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_E during its ascent. Inconveniently, the equivalent temperature of a saturated parcel is a complicated function of T_w 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 T_E to get an explicit expression for T_w. The general view has been that the problem is mathematically intractable, and that solutions for T_w 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.

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_E 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) θ_w 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 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_E, it uses θ_w, which is computed from the Wobus function, W_F(T_K), of absolute temperature 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_p, with mixing ratio 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_E (section 3). Over most of the atmospheric range of θ_W (−19° < θ_w < 29°C), a linear relationship is discovered between θ_w and the −λ power of θ_E, where λ ≡ 1/κ_d (=3.504) and 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_w and the −λ power of equivalent temperature T_E (K) is found in a significant region of a thermodynamic diagram. One iteration of the algorithm then gives a highly accurate solution for T_w. Next the algorithm is modified slightly for computation of temperature along 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_w caused by assuming that the Wobus function is independent of pressure is less than 1 K.

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/p₀)^1/λ.

In his Eqs. (28), (33), (35), (38), and (39), Bolton (1980) gives five formulas for the EPT. These are, respectively, the traditional but inaccurate Rossby formula, Bolton’s adjustment of the Simpson (1978) formula, his adaptation of the Betts and Dugan (1973) formula, and two new formulas. The formulas are written for an unsaturated parcel at the point (T_K, π) and involve its mixing ratio r, its vapor pressure e and T_L. 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 (T_W, π) simply by replacing T_L by T_W, r by saturation mixing ratio r_s(T_W, π), and e by saturation vapor pressure e_s(T_W). When this is done, the remaining formulas all have the following form:

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where

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and the constants are listed in Table 1. Note that we have used Bolton’s Eqs. (24) or (7) and the relationship T_E = θ_Eπ to write the formulas in the form in (2.1). We can find the temperature at pressure p along a given pseudoadiabat with EPT θ_E by solving for T_W. However, for the reason given below, it is generally more advantageous to solve (2.1) raised to the −λ power (i.e., to solve for given π and θ_E):

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where λ = 1/κ_d = c_pd/R_d = 3.504. In (2.3) T_W and T_E have been scaled by C simply to avoid large numbers. By Taylor series expansion about a temperature τ (K) at constant π,

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[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)^λ and T_W in this interval,

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where an expression for f ′(τ ; π)is provided in the appendix for the reader’s convenience. This anticipates our later finding that, with a good initial guess τ₀, one or two iterations of the algorithm,

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in Newton’s method always provide a precise numerical solution, T_W, of (2.3). We can accelerate the convergence by retaining the second-order term in the series expansion and solving the resulting quadratic equation:

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

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where 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_W 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.

A first guess τ₀ that is accurate to within 10 K is given by

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where θ_W is obtained from θ_E 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).

3. Computing θ_W from θ_E

We first look at the problem of computing WBPT from EPT to gain insight into the more general problem of computing temperature on a given pseudoadiabat at a given pressure. Using Bolton’s Eq. (39) for θ_E, we find from (2.3) applied at 1000 mb that θ_W is the solution of

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where

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One linear iteration of Newton’s method with a first guess of C provides the following solution:

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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_w;1). A plot (Fig. 2) of the actual (i.e., converged iterative) solution for θ_w as a function of (C/θ_E)^λ shows that the linear solution in (3.3) is approximately valid in the interval −19° < θ_w < 29°C. The minimax-polynomial approximation method (Scheid 1989) was used to obtain the minimax line in this interval,

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which fits the solution to 0.1°C. Note that (3.3) and (3.4) are very similar.

Is there a better linear relationship than (3.4) between θ_w and another power of (θ_E/C), say (θ_E/C)^−μ? To answer this question, the standard error of the θ_w predicted by linear regression over the interval −19°C < θ_w < 29°C was computed for different values of μ. The minimum standard error (0.06 K) occurred for μ ≈ 3.5, thus confirming that μ = λ produces the most linearity.

The linear fit naturally breaks down at large values of θ_E because θ_w cannot remain finite as θ_E tends to infinity and at cold values of θ_E because θ_W tends to θ_E as the saturation mixing ratio becomes small. For 377 ≤ θ_E < 674 K (28.2° ≤ θ_w < 50°C), a minimax polynomial was fitted to the difference between the actual and the linear solutions. For θ_E ≤ 257 K (θ_w ≤ −18.6°C), one iteration of Newton’s method applied directly to Bolton’s Eq. (35) version of (2.1)–(2.2) with T_E = θ_E, T_w = θ_w, and π = 1 suffices. After some minor approximations, the resulting solution is

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where 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_K, 1) = εe_s(T_K)/[p₀π^λ − e_s(T_K)], e_s(T_K) = e_s(C) exp[a(T_K − C)/(T_K − C + b)], d lne_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_w = 40°C (θ_E = 478.4 K).

For even greater overall accuracy a rational function was collocated to the actual solution of (3.1) at θ_w = −30°, −20°, . . . , 40°, 50°C. The resulting approximate solution is

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where X ≡ θ_E/C and a₀ = 7.101574, a₁ = −20.68208, a₂ = 16.11182, a₃ = 2.574631, a₄ = −5.205688, b₁ = −3.552497, b₂ = 3.781782, b₃ = −0.6899655, b₄ = −0.5929340. For θ_w ≤ 50°C, its maximum deviation from the actual solution (the relative error) is 0.02 K, and it is within 0.005 K for θ_w ∈ (−20°, 40°C), the range of θ_w 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 θ_W = θ_E with negligible error (≤10⁻⁴ K).

Associated with the absolute error δθ_E of up to 0.2 K in the value of θ_E provided by Bolton’s Eq. (39), there is a corresponding error in θ_W given, from (3.6), by

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For δθ_E = 0.2 K, δθ_W varies from 0.17 K at θ_E = 257 to 0.05 K at θ_E = 335 to 0.03 K at θ_E = 377 K. The absolute error in θ_W becomes quite small at the EPTs most often associated with atmospheric convection.

4. The solution for T_W as a function of T_E and p

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_w = −20°, −18°, . . . , 40°C. We then obtain the wet-bulb temperatures 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_d power. We refer to the array of 31 × 39 (=1209) points in (θ_w, p) space as the “grid.”

Plots (Fig. 3) of the actual solution of (2.3) as a function of (C/T_E)^λ 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 T_W approaches T_E, and the solution becomes asymptotic to the curve T_W = Cx^−1/λ, where x is the abscissa (C/T_E)^λ. 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 [(C/T_E)^λ < 0.4]. This is clearly visible at 1000 mb (Fig. 2), but hardly evident at 850 mb (Fig. 3).

From (2.3) and (2.4) it is evident that the equations of the straight-line portions are given by

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where

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and τ* is any value of T_w such that the horizontal line 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 k₁(π) and k₂(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:

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with correlation coefficients, r, of 1 (Figs. 5 and 6). Remarkably, k₂ is almost linear in π with the regression line:

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A similar procedure to that used in obtaining (3.5) gives the following good initial estimate for the solution at cold temperatures:

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The transition point S at each of the 39 pressure levels was located by evaluating the errors in the two approximate solutions [(4.1)–(4.2) and (4.6)] as a function of (C/T_E)^λ, and determining by linear interpolation the value D(p) of (C/T_E)^λ where the magnitudes of the errors are equal. Then a regression line was fitted to the reciprocal of the data for D(p). The resulting equation,

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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_E > 355 K), an additional term in (C/T_E)^−λ was added and the constant term adjusted to describe the “warm-side” nonlinearity.

The resulting initial guess for T_w is

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where A = 2675 K, C = 273.15 K, λ = c_pd/R_d, and k₁(π), k₂(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 k₁(π) and k₂(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 k₂ [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_E on T_w is shown in Fig. 8. Clearly the upper bound on the corresponding absolute error in T_w is also 0.2 K. Since the algorithm’s relative error after one iteration is much smaller, its overall error is ≤0.2 K.

5. Computation of temperature along reversible adiabats

The equation governing the temperature T_R along reversible water-saturation adiabats is

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(Saunders 1957). We make the usual assumption that c_W, the specific heat of liquid water, is 4190 J kg⁻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:

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where A₁ is a constant along a given reversible adiabat. The constant A₁ 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):

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where ν ≡ R_d/(c_pd + c_WQ).

It should be possible to find good initial guesses and identify near-linear relationships between T_R and (A₁π)^−λ 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 (Q), this has been left for future work. Instead, we used the same initial guess for T_r as the one for T_w [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 T_r − T_w 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.

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_S and θ_A (Fig. 10). The saturated WBPT, θ_S, is reached by saturating a parcel at (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_A, is attained by desiccating a parcel at (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_W and θ_S with height, 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_W of any parcel lifted pseudoadiabatically exceeds the θ_S of the unmodified environment at a higher level (see Fig. 12 in Browning and Donaldson 1963).

To test the claim that the Wobus function is a function solely of temperature, precise values of T_w and θ_W − θ_A 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 W_F(T_W), 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 T_w and θ_W − θ_A on a (θ_w, p) thermodynamic diagram, and maximum variation at constant temperature in θ_W − θ_A of almost 1 K.

To include the pressure variation, we define a generalized Wobus function of two arguments, W*, as

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For a saturated parcel, T_K is the same as its adiabatic wet-bulb temperature T_W and θ_S is its WBPT, θ_W. Therefore,

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If this parcel descends dry adiabatically to 1000 mb, it still has the same potential temperature θ (≡T_W/π here because Wobus imprecisely used the Poisson constant for dry air) so its θ_A is unchanged. However, its θ_S is now θ and its T_K becomes θ. Hence, by (6.1),

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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_K becomes equal to its equivalent potential temperature θ_E(T_W, π), its new θ_S is θ_E and its new θ_A is θ_W (a bijective function of θ_E). Thus, by (6.1),

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which indicates that at 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_E as a function of θ_W in the header of Table 78 of the 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_W computed from (6.4) with List’s (Bolton’s) values of θ_E have a maximum error of 0.66 K (0.53 K). There are similar errors in the computation of θ_A from 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).

Eliminating θ_A from (6.2) and (6.3) gives a formula for θ_W:

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The temperature T_W at a pressure p along a pseudoadiabat with WBPT θ_W can be found as the solution of (6.5). Note that for a saturated parcel at 1000 mb, π = 1 and (6.5) reduces to θ_W = T_W. The method gives the correct answer at 1000 mb despite the errors in θ_W as a function of θ_E and in θ_A as a function of θ that exist because (6.4) and (6.3) are not satisfied exactly.

In the special case when the parcel is initially unsaturated, then (6.5) applies at its saturation point (T_L, π_L) instead of at its initial location (T_W, π). In this case (6.5) becomes

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If the pressure dependency of W is disregarded and the original Wobus function is used in (6.5), we get the equation that Wobus solved for T_W,

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To solve (6.7), Wobus used the secant method (Gerald and Wheatley 1984), which, starting from his initial guess 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_W) − the true θ_W, where T_W 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 (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_w ≤ 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).

7. The generalized Wobus method reduces to the new method

We now show that the generalized Wobus method, which is obtained above by allowing the Wobus function to have some pressure dependency, reduces to the new method. First, note that Eqs. (3.5)–(3.7) or (3.8) can be written as 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),

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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 is

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If 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,

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Substituting (7.1) and (7.3) into (6.5) yields

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which 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

Despite its widespread use, no one seems to know why the Wobus function 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_s, not saturation mixing ratio r_s, that is a function of temperature alone. Although Wobus is the last author on the Doswell et al. paper, he apparently was aware that 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_W is possible by using two slightly different functions for the two arguments [i.e., θ_W = θ − W₁(θ) + W₂(T_W)]. This would permit the function to be tuned in favor of the lower value arguments as used for T_W and the other to be tuned in favor of the higher arguments appearing as 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_w ≤ 28.2°C) as follows. From (3.6) the function X is given to a very good approximation by

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where K₁ = 45.114 and K₂ = 51.489 K. We progress further by using the simplest formula for θ_E 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 yields

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where A (a surrogate for L/c_pd) = 2675 K when r_s is in units of grams per gram. Raising (8.2) to the −λ power and substituting the usual approximate expression for r_s gives us

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where ε = 0.6220 is the ratio of the gas constants for dry air and water vapor.

Substituting (8.1) and (8.3) into (7.2) provides us with a generalized Wobus function:

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Figure 15 shows that for EPTs less than 377 K, W*_G varies only slightly with pressure.

It is easily verified by series expansion that

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to second order in the arguments of the exponentials. Inserting this approximation into (8.4) gives us an expression for the pressure-independent Wobus function:

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which has an intrinsic error φ(T_W, π) ≈ W*_G(T_W, π) − W_G(T_W), where φ(T_W, 1) = 0 because W_G(T_W) = W*_G(T_W, 1). A corrected version of (6.7) that uses the pressure-independent Wobus function (8.6) is therefore

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The inclusion of φ reduces the maximum relative error in the computed θ_W from 0.87 to 0.25 K (Figs. 16 and 17) for θ_w ≤ 28.2°C. (Note that we can safely exclude the lower range limit θ_a ≥ −18.6°C because 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_w 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.

The new technique is based on Bolton’s (1980) formula for θ_E. The temperature T_w on a given pseudoadiabat at a given pressure is obtained from this formula by an iterative technique. A very good “initial-guess” formula for T_w is devised. In the pressure range 100 ≤ p ≤ 1050 mb and wet-bulb potential temperature range θ_W ≤ 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 θ_E in Bolton’s formula that is caused mostly by variation of c_pd with temperature and pressure. It is shown that the upper bound on the corresponding absolute error in T_w 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.

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.