( w 18.6°C), one C ␭ iteration of Newton’s method applied directly to Bol- ϭ ͩ ͪ ͑ ͒ ϭ ␪ ϭ ␪ , 3.1 ton’s Eq. (35) version of (2.1)–(2.2) with TE E, Tw E ␪ ␲ ϭ w, and 1 suffices. After some minor approxima- where tions, the resulting solution is Ar ͑␪ ,1͒ 3036 ␪ ϭ ␪ Ϫ Ϫ s E G͑␪ ,1͒ ϭ ͩ Ϫ 1.78͓ͪr ͑␪ ,1͒ ϩ 0.448r 2͑␪ ,1͔͒. w E C ϩ ͑␪ ͒ ͑␪ ͒ր ␪ for W ␪ s W s W 1 Ars E ,1d lnes E d E W ␪ Յ ͑ ͒ ͑3.2͒ E 257 K, 3.5 ␪ ϭ 45.114 Ϫ 51.489͑Cր␪ ͒␭ for One linear iteration of Newton’s method with a first w E Ͻ ␪ Ͻ ͑ ͒ guess of C provides the following solution: 257 E 377 K, 3.6

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␪ ␪ 3.504 FIG. 2. Graph of w as a function of (C/ E) . The center of the / mark indicates each point ␪ 3.504 ␪ ␪ ϭϪ Ϫ ␪ [(C/ E) , w] for w 100°, 99°,...,50°C and the corresponding E given by Bolton’s Eq. (39). The straight line (solid) is the minimax line in (3.6). The curve on the right (long dashes) is the graph of (3.5). The curve delineated by the short dashes is the graph of (3.7). The ␪ data points lie in turn on (3.5), (3.6), and (3.7) as E increases. Here E is the point on the line ␪ ϭ in (3.3) [not shown as it is practically coincident with the line in (3.6)], where W C [the WBPT where the Taylor series is evaluated to derive (3.6)]. Here S is the point on this line determined by the method in section 4 where the solution becomes nonlinear at cold tem- peratures.

␪ ϭ Ϫ ͑ ր␪ ͒␭ ϩ ͑␪ ր ͒␭ 0.6220, e (C) ϭ 6.112 mb, a ϭ 17.67, and b ϭ 243.5 K. w 43.380 51.489 C E 0.6069 E C s Remarkably, this solution fits the actual solution to Ϫ ͑␪ ր ͒2␭ 0.01005 E C for 0.1°C. It should be emphasized, however, that the ac- ␪ ϭ 377 Յ ␪ Ͻ 674 K, ͑3.7͒ curacy of Bolton’s Eq. (39) is not known beyond w E ␪ ϭ 40°C( E 478.4 K). where a small term has been neglected in (3.5) and A ϭ For even greater overall accuracy a rational function ϭ ␭ ϭ ϭ␧ ␪ ϭ 2675 K, C 273.15 K, cpd /Rd, rs(TK,1) es(TK)/ was collocated to the actual solution of (3.1) at w ␲␭ Ϫ ϭ Ϫ Ϫ Ϫ Ϫ [p0 es(TK)], es(TK) es(C) exp[a(TK C)/(TK 30°, 20°,...,40°,50°C. The resulting approximate ϩ ϭ Ϫ ϩ 2 ␧ϭ C b)], d lnes(TK)/dTK ab/(TK C b) , solution is

a ϩ a X ϩ a X 2 ϩ a X 3 ϩ a X 4 ␪ Ϫ C Ϫ ͫ 0 1 2 3 4 ͬ for ␪ Ն 173.15 K E exp E ␪ ϭ 1 ϩ b X ϩ b X 2 ϩ b X 3 ϩ b X 4 , ͑3.8͒ w Ά 1 2 3 4 ␪ Ϫ ␪ Ͻ E C for E 173.15 K

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ϵ ␪ ϭ ϭϪ where X E /C and a0 7.101574, a1 20.68208, From (2.3) and (2.4) it is evident that the equations of ϭ ϭ ϭϪ ϭ a2 16.11182, a3 2.574631, a4 5.205688, b1 the straight-line portions are given by Ϫ3.552497, b ϭ 3.781782, b ϭϪ0.6899655, b ϭ 2 3 4 ϭ ͑␲͒ Ϫ ͑␲͒͑ ր ͒␭ ͑ ͒ Ϫ ␪ Յ Tw k1 k2 C TE , 4.1 0.5929340. For w 50°C, its maximum deviation from the actual solution (the relative error) is 0.02 K, where and it is within 0.005 K for ␪ ∈ (Ϫ20°,40°C), the range w ͑␲͒ ϭ ␶ Ϫ ͑␶ ␲͒ր Ј͑␶ ␲͒ ͑␲͒ ϭϪ ր Ј͑␶ ␲͒ ␪ k1 * f *; f *; , k2 1 f *; , of w tabulated in the Smithsonian tables (List 1971, see his Table 78). The magnitude of the argument in the ͑4.2͒ exponential in (3.8) becomes large at temperatures less and ␶* is any value of T such that the horizontal line than Ϫ100°C, and the rational-function approximation w T ϭ ␶* intersects the linear portion of the curve. By fails. However, at these temperatures we can assume w Ϫ inspection ␶* ϭϪ50 (1 Ϫ ␲) always lies on the linear that ␪ ϭ ␪ with negligible error (Յ10 4 K). W E part of the solution curve (Figs. 4 and 1) and is used Associated with the absolute error ␦␪ of up to 0.2 K E here as the points E in Figs. 2 and 3, where k (␲) and in the value of ␪ provided by Bolton’s Eq. (39), there 1 E k (␲) are evaluated for the 39 different pressures. The is a corresponding error in ␪ given, from (3.6), by 2 W coefficient Ϫ50 was chosen because it is nearly optimal ␦␪ ϭ ␭͑ ր␪ ͒␭␪Ϫ1␦␪ Ͻ ␪ Ͻ for minimizing the maximum absolute error over the W 51.5 C E E E for 257 E 377 K. grid. Fitting quadratic regression curves to the results ͑ ͒ 3.9 yields the following expressions: ␦␪ ϭ ␦␪ ␪ ϭ For E 0.2 K, W varies from 0.17 K at E 257 to k ͑␲͒ ϭϪ38.5␲ 2 ϩ 137.81␲ Ϫ 53.737, ͑4.3͒ ␪ ϭ ␪ ϭ 1 0.05 K at E 335 to 0.03 K at E 377 K. The ␪ ͑␲͒ ϭϪ ␲ 2 ϩ ␲ Ϫ ͑ ͒ absolute error in W becomes quite small at the EPTs k2 4.392 56.831 0.384, 4.4 most often associated with . with correlation coefficients, r, of 1 (Figs. 5 and 6). ␲ Remarkably, k2 is almost linear in with the regression

4. The solution for TW as a function of TE and p line: ͑␲͒ ϭ ␲ ϩ ͑ ͒ The first step toward finding an efficient and accurate k2 49.896 2.2648. 4.5 algorithm for computing temperature along pseudoadi- A similar procedure to that used in obtaining (3.5) abats is to obtain the “true” values that are produced by gives the following good initial estimate for the solution precisely inverting Bolton’s Eq. (39). We apply this for- at cold temperatures: mula initially to saturated parcels at 1000 mb to acquire ͑ ␲͒ ␪ ␪ ϭϪ Ϫ Ars TE , the E values corresponding to w 20°, 18°,..., ϭ Ϫ Ϫ Tw TE C ϩ ͑ ␲͒ ͑ ͒ր . 40°C. We then obtain the wet-bulb temperatures 1 Ars TE , d lnes TE dTE ␪ TW( W, p) along these 31 pseudoadiabats at 25-mb in- ͑4.6͒ tervals from 1050 to 100 mb (39 different pressures) to a tolerance of 5 ϫ 10Ϫ5 K by Newton’s algorithm (2.6) The transition point S at each of the 39 pressure levels Ϫ ␬ applied to Bolton’s Eq. (39) raised to the 1/ d power. was located by evaluating the errors in the two approxi- ϫ ϭ ␪ We refer to the array of 31 39 ( 1209) points in ( w, mate solutions [(4.1)–(4.2) and (4.6)] as a function of ␭ p) space as the “grid.” (C/TE) , and determining by linear interpolation the ␭ Plots (Fig. 3) of the actual solution of (2.3) as a func- value D(p)of(C/TE) where the magnitudes of the ␭ tion of (C/TE) at selected pressures have some of the errors are equal. Then a regression line was fitted to the same characteristics as Fig. 2. In each plot, there is a reciprocal of the data for D(p). The resulting equation, range in which the solution (marked by Xs) is almost Ϫ D͑p͒ ϭ ͑0.1859pրp ϩ 0.6512͒ 1, ͑4.7͒ linear and an immediately adjacent range at cold 0

equivalent temperatures, where TW approaches TE, and fits the data adequately enough (Fig. 4). The transition ϭ the solution becomes asymptotic to the curve TW points at particular pressure levels are plotted in Figs. 2 Ϫ1/␭ ␭ Cx , where x is the abscissa (C/TE) . The slopes and and 3. intercepts of the linear parts, and the transition points S Two empirical corrections were applied to (4.1). For Ͼ where the solutions depart from linearity toward their TE C, the coefficients were adjusted slightly, and at Ͼ asymptotes all vary with pressure. The solution also warm equivalent temperatures (TE 355 K), an addi- Ϫ␭ becomes nonlinear at the warmest equivalent tempera- tional term in (C/TE) was added and the constant ␭ Ͻ tures [(C/TE) 0.4]. This is clearly visible at 1000 mb term adjusted to describe the “warm-side” nonlinearity. (Fig. 2), but hardly evident at 850 mb (Fig. 3). The resulting initial guess for Tw is

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3.504 FIG. 3. Graph of Tw as a function of (C/TE) at pressures of 850, 700, 500, and 300 mb. The centers of the / 3.504 ␪ ϭϪ Ϫ mark indicate the points [(C/TE) , Tw] for w 20°, 18°,...,40°C and the given pressure. The curve and

straight line are the graphs of (4.6) and (4.1) [with k1 and k2 given by (4.2)], respectively. For each pressure, E is 3.504 ϭ the point of evaluation for the straight line, and S is the “transition point” at which (C/TE) D(p) and the solution departs from linearity as it approaches its asymptote. Both E and S at 1000 mb are marked in Fig. 2.

Ar ͑T , ␲͒ where A ϭ 2675 K, C ϭ 273.15 K, ␭ ϭ c /R , and ϭ Ϫ Ϫ s E pd d Tw TE C ϩ ͑ ␲͒ ͑ ͒ր for ␲ ␲ 1 Ars TE , d lnes TE dTE k1( ), k2( ), and D(p) are given by (4.2) and (4.7). The regions in which the different parts of the initial solu- ͑ ր ͒␭ Ͼ ͑␲͒ ͑ ͒ C TE D , 4.8 tion apply are shown in Fig. 4. ϭ ͑␲͒ Ϫ ͑␲͒͑ ր ͒␭ The relative errors in the initial guess and that after Tw k1 k2 C TE for one iteration were computed. Finally, as a check the Յ ͑ ր ͒␭ Յ ͑␲͒ ͑ ͒ 1 C TE D , 4.9 EPT was computed again from Bolton’s Eq. (39) using ␭ ϭ ͓ ͑␲͒ Ϫ ͔ Ϫ ͓ ͑␲͒ Ϫ ͔͑ ր ͒ the one-iteration solution for Tw. The largest difference Tw k1 1.21 k2 1.21 C TE for between recomputed and original values of EPT was Յ ͑ ր ͒␭ Ͻ ͑ ͒ 0.4 C TE 1, 4.10 less than 0.002 K. ␭ The two empirical corrections reduce the maximum T ϭ ͓k ͑␲͒ Ϫ 2.66͔ Ϫ ͓k ͑␲͒ Ϫ 1.21͔͑CրT ͒ w 1 2 E relative error at any grid point in the initial guess from ϩ ͑ ր ͒Ϫ␭ ͑ ր ͒␭ Ͻ ͑ ͒ 0.58 C TE for C TE 0.4, 4.11 1.8 to 0.34 K (Fig. 7). One ordinary (accelerated) itera-

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␭ FIG. 4. The four regions in [p,(C/TE) ] space where the different parts in (4.8)–(4.11) of the initial solution are valid. The data points for D(p), the fitted curve that determines when (4.8) ␲ ␲ should be used instead of (4.9), are marked by X. The points at which k1( ) and k2( ) are evaluated are marked by plus signs. These points all lie in the two middle regions where the solution is linear.

␭ Ϫ␬ tion then reduces this relative error to less than 0.002 K ͕ ͓␲ Ϫ ͑ ͒ր ͔ d͖ ϩ cpdd TR es TR p0 cWQd lnTR ␲ (0.001 K), which is more than sufficient. When k1( ) and k (␲) are approximated by the quadratic regression ͑ ͒ ͑ ␲͒ 2 L TR rs TR, curves in (4.3) and (4.4), the maximum relative error in ϩ dͫ ͬ ϭ 0 ͑5.1͒ TR the initial solution increases to 0.47 K. When linear re-

gression is used for k2 [i.e., when (4.5) is used instead of (4.4)], the initial relative error is slightly larger (0.52 K). (Saunders 1957). We make the usual assumption that Ϫ1 Ϫ1 The overall accuracy of the algorithm is determined cW, the specific heat of liquid water, is 4190 J kg K almost entirely by its absolute error. The absolute error of (e.g., Saunders 1957; Bolton 1980; Emanuel 1994), even ␪ though it increases by 8%, 14%, and 30% over this up to 0.2 K in E in Bolton’s Eq. (39) is caused mostly by value at temperatures of Ϫ30°, Ϫ40°, Ϫ50°C, respec- variation of cpd with temperature and pressure. The effect ␪ tively (List 1971, see his Table 92). Equation (5.1) also of a 0.2-K error in E on Tw is shown in Fig. 8. Clearly the fits the mold in (2.1) and (2.2) with the constants listed upper bound on the corresponding absolute error in Tw is also 0.2 K. Since the algorithm’s relative error after in the last column of Table 1. Since the mixing ratio of one iteration is much smaller, its overall error is Յ0.2 K. all phases of water to dry air, Q, is conserved, this equa- tion has the following integral:

5. Computation of temperature along reversible ͑ ͒ ͑ ␲͒ ␭ Ϫ ր ϩ ͒ L TR rs TR, adiabats ͓␲ Ϫ ͑ ͒ր ͔ Rd ͑cpd cWQ ͫ ͬ TR es TR p0 exp ͑ ϩ ͒ cpd cWQ TR The equation governing the temperature TR along ϭ ͑ ͒ reversible water-saturation adiabats is A1, 5.2

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␲ FIG. 5. Quadratic regression for k1( ).

␭ ͑ ͒ ␭␯ where A1 is a constant along a given reversible adia- C es TR f͑T ; ␲͒ ϵ ͩ ͪ ͫ1 Ϫ ͬ R ␭ bat. The constant A1 is evaluated at a parcel’s satura- T ␲ R p0 tion point. It depends on Q so that through each ␭␯ L͑T ͒r ͑T , ␲͒ C ␭ point on a thermodynamic diagram there passes a ϫ Ϫ R s R ϭ expͫ ͬ ͩ ␲ͪ , unique pseudoadiabat and an infinity of reversible adia- Rd TR A1 bats, one for each value of Q (Saunders 1957). Raising ͑5.3͒ (5.2) to the Ϫ␭ power gives the following version of ␯ ϵ ϩ (2.3): where Rd /(cpd cWQ).

␲ FIG. 6. Quadratic and linear regression for k2( ). The regression line and regression curve are nearly collocated.

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␪ ␪ FIG. 7. The error of the initial guess for w (°C) as a function of w and p. The numbers in the parentheses at the bottom right are the minimum value, minimum contour value, the maximum contour value, the maximum value, and the contour interval, respectively.

It should be possible to find good initial guesses and qualitatively similar to Fig. 2 in Saunders (1957), but

identify near-linear relationships between TR and there are quantitative differences owing to the use of ␲ Ϫ␭ (A1 ) in a region of the parameter space by the pro- up-to-date data. cedures used above for pseudoadiabatic ascent. Be- The method also works for computation of the tem- cause of the complication of dealing with an additional perature along ice-saturation reversible adiabats with parameter (Q), this has been left for future work. In- trivial substitutions. The specific heat of ice replaces

stead, we used the same initial guess for Tr as the one that of water, the latent heat of sublimation supplants for Tw [see (4.8)–(4.11)], even though it is no longer that of vaporization (Saunders 1957), the saturation accurate because retention of liquid water during as- mixing ratio is now with respect to ice instead of water, cent from low levels can make a parcel warmer by as and a and b become Tetens’s (1930) ice coefficients much as6Kat100mb(Emanuel 1994, p. 133). Initially (a ϭ 21.87, b ϭ 265.5 K). and after one and two accelerated iterations, the maxi- mum error relative to the converged solution is 6.25 and 0.034 K, respectively. With two ordinary iterations, the 6. The Wobus method maximum relative error reduces from the initial 6.25 to 0.36 K and then 0.001 K. Figure 9 shows the difference We now show that the Wobus method has an intrin- Ϫ Tr Tw between the temperatures attained in revers- sic error, owing to it being supposedly a function of just ible adiabatic expansion and pseudoadiabatic expan- temperature, that makes it inferior to the new method sion from the same initial state at 900 mb. This figure is described above. The Wobus function is defined as fol-

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␪ ␪ FIG. 8. The error in the converged solution for Tw owing to a 0.2-K error in E as a function of w and p. The quantities inside the parentheses are the same as in Fig. 7.

␲ ␪ ␪ lows. At any point (TK, ) on a pseudoadiabatic right. The distribution of W and S with height, z, de- (Stüve) diagram, one can consider two hypothetical termine atmospheric stability. A layer is potentially Ͻ ץ ␪ץ ␪ ␪ wet-bulb potential temperatures S and A (Fig. 10). unstable if W / z 0 and conditionally unstable if Ͻ ץ ␪ץ ␪ The saturated WBPT, S, is reached by saturating S / z 0. The atmosphere is latently unstable if the ␲ ␪ a parcel at (TK, ), then bringing it down to 1000 mb W of any parcel lifted pseudoadiabatically exceeds the ␲ ϭ ␪ ( 1) pseudoadiabatically. [Just enough rain is as- S of the unmodified environment at a higher level (see sumed to fall into and evaporate in the descending Fig. 12 in Browning and Donaldson 1963). parcel to keep it saturated without leftover liquid wa- To test the claim that the Wobus function is a ␪ ter.] The dry WBPT, A, is attained by desiccating a function solely of temperature, precise values of Tw ␲ ␪ Ϫ ␪ parcel at (TK, ), lifting it dry adiabatically to a great and W A at the 1209 grid points were computed height and then bringing it down pseudoadiabatically and plotted on a scatter diagram (Fig. 11). The

to 1000 mb. The original Wobus function (WF)of points tend to lie on the curve of the Wobus func- just temperature is the difference between these two tion WF(TW), but there is some scatter, which indi- ϭ ␪ ␲ Ϫ ␪ ␲ temperatures, that is, WF(TK) S(TK, ) A(TK, ). cates minor dependence on pressure. The pressure It is evaluated using Wobus’s numerical fit to the dependency is also evident in Fig. 12, which shows the ␪ Ϫ ␪ data. slight misalignment of the contours of Tw and W A ␪ ␪ Incidentally, S is an important variable in its own ona(w, p) thermodynamic diagram, and maxi-

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Ϫ FIG. 9. The difference Tr Tw between the temperature attained in the water-saturation adiabatic expansion and that attained in the water-saturation pseudoadiabatic expansion from the same initial Ϫ ␪ state at 900 mb. Contours of Tr Tw at levels listed at the bottom are plotted on a w vs p diagram. Also shown are the Ϫ40°, Ϫ20°, and 0°C contours of temperature. This is an updated version of Fig. 2 in Saunders (1957).

␪ Ϫ ␪ ␪ ␪ mum variation at constant temperature in W A of for dry air) so its A is unchanged. However, its S is ␪ ␪ almost 1 K. now and its TK becomes . Hence, by (6.1), To include the pressure variation, we define a gen- ͑ ր␲ ͒ ϭ ր␲ Ϫ ␪ ͑ ր␲ ͒ eralized Wobus function of two arguments, W*, as W* TW ,1 TW A TW ,1 ͑ ␲͒ ϭ ␪ ͑ ␲͒ Ϫ ␪ ͑ ␲͒ ͑ ͒ ϭ ր␲ Ϫ ␪ ͑ ␲͒ ͑ ͒ W* TK, S TK, A TK, . 6.1 TW A TW, . 6.3

For a saturated parcel, TK is the same as its adiabatic If on the other hand, the parcel is lifted pseudoadiabati- ␪ ␪ wet-bulb temperature TW and S is its WBPT, W. cally to a great height and then brought down dry adia- Therefore, batically to 1000 mb, its TK becomes equal to its equiva- lent potential temperature ␪ (T , ␲), its new ␪ is ␪ W*͑T , ␲͒ ϭ ␪ ͑T , ␲͒ Ϫ ␪ ͑T , ␲͒. ͑6.2͒ E W S E W W W A W ␪ ␪ ␪ and its new A is W (a bijective function of E). Thus, If this parcel descends dry adiabatically to 1000 mb, it by (6.1), ␪ ϵ ␲ still has the same potential temperature ( TW / here ͑␪ ͒ ϭ ␪ Ϫ ␪ ͑␪ ͒ ͑ ͒ because Wobus imprecisely used the Poisson constant W* E ,1 E W E , 6.4

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FIG. 10. A pseudoadiabatic diagram showing the relationships between the temperatures ␪ ␪ ␪ ␪ ϭ ES, , S, and A for a parcel at point X (T, p). The horizontal dotted lines are selected isobars, the solid curves are pseudoadiabats (shown only from 1000 to 100 mb), and the diagonal dashed lines are the particular adiabat to which the pseudoadiabats are asymptotic. ␪ ␪ The variable is the parcel’s temperature if it is lowered dry adiabatically to 1000 mb and S is its temperature if it is hypothetically saturated at X and then lowered pseudoadiabatically ␪ to 1000 mb. The parcel’s saturated EPT, ES is its temperature if it is saturated at X, lifted pseudoadiabatically to a great height, and then lowered dry adiabatically to 1000 mb. The ␪ Ͻ␪ quantity A ( S) is its temperature if it is hypothetically desiccated at X, then raised dry adiabatically to a great height, and subsequently lowered pseudoadiabatically to 1000 mb. If ␪ ϭ ␪ ␪ ϭ ␪ ϭ the parcel is already saturated at X, S W and ES E. In the example shown, p 750 mb, ϭ ␪ ϭ ␪ ϭ ␪ ϭ ␪ ϭ T 23.1°C, s 32.0°C, a 16.0°C, ES 400.4 K, and 321.4 K.

which indicates that at ␲ ϭ 1 the Wobus function maps tion to the more general equation in (6.2) using data the EPT of a parcel to the difference between its EPT from parcels at many different pressure levels (not just and its WBPT. According to Doswell et al. (1982), Wo- at 1000 mb). ␪ bus evaluated the right side of (6.4) using the data that Eliminating A from (6.2) and (6.3) gives a formula ␪ ␪ ␪ gives E as a function of W in the header of Table 78 of for W: the Smithsonian Meteorological Tables (List 1971) and then essentially fitted a high-order polynomial to the ␪ ͑ ␲͒ ϭ ր␲ Ϫ ͑ ր␲ ͒ ϩ ͑ ␲͒ W TW, TW W* TW ,1 W* TW, . reciprocal of the right side to obtain an approximation ␪ ͑6.5͒ to the Wobus function. However, the values of W com- ␪ puted from (6.4) with List’s (Bolton’s) values of E have a maximum error of 0.66 K (0.53 K). There are similar The temperature TW at a pressure p along a pseudo- ␪ ␪ ␪ errors in the computation of A from using (6.3). Er- adiabat with WBPT W can be found as the solution of rors of these magnitudes suggest that Wobus did not (6.5). Note that for a saturated parcel at 1000 mb, ␲ ϭ ␪ ϭ seek an accurate fit to (6.4), but instead fitted his func- 1 and (6.5) reduces to W TW. The method gives the

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␪ Ϫ ␪ FIG. 11. Scatter diagram of adiabatic wet-bulb temperature Tw vs W A. The plus signs

mark points along the Wobus curve WF(TW).

␪ ␪ correct answer at 1000 mb despite the errors in W as a The error in the Wobus estimate of W at a grid point ␪ ␪ ␪ ␲ Ϫ ␲ Ϫ Ϫ function of E and in A as a function of that exist is defined as TW / WF(TW / ) WF(TW) the true ␪ because (6.4) and (6.3) are not satisfied exactly. W, where TW is the accurate value determined by the In the special case when the parcel is initially unsat- new method [not the one computed as the solution of

urated, then (6.5) applies at its saturation point (TL, (6.5) by the Wobus routines]. This error was computed ␲ ␲ ␪ L) instead of at its initial location (TW, ). In this case at each grid point and plotted on a ( w, p) diagram (Fig. (6.5) becomes 13). The maximum error is 0.58 K over the whole of the domain, and 0.53 K over the region defined by ␪ Յ ␪ ͑ ␲ ͒ ϭ ր␲ Ϫ ͑ ր␲ ͒ ϩ ͑ ␲ ͒ w W TL, L TL L W* TL L,1 W* TL, L . 28°C. The associated error in the temperature of a par- ͑6.6͒ cel lifted from 1000 mb adiabatically to its lifted con- densation level (LCL) and then pseudoadiabatically to If the pressure dependency of W is disregarded and 200 mb ranges up to 1.2 K (Fig. 14). the original Wobus function is used in (6.5), we get the

equation that Wobus solved for TW, 7. The generalized Wobus method reduces to the ␪ ͑ ␲͒ ϭ ր␲ Ϫ ͑ ր␲͒ ϩ ͑ ͒ W TW, TW WF TW WF TW . new method ͑6.7͒ We now show that the generalized Wobus method, To solve (6.7), Wobus used the secant method (Gerald which is obtained above by allowing the Wobus func- and Wheatley 1984), which, starting from his initial tion to have some pressure dependency, reduces to the ϭ ␪ ␲ guess TW W / , achieved convergence within a few new method. First, note that Eqs. (3.5)–(3.7) or (3.8) ␪ ␲ ϭ ␪ ␲ iterations. can be written as W(TW, ) X[ E(TW, )], where X

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␪ Ϫ ␪ FIG. 12. Contour lines of Tw (solid for positive and long dashes for negative values) and of W A ␪ ␪ Ϫ ␪ (short dashes) on a ( w, p) diagram. The parentheses at bottom left and right are for Tw and ( W A), respectively. The quantities inside the parentheses are the same as in Fig.7.

␲ is a function that maps EPT into the corresponding WBPT becomes TW / . Therefore, at the new location ␲ WBPT. Hence, for a parcel that is initially saturated at (TW / , 1) on the pseudoadiabatic diagram, (T , ␲), W ͑ ր␲ ͒ ϭ ր␲ Ϫ ␪ ͑ ␲͒ ͑ ͒ W* TW ,1 TW A TW, . 7.3 ͑ ␲͒ ϭ ͓␪ ͑ ␲͔͒ Ϫ ␪ ͑ ␲͒ ͑ ͒ W* TW, X E TW, A TW, . 7.1 Substituting (7.1) and (7.3) into (6.5) yields ␪ ␲ ␪ ϭ ͓␪ ͑ ␲͔͒ ͑ ͒ Since A(TW, ) is by definition the WBPT associated W X E TW, , 7.4 with an EPT T /␲ (the parcel’s potential temperature), W which is the basis of the new method. Thus, the gener- we also have ␪ (T , ␲) ϭ X(T , ␲). Therefore, the A W W alized Wobus method is equivalent to, but more con- relationship between the generalized Wobus function voluted than, the new method. and the X function is

͑ ␲͒ ϭ ͓␪ ͑ ␲͔͒ Ϫ ͑ ր␲͒ ͑ ͒ W* TW, X E TW, X TW . 7.2 8. How the original Wobus method works to a degree If this parcel is brought down dry adiabatically to 1000 ␲ mb, its new temperature is TW / and its desiccated Despite its widespread use, no one seems to know ␪ ␲ WBPT remains A(TW, ). However, its saturated why the Wobus function W is primarily a function of

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␪ FIG. 13. The error in the unmodified Wobus-method determination of W from (6.7) as a function of ␪ w and p. Contour lines with positive (negative) values are solid (dashed). The quantities inside the parentheses are the same as in Fig. 7.

temperature and consequently why the Wobus method tuned in favor of the higher arguments appearing as ␪.” works to about 1-K precision. Doswell et al. (1982) The Wobus method works reasonably well only if the claim that W is a function only of temperature because pressure dependency of the Wobus function is small. “the amount of needed to saturate a parcel We can show this over the WBPTs most likely to occur ␪ Յ is dependent only upon its temperature,” forgetting in the atmosphere ( w 28.2°C) as follows. From (3.6) that it is saturation vapor pressure es, not saturation the function X is given to a very good approximation by mixing ratio rs, that is a function of temperature alone. ͑␪ ͒ ϭ ϩ Although Wobus is the last author on the Doswell et al. X E K1 C paper, he apparently was aware that W had a slight ␭ C for 257 Յ ␪ Յ 377 K dependence on pressure because, in a letter to Dr. Jo- Ϫ ͩ ͪ ͭ E K2 ␪ Ϫ Յ ␪ Յ Њ , seph Schaefer dated 3 November 1975, he stated that E or 18.6 w 28.2 C ␪ “A more accurate approximation of W is possible by ͑8.1͒ using two slightly different functions for the two argu- ␪ ϭ ␪ Ϫ ␪ ϩ ϭ ϭ ments [i.e., W W1( ) W2(TW)]. This would where K1 45.114 and K2 51.489 K. We progress ␪ permit the function to be tuned in favor of the lower further by using the simplest formula for E that is still value arguments as used for TW and the other to be quite precise. This is Bolton’s (1980) Eq. (35), which is

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FIG. 14. Error of Wobus’s method in the temperature of a parcel lifted from 1000 to 200 mb as a function of initial temperature and dewpoint depression. Conventions are the same as in Fig. 13.

a slight modification of Betts and Dugan’s (1973) for- C ␭ *͑ ␲͒ ϭ ͩ ͪ ␲ ␭ mula. Applying it to any saturated parcel yields W G TW, K2 TW Ϫ␭ ␧ ͑ ͒ ␪ ϭ ␪ ͓ ͑ ͒ր ͔ ͑ ͒ A es TW Ϫ␭ E exp Ars TW, p TW , 8.2 ϫ ͭ1 Ϫ expͫ ␲ ͬͮ for p0TW where A (a surrogate for L/c ) ϭ 2675 K when r is in ␪ Ն Ϫ Њ ␪ Յ Њ ͑ ͒ pd s a 18.6 C, w 28.2 C. 8.4 units of grams per gram. Raising (8.2) to the Ϫ␭ power and substituting the usual approximate expression for rs Figure 15 shows that for EPTs less than 377 K, W*G gives us varies only slightly with pressure. It is easily verified by series expansion that ͑ ր␪ ͒␭ Ϸ ͑ ␲ր ͒␭ ͓Ϫ␭ ␧ ͑ ͒ր ͔ ͑ ͒ C E C TW exp A es TW pTW , 8.3 Ϫ␭A␧e ͑T ͒ ␲ ␭ Ϫ s W ␲ Ϫ␭ where ␧ϭ0.6220 is the ratio of the gas constants for dry ͭ1 expͫ ͬͮ p0TW air and water vapor. Ϫ␭ ␧ ͑ ͒ Substituting (8.1) and (8.3) into (7.2) provides us with A es TW Ϸ 1 Ϫ expͫ ͬ ͑8.5͒ a generalized Wobus function: p0TW

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␲ FIG. 15. The Wobus function W*G(TW, )vsTw at constant pressures of 1000, 900, ...,300 mb. The curves are truncated at the data points marked by | signs by excluding data from ␪ Ն ␪ Ն points where E 377K( w 28.2°C).

to second order in the arguments of the exponentials. The inclusion of ␸ reduces the maximum relative error ␪ Inserting this approximation into (8.4) gives us an ex- in the computed W from 0.87 to 0.25 K (Figs. 16 and ␪ Յ pression for the pressure-independent Wobus function: 17) for w 28.2°C. (Note that we can safely exclude ␪ Ն Ϫ ␸ the lower range limit a 18.6°C because is negli- C ␭ gible at cold WBPTs.) Thus, the estimated pressure cor- ͑ ͒ ϭ ͩ ͪ WG TW K2 rection is indeed quite small, which explains why the TW original Wobus method works fairly well. Ϫ␭ ␧ ͑ ͒ A es TW ϫ ͭ1 Ϫ expͫ ͬͮ for p0TW 9. Conclusions ␪ Ն Ϫ Њ ␪ Յ Њ ͑ ͒ a 18.6 C, w 28.2 C, 8.6 ␪ A new method for computing w and the adiabatic ␸ ␲ Ϸ ␲ Ϫ wet-bulb temperature along pseudoadiabats is pre- which has an intrinsic error (TW, ) W*G(TW, ) ␸ ϭ ϭ sented. Currently Wobus’s method is widely used for WG(TW), where (TW,1) 0 because WG(TW) these purposes. It is based on a Wobus function W that W*G(TW, 1). A corrected version of (6.7) that uses the pressure-independent Wobus function (8.6) is is supposedly a function only of temperature. However, therefore W has a slight dependency on pressure, which gives rise ␪ to errors of over half a degree in w and to errors up to ␪ ͑ ␲͒ ϭ ր␲ Ϫ ͑ ր␲͒ ϩ ͑ ͒ 1.2 K in the temperature of parcels that are lifted adia- W TW, TW WG TW WG TW batically and then pseudoadiabatically to 200 mb. Al- ϩ ␸͑ ␲͒ ͑ ͒ TW, . 8.7 though a new Wobus function of both temperature and

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␪ FIG. 16. The error in kelvins in the determination of W from (6.7) using the pressure-independent ␪ ␪ ϭ Wobus function WG as a function of w and p. The vertical line marks w 28°C to facilitate comparison with Fig. 17.

pressure is devised in this paper, the resulting modified Since the algorithm has a relative error after one itera- Wobus method is then just a convoluted version of the tion that is much smaller, its overall error is Յ0.2 K. new method. With a few minor changes, the procedure also finds the The new technique is based on Bolton’s (1980) for- temperature on water- or ice-saturation reversible adia- ␪ mula for E. The temperature Tw on a given pseudo- bats. adiabat at a given pressure is obtained from this for- Part of the initial solution is a linear relationship (4.9) mula by an iterative technique. A very good “initial- or (4.10) between wet-bulb temperature and equivalent Ϫ ␬ guess” formula for Tw is devised. In the pressure range temperature raised to the 1/ d power in a significant 100 Յ p Յ 1050 mb and wet-bulb potential temperature region of a thermodynamic diagram. This appears to be ␪ Յ range W 40°C, this formula is accurate to within 0.34 an interesting new discovery. K of the iterated solution. With only one iteration, the relative error is reduced to less than 0.02 K. There is an Acknowledgments. I acknowledge the ingenuity of ␪ absolute error of up to 0.2 K in E in Bolton’s formula the Wobus method, which was ahead of its time. My

that is caused mostly by variation of cpd with tempera- sporadic attempts over the years to discover how it ture and pressure. It is shown that the upper bound on worked enabled me to invent the new method. Valu-

the corresponding absolute error in Tw is also 0.2 K. able suggestions from the two anonymous reviewers led

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FIG. 17. Same as in Fig. 16, but the error has been reduced by the derived correction ␸ in the region ␪ Յ of the grid for which the correction is either valid or negligible ( w 28°C). to significant improvements in the paper. This work ѨG k k ͩ ͪ ϭϪ 0 ͑ ϩ 2͒ ϩ ͩ 0 Ϫ ͪ͑ ϩ ͒ 2 rs k2rs k1 1 2k2rs was supported in part by NSF Grant ATM-0340693. Ѩ␶ ␲ ␶ ␶ Ѩ rs APPENDIX ϫ ͩ ͪ , ͑A.3͒ Ѩ␶ ␲ The Derivatives of f Ѩ ␧ rs p des The first derivative of f(␶; ␲) at fixed pressure is ͩ ͪ ϭ , ͑A.4͒ Ѩ␶ ͑ Ϫ ͒2 d␶ given by ␲ p es Ј͑␶ ␲͒ ϵ ͑Ѩ րѨ␶͒ ϭ ͑Ѩ րѨ␶͒ ͑ ͒ f ; f ␲ f lnf ␲, A.1 des ab ϭ e . ͑A.5͒ ␶ s 2 where ␶ is in kelvins and d ͑␶ Ϫ C ϩ b͒

Ѩ lnf 1 ␯ de p Ѩr ͩ ͪ ϭϪ␭ͫ ϩ s ϩ ␬ ͩ ͪͩ sͪ The second derivative of f(␶;␲) at fixed ␲ is given by Ѩ␶ ␶ Ϫ ␶ k3 d ln Ѩ␶ ␲ p es d p0 ␲ 2 2 2 2 ѨG f Љ͑␶; ␲͒ ϵ ͑Ѩ fրѨ␶ ͒␲ ϭ fЈ͑Ѩ lnfրѨ␶͒␲ ϩ f͑Ѩ lnfրѨ␶ ͒␲, ϩ ͑ ͒ ͩ Ѩ␶ ͪ ͬ, A.2 ␲ ͑A.6͒

Unauthenticated | Downloaded 09/25/21 02:54 PM UTC JULY 2008 DAVIES-JONES 2785 where for adiabatic condensation temperature. Mon. Wea. Rev., 111, 1119–1121. Ѩ2 lnf 1 ␯ de 2 ␯ d 2e Doswell, C. A., III, J. T. Schaefer, D. W. McCann, T. W. Schlat- ͩ ͪ ϭ ␭ͫ Ϫ ͩ sͪ Ϫ s Ѩ␶ 2 ␶ 2 ͑ Ϫ ͒2 d␶ p Ϫ e ␶ 2 ter, and H. B. Wobus, 1982: Thermodynamic analysis proce- ␲ p es s d dures at the National Severe Storms Forecast Center. Pre- prints, Ninth Conf. on Weather Forecasting and Analysis, Se- p Ѩ2r Ѩ2G Ϫ ␬ ͩ ͪͩ sͪ Ϫ ͩ ͪ ͬ attle, WA, Amer. Meteor. Soc., 304–309. k3 d ln 2 2 , p0 Ѩ␶ ␲ Ѩ␶ ␲ Emanuel, K. A., 1994: Atmospheric Convection. Oxford Univer- sity Press, 580 pp. ͑A.7͒ Gerald, C., and P. Wheatley, 1984: Applied Numerical Analysis. 3rd ed. Addison-Wesley, 576 pp. Ѩ2 Ѩ G 2k0 2k0 rs Hamill, T. M., and A. T. Church, 2000: Conditional probabilities ͩ ͪ ϭ ͑r ϩ k r 2͒ Ϫ ͑1 ϩ 2k r ͒ͩ ͪ Ѩ␶ 2 ␶ 3 s 2 s ␶ 2 2 s Ѩ␶ of significant tornadoes from RUC-2 forecasts. Wea. Fore- ␲ ␲ casting, 15, 461–475. k Ѩr 2 k Hart, J. A., J. Whistler, R. Lindsay, and M. Kay, 1999: NSHARP, ϩ ͩ 0 Ϫ ͪ ͩ sͪ ϩ ͩ 0 Ϫ ͪ version 3.10. Storm Prediction Center, National Centers for ␶ k1 2k2 Ѩ␶ ␶ k1 ␲ Environmental Prediction, Norman, OK, 33 pp. Henrici, P., 1964: Elements of Numerical Analysis. Wiley, 328 pp. Ѩ2 rs List, R. J., 1971: Smithsonian Meteorological Tables. 6th ed. ϫ ͑1 ϩ 2k r ͒ͩ ͪ , ͑A.8͒ 2 s Ѩ␶ 2 Smithsonian Institute Press, 527 pp. ␲ Prosser, N. E., and D. S. Foster, 1966: Upper air sounding analysis Ѩ2 ␧ 2 2 by use of an electronic computer. J. Appl. Meteor., 5, 296–300. rs p d es 2 des ͩ ͪ ϭ ͫ Ϫ ͩ ͪ ͬ, ͑A.9͒ Rasmussen, E. N., 2003: Refined supercell and tornado forecast Ѩ␶ 2 ͑ Ϫ ͒2 ␶ 2 Ϫ ␶ ␲ p es d p es d parameters. Wea. Forecasting, 18, 530–535. Saunders, P. M., 1957: The thermodynamics of saturated air: A 2 d es ab des 2es contribution to the classical theory. Quart. J. Roy. Meteor. ϭ ͩ Ϫ ͪ. ͑A.10͒ ␶ 2 ͑␶ Ϫ ϩ ͒2 d␶ ␶ Ϫ C ϩ b Soc., 83, 342–350. d C b Scheid, F., 1989: Schaum’s Outline of Theory and Problems of Numerical Analysis. 2nd ed. McGraw-Hill, 471 pp. REFERENCES Simpson, R. H., 1978: On the computation of equivalent potential temperature. Mon. Wea. Rev., 106, 124–130. Betts, A. K., and F. J. Dugan, 1973: Empirical formula for satu- Stackpole, J. D., 1967: Numerical analysis of atmospheric sound- ration pseudoadiabats and saturation equivalent potential ings. J. Appl. Meteor., 6, 464–467. temperature. J. Appl. Meteor., 12, 731–732. Tetens, O., 1930: Über einige meteorologische Begriffe. Z. Geo- Bolton, D., 1980: The computation of equivalent potential tem- phys., 6, 297–309. perature. Mon. Wea. Rev., 108, 1046–1053. Thompson, R. L., R. Edwards, J. A. Hart, K. L. Elmore, and P. Browning, K. A., and R. J. Donaldson Jr., 1963: Airflow and Markowski, 2003: Close proximity soundings within supercell structure of a tornadic storm. J. Atmos. Sci., 20, 533–545. environments obtained from the Rapid Update Cycle. Wea. Davies-Jones, R. P., 1983: An accurate theoretical approximation Forecasting, 18, 1243–1261.

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