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    The second derivative f ″ of f (TW; π) w.r.t. TW vs Tw at pressures of 1000, 700, 500, and 300 mb. (At 1000 mb, Tw is equal to θw.) The Taylor series expansion to first order of f (TW; π) at constant pressure, evaluated at the temperature τ* = −50(1 − π) (marked by the asterisk), is written at the upper left of each grid window. Throughout the figure | f ″| is less than 0.0002 within 20°C of τ. At each pressure therefore, the upper bound for the remainder term in (2.4) within the temperature interval (τ* − 20°C, τ* + 20°C) is an order of magnitude smaller than the first-order term in the written expansion, proving that f (TW; π) is almost a linear function of Tw in this interval.

  • View in gallery

    Graph of θw as a function of (C/θE)3.504. The center of the / mark indicates each point [(C/θE)3.504, θw] for θw = −100°, −99°, . . . , 50°C and the corresponding θE 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 θE increases. Here E is the point on the line in (3.3) [not shown as it is practically coincident with the line in (3.6)], where θW = 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 temperatures.

  • View in gallery

    Graph of Tw as a function of (C/TE)3.504 at pressures of 850, 700, 500, and 300 mb. The centers of the / mark indicate the points [(C/TE)3.504, Tw] for θw = −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 the point of evaluation for the straight line, and S is the “transition point” at which (C/TE)3.504 = 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.

  • View in gallery

    The four regions in [p, (C/TE)λ] 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(π) and k2(π) are evaluated are marked by plus signs. These points all lie in the two middle regions where the solution is linear.

  • View in gallery

    Quadratic regression for k1(π).

  • View in gallery

    Quadratic and linear regression for k2(π). The regression line and regression curve are nearly collocated.

  • View in gallery

    The error of the initial guess for θw (°C) as a function of θw 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.

  • View in gallery

    The error in the converged solution for Tw owing to a 0.2-K error in θE as a function of θw and p. The quantities inside the parentheses are the same as in Fig. 7.

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    The difference TrTw between the temperature attained in the water-saturation adiabatic expansion and that attained in the water-saturation pseudoadiabatic expansion from the same initial state at 900 mb. Contours of TrTw at levels listed at the bottom are plotted on a θw vs p diagram. Also shown are the −40°, −20°, and 0°C contours of temperature. This is an updated version of Fig. 2 in Saunders (1957).

  • View in gallery

    A pseudoadiabatic diagram showing the relationships between the temperatures θES, θ, θS, and θA 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 θ is the parcel’s temperature if it is lowered dry adiabatically to 1000 mb and θS is its temperature if it is hypothetically saturated at X and then lowered pseudoadiabatically to 1000 mb. The parcel’s saturated EPT, θES is its temperature if it is saturated at X, lifted pseudoadiabatically to a great height, and then lowered dry adiabatically to 1000 mb. The quantity θA (<θS) 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 the parcel is already saturated at X, θS = θW and θES = θE. In the example shown, p = 750 mb, T = 23.1°C, θs = 32.0°C, θa = 16.0°C, θES = 400.4 K, and θ = 321.4 K.

  • View in gallery

    Scatter diagram of adiabatic wet-bulb temperature Tw vs θWθA. The plus signs mark points along the Wobus curve WF(TW).

  • View in gallery

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

  • View in gallery

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

  • View in gallery

    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.

  • View in gallery

    The Wobus function W*G(TW, π) vs Tw at constant pressures of 1000, 900, . . . , 300 mb. The curves are truncated at the data points marked by | signs by excluding data from points where θE ≥ 377 K (θw ≥ 28.2°C).

  • View in gallery

    The error in kelvins in the determination of θW from (6.7) using the pressure-independent Wobus function WG as a function of θw and p. The vertical line marks θw = 28°C to facilitate comparison with Fig. 17.

  • View in gallery

    Same as in Fig. 16, but the error has been reduced by the derived correction φ in the region of the grid for which the correction is either valid or negligible (θw ≤ 28°C).

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An Efficient and Accurate Method for Computing the Wet-Bulb Temperature along Pseudoadiabats

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  • 1 NOAA/National Severe Storms Laboratory, Norman, Oklahoma
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Abstract

A new technique for computing the wet-bulb potential temperature of a parcel and its temperature after pseudoadiabatic ascent or descent to a new pressure level is presented. It is based on inverting Bolton’s most accurate formula for equivalent potential temperature θE 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 function of equivalent temperature raised to the −1/κd (i.e., −3.504) power, where κd is the Poisson constant for dry air, in a significant region of a thermodynamic diagram. Consequently, Bolton’s formula is raised to the −1/κd power prior to the solving. A good “initial-guess” formula for Tw is devised. In the pressure range 100 ≤ p ≤ 1050 mb, this guess is within 0.34 K of the converged solution for wet-bulb potential temperatures θw ≤ 40°C. Just one iteration reduces this relative error to less than 0.002 K for −20° ≤ θw ≤ 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.

Corresponding author address: Dr. Robert Davies-Jones, National Severe Storms Laboratory, National Weather Center, 120 David L. Boren Blvd., Norman, OK 73072. Email: bob.davies-jones@noaa.gov

Abstract

A new technique for computing the wet-bulb potential temperature of a parcel and its temperature after pseudoadiabatic ascent or descent to a new pressure level is presented. It is based on inverting Bolton’s most accurate formula for equivalent potential temperature θE 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 function of equivalent temperature raised to the −1/κd (i.e., −3.504) power, where κd is the Poisson constant for dry air, in a significant region of a thermodynamic diagram. Consequently, Bolton’s formula is raised to the −1/κd power prior to the solving. A good “initial-guess” formula for Tw is devised. In the pressure range 100 ≤ p ≤ 1050 mb, this guess is within 0.34 K of the converged solution for wet-bulb potential temperatures θw ≤ 40°C. Just one iteration reduces this relative error to less than 0.002 K for −20° ≤ θw ≤ 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.

Corresponding author address: Dr. Robert Davies-Jones, National Severe Storms Laboratory, National Weather Center, 120 David L. Boren Blvd., Norman, OK 73072. Email: bob.davies-jones@noaa.gov

Keywords: Temperature

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, Tw, 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) θE (K) so we can easily calculate the parcel’s equivalent temperature TE during its ascent. Inconveniently, the equivalent temperature of a saturated parcel is a complicated function of Tw 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 TE to get an explicit expression for Tw. The general view has been that the problem is mathematically intractable, and that solutions for Tw 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, TL, which is needed for computation of θE 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) θw 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 θE, it uses θw, which is computed from the Wobus function, WF(TK), of absolute temperature TK 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 θE with this error mostly owing to variation of cpd, the specific heat of dry air at constant pressure, with temperature and pressure (List 1971, his Table 88). Note that, although cpd is treated as a constant, the variation of the specific heat of moist air at constant pressure, cp, 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 θw as a function of θE (section 3). Over most of the atmospheric range of θW (−19° < θw < 29°C), a linear relationship is discovered between θw and the −λ power of θE, where λ ≡ 1/κd (=3.504) and κd = Rd/cpd (=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 Tw and the −λ power of equivalent temperature TE (K) is found in a significant region of a thermodynamic diagram. One iteration of the algorithm then gives a highly accurate solution for Tw. 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 Tw 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, TK, 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 (TK, π) and involve its mixing ratio r, its vapor pressure e and TL. 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 (TW, π) simply by replacing TL by TW, r by saturation mixing ratio rs(TW, π), and e by saturation vapor pressure es(TW). When this is done, the remaining formulas all have the following form:
i1520-0493-136-7-2764-e21
where
i1520-0493-136-7-2764-e22
and the constants are listed in Table 1. Note that we have used Bolton’s Eqs. (24) or (7) and the relationship TE = θEπ to write the formulas in the form in (2.1). We can find the temperature at pressure p along a given pseudoadiabat with EPT θE by solving for TW. 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 θE):
i1520-0493-136-7-2764-e23
where λ = 1/κd = cpd/Rd = 3.504. In (2.3) TW and TE have been scaled by C simply to avoid large numbers. By Taylor series expansion about a temperature τ (K) at constant π,
i1520-0493-136-7-2764-e24
[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 TW ∈ [τ* − 20°C, τ* + 20°C] (this can be deduced from Fig. 1). In other words, there is the almost linear relationship between (C/TE)λ and TW in this interval,
i1520-0493-136-7-2764-e25
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,
i1520-0493-136-7-2764-e26
in Newton’s method always provide a precise numerical solution, TW, of (2.3). We can accelerate the convergence by retaining the second-order term in the series expansion and solving the resulting quadratic equation:
i1520-0493-136-7-2764-e27
in a form that is accurate for a small second-order term (see Henrici 1964, p. 199). This gives
i1520-0493-136-7-2764-e28
where f ″(τ ;π) is given in the appendix. We choose the root that is closest to the linear τn+1 provided by (2.6).

In this paper, we use Bolton’s Eq. (39) as the basis for the computation of TW. However, we can compute TW 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
i1520-0493-136-7-2764-e29
where θW is obtained from θE 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 θW from θE

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 θE, we find from (2.3) applied at 1000 mb that θW is the solution of
i1520-0493-136-7-2764-e31
where
i1520-0493-136-7-2764-e32
One linear iteration of Newton’s method with a first guess of C provides the following solution:
i1520-0493-136-7-2764-e33
which is valid in some interval around C. This interval turns out to be fairly large owing to the small second derivative of f (θw;1). A plot (Fig. 2) of the actual (i.e., converged iterative) solution for θw as a function of (C/θE)λ shows that the linear solution in (3.3) is approximately valid in the interval −19° < θw < 29°C. The minimax-polynomial approximation method (Scheid 1989) was used to obtain the minimax line in this interval,
i1520-0493-136-7-2764-e34
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 θw and another power of (θE/C), say (θE/C)μ? To answer this question, the standard error of the θw predicted by linear regression over the interval −19°C < θw < 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 θE because θw cannot remain finite as θE tends to infinity and at cold values of θE because θW tends to θE as the saturation mixing ratio becomes small. For 377 ≤ θE < 674 K (28.2° ≤ θw < 50°C), a minimax polynomial was fitted to the difference between the actual and the linear solutions. For θE ≤ 257 K (θw ≤ −18.6°C), one iteration of Newton’s method applied directly to Bolton’s Eq. (35) version of (2.1)–(2.2) with TE = θE, Tw = θw, and π = 1 suffices. After some minor approximations, the resulting solution is
i1520-0493-136-7-2764-e35
i1520-0493-136-7-2764-e36
i1520-0493-136-7-2764-e37
where a small term has been neglected in (3.5) and A = 2675 K, C = 273.15 K, λ = cpd/Rd, rs(TK, 1) = εes(TK)/[p0πλes(TK)], es(TK) = es(C) exp[a(TKC)/(TKC + b)], d lnes(TK)/dTK = ab/(TKC + b)2, ε = 0.6220, es(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 θw = 40°C (θE = 478.4 K).
For even greater overall accuracy a rational function was collocated to the actual solution of (3.1) at θw = −30°, −20°, . . . , 40°, 50°C. The resulting approximate solution is
i1520-0493-136-7-2764-e38
where XθE/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 θw ≤ 50°C, its maximum deviation from the actual solution (the relative error) is 0.02 K, and it is within 0.005 K for θw ∈ (−20°, 40°C), the range of θw 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 θW = θE with negligible error (≤10−4 K).
Associated with the absolute error δθE of up to 0.2 K in the value of θE provided by Bolton’s Eq. (39), there is a corresponding error in θW given, from (3.6), by
i1520-0493-136-7-2764-e39
For δθE = 0.2 K, δθW varies from 0.17 K at θE = 257 to 0.05 K at θE = 335 to 0.03 K at θE = 377 K. The absolute error in θW becomes quite small at the EPTs most often associated with atmospheric convection.

4. The solution for TW as a function of TE 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 θE values corresponding to θw = −20°, −18°, . . . , 40°C. We then obtain the wet-bulb temperatures TW(θW, 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/κd power. We refer to the array of 31 × 39 (=1209) points in (θw, p) space as the “grid.”

Plots (Fig. 3) of the actual solution of (2.3) as a function of (C/TE)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 TW approaches TE, and the solution becomes asymptotic to the curve TW = Cx−1/λ, where x is the abscissa (C/TE)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/TE)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
i1520-0493-136-7-2764-e41
where
i1520-0493-136-7-2764-e42
and τ* is any value of Tw such that the horizontal line Tw = 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:
i1520-0493-136-7-2764-e43
i1520-0493-136-7-2764-e44
with correlation coefficients, r, of 1 (Figs. 5 and 6). Remarkably, k2 is almost linear in π with the regression line:
i1520-0493-136-7-2764-e45
A similar procedure to that used in obtaining (3.5) gives the following good initial estimate for the solution at cold temperatures:
i1520-0493-136-7-2764-e46
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/TE)λ, and determining by linear interpolation the value D(p) of (C/TE)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,
i1520-0493-136-7-2764-e47
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 TE > C, the coefficients were adjusted slightly, and at warm equivalent temperatures (TE > 355 K), an additional term in (C/TE)λ was added and the constant term adjusted to describe the “warm-side” nonlinearity.

The resulting initial guess for Tw is
i1520-0493-136-7-2764-e48
i1520-0493-136-7-2764-e49
i1520-0493-136-7-2764-e410
i1520-0493-136-7-2764-e411
where A = 2675 K, C = 273.15 K, λ = cpd/Rd, 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 Tw. 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 θE in Bolton’s Eq. (39) is caused mostly by variation of cpd with temperature and pressure. The effect of a 0.2-K error in θE on Tw is shown in Fig. 8. Clearly the upper bound on the corresponding absolute error in Tw 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 TR along reversible water-saturation adiabats is
i1520-0493-136-7-2764-e51
(Saunders 1957). We make the usual assumption that cW, 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:
i1520-0493-136-7-2764-e52
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):
i1520-0493-136-7-2764-e53
where νRd/(cpd + cWQ).

It should be possible to find good initial guesses and identify near-linear relationships between TR 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 Tr as the one for Tw [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 TrTw 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 (TK, π) on a pseudoadiabatic (Stüve) diagram, one can consider two hypothetical wet-bulb potential temperatures θS and θA (Fig. 10). The saturated WBPT, θS, is reached by saturating a parcel at (TK, 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, θA, is attained by desiccating a parcel at (TK, π), lifting it dry adiabatically to a great height and then bringing it down pseudoadiabatically to 1000 mb. The original Wobus function (WF) of just temperature is the difference between these two temperatures, that is, WF(TK) = θS(TK, π) − θA(TK, π). It is evaluated using Wobus’s numerical fit to the data.

Incidentally, θS is an important variable in its own right. The distribution of θW and θS with height, z, determine atmospheric stability. A layer is potentially unstable if ∂θW/∂z < 0 and conditionally unstable if ∂θS/∂z < 0. The atmosphere is latently unstable if the θW of any parcel lifted pseudoadiabatically exceeds the θS 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 Tw and θWθA 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 WF(TW), 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 Tw and θWθA on a (θw, p) thermodynamic diagram, and maximum variation at constant temperature in θWθA of almost 1 K.

To include the pressure variation, we define a generalized Wobus function of two arguments, W*, as
i1520-0493-136-7-2764-e61
For a saturated parcel, TK is the same as its adiabatic wet-bulb temperature TW and θS is its WBPT, θW. Therefore,
i1520-0493-136-7-2764-e62
If this parcel descends dry adiabatically to 1000 mb, it still has the same potential temperature θ (≡TW/π here because Wobus imprecisely used the Poisson constant for dry air) so its θA is unchanged. However, its θS is now θ and its TK becomes θ. Hence, by (6.1),
i1520-0493-136-7-2764-e63
If on the other hand, the parcel is lifted pseudoadiabatically to a great height and then brought down dry adiabatically to 1000 mb, its TK becomes equal to its equivalent potential temperature θE(TW, π), its new θS is θE and its new θA is θW (a bijective function of θE). Thus, by (6.1),
i1520-0493-136-7-2764-e64
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 θE as a function of θW 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 θW computed from (6.4) with List’s (Bolton’s) values of θE have a maximum error of 0.66 K (0.53 K). There are similar errors in the computation of θA 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 θA from (6.2) and (6.3) gives a formula for θW:
i1520-0493-136-7-2764-e65
The temperature TW at a pressure p along a pseudoadiabat with WBPT θW 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 θW = TW. The method gives the correct answer at 1000 mb despite the errors in θW as a function of θE and in θA 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 (TL, πL) instead of at its initial location (TW, π). In this case (6.5) becomes
i1520-0493-136-7-2764-e66
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 TW,
i1520-0493-136-7-2764-e67
To solve (6.7), Wobus used the secant method (Gerald and Wheatley 1984), which, starting from his initial guess TW = θW/π, achieved convergence within a few iterations.

The error in the Wobus estimate of θW at a grid point is defined as TW/πWF(TW/π) − WF(TW) − the true θW, where TW 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 (θw, 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 θw ≤ 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 θW(TW, π) = X[θE(TW, π)], where X is a function that maps EPT into the corresponding WBPT. Hence, for a parcel that is initially saturated at (TW, π),
i1520-0493-136-7-2764-e71
Since θA(TW, π) is by definition the WBPT associated with an EPT TW/π (the parcel’s potential temperature), we also have θA(TW, π) = X(TW, π). Therefore, the relationship between the generalized Wobus function and the X function is
i1520-0493-136-7-2764-e72
If this parcel is brought down dry adiabatically to 1000 mb, its new temperature is TW/π and its desiccated WBPT remains θA(TW, π). However, its saturated WBPT becomes TW/π. Therefore, at the new location (TW/π, 1) on the pseudoadiabatic diagram,
i1520-0493-136-7-2764-e73
Substituting (7.1) and (7.3) into (6.5) yields
i1520-0493-136-7-2764-e74
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 es, not saturation mixing ratio rs, 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 θW is possible by using two slightly different functions for the two arguments [i.e., θW = θW1(θ) + W2(TW)]. This would permit the function to be tuned in favor of the lower value arguments as used for TW 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 (θw ≤ 28.2°C) as follows. From (3.6) the function X is given to a very good approximation by
i1520-0493-136-7-2764-e81
where K1 = 45.114 and K2 = 51.489 K. We progress further by using the simplest formula for θE 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
i1520-0493-136-7-2764-e82
where A (a surrogate for L/cpd) = 2675 K when rs is in units of grams per gram. Raising (8.2) to the −λ power and substituting the usual approximate expression for rs gives us
i1520-0493-136-7-2764-e83
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:
i1520-0493-136-7-2764-e84
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
i1520-0493-136-7-2764-e85
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:
i1520-0493-136-7-2764-e86
which has an intrinsic error φ(TW, π) ≈ W*G(TW, π) − WG(TW), where φ(TW, 1) = 0 because WG(TW) = W*G(TW, 1). A corrected version of (6.7) that uses the pressure-independent Wobus function (8.6) is therefore
i1520-0493-136-7-2764-e87
The inclusion of φ reduces the maximum relative error in the computed θW from 0.87 to 0.25 K (Figs. 16 and 17) for θw ≤ 28.2°C. (Note that we can safely exclude the lower range limit θa ≥ −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 θw 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 θw 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 θE. The temperature Tw on a given pseudoadiabat at a given pressure is obtained from this formula by an iterative technique. A very good “initial-guess” formula for Tw is devised. In the pressure range 100 ≤ p ≤ 1050 mb and wet-bulb potential temperature range θW ≤ 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 θE in Bolton’s formula that is caused mostly by variation of cpd with temperature and pressure. It is shown that the upper bound on the corresponding absolute error in Tw 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/κd power in a significant region of a thermodynamic diagram. This appears to be an interesting new discovery.

Acknowledgments

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.

REFERENCES

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APPENDIX

The Derivatives of f

The first derivative of f (τ ; π) at fixed pressure is given by
i1520-0493-136-7-2764-ea1
where τ is in kelvins and
i1520-0493-136-7-2764-ea2
i1520-0493-136-7-2764-ea3
i1520-0493-136-7-2764-ea4
i1520-0493-136-7-2764-ea5
The second derivative of f (τ ;π) at fixed π is given by
i1520-0493-136-7-2764-ea6
where
i1520-0493-136-7-2764-ea7
i1520-0493-136-7-2764-ea8
i1520-0493-136-7-2764-ea9
i1520-0493-136-7-2764-ea10
Fig. 1.
Fig. 1.

The second derivative f ″ of f (TW; π) w.r.t. TW vs Tw at pressures of 1000, 700, 500, and 300 mb. (At 1000 mb, Tw is equal to θw.) The Taylor series expansion to first order of f (TW; π) at constant pressure, evaluated at the temperature τ* = −50(1 − π) (marked by the asterisk), is written at the upper left of each grid window. Throughout the figure | f ″| is less than 0.0002 within 20°C of τ. At each pressure therefore, the upper bound for the remainder term in (2.4) within the temperature interval (τ* − 20°C, τ* + 20°C) is an order of magnitude smaller than the first-order term in the written expansion, proving that f (TW; π) is almost a linear function of Tw in this interval.

Citation: Monthly Weather Review 136, 7; 10.1175/2007MWR2224.1

Fig. 2.
Fig. 2.

Graph of θw as a function of (C/θE)3.504. The center of the / mark indicates each point [(C/θE)3.504, θw] for θw = −100°, −99°, . . . , 50°C and the corresponding θE 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 θE increases. Here E is the point on the line in (3.3) [not shown as it is practically coincident with the line in (3.6)], where θW = 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 temperatures.

Citation: Monthly Weather Review 136, 7; 10.1175/2007MWR2224.1

Fig. 3.
Fig. 3.

Graph of Tw as a function of (C/TE)3.504 at pressures of 850, 700, 500, and 300 mb. The centers of the / mark indicate the points [(C/TE)3.504, Tw] for θw = −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 the point of evaluation for the straight line, and S is the “transition point” at which (C/TE)3.504 = 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.

Citation: Monthly Weather Review 136, 7; 10.1175/2007MWR2224.1

Fig. 4.
Fig. 4.

The four regions in [p, (C/TE)λ] 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(π) and k2(π) are evaluated are marked by plus signs. These points all lie in the two middle regions where the solution is linear.

Citation: Monthly Weather Review 136, 7; 10.1175/2007MWR2224.1

Fig. 5.
Fig. 5.

Quadratic regression for k1(π).

Citation: Monthly Weather Review 136, 7; 10.1175/2007MWR2224.1

Fig. 6.
Fig. 6.

Quadratic and linear regression for k2(π). The regression line and regression curve are nearly collocated.

Citation: Monthly Weather Review 136, 7; 10.1175/2007MWR2224.1

Fig. 7.
Fig. 7.

The error of the initial guess for θw (°C) as a function of θw 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.

Citation: Monthly Weather Review 136, 7; 10.1175/2007MWR2224.1

Fig. 8.
Fig. 8.

The error in the converged solution for Tw owing to a 0.2-K error in θE as a function of θw and p. The quantities inside the parentheses are the same as in Fig. 7.

Citation: Monthly Weather Review 136, 7; 10.1175/2007MWR2224.1

Fig. 9.
Fig. 9.

The difference TrTw between the temperature attained in the water-saturation adiabatic expansion and that attained in the water-saturation pseudoadiabatic expansion from the same initial state at 900 mb. Contours of TrTw at levels listed at the bottom are plotted on a θw vs p diagram. Also shown are the −40°, −20°, and 0°C contours of temperature. This is an updated version of Fig. 2 in Saunders (1957).

Citation: Monthly Weather Review 136, 7; 10.1175/2007MWR2224.1

Fig. 10.
Fig. 10.

A pseudoadiabatic diagram showing the relationships between the temperatures θES, θ, θS, and θA 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 θ is the parcel’s temperature if it is lowered dry adiabatically to 1000 mb and θS is its temperature if it is hypothetically saturated at X and then lowered pseudoadiabatically to 1000 mb. The parcel’s saturated EPT, θES is its temperature if it is saturated at X, lifted pseudoadiabatically to a great height, and then lowered dry adiabatically to 1000 mb. The quantity θA (<θS) 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 the parcel is already saturated at X, θS = θW and θES = θE. In the example shown, p = 750 mb, T = 23.1°C, θs = 32.0°C, θa = 16.0°C, θES = 400.4 K, and θ = 321.4 K.

Citation: Monthly Weather Review 136, 7; 10.1175/2007MWR2224.1

Fig. 11.
Fig. 11.

Scatter diagram of adiabatic wet-bulb temperature Tw vs θWθA. The plus signs mark points along the Wobus curve WF(TW).

Citation: Monthly Weather Review 136, 7; 10.1175/2007MWR2224.1

Fig. 12.
Fig. 12.

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

Citation: Monthly Weather Review 136, 7; 10.1175/2007MWR2224.1

Fig. 13.
Fig. 13.

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

Citation: Monthly Weather Review 136, 7; 10.1175/2007MWR2224.1

Fig. 14.
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.

Citation: Monthly Weather Review 136, 7; 10.1175/2007MWR2224.1

Fig. 15.
Fig. 15.

The Wobus function W*G(TW, π) vs Tw at constant pressures of 1000, 900, . . . , 300 mb. The curves are truncated at the data points marked by | signs by excluding data from points where θE ≥ 377 K (θw ≥ 28.2°C).

Citation: Monthly Weather Review 136, 7; 10.1175/2007MWR2224.1

Fig. 16.
Fig. 16.

The error in kelvins in the determination of θW from (6.7) using the pressure-independent Wobus function WG as a function of θw and p. The vertical line marks θw = 28°C to facilitate comparison with Fig. 17.

Citation: Monthly Weather Review 136, 7; 10.1175/2007MWR2224.1

Fig. 17.
Fig. 17.

Same as in Fig. 16, but the error has been reduced by the derived correction φ in the region of the grid for which the correction is either valid or negligible (θw ≤ 28°C).

Citation: Monthly Weather Review 136, 7; 10.1175/2007MWR2224.1

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 and to Saunders’s Eq. (3) for water-saturation reversible adiabats. The latent heat of vaporization is given by L = L0L1T, where the constants L0 = 2.501 × 106 J kg−1 and L1 = 2.37 × 103 J kg−1 K−1. In the last column, cW is the specific heat of water and Q is the mixing ratio of total water to dry air.

Table 1.
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