2270 JOURNAL OF APPLIED METEOROLOGY VOLUt~35A New Formulation for the Critical Temperature for Contrail Formation PaCH F. COLEMANThe Aerospace Corporation, Silver Spring, Maryland(Manuscript received 16 August 1995, in final form 30 May 1996)ABSTRACT A new formulation of the equations describing the conditions necessary foi aircraft exhaust contrail formationis derived from the fundamental necessary condition. First, the original solution of Appleman is derived fromthe necessary condition to illustrate the continuity of the new formulation. Then the new formulation offers ananalytic solution for the critical temperature Tc expressed in terms of water vapor mixing ratio and atmosphericpressure, rather than in terms of relative humidity and pressure, thus avoiding potential forecast errors associatedwith the temperature sensitivity inherent in relative humidity. A variety of results is presented, including acomparison with the seminal results of Appleman, a comparison of the sensitivity of T- to perturbations inrelative humidity versus perturbations in mixing ratio, and some typical results for actual atmospheric conditions.The clear superiority of a formulation based on mixing ratio rather than relative humidity is seen in the reducedsensitivity of Tc to errors or uncertainties in the input atmospheric variables.1. Introduction Contrail formation has been a subject of interest inatmospheric physics and to high-altitude aircraft pilotssince World War II, when a new generation of highaltitude bombers began producing contrails routinely.Contrail forecasting remains a subject of considerableinterest to military weather forecasters and pilots sincecontrails unequivocally indicate the presence of aircraft. Since the presence of contrails can significantlyincrease the probability that an aircraft will be detected,either by ground-based observers or by other aircraft,mission planners adjust routes and flight levels basedon contrail forecasts in an attempt to minimize thechance of detection. As recently as 1989 (cf. Peters1993), the Deputy Chief of Staff, Operations, for Strategic Air Command (SAC) expressed concern over theaccuracy of the contrail forecasts provided to SAC aircrews by Air Weather Service (AWS). This concernhas resulted in new activity in contrail forecasting research and has offered an opportunity to revisit theoriginal analysis of the problem. Considerable research on the subject of contrails produced by propeller-driven aircraft was done in the1940s, as cited by Appleman (1953). The fundamentalpaper on the formation of contrails by jet aircraft (A/.C) was published by Appleman (1953). In that paperAppleman described the thermodynamics of contrail Corresponding author address: Rich F. Coleman, Sr. Project Engineer, NPOESS Integrated Program Office, The Aerospace Corporation, 8455 Colesvillc Road, Suite 1450, Silver Spring, MD 20910.E-mail: rcoleman @ipo.noaa.govformation and provided a method of forecasting theoccurrence of contrails based on forecasts of relativehumidity and pressure. Appleman's forecasting methodrests on four assumptions associated with contrail phenomena and with his analysis of their formation: (i) contrails are composed of ice crystals; (ii) ice crystals do not nucleate directly from thevapor state, so contrails cannot form unless the A/Cwake reaches saturation with respect to water; (iii) from combustion of A/C fuel the ratio of theamount of water injected into the wake to the amountof heat injected into the wake is a constant for all jetA/C (this ratio is called the contrail factor); (iv) for a given ambient relative humidity and pressure, there is a critical temperature below which contrails will always form (but not necessarily persist) inA/C wakes.Appleman also supplied a criterion for the water content needed for contrails to persist and remain visible,and several researchers have addressed that subsequently. However, this paper is concerned only withcontrail onset since saturation with respect to watersures enough water is available to form a visible icecrystal contrail, even if it sublimes shortly thereafter.Subsequent to Appleman's original paper, Air WeatherService and others conducted and/or funded studies intended to validate the forecasting method, to identifypotential sources of error in the method, and to developalternate forecasting methods. Appleman continued to work on contrails underProject CLOUD TRAIL, which found discrepanciesbetween observed and predicted contrail formationconditions. He examined the effects of A/C engine setc 1996 American Meteorological SocietyDECEMBER 1996 C O L E M A N 2271ting, that is, assumption (iii), on contrail formation(Appleman 1954). In an AWS report in 1957, Appleman consolidated work from both previous papers, plusadditional work from CLOUD TRAIL, and providedthe basis for the AWS standard forecasting manual stillin use today (cf. AWS 1981 ). In that report, Applemanalso developed an empirical forecasting method basedon the frequency of occurrence of contrails reported bypilots (PIREPS), which he provided as an adjunct tohis main forecasting method, modified his original conclusions as to the amount of ambient water necessaryto allow contrails to persist and be seen, and providedrepresentative values of relative humidity (RH) to beused by forecasters when no reliable forecast RH valuewas available. An interesting observational result wasthat horizontal temperature gradients at or near the tropopause could average as high as 0.099-C km-~. Assumptions (i) and (ii) have been examined repeatedly. Pilie and Jiusto (1958) examined contrailsformed in a laboratory cloud chamber and found evidence of liquid particles in some contrails at T< -40-C, but that finding has not been replicated. Murcray (1970) studied contrails at ground level in interiorAlaska winter at temperatures less than or equal to-40-C, mostly from Boeing 727 A/C. He found theliquid phase always occurred first, followed within 1 sby freezing. He found no occurrence of long-term liquid phase contrail particles. Initial contrails showedspherical ice particles with a mode diameter of order 2/~m and with at least 80% having a diameter 4/~m orsmaller. Knollenberg (1972) made flight test observations of contrails that appeared to occur below watersaturation; however, it is not clear if the meteorologicalconditions presented were measured in situ or weretaken from the closest recent sounding. Also, the A/Cwas operated at an artificially low airspeed due to flapsbeing set to partial approach; this was to aid particlecollection efforts. However, the unusual engine settingalmost certainly altered the contrail factor (cfi Saatzer1995), which also could explain Knollenberg's observation. Knollenberg's particle counts tend to confirmMurcray's finding that all liquid particles freeze rapidlyrather than persisting to be scavenged by ice particles.Sassen (1979) photographed iridescence in an A/Ccontrail, a phenomenon typically assumed to confirmthe presence of water droplets. However, iridescence isassociated with diffraction from small spherical particles, and Murcray's results show that ice particles ofTABLE 1. Values of the contrail factor CF for several engine types. Engine type Cr Nonbypass 0.036 Low bypass 0.040 High bypass 0.049 Original Appleman 0.0336TABLE 2. Values for constants for saturationvapor pressure expression. a~ = 54.8758 a2 = 58.0691 a3 = 6790.51a4 = 1.44078 x 10 as = 5.02808 b~ = 2999.992 b2 = 0.069998that type are present in initial contrails. Interestingly,Sassen's results require particles with diameters of 13 /~m, corresponding almost exactly with Murcray'sobservations. At present them remains no compellingevidence that either assumption (i) or (ii) is invalid. Several scientists have addressed assumption (iii).Appleman (1954) found no change in contrail factorwith engine setting, altitude, or airspeed. Scorer andDavenport (1970) noted, however, that fuels with different proportions of hydrogen than kerosene (JP-4)would yield greater or lesser amounts of water for essentially the same amount of heat, thus violating assumption (iii). Work by Peters (1993) and Saatzer(1995) has shown conclusively that different enginetypes, as determined by the amount of ambient air injected into the exhaust, give rise to different contrailfactors, requiring contrail factor to be treated as an independent variable in contrail formation. Saatzer additionally showed that engine setting has a significanteffect on contrail factor. A number of scientists subsequent to Appleman haveaddressed the problem of determining the critical temperature and forecasting the onset of contrails. Pilie andJiusto (1958) presented a major simplification of Appleman's approach to finding the critical temperature,discussed further in section 2. Scorer and Davenport(1970) extended the approach of Pilie and Jiusto toinclude some alternate fuel types. All of these approaches, including Appleman's original formulation,yield a solution graphically. Appleman (1957), Miller(1990), Bjornson (1992), and Peters (1993) have presented empirically based solutions for the critical temperature. However, the standard forecasting methodstill in use, as documented in AWS (1981) and usedin validation studies by Miller (1990), Bjornson(1992), and Peters (1993), is the Appleman method,modified to allow different values of the contrail factor.This present paper offers a new formulation of theequation for the critical temperature for contrail formation in terms of a variable contrail factor, mixingratio, and pressure rather than RH, pressure, and a constant contrail factor. In addition, the new formulationis an analytic solution for critical temperature and thuswill offer an easier implementation for computationalforecasting than previous graphical solution methods.The new formulation also will result in increased fore2272 JOURNAL OF APPLIED METEOROLOGY VOLUME35cast accuracy since errors in RH depend on both errorsin ambient temperature and in mixing ratio. Rather than derive the new formulation in the traditional way based on the thermodynamics of jet aircraft exhaust, a fundamental necessary condition forcontrail formation is presented and then shown to leadto the Appleman curves. The new formulation for critical temperature is then derived from the fundamentalcondition, followed by a determination of the sensitivity of the two formulations to small perturbations in theindependent variables. Finally, the inherent accuracy ofthe two formulations is discussed in light of currentforecasting methods, followed by some representativeresults.2. Fundamental condition for contrail formation The fundamental condition for contrail fom~ationmay be stated as follows. If the water vapor mixingratio w, at some point in the A/C wake equals or exceeds the saturation mixing ratio w, at the temperatureT, anywhere in the exhaust plume, then a contrail canoccur: wp~>ws at rp, (1)where, of course, the conditions wp and Tp occur atpoint p in the wake. This equation, if the mixing ratioand coincident temperature are available, is the easiestapproach to determining the critical temperature Tc,that is, the ambient temperature for which Eq. (1) becomes an equality. This condition was noted by Appleman (1953) butused only to determine the amount of water requiredbeyond the ambient RH values necessary to maintainsaturation in the wake in the presence of the added heatfrom combustion. Pilie and Jiusto (1958) made use ofit and the contrail factor to greatly simplify Appleman' sgraphical method of determining Tc. They made use ofthe fact that the ratio of water added to the wake to theheat added to the wake (the contrail factor) can be represented by a line with slope equal to its value on aphase diagram of mixing ratio versus temperature. Ifthe saturation mixing ratio is plotted on the same diagram, then the tangent point of the two curves is Tc,that is, the temperature above which saturation in thewake cannot occur. They used this method to determineTc for a variety of pressures and RH values and forvapor pressure values, producing modified forecast tables. However, it is possible to go further using this condition: it is possible to derive nongraphical expressionsfor To. A transcendental expression that replicates theoriginal Appleman results may be derived, and further,an analytic expression for Tc may be derived that avoidsthe use of RH entirely. This latter expression has a variety of positive consequences, as will be shown in subsequent sections.3. Appleman curves derived from fundamental condition Appleman' s original curves are derived from Eq. ( 1 )by first subtracting the ambient mixing ratio wa from20 i ,15 5E~ 10<~ Relotive Humidity0 6O 9O 100 -60 -50 -40 -30 Temperoture (C)FIG. 1. Replicating Appleman's critical temperatures for contrail formation at various relative humidities and altitudes.DECEMBER 1996 C O L E M A N 227320 ' o.32 ~) Mixing Ratio ........ Relative Humidity15 ,X. ,,% - -60 -50 -40 -30 Temperature (C)FI6. 2. Comparison of T~ for formulations based on mixing ratio and on relative humidity.both sides, then dividing both sides by the differencebetween the temperature of the entrained air in thewake at point p(Tp) and the ambient temperature Ta.Then the inequality becomes we - w,~ ws(Te) - w,~ --~> (2) rp- Ta Tl,- raThe left-hand side is now the ratio of the change inwater vapor mixing ratio to the change in temperaturedue to exhaust. These quantities are defined as followsin Appleman's original paper (1953). The change in the water vapor mixing ratio of an airparcel, resulting from the combustion of a unit mass offuel in an A/C jet engine, which mixes with an ambientparcel of mass N, may be written as (following Appleman) mw Aw = Wp -- w~ = 1000 m--~ (g kg-~)' (3)where wp is the water vapor mixing ratio at point p inthe plume, wa is the ambient water vapor mixing ratio,N is the mass of ambient air parcel that mixes with unitmass of exhaust, m,~ is the mass of water vapor in A/C exhaust, and me is the mass of exhaust parcel. Notethat Appleman (1953) gives Aw = 1000(1.4/12N) forall fuels and engines. The change in temperature due to the energy releasedin the A/C exhaust by combustion of a unit mass offuel (again following Appleman) is given as E AT = T, - Ta - mecpN' (4)where E is the energy released, Cp is the specific heatof air at constant pressure, Tp is the temperature of themixed air parcel at point p, and Ta is the ambient temperature. Note that Appleman (1953) gives AT= 104(0.24 x 12N) -~ for all fuels and engines. The ratio of these two quantities was assumed byAppleman to be a constant based on early work on jetengines. This quantity has come to be called the contrail factor CF and, while not a constant, still has onlya limited range of values. Unfortunately the individualvalues for the increase in the mixing ratio and the temperature are typically not available for jet engines; instead this ratio is the only available quantity. The ratioitself is CF -- AW _ 1000 mw% (5) AT EThus the left side of Eq. (2) is just Appleman's contrailfactor. A set of values for CF at 35 000 ft, grouped byengine type, is given by Peters (1993) and is shown inTable 1. To address the right side of Eq. (2), it is noted thatthe water vapor mixing ratio for a saturated parcel w.~is a function of the temperature and pressure of theparcel. The definition of mixing ratio and the equationsof state for dry air and water vapor give es M~ e~M~ w~ = 1000- ~ 1000----, (6) p-e~Ma pM~where p is pressure, e~ is saturation vapor pressure, M~,is molecular weight of dry air, and M~ is molecularweight of vapor. The final approximation is appropriate2274 JOURNAL OF APPLIED METEOROLOGY VOLUME35since for temperatures T below freezing, p >> e~. Thesaturation mixing ratio and the ambient mixing ratioare related through the relative humidity as follows: RH Wa = Ws 100' (7)where RH is relative humidity. The saturation vapor pressure is given by [afterGoff-Gratch as shown in List (1968) with some manipulation] 6790.51.ln(es) = 54.8758 + 58.0691e-2999'992/r T + 1.44078 x 10-~Se-'-69998r + 5.02808 ln(T). (8)This may be rewritten as follows: [ a3es = exp a~ + a2e--~/r r + a4e~2r + as ln(T)l , (9)where the values for the constants are given in Table 2. Returning to the fundamental condition, as expressedin Eq. (2), the inequality may now be rewritten as follows, after substituting Eq. (7) for RH: 100w~(T.) - w~(T.)RH CF >~ (10) iOOAT The critical temperature Tc is defined as the ambienttemperature T, at which Eq. (10) becomes an equality:100ATCF - 100ws(T~. + AT) + w~(Tc)RH = 0, (11)noting that Tp = T, + AT. Substituting the value for the ratio of the vapor andair molecular weights, 0.621979, givesiOOpATCr - 62197.9es(Tc + AT) + 621.979e~(Tc)RH = 0. (12) As a final step, substituting Eq. (9) into Eq. (12)givesK~pATCr - K~K2 exp [a~ + a2ea3Tc -4- AT---- + a4eo2(rc+~xr) + as ln(Tc + AT)] [ a3 ]+ RHK2 exp a~ + a2e-o~/r~ - ~ + a4eo~r~ + as ln(Tc) = 0,(13)where K~ = 100, K2 = 621.979, and the const~mts aand b have been given in Table 2. 'In Eq. (13) the pressure, relative humidity, andtemperature dependence may be seen directly'. Theimpact of the extent to which the A/C exhaust parcel mixes with the ambient air also may be seen directly since AT is inversely proportional to this mixing. This equation, being transcendental in Tc, hasno analytic solution. Previous methods for forecasting contrail formation, implicitly based on thisequation, have used graphical approaches to obtaining a solution. They also have ignored the effects ofAT by taking the highest Tc at which a contrail canform and assuming that at some point in the contrailthe mixing ratio associated with that particular ATwill occur. Equation (13) is equivalent to Appleman's original result, which can be shown by comparing results from Eq. (13) with his numerical results. To replicate the data from Appleman's (1953)Fig. 4, T~ was computed at the same pressure levelsand relative humidity values using the AT value thatgives the highest critical temperature at each pressure level. The resulting data are shown in Fig. 1,where the zeros of Eq. (13) were determined numerically rather than using graphical methods. Thedata are plotted as a function of altitude rather thanpressure. Comparison of the values from this approach with the original Appleman tables shows themaximum difference to be less than 0.4-C across allvalues of RH and p.4. New expression for critical temperature The equation for the critical temperature may be rewritten to remove the relative humidity by using thefundamental condition given in Eq. ( 1 ). As before, thefirst step is to derive Eq. (2): As noted before, the lefthand side of Eq. (2) is just the contrail factor, and aftersubstitution the equation becomes w~(Tp) - w~ CF >~ (14) Tp- T~This may be rewritten as an equality in terms of AT[as defined in Eq. (4)] and Tc asDECEMBER 1996 C O L E M A N 2275ATCr - ws(Tc + AT) + wa = 0. (15)Substitution of Eq. (6) for ws into Eq. (15)gives e,(Tc + AT) MyATCr+wa- 1000 =0. (16) p maAt this point, the equation may be treated as beforeby using the Goff-Gratch expression for saturationvapor pressure. However, unlike the case of the relative humidity formulation, here the use of Tetens'sexpression (Bolton 1980) for vapor pressure allowsan analytic solution to be obtained. Tetens's expression ise,(T) = 6.1078 exp ~ 17.26939 T- 273.15 '~ 1T- 273.15 + 237.3] ' (17)Substituting into Eq. (16) gives 621.979ATCr + wa -- 6.1078Px exp ( 17.26939 Tc + AT- 273.15 hTc + AT -- ~.~-~ ~ 537.3 J= o. (18)Equation (18) may be solved for Tc to give - [p(ATCr + wa)]f. [p(/XTCF + Wa)-[ }-'.Tc = 273.15 - AT- 237.3 tn[ ~..1-'~'~ j~tn[ 6.--i~i ] - 17.26939(19)Equation (19) is a fundamental equation for criticaltemperature and represents an alternate formulation tothe original Appleman result. The two are comparedbelow. Figure 2 shows a comparison of the two formulationsfor critical temperature for the case of a low-bypassengine (Cr = 0.034) and an exhaust mixing resultingin a temperature difference of 9-C (AT = 9-C). Notethat the zero lines are parallel to each other but notexactly equal. This is due to the use of two differentexpressions for vapor pressure (Goff-Gratch versusTetens). For situations where water is present (i.e., w,~ 0) in the ambient atmosphere, lines of constant mixing ratio do not run parallel to lines of constant relativehumidity, as is to be expected. Note that as in the caseof Fig. 1, the data are plotted as a function of altitude,but the calculations are made for specific values ofpressure rather than altitude. The conversion to altitudeis made based on the standard approximation to thehypsometric equation 2g(h2 - h~) - ln(P2'~ (20) R(T2 + T~) \p~ / 'where g is the acceleration of gravity, R is the ideal gasconstant, and h~ is the height of ith point.5. Effects of differences in exhaust mixing As an exhaust parcel moves downstream from theengine nozzle it mixes with the ambient atmosphereand as a result it cools and dries. This is a competingpair of processes in terms of creating a condition ofsupersaturation, and this is seen in the behavior of thecritical temperature as a function of mixing. The parameter that indicates the extent of mixing is the temperature difference between the ambient air and theparcel, AT. When the parcel leaves the nozzle, thevalue of A T is large, and the parcel has not mixed muchwith the surrounding air. As it moves downstream, ATdecreases as the parcel mixes with the ambient air. The plot in Fig. 3 shows the effects of mixing on thecritical temperature in the cases where the ambient water vapor mixing ratio is zero, where it is 0.12 g kgand where it is 0.42 g kg-~ at 300 mb (about 9-kmaltitude) and Cr = 0.04 (low-bypass). Similar behavior is observed at other pressure levels and contrailfactors. As can be seen, as the value of AT changes, thecritical temperature passes through a maximum foreach water vapor mixing ratio except for the onethat corresponds to saturation (RH = 100), whichin this case is 0.42 g kg-~. It is also apparent thatthe AT that gives a maximum in T~ changes withmixing ratio, but not very much, and that the valueof AT where the maximum is reached decreaseswith increasing w~. Because exhaust parcels passthrough all of the values of AT in each plume thiscannot really be described as a sensitivity; it is appropriate to select the maximum value of Tc sincethe contrail need only form at one point in the exhaust. Unfortunately, the derivative of Eq. (19) with respect to A T gives an equation with no analytic solution,so determining the AT for which Tc is a maximum mustbe done numerically. However, numerical analysis re2276 JOURNAL OF APPLIED METEOROLOGY VOLUME35vcElOO , .,... ~.._:.~ Low-bypass ' '" "'" ' "" ' ~ ' "':' ' '~' ' '~' ' "" '~' '~' ' ~ ' ~ ' ~ ' '~' ' "" ' '~' '~'i ',. lO wo -_ ..o......' -_o.? /wo: // / ......... , ......... ....... ,/ ........ ,I .... -8 -70 -60 -50 -40 -30 Critical lernperoture (C)FIG. 3. Critical temperature at several mixing ratios as a function of temperature difference.veals that if the derivative is set equal to zero and thensolved for AT, that is, Ore AT for which ~-~ = O(~ATm), (21)then ATm may be fit with an equation of the form ATm = aw, + b ln(p) + c, (22)where the values of the coefficients a, b, and c, forseveral values of Cr, are given in Table 3. Note thatthis fit allows ATm tO become negative for larger valuesof Wa, even though the actual solutions (zeros) of Eq.(21) do not. Thus to use Eq. (22), it is necessary toset it to zero at the points where it goes negative whenmaking calculations. Term ATm is explicitly includedin Eq. (19) as follows: 17.26939}Tc = 273.15 - /XT,~ - 237.3 InL ~-A-~--~2 J[ [ 6.-~0-~8~2' J(23)All of the calculations in this paper for critical temperature use this equation, suitably restricted to nonnegative values.6. Effects of differences in contrail factor The effects of the contrail factor on the critical temperature are shown in Fig. 4. In the figure, critical temperature is shown for three values of Cr at each of threemixing ratios. There are two effects of interest asshown here. First, at any given mixing ratio, the higherthe contrail factor the higher the critical temperature.This indicates that the lower the contrail factor, thehigher the altitude at which contrails will forth for agiven mixing ratio. Since higher-bypass engines havehigher contrail factors, they will form contrails at loweraltitudes than lower-bypass engines. The second effect to note is that as the mixing ratioincreases, the effect of the contrail factor decreases.When there is no water available in the ambient atmosphere, the difference between low~ and high-contrail-factor engines can be 3.5-C, whereas at a mixingratio of 0.24 g kg-~ the difference between low- andhigh-contrail-factor engines is only about 2-C. In otherwords, as the amount of water available in the atmosphere increases, there is a decline in the effect of differences in the amount of water provided by the combustion process.7. Sensitivity of critical temperature to perturbations in mixing ratio and relative humidity The effects of small perturbations in the amount ofmoisture, as represented by the mixing ratio and byDECEMBER 1996 C O L E M A N 2277TABLE 3. Values of the coefficients for ATfor three values of contrail factor Cr. Cr= 0.03 Cv= 0.034 CF= 0.039a -33.3333 -29.4118 -25.641b 0.940286 0.959569 0.981363c 3.92056 3.93463 3.94932the relative humidity, are of interest in that they arerepresentative of the sensitivity of the resulting critical temperature to errors or uncertainties (e.g., inforecasts) in the input value. It is of interest for eachrepresentation individually and as a comparison ofthe two representations. To examine these effects forrelative humidity requires the use of the polynomialfit to the numerical solution to Eq. (13) for To. Thefit was used to make comparisons with the originalAppleman curves. This fit is of the form 4 5 Tc = a + '~ fi~pi + ~ .yjRH~+r/PRH. (24) i=1 j=lThe sensitivity to RH may be seen by taking the partialderivative of Eq. (24) with respect to RH and multiplying by a small change in relative humidity. This inturn gives a corresponding change in the critical temperature. The partial of Eq. (24) is OTc ~ 0RH = ~ JyjRHJ-1 + ~/p' (25) j=l The result of applying a 10% perturbation in RH isshown in Fig. 5. The result in Fig. 5 is for pressure of1000 mb. Inspection of Eq. (22) and Eq. (24) showsthat higher pressure will give a greater perturbation inTc due to the fact that the atmosphere can contain morewater vapor. Thus the result shown in Fig. 5 for 1000mb is the maximum effect. As can be seen from thefigure, the sensitivity of Tc to perturbations in RH isquite small until RH values exceed 80%. At high RH,the effect is severe, with errors of several degrees possible for RH > 90%. These perturbations are representative of the errors and uncertainties to be expected athigh values of RH, where accurate measurements andforecasts are both are difficult to achieve. The same technique may be applied to perturbations in Wa by taking the partial of Eq. (23) withrespect to wa and multiplying by a small change inmixing ratio. This in turn gives a correspondingchange in the critical temperature. The partial derivative is given byOwa = 237.3 x 17.26939 (CrATm + wa)~[,n[ 6.-~-q-8~2 ] - 17.26939 .(26) The results of applying a 10% perturbation in wato Eq. (26) are shown in Fig. 6. This figure showsseveral interesting points. First, note that, as for theperturbation applied to RH, the 10% perturbation isrepresentative of errors in mixing ratio. Thus, theperturbations shown in Fig. 5, which are less than1.1-C, are representative for the altitudes of interestfor the contrail forecasting problem, that is, fromabout 5-km altitude and up. For comparative purposes recall that for relative humidity above 80% theeffect of a 10% perturbation in RH ranges from1.25-C at 80% to 5-C at 95%. As can be seen in Fig.6, the effect of errors in wa on Tc is small and evenat high values of mixing ratio the change in Tc is onlyslightly more than i-C. As a result, as anticipated,the conversion from a formulation for Tc with a relative humidity dependence to a mixing ratio dependence significantly reduces the effect of errors in Wa.Use of this formulation should significantly reducethe effect of forecast errors in atmospheric water vapor on forecasts of Tc. A second point of interest is the linear behavior ofthe perturbations in mixing ratio. This results fromthe fact that at ATm, where the effects of w~ are actually greatest, the partial of T~ given in Eq. (26)turns out to be a constant (29.5 _+ 0.2, depending onCr). As a result, regardless of altitude (i.e., p) theeffect of wa perturbations is the same, and the smallerthe mixing ratio the smaller the sensitivity. Anotherinteresting point is that the sensitivity to mixing ratiodecreases with increased Cv, although not greatly.Finally, note that at high values of w~, the curve flattens. This is a result of reaching the limit of physically realizable values of AT,~ (i.e., >~ 0). The limitcorresponds to reaching saturation for the pressureand temperature.8. Sensitivity of critical temperature to perturbations in pressure As in the case of w~, the effects of small perturbations in atmospheric pressure are of interest in2278 JOURNAL OF APPLIED METEOROLOGY VOLUME35that they are representative of errors or uncertainties(e.g., in forecasts) in the actual value for the pressure at a given altitude. These effects may be examined by taking the partial derivative of Eq. (23)with respect to p and multiplying by a smallchange in p. This gives a corresponding change inthe critical temperature. The partial derivative isgiven by <~'r~(cF/'~Tm-~14/a)l P L 6 "--'~'~8 ~'2 J0Tc__ 237.3 x 17.26939 ,n, - 17.26939Op(27)The results of applying a perturbation in p equal to a1-km change in height to Eq. (27) are shown inFig. 7. Figure 7 shows results for a C~ = 0.4. Inspection ofEq. (27) indicates that variations due to different C~are minor since it is within a logarithm. As a result,only the low-bypass case has been plotted as it providesan intermediate value. As can be seen in the figure, Tcis more sensitive to perturbations in p than to Wa. A 1km perturbation causes approximately a factor of 2greater change in Tc than a 10% change in wa except atvery high values of mixing ratio. Also of note is thelarge variation with Wa, coupled with the fact that thelower the ambient mixing ratio, the greater the effectof perturbations in the pressure. This is of interest bothin terms of forecast errors in pressure and in terms ofvariations in the pressure altitude measured in an aircraft versus the actual pressure at the aircraft altitude.9. Implications for forecasting As currently practiced by the Air Force Air WeatherService, contrail forecasting attempts to predict threethings: (i) the occurrence of a contrail, (ii) the persistence of a contrail, and (iii) whether to ascend or descend if a contrail does occur at a particular altitude.Two types of forecast products are available, bothbased on Appleman's work (1957) as described in theAWS contrail forecast manual (AWS 1981 ). The firstis a global product generated at Air Force GlobalWeather Central (AFGWC). The global product usescurrent analysis data for tropopause height, p, and T,for levels at 500, 400, 300, 250, 200, 150, 100, 50, and30 mb as the initial state. This state is persisted for p> 100 mb out through 72 h ahead. For each forecasttime in that interval (e.g., 6 h, 12 h, etc.) the forecastlapse rate is compared to the critical lapse rate fromAppleman's curves, assuming 70% RH for +300 m ofthe tropopause height, 40% RH below this altitude, and10% above it. This comparison starts at 500 mb andmoves upward until T falls below T-, which is the bottom of the lowest layer. This search continues upwarduntil T rises above Tc, which is the top of the first layer.Subsequent layers are identified in the same way. Layers less than 600 m deep are discarded, and layers lessthan 400 m apart are consolidated. The two deepestlayers are retained for the global forecast product. The second type of forecast is generated by AWSforecasters in the field for specific flight operations.They may use the global product or may use the Appleman curves together with a local forecast of RH,or usually a combination of both, to generate a tailored forecast. Recently, Peters (1993) has releasednew Appleman tables that take the known variationsin Ce into account, that is, there are tables for eachstandard engine type except the superbypass (ductedfan) engine, as described in Table 1. These new tables are being used in the field but have yet to beimplemented in the global long-term product fromAFGWC. Note that long-range and/or long-term mission planning is based on the global forecast product, and in thisapplication it has proven to be insufficiently accurate(Peters 1993). Clearly, a number of improvements in implementation at GWC are possible, for example, using anactual forecast of RH instead of the assumed constantvalues noted above. However, the fundamental sensitivityto errors in forecast RH will remain, as shown by recentvalidation studies (Miller 1990; Bjomson 1992; Peters1993). The formulation presented in this paper overcomes this sensitivity, and at the same time makes use ofthe water-mass variable (wa) actually used in current forecast models, as well as output by satellite-sounding retrievals. In fact, in forecast applications Eq. (23) is inherently more accurate than is Eq. (13). When used as a diagnostic equation, that is, when all.the independent variables needed to compute the criticaltemperature are known accurately, Eq. (23) will give thesame answer as Eq. (13). However, when used as a forecasting equation, it is more accurate to use the formulationbased on mixing ratio (wa). This is most easily seen bycomparing the two formulations in their fundamentalform before explicitly solving for Tc. The formulation,using RH given previously as Eq. (11 ) in the paper, is100ATCr - lOOws(Tc + AT) + ws(Tc)RH = O, (28)while the formul~ttion using wa, given previously as Eq.(15), is /XTCr - ws(Tc + AT) + w~ = 0. (29)DECEMBER 1996 C O L E M A N 2279201510 , , ~ i I ' ' ' ' I i i ~ ~ I ' ' ' ' I ' ~ ' ' I ' ~ ' ' I ' ' ' ' ~.*%.~,. ~,, "::,,",, ~-*'- wo = . ..-.--. ........ wo ~ :~ '-~,.'~.. 'E~.~,iE]. ~,,.,. - 6- "::'.'". --- - 'Ci.' '.. '.i-, ", xx'-, - - ,,, ,, \\\ - ,,,., -,, \\\ -,.',. %, ~.\\ -~ .. .. \,., \ ',,'% '*, ~N '% -, .. \\ \ NN %X. ~. *~ a XX0 , ~ , I .... I , ~ , I , , , , I ~ , , , I ,'~,"A, , 'A, 'F~'~'"AD-65 -60 -55 -50 -45 -40 -35 -30 Temperoture (C) FIG. 4. Effects of different values of the contrail factor at different mixing ratios.As can be seen, the first two terms of Eq. (28) areidentical to the first two terms in Eq. (29) except forthe constant multiplier, and thus show the same behavior for errors in forecasts of the input values. However,'in Eq. (28), the RH in the third term depends on temperature as well as atmospheric moisture content and,thus, is susceptible to errors in temperature. In Eq.(29), the third term depends only on moisture content.Thus unless a temperature forecast is perfect (no error), the Wa formulation must give a more accurate prediction since all other variables are common to bothformulations. As an example of applying the new equation for To,three examples are included here. Figure 8 shows thecritical temperature for an arctic atmospheric temperature and mixing ratio RAOB profile. The site was at67.7-S, 62.88-E. Several interesting features are evident in the figure. First note that the atmosphere is quite.3 2_ Chonge in Tc due to o 10% Perturbotion in Relotive Humidit~ p = 1000 '2- 0 20 40 60 80 100 Relotive Humidity (~) FIG. 5. Sensitivity of Tc to perturbations in RH for a nonbypass engine.2280 JOURNAL OF APPLIED METEOROLOGY VOLUME351.21.00.80.60.40.20.0 0.00 Change in Tc due to o 10% Perturbetion in Mixing Redo p=lO00 .,,~,,, ..' '////// .,' / / ____ Cf=0.036 ........ Cf=O.04 / .,,-" ,," f " ,/ _ _ _ - =0.049 J ...'" ,," .~ , , ~ , , , , , I , , , , , ~ , , , I , , , , , , ~ ~ , I , , , ~ , , ~ , , 0.10 0.20 0.30 0.40 Mixing Ratio (g/kg) FIG. 6. Sensitivity of Tc to perturbations in w. for several contrail factors.dry and exhibits several mixing ratio inversion layers,the lowest of which peaks at around 6 km. Comparethat with the temperature profile, which shows two inversions, with the lowest peaking at 1.5 km, corresponding to the minimum of the lowest mixing: ratioinversion layer. As would be expected, this combination has the effect of depressing the critical temperatureand creating an inversion for Tc as well. Also, as expected, the Tc profile for low-bypass engines is lowerthan for high-bypass engines. Both types of engineswould generate contrails in this situation starting ataround 7.5 km. A very interesting feature of this sounding is that Tcfor the low-bypass engine drops below the ambienttemperature at around 14.5 km and remains below ituntil about 16.5 km. The effect of this is to produce2.52.01.51.00.50.0 0Change in Tc due to a 1 km PerturbQtion in PressureCf=O.04__ wo=O...... wo=0.2__ wa=0.4 200 400 600 800 Pressure (mb)FIG. 7. Sensitivity of Tc to perturbations in p for several mixing ratios.1000DECEMBER 1996 C O L E M A N 2281 Mixing Ratio (g/kg)0.0001 0.0010 0.01 O0 O. 1000 1.0000201 ....... ~ ....... ~ ....... ~ ....... x / '\ \ .' '\ \ ~ 'x \ '"'-, Temperature '"%~xx\ Critical Temperature High Bypass Engines ~ ~~ ~ for t, x ~'_ .... Critical Temperature for Low Bypass Engine~1 5 k ~ N , ~ . ~ 'x x : ..... M~mng Ratio ~ 'X X ~-..~ / '.~, .... , 'X X % 'X,X X '~ ' -. ~ ~'NN ~ *.. 'x x ~-. ./" '% % '~. -.. %,~ '-~ ' ~ '; 5 / ~ ~ ~' '0 ,,,, ......... , ......... ,,, ,, ....... ......... -~0 -50 -40 -30 -20 -10 0 lemperoture (C) Fro. 8. Critical temperature for an ~ctic atmosphere for low- and high-bypass engines.~o 10g-~two distinct contrail formation layers for low-bypassengines, as distinct from the high-bypass-engine profile, which shows contrail formation conditions from7.5 km up through 20 km without any break. Thus, anaircraft with low-bypass engines could find a high-altitude layer where contrails would be avoidable, whileaircraft with high-bypass engines would have to remainbelow 7.5 km to avoid generating contrails. Figures 9 and 10 show the difference in critical temperature between a relatively dry and a relatively wet midlatitude atmosphere taken from a RAOB for a low-bypassengine. The dry profile was taken at 40.65-N, 17.95-E,while the wet profile was taken at 41.65-N, 12.43-E. Thetemperature profile in both cases is quite similar, showingsignificant differences only above 10 km. The dry profileshows a distinctly dryer but colder tropopause relative toEv Mixing Ratio (g/kg) 0.001 0.010 O. 1 O0 1.000 10.000 20 ...... ~ ...... ~ ...... i ...... I ./ t ,...:' x Temperature x / x Critical Temperature ,, x / ..... Mixing Ratio 15 x - / \ ,, / "'x. - '"'-, '-~. - - -..~. --~ '-~.. "'-- -- -.. .5 ~ ~"~ '-x... 0 ~,1,,,,~,,,~1,,,, .... ,I,,,,,,,,~1 .... , ....I~ ,~ , ,~ ~ , , I ,~ ~ , ~ , ,'~., -60 -50 -40 -30 -20 -10 0 Temperature (C}FzO. 9. Critical temperature for a relatively dry midlatitude atmosphere for a nonbypass engine.2282 JOURNAL OF APPLIED METEOROLOGY VOLUME35 Mixing Ratio (g/kg) 0.001 0.010 O. 1 O0 1.000 10.000 20 ~ ....... ~ ........ ~ / ....... i ....... ~ x ! Temperature x ~ :~ Critical Temperature ~xx ~ ... _ .... M~x~ng Raho 15 x . ,:f ~ 10 xx_~ -60 -50 -40 -30 -20 - 10 0 lemperature (C)FIo. 10. Critical temperature for a relatively wet midlatitude atmosphere for a nonbypass engine.the wet profile, with the result that a contrail formationlayer exists in the "dry" sounding but is impossible inthe "wet." This is somewhat counterintuitive but demonstrates that for a layer with sufficiently low temperature, no ambient water is necessary for contrail formation.Note also in the wet profile that between 3 and 4 km, Tcis equal to T, due to the fact that mixing ratio is high atthat point relative to the dry profile. In fact the mixingratio equals the saturation mixing ratio at 750 mb for thisprofile. The' result is that for this very shallow layer acontrail might form but is unlikely.10. Conclusions A formulation of the contrail formation expression hasbeen provided that is analytic and is expressed in termsof mixing ratio and pressure rather than relative hu:midityand pressure. Both represent improvements over previousformulations from the standpoint of actual forecastingtechnique and accuracy. The analytic formulation improves computational implementation. The formulation interms of mixing ratio significantly reduces the sensitivityof the resulting critical temperature forecasts to errors oruncertainties in the moisture input by removing the effectof ambient temperature errors or uncertainties that areinherent in relative humidity. Acknowledgments. The research for this paper wasaccomplished as part of support to the Defense NuclearAgency under Contract F04701-93-C-0094. I am grateful to Major Robert Cox for consultations regardingforecasting aspects of the research described in thispaper. REFERENCESAir Weather Service, 1981: Forecasting aircraft condensation trails. Air Weather Service Tech. Rep. AWS/TR-81/001, 46 pp. [ DTIC AD-A 111876.]Appleman, H. S., 1953: The formation of exhaust condensation trails by jet aircraft. Bull. Amer. Meteor. Sac., 34~ 14-20.---, 1954: Memorandum on the effect of engine power setting on contrail formation and intensity. Air Weather Service Tech. Rep. AWS/TR-105-126, 4 pp. [DTIC AD-074310.]---, 1957: Derivation of jet-aircraft contrail-formation curves. Air Weather Service Tech. Rep. AWS/TR-105-145, 46 pp. [DTIC AD-125760.]Bjornson, B. M., 1992: SAC contrail formation study. USAF Envi ronmental Technical Applications Center Project Rep. USA FETAC/PR-92/004, 48 pp. [DTIC AD~A254410.]Bolton, D., 1980: The computation of equivalent potential temperature. Man. Wea. Rev., 108, 1046-1053.Knollenberg, R. G., 1972: Measurements of the growth of the ice budget in a persisting contrail. J. Atmos. Sci., 29, 1367-1374.List, R. J., Ed., 1963: Smithsonian Meteorological Tables. 6th ed. Smithsonian Institution, 527 pp.Miller, W. F., 1990: SAC contrail forecasting algorithm validationstudy. USAF Environmental Technical Applications CenterProject Rep. USAFETAC/PR-90/003, 28 pp. [DTIC ADB152198.]Murcray, W. B., 1970: On the possibility of weather modification by aircraft contrails. Man. Wea. Rev., 98~ 745-748.Peters, J. L., 1993: New techniques for contrail forecasting. Air Weather Service Tech. Rep. AWS/TR-93/001, 35 pp. [DTIC AD-A269686.]Pilie, R. J., and J. E. Jiusto, 1958: A laboratory study of contrails. J. Meteor., 15, 149-154.Saatzer, P., 1995: Pilot alert system flight test. Final Tech. Rep. for Period November 1988-May 1993, 241 pp. [Available from Northrup Grumman, B-2 Division, Northrop Grumman Corp., 8900 East Washington Blvd., Pica Rivera, CA 90660-3783.]Sassen, K., 1979: Iridescence in an aircraft contrail. J. Opt. Sac. Amer., 69, 1080-1083.Scorer, R. S., and L. J. Davenport, 1970: Contrails and aircraft down wash. J. Fluid Mech., 43, 451-464.

## Abstract

A new formulation of the equations describing the conditions necessary for aircraft exhaust contrail formation is derived from the fundamental necessary condition. First, the original solution of Appleman is derived from the necessary condition to illustrate the continuity of the new formulation. Then the new formulation offers an analytic solution for the critical temperature *T _{c}* expressed in terms of water vapor mixing ratio and atmospheric pressure, rather than in terms of relative humidity and pressure, thus avoiding potential forecast errors associated with the temperature sensitivity inherent in relative humidity. A variety of results is presented, including a comparison with the seminal results of Appleman, a comparison of the sensitivity of

*T*to perturbations in relative humidity versus perturbations in mixing ratio, and some typical results for actual atmospheric conditions. The clear superiority of a formulation based on mixing ratio rather than relative humidity is seen in the reduced sensitivity of

_{c}*, to errors or uncertainties in the input atmospheric variables.*

_{c}