## 1. Introduction

Numerical analyses of actual and model-output upper-air soundings (e.g., Prosser and Foster 1966; Stackpole 1967; Doswell et al. 1982) are used to determine several weather forecast parameters [e.g., convective available potential energy (CAPE), CAPE in the lowest 3 km of the sounding, convective inhibition, level of free convection, height of the wet-bulb zero, bulk Richardson number, energy–helicity index, the height to which penetrative convection can reach, etc.] that identify environments that support various types of severe weather (e.g., Rasmussen 2003; Thompson et al. 2003) and that may factor in the forecast likelihood that a thunderstorm will produce a significant tornado in probabilistic models (e.g., Hamill and Church 2000). These parameters all require the computation of adiabatic wet-bulb temperature, *T _{w}*, along water-saturation pseudoadiabats. They should be calculated as accurately as possible because errors affect statistical measures of their forecast skill and also conditional tornado probabilities.

Given the initial state of a parcel, there is no simple way to compute its temperature during undiluted pseudoadiabatic ascent. In contrast, there are precise explicit formulas for equivalent potential temperature (EPT) *θ _{E}* (K) so we can easily calculate the parcel’s equivalent temperature

*T*during its ascent. Inconveniently, the equivalent temperature of a saturated parcel is a complicated function of

_{E}*T*both explicitly and implicitly through the dependence of the parcel’s saturation mixing ratio on its temperature. This has discouraged meteorologists from trying to invert a formula for

_{w}*T*to get an explicit expression for

_{E}*T*. The general view has been that the problem is mathematically intractable, and that solutions for

_{w}*T*can be obtained only through numerical integration, using small vertical steps, of the differential equation governing the pseudoadiabat or through iterative numerical techniques (e.g., Doswell et al. 1982). This paper demonstrates that there is in fact an explicit solution if errors up to 0.34 K relative to a converged solution are permitted. If greater accuracy is desired, this solution is an excellent first guess for an iterative method.

_{w}A variety of numerical techniques have been used to derive the temperature of a parcel lifted adiabatically (if initially unsaturated), then pseudoadiabatically (i.e., with all condensate instantly falling out) to some lower pressure, *p*, (e.g., Prosser and Foster 1966; Stackpole 1967; Doswell et al. 1982). In these procedures for the automated analyses of soundings, condensation temperature, *T _{L}*, which is needed for computation of

*θ*if the parcel is unsaturated initially, was determined by either a search technique (Prosser and Foster), by iteration (Stackpole), or by curve fitting (Doswell et al.). To compute the temperature along pseudoadiabats, Prosser and Foster used a computationally fast, but error-prone, scheme. First, they approximated the temperatures along three specific pseudoadiabats (the ones with wet-bulb potential temperatures of 10°, 20°, and 30°C) by third-order polynomials. Then they obtained the temperature of the lifted parcel by linear interpolation, after computing its wet-bulb potential temperature (WBPT)

_{E}*θ*from a crude empirical formula. Stackpole computed the difference between the EPT (via the imprecise Rossby formula) of the pseudoadiabat and that of a parcel at pressure

_{w}*p*with temperature given by the latest iterative solution. He also used the inefficient

*interval-halving*numerical procedure (Gerald and Wheatley 1984). Doswell et al. reported a similar technique, due to Hermann Wobus, with some important differences. The Wobus method employs the much faster secant method (Gerald and Wheatley 1984), which converges within a few iterations. It also uses the Wobus function, which was devised by Wobus in 1968. At the time of its invention, the Wobus method was much more efficient and faster than other methods. In lieu of

*θ*, it uses

_{E}*θ*, which is computed from the Wobus function,

_{w}*W*(

_{F}*T*), of absolute temperature

_{K}*T*

_{K}only.

The Wobus method is little known because it has never been documented previously in the formal literature. However, it is widely used because it is utilized unseen in the National Centers Skew–*T*/Hodograph Analysis and Research Program (NSHARP; Hart et al. 1999), which is the interactive software for upper-air profiles in the National Centers’ Advanced Weather Interactive Processing System/General Meteorological Package (N-AWIPS/GEMPAK; J. Hart 2007, personal communication). Since it is in wide use, its errors should be evaluated. Despite the advent of more recent empirical data and Bolton’s creation in 1980 of a highly accurate formula for EPT, the Wobus method has never been upgraded since its invention.

Bolton (1980) first obtained new empirical formulas for saturation vapor pressure and condensation temperature. With the use of these formulas, he accurately determined EPT as a function of condensation temperature and pressure by numerically integrating the differential equation for the pseudoadiabatic process from the saturation point to a great height. He then used the numerical results to obtain accurate formulas for EPT, of which his Eq. (39) is the most precise. Apart from the more accurate, but more complicated, formula for condensation temperature developed by Davies-Jones (1983), Bolton’s formulas are the most exact of their type. For initially saturated air, Bolton’s Eq. (39) is accurate to within 0.2 K in *θ _{E}* with this error mostly owing to variation of

*c*

_{pd}, the specific heat of dry air at constant pressure, with temperature and pressure (List 1971, his Table 88). Note that, although

*c*

_{pd}is treated as a constant, the variation of the specific heat of moist air at constant pressure,

*c*, with mixing ratio

_{p}*is*parameterized.

This paper devises a new accurate method for computing temperature along pseudoadiabats, and hence for reducing the errors involved in evaluating the above forecast parameters. First an efficient algorithm for inverting Bolton’s Eq. (39) to obtain the wet-bulb temperature along a given pseudoadiabat at a given pressure *p* is formulated (section 2). The output at 1000 mb from this algorithm is then used to determine empirical formulas for *θ _{w}* as a function of

*θ*(section 3). Over most of the atmospheric range of

_{E}*θ*(−19° <

_{W}*θ*< 29°C), a linear relationship is discovered between

_{w}*θ*and the −

_{w}*λ*power of

*θ*, where

_{E}*λ*≡ 1/

*κ*(=3.504) and

_{d}*κ*=

_{d}*R*/

_{d}*c*

_{pd}(=0.2854) is the Poisson constant for dry air. In section 4, highly accurate initial guesses for the computation of wet-bulb temperature along pseudoadiabats are derived. A new linear relationship between

*T*and the −

_{w}*λ*power of equivalent temperature

*T*(K) is found in a significant region of a thermodynamic diagram. One iteration of the algorithm then gives a highly accurate solution for

_{E}*T*. Next the algorithm is modified slightly for computation of temperature along

_{w}*reversible*adiabats (section 5). The Wobus method is described in section 6 and its intrinsic errors are evaluated and found to be quite large. Although the Wobus function is supposedly only a function of temperature, it in fact has a slight dependence on pressure. The linear relationship discovered in section 3 is used in section 7 to formulate a new more accurate Wobus function of both temperature and pressure. The modified Wobus method thus obtained is shown to be simply a convoluted version of the new method. The reason why the original Wobus method works fairly well is addressed in section 8 where it is shown that the error in

*T*caused by assuming that the Wobus function is independent of pressure is less than 1 K.

_{w}Before proceeding, we explain our terminology of errors. “Relative error” denotes the error relative to the converged solution of Bolton’s Eq. (39). “Absolute error” also includes the error inherent in Bolton’s Eq. (39) itself. The “intrinsic error” of the Wobus method refers to the error resulting from the assumption that the Wobus function is a function of just temperature.

## 2. Mathematical formulation of the new method

We start by developing the new method. We use Bolton’s nomenclature here, including his convention that a temperature with a capital subscript is in kelvins and one with a small subscript is in degrees Celsius. Temperatures with the same letter subscript but different case are the same variable in different units. The only departures from these rules are *T*, the temperature in degrees Celsius, *T _{K}*, the absolute temperature (=

*T*+

*C*, where

*C*= 273.15 K), and

*θ*, the potential temperature in kelvins. The one exception to Bolton’s nomenclature in this paper is the unit of mixing ratio, which is grams per gram instead of grams per kilogram. Any variable that can be determined uniquely from a thermodynamic diagram is a function of just two independent variables, chosen in the following analyses to be temperature and the nondimensional pressure

*π*≡ (

*p*/

*p*

_{0})

^{1/}

*.*

^{λ}*T*,

_{K}*π*) and involve its mixing ratio

*r*, its vapor pressure

*e*and

*T*. Bolton’s Eq. (39) is the most accurate. Simpson’s formula does not fit the same mathematical mould as the other formulas and so is not considered further here. We can apply the formulas to any saturated parcel at (

_{L}*T*,

_{W}*π*) simply by replacing

*T*by

_{L}*T*,

_{W}*r*by saturation mixing ratio

*r*(

_{s}*T*,

_{W}*π*), and

*e*by saturation vapor pressure

*e*(

_{s}*T*). When this is done, the remaining formulas all have the following form:whereand the constants are listed in Table 1. Note that we have used Bolton’s Eqs. (24) or (7) and the relationship

_{W}*T*=

_{E}*θ*to write the formulas in the form in (2.1). We can find the temperature at pressure

_{E}π*p*along a given pseudoadiabat with EPT

*θ*by solving for

_{E}*T*. However, for the reason given below, it is generally more advantageous to solve (2.1) raised to the −

_{W}*λ*power (i.e., to solve for given

*π*and

*θ*):where

_{E}*λ*= 1/

*κ*=

_{d}*c*

_{pd}/

*R*= 3.504. In (2.3)

_{d}*T*and

_{W}*T*have been scaled by

_{E}*C*simply to avoid large numbers. By Taylor series expansion about a temperature

*τ*(K) at constant

*π*,[The notation

*f*(

*τ*;

*π*) indicates that

*f*is a function of

*τ*with

*π*fixed.] At each pressure ≥300 mb, we can choose

*τ*=

*τ** such that the remainder (the last term) is much smaller than the first-order term for

*T*∈ [

_{W}*τ** − 20°C,

*τ** + 20°C] (this can be deduced from Fig. 1). In other words, there is the almost linear relationship between (

*C*/

*T*)

_{E}*and*

^{λ}*T*in this interval,where an expression for

_{W}*f*′(

*τ*;

*π*)is provided in the appendix for the reader’s convenience. This anticipates our later finding that, with a good initial guess

*τ*

_{0}, one or two iterations of the algorithm,in Newton’s method always provide a precise numerical solution,

*T*, of (2.3). We can accelerate the convergence by retaining the second-order term in the series expansion and solving the resulting quadratic equation:in a form that is accurate for a small second-order term (see Henrici 1964, p. 199). This giveswhere

_{W}*f*″(

*τ*;

*π*) is given in the appendix. We choose the root that is closest to the linear

*τ*

_{n}_{+1}provided by (2.6).

In this paper, we use Bolton’s Eq. (39) as the basis for the computation of *T _{W}*. However, we can compute

*T*from Bolton’s Eqs. (28), (35), or (38), or even compute temperature along water- or ice-saturation reversible adiabats from Saunders’s (1957) Eqs. (3) or (4), simply by changing a few parameters in the computer code as dictated by Table 1.

_{W}*τ*

_{0}that is accurate to within 10 K is given bywhere

*θ*is obtained from

_{W}*θ*via a formula obtained in section 3. This is sufficient for convergence of the algorithm, but not optimal. A far more accurate initial estimate is based on results presented below and so is supplied later in Eqs. (4.8)–(4.11).

_{E}## 3. Computing *θ*_{W} from *θ*_{E}

_{W}

_{E}

*θ*, we find from (2.3) applied at 1000 mb that

_{E}*θ*is the solution ofwhereOne linear iteration of Newton’s method with a first guess of

_{W}*C*provides the following solution:which is valid in some interval around

*C*. This interval turns out to be fairly large owing to the small second derivative of

*f*(

*θ*;1). A plot (Fig. 2) of the actual (i.e., converged iterative) solution for

_{w}*θ*as a function of (

_{w}*C*/

*θ*)

_{E}*shows that the linear solution in (3.3) is approximately valid in the interval −19° <*

^{λ}*θ*< 29°C. The minimax-polynomial approximation method (Scheid 1989) was used to obtain the minimax line in this interval,which fits the solution to 0.1°C. Note that (3.3) and (3.4) are very similar.

_{w}Is there a better linear relationship than (3.4) between *θ _{w}* and another power of (

*θ*/

_{E}*C*), say (

*θ*/

_{E}*C*)

^{−}

*? To answer this question, the standard error of the*

^{μ}*θ*predicted by linear regression over the interval −19°C <

_{w}*θ*< 29°C was computed for different values of

_{w}*μ*. The minimum standard error (0.06 K) occurred for

*μ*≈ 3.5, thus confirming that

*μ*=

*λ*produces the most linearity.

*θ*because

_{E}*θ*cannot remain finite as

_{w}*θ*tends to infinity and at cold values of

_{E}*θ*because

_{E}*θ*tends to

_{W}*θ*as the saturation mixing ratio becomes small. For 377 ≤

_{E}*θ*< 674 K (28.2° ≤

_{E}*θ*< 50°C), a minimax polynomial was fitted to the difference between the actual and the linear solutions. For

_{w}*θ*≤ 257 K (

_{E}*θ*≤ −18.6°C), one iteration of Newton’s method applied directly to Bolton’s Eq. (35) version of (2.1)–(2.2) with

_{w}*T*=

_{E}*θ*,

_{E}*T*=

_{w}*θ*, and

_{w}*π*= 1 suffices. After some minor approximations, the resulting solution iswhere a small term has been neglected in (3.5) and

*A*= 2675 K,

*C*= 273.15 K,

*λ*=

*c*

_{pd}/

*R*,

_{d}*r*(

_{s}*T*, 1) = ε

_{K}*e*(

_{s}*T*)/[

_{K}*p*

_{0}

*π*−

^{λ}*e*(

_{s}*T*)],

_{K}*e*(

_{s}*T*) =

_{K}*e*(

_{s}*C*) exp[

*a*(

*T*−

_{K}*C*)/(

*T*−

_{K}*C*+

*b*)],

*d*ln

*e*(

_{s}*T*)/

_{K}*dT*=

_{K}*ab*/(

*T*−

_{K}*C*+

*b*)

^{2}, ε = 0.6220,

*e*(

_{s}*C*) = 6.112 mb,

*a*= 17.67, and

*b*= 243.5 K. Remarkably, this solution fits the actual solution to 0.1°C. It should be emphasized, however, that the accuracy of Bolton’s Eq. (39) is not known beyond

*θ*= 40°C (

_{w}*θ*= 478.4 K).

_{E}*θ*= −30°, −20°, . . . , 40°, 50°C. The resulting approximate solution iswhere

_{w}*X*≡

*θ*/

_{E}*C*and

*a*

_{0}= 7.101574,

*a*

_{1}= −20.68208,

*a*

_{2}= 16.11182,

*a*

_{3}= 2.574631,

*a*

_{4}= −5.205688,

*b*

_{1}= −3.552497,

*b*

_{2}= 3.781782,

*b*

_{3}= −0.6899655,

*b*

_{4}= −0.5929340. For

*θ*≤ 50°C, its maximum deviation from the actual solution (the relative error) is 0.02 K, and it is within 0.005 K for

_{w}*θ*∈ (−20°, 40°C), the range of

_{w}*θ*tabulated in the Smithsonian tables (List 1971, see his Table 78). The magnitude of the argument in the exponential in (3.8) becomes large at temperatures less than −100°C, and the rational-function approximation fails. However, at these temperatures we can assume that

_{w}*θ*=

_{W}*θ*with negligible error (≤10

_{E}^{−4}K).

*δθ*of up to 0.2 K in the value of

_{E}*θ*provided by Bolton’s Eq. (39), there is a corresponding error in

_{E}*θ*given, from (3.6), byFor

_{W}*δθ*= 0.2 K,

_{E}*δθ*varies from 0.17 K at

_{W}*θ*= 257 to 0.05 K at

_{E}*θ*= 335 to 0.03 K at

_{E}*θ*= 377 K. The absolute error in

_{E}*θ*becomes quite small at the EPTs most often associated with atmospheric convection.

_{W}## 4. The solution for *T*_{W} as a function of *T*_{E} and *p*

_{W}

_{E}

The first step toward finding an efficient and accurate algorithm for computing temperature along pseudoadiabats is to obtain the “true” values that are produced by precisely inverting Bolton’s Eq. (39). We apply this formula initially to saturated parcels at 1000 mb to acquire the *θ _{E}* values corresponding to

*θ*= −20°, −18°, . . . , 40°C. We then obtain the wet-bulb temperatures

_{w}*T*(

_{W}*θ*,

_{W}*p*) along these 31 pseudoadiabats at 25-mb intervals from 1050 to 100 mb (39 different pressures) to a tolerance of 5 × 10

^{−5}K by Newton’s algorithm (2.6) applied to Bolton’s Eq. (39) raised to the −1/

*κ*power. We refer to the array of 31 × 39 (=1209) points in (

_{d}*θ*,

_{w}*p*) space as the “grid.”

Plots (Fig. 3) of the actual solution of (2.3) as a function of (*C/T _{E}*)

*at selected pressures have some of the same characteristics as Fig. 2. In each plot, there is a range in which the solution (marked by Xs) is almost linear and an immediately adjacent range at cold equivalent temperatures, where*

^{λ}*T*approaches

_{W}*T*, and the solution becomes asymptotic to the curve

_{E}*T*=

_{W}*Cx*

^{−1/}

*, where*

^{λ}*x*is the abscissa (

*C*/

*T*)

_{E}*. The slopes and intercepts of the linear parts, and the transition points S where the solutions depart from linearity toward their asymptotes all vary with pressure. The solution also becomes nonlinear at the warmest equivalent temperatures [(*

^{λ}*C*/

*T*)

_{E}*< 0.4]. This is clearly visible at 1000 mb (Fig. 2), but hardly evident at 850 mb (Fig. 3).*

^{λ}*τ** is any value of

*T*such that the horizontal line

_{w}*T*=

_{w}*τ** intersects the linear portion of the curve. By inspection

*τ** = −50 (1 −

*π*) always lies on the linear part of the solution curve (Figs. 4 and 1) and is used here as the points E in Figs. 2 and 3, where

*k*

_{1}(

*π*) and

*k*

_{2}(

*π*) are evaluated for the 39 different pressures. The coefficient −50 was chosen because it is nearly optimal for minimizing the maximum absolute error over the grid. Fitting quadratic regression curves to the results yields the following expressions:with correlation coefficients,

*r*, of 1 (Figs. 5 and 6). Remarkably,

*k*

_{2}is almost linear in

*π*with the regression line:

*C*/

*T*)

_{E}*, and determining by linear interpolation the value*

^{λ}*D*(

*p*) of (

*C*/

*T*)

_{E}*where the magnitudes of the errors are equal. Then a regression line was fitted to the reciprocal of the data for*

^{λ}*D*(

*p*). The resulting equation,fits the data adequately enough (Fig. 4). The transition points at particular pressure levels are plotted in Figs. 2 and 3.

Two empirical corrections were applied to (4.1). For *T _{E}* >

*C*, the coefficients were adjusted slightly, and at warm equivalent temperatures (

*T*> 355 K), an additional term in (

_{E}*C*/

*T*)

_{E}^{−}

*was added and the constant term adjusted to describe the “warm-side” nonlinearity.*

^{λ}*T*iswhere

_{w}*A*= 2675 K,

*C*= 273.15 K,

*λ*=

*c*

_{pd}/

*R*, and

_{d}*k*

_{1}(

*π*),

*k*

_{2}(

*π*), and

*D*(

*p*) are given by (4.2) and (4.7). The regions in which the different parts of the initial solution apply are shown in Fig. 4.

The relative errors in the initial guess and that after one iteration were computed. Finally, as a check the EPT was computed again from Bolton’s Eq. (39) using the one-iteration solution for *T _{w}*. The largest difference between recomputed and original values of EPT was less than 0.002 K.

The two empirical corrections reduce the maximum relative error at any grid point in the initial guess from 1.8 to 0.34 K (Fig. 7). One ordinary (accelerated) iteration then reduces this relative error to less than 0.002 K (0.001 K), which is more than sufficient. When *k*_{1}(*π*) and *k*_{2}(*π*) are approximated by the quadratic regression curves in (4.3) and (4.4), the maximum relative error in the initial solution increases to 0.47 K. When linear regression is used for *k*_{2} [i.e., when (4.5) is used instead of (4.4)], the initial relative error is slightly larger (0.52 K).

The overall accuracy of the algorithm is determined almost entirely by its absolute error. The absolute error of up to 0.2 K in *θ _{E}* in Bolton’s Eq. (39) is caused mostly by variation of

*c*

_{pd}with temperature and pressure. The effect of a 0.2-K error in

*θ*on

_{E}*T*is shown in Fig. 8. Clearly the upper bound on the corresponding absolute error in

_{w}*T*is also 0.2 K. Since the algorithm’s relative error after one iteration is much smaller, its overall error is ≤0.2 K.

_{w}## 5. Computation of temperature along reversible adiabats

*T*along reversible water-saturation adiabats is(Saunders 1957). We make the usual assumption that

_{R}*c*, the specific heat of liquid water, is 4190 J kg

_{W}^{−1}K

^{−1}(e.g., Saunders 1957; Bolton 1980; Emanuel 1994), even though it increases by 8%, 14%, and 30% over this value at temperatures of −30°, −40°, −50°C, respectively (List 1971, see his Table 92). Equation (5.1) also fits the mold in (2.1) and (2.2) with the constants listed in the last column of Table 1. Since the mixing ratio of all phases of water to dry air,

*Q*, is conserved, this equation has the following integral:where

*A*

_{1}is a constant along a given reversible adiabat. The constant

*A*

_{1}is evaluated at a parcel’s saturation point. It depends on

*Q*so that through each point on a thermodynamic diagram there passes a unique pseudoadiabat and an infinity of reversible adiabats, one for each value of

*Q*(Saunders 1957). Raising (5.2) to the −

*λ*power gives the following version of (2.3):where

*ν*≡

*R*/(

_{d}*c*

_{pd}+

*c*).

_{W}QIt should be possible to find good initial guesses and identify near-linear relationships between *T _{R}* and (

*A*

_{1}

*π*)

^{−}

*in a region of the parameter space by the procedures used above for pseudoadiabatic ascent. Because of the complication of dealing with an additional parameter (*

^{λ}*Q*), this has been left for future work. Instead, we used the same initial guess for

*T*as the one for

_{r}*T*[see (4.8)–(4.11)], even though it is no longer accurate because retention of liquid water during ascent from low levels can make a parcel warmer by as much as 6 K at 100 mb (Emanuel 1994, p. 133). Initially and after one and two accelerated iterations, the maximum error relative to the converged solution is 6.25 and 0.034 K, respectively. With two ordinary iterations, the maximum relative error reduces from the initial 6.25 to 0.36 K and then 0.001 K. Figure 9 shows the difference

_{w}*T*−

_{r}*T*between the temperatures attained in reversible adiabatic expansion and pseudoadiabatic expansion from the same initial state at 900 mb. This figure is qualitatively similar to Fig. 2 in Saunders (1957), but there are quantitative differences owing to the use of up-to-date data.

_{w}The method also works for computation of the temperature along ice-saturation reversible adiabats with trivial substitutions. The specific heat of ice replaces that of water, the latent heat of sublimation supplants that of vaporization (Saunders 1957), the saturation mixing ratio is now with respect to ice instead of water, and *a* and *b* become Tetens’s (1930) ice coefficients (*a* = 21.87, *b* = 265.5 K).

## 6. The Wobus method

We now show that the Wobus method has an intrinsic error, owing to it being supposedly a function of just temperature, that makes it inferior to the new method described above. The Wobus function is defined as follows. At any point (*T _{K}*,

*π*) on a pseudoadiabatic (Stüve) diagram, one can consider two hypothetical wet-bulb potential temperatures

*θ*and

_{S}*θ*(Fig. 10). The saturated WBPT,

_{A}*θ*, is reached by saturating a parcel at (

_{S}*T*,

_{K}*π*), then bringing it down to 1000 mb (

*π*= 1) pseudoadiabatically. [Just enough rain is assumed to fall into and evaporate in the descending parcel to keep it saturated without leftover liquid water.] The dry WBPT,

*θ*, is attained by desiccating a parcel at (

_{A}*T*,

_{K}*π*), lifting it dry adiabatically to a great height and then bringing it down pseudoadiabatically to 1000 mb. The original Wobus function (

*W*

_{F}) of just temperature is the difference between these two temperatures, that is,

*W*

_{F}(

*T*) =

_{K}*θ*(

_{S}*T*,

_{K}*π*) −

*θ*(

_{A}*T*,

_{K}*π*). It is evaluated using Wobus’s numerical fit to the data.

Incidentally, *θ _{S}* is an important variable in its own right. The distribution of

*θ*and

_{W}*θ*with height,

_{S}*z*, determine atmospheric stability. A layer is potentially unstable if ∂

*θ*/∂

_{W}*z*< 0 and conditionally unstable if ∂

*θ*/∂

_{S}*z*< 0. The atmosphere is latently unstable if the

*θ*of any parcel lifted pseudoadiabatically exceeds the

_{W}*θ*of the unmodified environment at a higher level (see Fig. 12 in Browning and Donaldson 1963).

_{S}To test the claim that the Wobus function is a function solely of temperature, precise values of *T _{w}* and

*θ*−

_{W}*θ*at the 1209 grid points were computed and plotted on a scatter diagram (Fig. 11). The points tend to lie on the curve of the Wobus function

_{A}*W*

_{F}(

*T*), but there is some scatter, which indicates minor dependence on pressure. The pressure dependency is also evident in Fig. 12, which shows the slight misalignment of the contours of

_{W}*T*and

_{w}*θ*−

_{W}*θ*on a (

_{A}*θ*,

_{w}*p*) thermodynamic diagram, and maximum variation at constant temperature in

*θ*−

_{W}*θ*of almost 1 K.

_{A}*W**, asFor a

*saturated*parcel,

*T*is the same as its adiabatic wet-bulb temperature

_{K}*T*and

_{W}*θ*is its WBPT,

_{S}*θ*. Therefore,If this parcel descends dry adiabatically to 1000 mb, it still has the same potential temperature

_{W}*θ*(≡

*T*/

_{W}*π*here because Wobus imprecisely used the Poisson constant for

*dry*air) so its

*θ*is unchanged. However, its

_{A}*θ*is now

_{S}*θ*and its

*T*becomes

_{K}*θ*. Hence, by (6.1),If on the other hand, the parcel is lifted pseudoadiabatically to a great height and then brought down dry adiabatically to 1000 mb, its

*T*becomes equal to its equivalent potential temperature

_{K}*θ*(

_{E}*T*,

_{W}*π*), its new

*θ*is

_{S}*θ*and its new

_{E}*θ*is

_{A}*θ*(a bijective function of

_{W}*θ*). Thus, by (6.1),which indicates that at

_{E}*π*= 1 the Wobus function maps the EPT of a parcel to the difference between its EPT and its WBPT. According to Doswell et al. (1982), Wobus evaluated the right side of (6.4) using the data that gives

*θ*as a function of

_{E}*θ*in the header of Table 78 of the

_{W}*Smithsonian Meteorological Tables*(List 1971) and then essentially fitted a high-order polynomial to the reciprocal of the right side to obtain an approximation to the Wobus function. However, the values of

*θ*computed from (6.4) with List’s (Bolton’s) values of

_{W}*θ*have a maximum error of 0.66 K (0.53 K). There are similar errors in the computation of

_{E}*θ*from

_{A}*θ*using (6.3). Errors of these magnitudes suggest that Wobus did not seek an accurate fit to (6.4), but instead fitted his function to the more general equation in (6.2) using data from parcels at many different pressure levels (not just at 1000 mb).

*θ*from (6.2) and (6.3) gives a formula for

_{A}*θ*The temperature

_{W}:*T*at a pressure

_{W}*p*along a pseudoadiabat with WBPT

*θ*can be found as the solution of (6.5). Note that for a saturated parcel at 1000 mb,

_{W}*π*= 1 and (6.5) reduces to

*θ*=

_{W}*T*. The method gives the correct answer at 1000 mb despite the errors in

_{W}*θ*as a function of

_{W}*θ*and in

_{E}*θ*as a function of

_{A}*θ*that exist because (6.4) and (6.3) are not satisfied exactly.

*T*,

_{L}*π*) instead of at its initial location (

_{L}*T*,

_{W}*π*). In this case (6.5) becomes

*W*is disregarded and the original Wobus function is used in (6.5), we get the equation that Wobus solved for

*T*,To solve (6.7), Wobus used the secant method (Gerald and Wheatley 1984), which, starting from his initial guess

_{W}*T*=

_{W}*θ*/

_{W}*π*, achieved convergence within a few iterations.

The error in the Wobus estimate of *θ _{W}* at a grid point is defined as

*T*/

_{W}*π*−

*W*

_{F}(

*T*/

_{W}*π*) −

*W*

_{F}(

*T*) − the true

_{W}*θ*, where

_{W}*T*is the accurate value determined by the new method [not the one computed as the solution of (6.5) by the Wobus routines]. This error was computed at each grid point and plotted on a (

_{W}*θ*,

_{w}*p*) diagram (Fig. 13). The maximum error is 0.58 K over the whole of the domain, and 0.53 K over the region defined by

*θ*≤ 28°C. The associated error in the temperature of a parcel lifted from 1000 mb adiabatically to its lifted condensation level (LCL) and then pseudoadiabatically to 200 mb ranges up to 1.2 K (Fig. 14).

_{w}## 7. The generalized Wobus method reduces to the new method

*θ*(

_{W}*T*,

_{W}*π*) =

*X*[

*θ*(

_{E}*T*,

_{W}*π*)], where

*X*is a function that maps EPT into the corresponding WBPT. Hence, for a parcel that is initially saturated at (

*T*,

_{W}*π*),Since

*θ*(

_{A}*T*,

_{W}*π*) is by definition the WBPT associated with an EPT

*T*/

_{W}*π*(the parcel’s potential temperature), we also have

*θ*(

_{A}*T*,

_{W}*π*) =

*X*(

*T*,

_{W}*π*). Therefore, the relationship between the generalized Wobus function and the

*X*function isIf this parcel is brought down dry adiabatically to 1000 mb, its new temperature is

*T*/

_{W}*π*and its desiccated WBPT remains

*θ*(

_{A}*T*,

_{W}*π*). However, its saturated WBPT becomes

*T*/

_{W}*π*. Therefore, at the new location (

*T*/

_{W}*π*, 1) on the pseudoadiabatic diagram,Substituting (7.1) and (7.3) into (6.5) yieldswhich is the basis of the new method. Thus, the generalized Wobus method is equivalent to, but more convoluted than, the new method.

## 8. How the original Wobus method works to a degree

*W*is primarily a function of temperature and consequently why the Wobus method works to about 1-K precision. Doswell et al. (1982) claim that

*W*is a function

*only*of temperature because “the amount of water vapor needed to saturate a parcel is dependent only upon its temperature,” forgetting that it is saturation vapor pressure

*e*, not saturation mixing ratio

_{s}*r*, that is a function of temperature alone. Although Wobus is the last author on the Doswell et al. paper, he apparently was aware that

_{s}*W*had a slight dependence on pressure because, in a letter to Dr. Joseph Schaefer dated 3 November 1975, he stated that “A more accurate approximation of

*θ*is possible by using two slightly different functions for the two arguments [i.e.,

_{W}*θ*=

_{W}*θ*−

*W*

_{1}(

*θ*) +

*W*

_{2}(

*T*)]. This would permit the function to be tuned in favor of the lower value arguments as used for

_{W}*T*and the other to be tuned in favor of the higher arguments appearing as

_{W}*θ*.” The Wobus method works reasonably well only if the pressure dependency of the Wobus function is small. We can show this over the WBPTs most likely to occur in the atmosphere (

*θ*≤ 28.2°C) as follows. From (3.6) the function

_{w}*X*is given to a very good approximation bywhere

*K*

_{1}= 45.114 and

*K*

_{2}= 51.489 K. We progress further by using the simplest formula for

*θ*that is still quite precise. This is Bolton’s (1980) Eq. (35), which is a slight modification of Betts and Dugan’s (1973) formula. Applying it to any saturated parcel yieldswhere

_{E}*A*(a surrogate for

*L*/

*c*

_{pd}) = 2675 K when

*r*is in units of grams per gram. Raising (8.2) to the −

_{s}*λ*power and substituting the usual approximate expression for

*r*gives uswhere ε = 0.6220 is the ratio of the gas constants for dry air and water vapor.

_{s}*W**

_{G}varies only slightly with pressure.

*φ*(

*T*,

_{W}*π*) ≈

*W**

_{G}(

*T*,

_{W}*π*) −

*W*

_{G}(

*T*), where

_{W}*φ*(

*T*, 1) = 0 because

_{W}*W*

_{G}(

*T*) =

_{W}*W**

_{G}(

*T*, 1). A corrected version of (6.7) that uses the pressure-independent Wobus function (8.6) is thereforeThe inclusion of

_{W}*φ*reduces the maximum relative error in the computed

*θ*from 0.87 to 0.25 K (Figs. 16 and 17) for

_{W}*θ*≤ 28.2°C. (Note that we can safely exclude the lower range limit

_{w}*θ*≥ −18.6°C because

_{a}*φ*is negligible at cold WBPTs.) Thus, the estimated pressure correction is indeed quite small, which explains why the original Wobus method works fairly well.

## 9. Conclusions

A new method for computing *θ _{w}* and the adiabatic wet-bulb temperature along pseudoadiabats is presented. Currently Wobus’s method is widely used for these purposes. It is based on a Wobus function

*W*that is supposedly a function only of temperature. However,

*W*has a slight dependency on pressure, which gives rise to errors of over half a degree in

*θ*and to errors up to 1.2 K in the temperature of parcels that are lifted adiabatically and then pseudoadiabatically to 200 mb. Although a new Wobus function of both temperature and pressure is devised in this paper, the resulting modified Wobus method is then just a convoluted version of the new method.

_{w}The new technique is based on Bolton’s (1980) formula for *θ _{E}*. The temperature

*T*on a given pseudoadiabat at a given pressure is obtained from this formula by an iterative technique. A very good “initial-guess” formula for

_{w}*T*is devised. In the pressure range 100 ≤

_{w}*p*≤ 1050 mb and wet-bulb potential temperature range

*θ*≤ 40°C, this formula is accurate to within 0.34 K of the iterated solution. With only one iteration, the relative error is reduced to less than 0.02 K. There is an absolute error of up to 0.2 K in

_{W}*θ*in Bolton’s formula that is caused mostly by variation of

_{E}*c*

_{pd}with temperature and pressure. It is shown that the upper bound on the corresponding absolute error in

*T*is also 0.2 K. Since the algorithm has a relative error after one iteration that is much smaller, its overall error is ≤0.2 K. With a few minor changes, the procedure also finds the temperature on water- or ice-saturation reversible adiabats.

_{w}Part of the initial solution is a linear relationship (4.9) or (4.10) between wet-bulb temperature and equivalent temperature raised to the −1/*κ _{d}* power in a significant region of a thermodynamic diagram. This appears to be an interesting new discovery.

I acknowledge the ingenuity of the Wobus method, which was ahead of its time. My sporadic attempts over the years to discover how it worked enabled me to invent the new method. Valuable suggestions from the two anonymous reviewers led to significant improvements in the paper. This work was supported in part by NSF Grant ATM-0340693.

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# APPENDIX

## The Derivatives of f

*f*(

*τ*;

*π*) at fixed pressure is given bywhere

*τ*is in kelvins andThe second derivative of

*f*(

*τ*;

*π*) at fixed

*π*is given bywhere

Parameters in (2.1) and (2.2) as they apply to Bolton’s Eqs. (28), (35), (38), and (39) for water-saturation pseudoadiabats and to Saunders’s Eq. (3) for water-saturation reversible adiabats. The latent heat of vaporization is given by *L* = *L*_{0} − *L*_{1}*T*, where the constants *L*_{0} = 2.501 × 10^{6} J kg^{−1} and *L*_{1} = 2.37 × 10^{3} J kg^{−1} K^{−1}. In the last column, *c _{W}* is the specific heat of water and

*Q*is the mixing ratio of total water to dry air.