Exact Expression for the Lifting Condensation Level

VOLUME 74 JOURNAL OF THE ATMOSPHERIC SCIENCES DECEMBER 2017

DAVID M. ROMPS Department of Earth and Planetary Science, University of California, Berkeley, and Climate and Ecosystem Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, California

(Manuscript received 4 April 2017, in ﬁnal form 31 August 2017)

ABSTRACT

Many analytic, but approximate, expressions have been proposed for the height of the lifting condensation level (LCL), including the popular expressions by Espy, Bolton, and Lawrence. Here, the exact, explicit, analytic expression is derived for an air parcel’s LCL as a function of its temperature and relative humidity. Unlike previous analytic expressions, some of which can have errors as high as hundreds or thousands of meters, this exact expression is accurate to within the uncertainty of empirical vapor pressure measurements: this translates into an uncertainty of around 5 m for all temperatures and relative humidities. An exact, explicit, analytic expression for the lifting deposition level (LDL) is also derived, and its behavior is compared to the LCL. At sufﬁciently cold temperatures, aerosols freeze homogeneously below the LCL; an approximate, implicit, analytic expression is given for this lifting freezing level (LFL). By comparing the LCL, LDL, and LFL, it is found that a well-mixed boundary layer can have an ice-supersaturated layer that is no thicker than 400 m.

1. Introduction height, temperature, and dewpoint temperature, respectively. Subsequent studies established that the co- The lifting condensation level (LCL) is the height at 2 efficient should be closer to 137 m K 1 (Davis 1889), and which an air parcel would saturate if lifted adiabatically. 2 the coefficient was later revised down to 123 m K 1 The LCL is a key concept in the prediction of cloud (McDonald 1963). Most recently, it has been suggested cover (e.g., Wetzel 1990), the parameterization of con- 2 by Lawrence (2005) that the optimal value is 125 m K 1, vection and precipitation in weather and climate models giving (e.g., Emanuel and Zivkovic ´-Rothman 1999), and the interpretation of atmospheric dynamics on other planets z 5 z 1 (125 m K21)(T 2 T ) (Espy). (1) (e.g., Atreya et al. 2006). Over the past 180 years, many LCL d explicit, analytic expressions have been proposed to In the years since Espy’s original work, more compli- approximate the LCL as a function of temperature and cated formulas were proposed, including the oft-used humidity, but none of those expressions is exact, and Eq. (22) of Bolton (1980) for the temperature of the none of them is expressed in terms of fundamental LCL, which we can convert to a height as physical constants. The first equation for the LCL was given by Espy 21 cpm 1 log(RH ) (1836, p. 244), who wrote that, as soon as the ascending z 5 z 1 T 2 55 K 2 2 l , LCL g T 2 55 K 2840 K air was ‘‘as many hundred yards high as the temperature of the air on the ground was above the dew-point (2) in degrees of Fahrenheit, the cold produced by the ex- where z is the parcel’s height, T is its absolute pansion of the air would begin to condense the vapour temperature, K denotes units of kelvins, RH is the par- and form clouds.’’ Since 1 yd equals 0.9144 m and 18F l cel’s relative humidity with respect to liquid, which ranges equals (5/9)8C, this means that the LCL is given by 21 21 from 0 to 1, and cpm/g is the inverse of the dry adiabatic z 5 z 1 (100 yards 8F )(T 2 T ) 5 z 1 (165 m K ) 2 LCL d lapse rate with g 5 9.81 m s 2 the gravitational accelera- (T 2 Td), where z, T, and Td are the parcel’s initial tion and cpm the parcel’s heat capacity at constant pressure (a precise definition of cpm is given in section 3). Corresponding author: David M. Romps, [email protected] Bolton (1980) also gives an implicit expression for the

DOI: 10.1175/JAS-D-17-0102.1 Ó 2017 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses). 3891 Unauthenticated | Downloaded 10/02/21 05:47 AM UTC 3892 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 74

LCL in his Eq. (18), but that equation is not analytic; it constant volume, cpy 5 cyy 1 Ry is the specific heat ca- must be solved using a numerical root solver. Finally, pacity of water vapor at constant pressure, cyl is the more recently, Lawrence (2005) proposed a formula of specific heat capacity of liquid water, ptrip is the triple- intermediate complexity, which is point vapor pressure, Ttrip is the triple-point temperature, and E y is the difference in specific internal energy 2 : 0 5 1 1 T 273 15 K 2 between water vapor and liquid at the triple point. zLCL z 20 (100 m)(1 RHl), (3) ,s 5K Similarly, defining p*y (T) to be the saturation vapor pressure of solid water (i.e., ice) at temperature where z is the parcel’s height, T is its absolute ,s T, p*y (T) is given by temperature, K denotes units of kelvins, m denotes units ! 2 of meters, and RHl ranges from 0 to 1. The intention of (cpy cys)/Ry T Lawrence (2005) is for this expression to be applied to p*,s 5 p y trip T parcels with 0:5 # RHl # 1 and 273 , T , 303 K. trip " !# The present study eliminates the need for this pro- 1 2 2 E y E (cyy cy )T 1 1 liferation of formulas by deriving an exact, explicit, an- 3 exp 0 0s s trip 2 , alytic expression for the LCL in the context of constant Ry Ttrip T heat capacities. The tiny errors introduced by assuming (5) constant heat capacities are also quantified: the uncertainty in this exact expression is less than or equal to where cys is the specific heat capacity of solid water and about 5 m. For those eager to use the exact expression E0s is the difference in specific internal energy between for the LCL, it may be found in Eq. (22). The analogous liquid and solid at the triple point. The values of cyy, ptrip, equation for the lifting deposition level (LDL) may be Ttrip, E0y, and E0s used here are found in Eq. (23). These functions are available for 5 21 21 download from the author’s website in a variety of cyy 1418 J kg K , (6) programming languages (currently R, Python, Fortran, p 5 611:65 Pa, (7) and MATLAB). trip 5 : Ttrip 273 16 K, (8) 2. Constant heat capacities 5 : 3 6 21 E0y 2 3740 10 Jkg , (9) To derive an exact, explicit, analytic expression for 5 : 3 6 21 E0s 0 3337 10 Jkg . (10) the LCL, we will need to use constant values for the heat capacities of water. Although heat capacities do Rather than specify the values of Ry, cyl, and cys a vary somewhat with temperature, neglecting this var- priori, we will choose their values to minimize the iation is standard practice. In this section, we show fractional difference between the saturation vapor that a careful choice of gas constant and heat capacities pressures given in Eqs. (4) and (5) and the saturation can put the analytic expressions for saturation vapor vapor pressures given by Wagner and Pruß (2002) and pressure into excellent agreement with laboratory Wagner et al. (2011) based on laboratory data. Allowing measurements. The reader who is comfortable with those three parameters to vary, the optimization method the use of constant heat capacities may skip to the next of Nelder and Mead (1965) is used to minimize the fol- section. lowing objective function: ,l Let p*y (T) be the saturation vapor pressure of liquid " # ð 2 273 K ,s water at temperature T. For an arbitrary temperature T, 1 p*y (T) ,l dT 2 1 and assuming constant heat capacities, p*y (T) is given 2 ,s,Wagner 273 K 180 K 180 K p*y (T) by (Romps 2008; Romps and Kuang 2010; Romps 2015) " # ð 2 330 K *,l ! 2 1 p y (T) (cpy cyl)/Ry 1 dT 2 1 . T 330 K 2 230 K ,l,Wagner *,l 5 230 K p*y (T) p y ptrip Ttrip " !# This identifies the optimal values, which are 2 2 E0v (cyy cyl)Ttrip 1 1 3 exp 2 , (4) 21 2 R 5 461 J kg K 1 , (11) Ry Ttrip T y 5 21 21 cyl 4119 J kg K , (12) where Ry is the specific gas constant for water vapor, 2 c 5 1 21 cyy is the specific heat capacity of water vapor at ys 1861 J kg K . (13)

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FIG. 1. (a) The saturation vapor pressure over ice as represented by Eq. (A1) (from Sonntag 1990) (dashed blue), Eq. (A2) (from Murphy and Koop 2005) (dotted red), Eq. (A3) (from Wagner et al. 2011) (dash–dotted green), and Eq. (5) (solid orange), all plotted as their fractional deviation (%) from the average of all four. The circles and associated error bars come from the laboratory measurements of Bielska et al. (2013). The solid black curves plot the estimated upper bound on the uncertainty from Eq. (14). (b) As in (a), but for the saturation vapor pressure over liquid as represented by Eq. (A4) (from Sonntag and Heinze 1982) (dashed blue), Eq. (A5) (from Murphy and Koop 2005) (dotted red), Eq. (A6) (from Wagner and Pruß 2002) (dash–dotted green), and Eq. (4) (solid orange).

,s Figure 1a compares p*y from Eq. (5) with the ice- of water from 180 to 330 K. To quantify the remaining saturation expressions given by Sonntag (1990), error or uncertainty, we can construct a simple MurphyandKoop(2005),andWagner et al. (2011), function U(T) such that 6U bounds the expressions which are ﬁts to laboratory data; see the appendix for and data in Fig. 1. The fractional uncertainty U is the formulas. Each of these four expressions is plotted modeled as as the fractional deviation (%) from the average of all 8 four. Also included are the recent laboratory mea- < 0:002 T $ 260 K 5 surements of Bielska et al. (2013), plotted as circles U : T 2 180 K , 0:01 1 (0:002 2 0:01) T , 260 K with error bars; these are also plotted as the fractional 260 K 2 180 K deviation from the mean of the four expressions. As is (14) evident, all of these expressions and data agree with each other to within 61% over the full range of tem- and 1U and 2U are plotted in Fig. 1 as the solid black perature from 180 to 273 K. Figure 1b compares the lines. Note that the difference between the vapor pres- ,l p*y from Eq. (4) with the liquid-saturation expressions sures derived here and the expressions from the other given by Sonntag and Heinze (1982), Murphy and studies is comparable to the differences among the ex- Koop (2005),andWagner and Pruß (2002),whichare pressions from the other studies. Therefore, U can be also ﬁts to laboratory data; see the appendix for the thought of as both an upper bound on the uncertainty formulas. Here, the agreement is better than 60.5% (i.e., the empirical uncertainty as to the true saturation over the full range of temperature from 233 to 330 K. vapor pressure) and an upper bound on the error This agreement is not guaranteed for other choices of (i.e., the deviation from the true vapor pressure caused Ry, cyl,andcys. For example, if we were to use a value by using analytical vapor pressure expressions with 21 21 of cys 5 2106 J kg K , which is listed in many text- constant heat capacities). books (e.g., Riegel 1992; Tsonis 2002; Wallace and Hobbs 2006; Cotton et al. 2011; Brasseur and Jacob ,s 3. Exact expression 2017), we would get a p*y that deviates from empirical measurements by several percent at a temperature In this section, we will derive an exact expression for of 180 K. an air parcel’s LCL height using the analytic saturation From Fig. 1, we conclude that the values in Eqs. (6)–(13) vapor pressures. Let us denote the air parcel’s initial do an excellent job of replicating the thermodynamics pressure and temperature by p and T, respectively. Let

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,l us also denote the air parcel’s pressure and temperature The next step is to recognize that py,LCL 5 p*y (TLCL), at its LCL by pLCL and TLCL, respectively. As a parcel of so the left-hand sides of Eqs. (20) and (21) are equal. air is transported adiabatically to its LCL, its potential Dividing Eq. (20) by Eq. (21), we get temperature is exactly conserved, which allows us to c /R 2(c 2c )/R relate its LCL pressure and temperature ( pLCL and TLCL) T pm m py yl y 1 5 RH LCL to its initial pressure and temperature ( p and T)by l T " # c /R E 2 (c 2 c )T T pm m 3 2 0y yy yl trip 2 T p 5 p LCL , (15) exp 1 , LCL T RyT TLCL 5 ,l where Rm is the air parcel’s specific gas constant and cpm where RHl py/p*y is the air parcel’s initial relative hu- is its specific heat capacity at constant pressure. The midity with respect to liquid. Defining x 5 T/TLCL,this 2a b(12x) subscript m denotes that these are the appropriate equation is of the form 1 5 RHlx e ,wherea, b,and values for moist air, that is, RHl are all independent of TLCL: they depend only on fundamental parameters and the initial properties of the 5 2 1 R (1 qy)R qyRy , (16) air parcel. Taking this equation to the power 1/a and de- m a 5 5 2 1 fining c b/a, we can then rearrange the equation to get cpm (1 qy)cpa qycpy , (17) cx 5 1/a c cxe RHl ce . where qy is the mass fraction of water vapor, Ra is the 5 1 specific gas constant of dry air, cpa cya Ra is the This can be solved using the Lambert W function, which specific heat capacity at constant pressure for dry air, is defined by W(yey) 5 y. The Lambert W function is and Ry and cpy are as defined in section 2. The values double valued when its argument is negative; since c , 0 used for Ra and cya are and x $ 1, we want the 21 branch, which is denoted by W2 . Taking W2 of the above equation, we get 5 : 21 21 1 1 Ra 287 04 J kg K , (18) 5 21 21 cx 5 W (RH1/acec). cya 719 J kg K . (19) 21 l By the ideal gas law, the partial pressure of water In a boundary layer with a dry adiabatic lapse rate, vapor py and the total pressure p are related by the dry static energy of a lifted parcel will be con- py 5 (Ryqy/Rm)p. Adiabatic lifting of a parcel from its served, implying that the height above the ground at which the parcel’s temperature equals T will be initial p to pLCL does not change its qy, and therefore, LCL 5 1 2 Ryqy/Rm is the same at pressures p and pLCL. Therefore, zLCL z cpm(T TLCL)/g. In fact, the parcel’s dry we can multiply the left-hand and right-hand sides of static energy will be very nearly conserved even in a Eq. (15) by the unique value of Ryqy/Rm to get boundary layer without a dry adiabatic lapse rate because the change in dry static energy for an adiabatically c /R T pm m lifted parcel is proportional to its buoyancy (Romps p 5 p LCL . (20) y,LCL y T 2015), and the typical buoyancy of parcels in a boundary layer is small. Therefore, the LCL temperature TLCL, the y This relates a parcel’s vapor pressure p ,LCL at its LCL to LCL pressure pLCL, and the LCL height zLCL are its initial vapor pressure py and temperature T. Next, we can use Eq. (4) to relate the parcel’s satu- 5 1/a c 21 TLCL c[W21(RHl ce )] T , (22a) ration vapor pressure with respect to liquid at its LCL c /R ,l T pm m temperature p*y (TLCL) to its saturation vapor pressure p 5 p LCL , (22b) ,l LCL with respect to liquid at its initial temperature p*y (T). T This yields c z 5 z 1 pm (T 2 T ), (22c) LCL g LCL (c 2c )/R T py yl y 2 ,l ,l LCL cpm cyl cpy p*y (T ) 5 p*y (T) 5 1 LCL T a , (22d) Rm Ry " # 2 2 2 2 E y (cyy cy )T T E y (cyy cy )T 3 exp 0 l trip 1 2 . b 52 0 l trip , (22e) RyT TLCL RyT (21) c 5 b/a. (22f)

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These equations give the parcel’s pressure, temperature, the homogeneous freezing temperature for aqueous aerosols is a function primarily of the activity of the and height at its LCL (pLCL, TLCL, and zLCL) in terms of its initial pressure, temperature, and height (p, T, and z). aerosol solution: the temperature of homogeneous Since W is a well-known special function, these are an- freezing for pure water (an activity of one) is 235 K alytic expressions for the properties of the LCL. Equa- (2388C), and that temperature decreases with de- tions (22a) and (22b) give the exact temperature and creasing activity (i.e., with decreasing liquid relative pressure of the parcel when it reaches saturation humidity). As an air parcel rises, its temperature de- through adiabatic expansion. If that adiabatic expansion creases (according to the dry adiabatic lapse rate) and takes place in a well-mixed layer, then Eq. (22c) gives the activity of its aerosols increases (to match the in- the exact height to which the parcel must ascend to creasing liquid relative humidity). Both of these effects saturate. We can easily check the limiting behaviors of bring the aerosols closer to freezing homogeneously. Eq. (22a). When RHl 5 1, we can use the fact that At a particular height, which we will refer to as the lifting c W21(ce ) 5 c to confirm that TLCL 5 T.AsRHl / 0, we freezing level (LFL), the parcel’s temperature equals can use the fact that W21(x) / 2log(2x)asjxj / 0to the homogeneous freezing temperature for the parcel’s confirm that TLCL / 0. For a well-mixed layer, this liquid relative humidity; at this level, the aerosols freeze would imply that the well-mixed subcloud layer occupies homogeneously. m the entire atmosphere up to an altitude of cpaT/g. Note From Fig. 3 of Koop et al. (2000), we note that a 1- m that Eq. (22) describes the pressure, temperature, and drop hits an ice supersaturation of 1.67 at 175 K. Since altitude of the lifting condensation level (i.e., the height the homogeneous freezing of pure water occurs at which the parcel’s vapor pressure equals the satura- at 2388C (235 K) (Hoose and Möhler 2012; Koop 2015), tion vapor pressure over liquid water). we can parameterize the relative humidity of homoge- Similarly, we can define the lifting deposition level neous freezing RHs,freeze as the following linear function (LDL) as the height at which the parcel’s vapor pres- of temperature: sure equals the saturation vapor pressure over solid water (i.e., ice). Above the LDL, the air parcel may p*,l(235 K) RH (T) 5 y form ice by heterogeneous deposition nucleation. s,freeze ,s p*y (235 K) Proceeding as in the derivation of the LCL, the LDL is 2 *,l found to be 1 235 K T : 2 p y (235 K) 1 67 ,s . 235 K 2 175 K p*y (235 K) 5 1/a c 21 TLDL c[W21(RHs ce )] T , (23a) (24) cpm/Rm TLDL p 5 p , (23b) At the LFL, py,LFL will be equal to RHs,freeze(TLFL) LDL T ,s p*y (TLFL). Proceeding as in the derivation of the LDL, c z 5 z 1 pm (T 2 T ), (23c) we get LDL g LDL 2 (" # )!2 c cy c y 1/a 1 a 5 pm 1 s p , (23d) RH T 5 c W s cec T , (25a) Rm Ry LFL 21 RHs,freeze(TLFL) E 1 E 2 (c 2 c )T 0y 0s yy ys trip c /R b 52 , (23e) pm m TLFL RyT p 5 p , (25b) LFL T 5 c b/a. (23f) c z 5 z 1 pm (T 2 T ), (25c) LFL g LFL Comparing to Eq. (22), we see that RHl has been ,s 2 replaced by RH 5 py/p* (T), which is the initial rel- cpm cys cpy s y a 5 1 , (25d) ative humidity with respect to solid (i.e., ice); cyl has Rm Ry been replaced by cys;andE0y has been replaced by 1 2 2 E0y E0s (cyy cys)Ttrip E y 1 E . b 52 , (25e) 0 0s R T If the temperature of the LCL is below 235 K (2388C), y then there is a height between the LDL and LCL at c 5 b/a. (25f) which aqueous aerosols freeze homogeneously. As an air parcel rises up below the LCL, the aerosols absorb Equation (25a) can be solved for TLFL using a root water so that, in equilibrium, their activity matches the solver. Note that, in contrast to the LCL and LDL, the liquid relative humidity. As shown by Koop et al. (2000), approximate temperature dependence of the LFL’s

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4. Physical implications of the LCL, LDL, and LFL For a rising air parcel, what are the physical implications of the LCL, LDL, and LFL? The LDL is the height above which the formation of an ice cloud is possible but not guaranteed. The LCL and the LFL are the heights at which an air parcel is guaranteed to become cloudy if it has not already done so. When the temperature of the LCL is between 2388 and 08C, the rising parcel can potentially form an ice cloud anywhere between the LDL and LCL through heterogeneous deposition nucleation (i.e., the formation of ice particles by deposition of water vapor onto ice nuclei) or immersion freezing (i.e., the freezing of aqueous aerosols onto an immersed ice nuclei) (Hoose and Möhler 2012). On the other hand, in the absence of any ice nuclei, the parcel can rise up past the LCL as a FIG. 2. For surface air with an RH of 50% with respect to liquid supercooled liquid cloud and still not freeze until the for T $ 08 C and with respect to solid (i.e., ice) for T , 08 C, the cloud reaches the homogeneous freezing temperature LCL from Eq. (22) (dashed), LDL from Eq. (23) (dotted), and LFL of 2388C, at which point it is obligated to become an from Eq. (25) (dash–dotted). The solid curves labeled 08 and 2388C ice cloud. are the parcel isotherms calculated assuming liquid-cloud moist adiabatic ascent above the LCL. When the temperature of the LCL is below 2388C, therisingparcelcanpotentiallyformanicecloud anywhere between the LDL and LFL through het- get a feel for the behavior under conditions that erogeneous deposition nucleation or immersion demonstrate the various characteristic domains of the freezing. In the absence of ice nuclei, however, neither phase diagram. of these processes is available, and the parcel will fail to form a cloud until it gets to the LFL. At the LFL, 220 K: A parcel that rises from the surface with a the aqueous aerosols are forced to freeze homoge- temperature of 220 K and a relative humidity of neously, and an ice cloud is born. In this way, the ex- 50% hits its LDL at a height of 620 m. If there are istence of an LFL renders the LCL irrelevant; this is sufﬁcient ice nuclei for heterogeneous deposition, the case whenever the temperature of the LCL is this parcel forms an ice cloud at or above that below 2388C. height. In the absence of ice nuclei, the parcel An application of the new expressions is shown in remains devoid of condensates until it reaches Fig. 2,inwhichtheLCL,LDL,andLFLareplottedas 990 m, which is its LFL. At this height, aerosols functions of surface air temperature for a surface air freeze homogeneously to form an ice cloud. relative humidity (RH) of 50%. Here, we deﬁne RH as 260 K: A parcel that rises from the surface with a

RHs when T , Ttrip and as RHl when T $ Ttrip.Also temperature of 260 K and a relative humidity of plotted on this diagram are the 2388 and 08Cisotherms 50% hits its LDL at a height of 885 m. As in the constructed by calculating liquid-cloud moist adiabats previous case, this parcel forms an ice cloud at or above the LCL. Note that the LCL curve is continued above its LDL if there are sufﬁcient ice nuclei. to temperatures well below 2388C; homogeneous With insufﬁcient ice nuclei, the parcel will not freezing prevents the possibility of an LCL at those nucleate any condensates until it reaches its LCL cold temperatures, but the LCL is plotted there to il- at 1180 m, at which point it forms a liquid cloud. lustrate its behavior in a hypothetical world with With no ice nuclei, the parcel’s condensates remain no ice. as supercooled liquid. If some ice nuclei are Using this diagram, we can track the evolution of present, then, depending on the number concen- an air parcel as it rises up from the surface. We can tration of ice nuclei and the elapsed time above the walk through three examples—corresponding to sur- LCL, the parcel forms either a mixed-phase cloud face air temperatures of 220, 260, and 300 K—to or an ice cloud.

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5 5 FIG. 3. (a) Contours of zLCL from the exact Eq. (22) plotted for z 0, p 1 bar, for all temperatures between 230 and 330 K, and all relative humidities from 0% to 100% with respect to liquid water. The gray region marks the ,l combinations of T and RHl for which TLCL , 230 K, below which we do not have empirical measurements of p*y because of homogeneous freezing at around 235 K. (b) The uncertainties from Eq. (26). (c) The error in the numerical solution of Eq. (18) in Bolton (1980), calculated as the difference from Eq. (22). (d) As in (c), but for the analytic Eq. (22) in Bolton (1980), replicated here as Eq. (2). (e) As in (c), but for the analytic expression proposed by Espy (1836), replicated here as Eq. (1). (f) As in (c), but for Eq. (24) in Lawrence (2005), replicated here as Eq. (3). The dashed box encloses the ranges of temperature (08–308C) and RH (0.5–1) over which Lawrence (2005) intended for the expression to be used.

300 K: A parcel that rises from the surface with a LCL is very weak, but the pressure is needed in order to temperature of 300 K and a relative humidity of calculate qy,whichentersintoRm and cpm.) The gray 50% hits its LCL at a height of 1435 m. At the LCL, region denotes where the LCL temperature would be the parcel forms a liquid cloud. less than 230 K, which is the lowest temperature for ,l whichwehavereliabledataonp*y ;thisisalsothe lowest temperature for which it makes much sense to 5. Comparison think about an LCL since water freezes homoge-

Figure 3a plots zLCL as given by Eq. (22) for all initial neously at 235 K. Not surprisingly, at ﬁxed tempera- temperatures from 230 to 330 K, all initial relative hu- ture, the LCL lowers as the relative humidity increases. midities RHl, an initial height of zero, and an initial Also, at ﬁxed relative humidity, the LCL lifts as the pressure of 1 bar. (The pressure dependence of the temperature increases. This is due to the fact that

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5 5 FIG. 4. (a) Contours of zLDL from the exact Eq. (23) plotted for z 0, p 1 bar, for all temperatures between 180 and 273 K, and all surface relative humidities from 0% to 100% with respect to solid water. The gray region marks the combinations of T and RHs for which TLDL , 180 K, below which we do not have empirical measurements of ,s p*y . (b) The uncertainties from Eq. (27).

,l 2 dlog(p*y )/dT is proportional to 1/T : as temperature The remaining panels of Fig. 3 plot the errors in the increases at fixed relative humidity, a larger tempera- previous approximations to the LCL, calculated as the ture change (and, therefore, a larger dry adiabatic as- difference between their prediction for the LCL and cent) is needed to get to saturation. the actual value from Eq. (22). Figure 3c plots the error Figure 3b plots the uncertainty in the exact LCL ex- in Eq. (18) of Bolton (1980), which is solved using a pression due to the nonzero U given by Eq. (14). These numerical root finder. Over all possible LCLs, its maxi- uncertainties in the LCL are calculated as mum error is 20 m, which occurs at the highest temperatures and relative humidities in Fig. 3c; this maximum Uncertainty in z 5 z (z, p, T,RH) LCL LCL l error exceeds the uncertainty of 5 m. Figure 3d plots the 2 1 zLCLfz, p, T,[1 U(T)]RHlg error in Eq. (22) of Bolton (1980), which is printed here in (26) Eq. (2). Its maximum error is 40 m. Figure 3e plots the error in the equation of Espy (1836) as subsequently since the fractional uncertainty in the true saturation modifiedandprintedhereinEq.(1).Itsmaximumerror vapor pressure translates into a fractional uncertainty in is 665 m. Finally, Fig. 3f plots the error in the Eq. (24) of the initial RH. For all combinations of T and RH in the Lawrence (2005), which is printed here in Eq. (3).Its ranges considered, this error in the predicted LCL lies in maximum error is 7130 m. The dashed box encloses the the range of 4–6 m (i.e., the uncertainty is about 5 m). ranges of temperature (08–308C) and relative humidity

FIG. 5. (a) Plot of zLCL minus zLDL as a function of surface air temperature and surface air relative humidity with respect to ice. Negative values are not plotted because they are not of interest (if the LCL is lower than the LDL, then the LDL is irrelevant). The two contours denote the 08 and 2388C isotherms of the LCL. The gray region

marks the combinations of T and RHs for which TLCL , 230 K. (b) As in (a), but for min(zLCL, zLFL) minus zLDL, where the LFL is the level of homogeneous freezing of aerosols. The gray region marks the combinations of T and RHs for which TLFL , 180 K.

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(0.5–1) over which Lawrence (2005) intended for the derived that are exact, explicit, and analytic; they are expression to be used; in those ranges of temperature and giveninEqs.(22) and (23). These expressions are given humidity, the maximum error is 170 m. in terms of fundamental constants, which may be Figure 4a plots the LDL given by the exact analytic adapted to extraterrestrial atmospheres with a con- Eq. (23) for all temperatures from 180 to 273 K, all rel- densible gas that can be adequately described with ative humidities RHs, and a pressure of 1 bar. The gray constant heat capacities. An expression is also given for region denotes where the LDL temperature would be the lifting freezing level (LFL), defined as the height at less than 180 K, which is the lowest temperature for which aqueous aerosols freeze homogeneously. That ,s which we have reliable data on p*y . Figure 3b plots the expression, given in Eq. (25), depends on an approxi- uncertainty in the exact LDL expression due to the mate equation for the homogeneous freezing activity; nonzero U given by Eq. (14). As for the LCL, the un- this makes the expression implicit, thereby requiring a certainties in the LDL are calculated as numerical root solver for evaluation. These expressions then allow for a quantification of the maximum thick- 5 Uncertainty in zLDL zLDL(z, p, T,RHs) ness of ice-supersaturated layers underlying liquid 2 1 cloud–topped boundary layers. On Earth, the maxi- zLDLfz, p, T,[1 U(T)]RHsg . mum potential thickness is about 400 m, which can be (27) attained when the LCL is at the homogeneous freezing For all possible LDLs, the uncertainties lie in the range temperature of 235 K. of 4–6 m (i.e., the uncertainty is about 5 m). Acknowledgments. This work was supported primarily by the U.S. Department of Energy’s (DOE) Atmo- 6. The ice-supersaturated layer spheric System Research, an Office of Science, Office of Figure 5a plots the LCL minus the LDL as a function Biological and Environmental Research program; of temperature T and relative humidity with respect to Lawrence Berkeley National Laboratory is operated for ice RHs. This serves as an upper bound on the thickness the DOE by the University of California under Contract of the ice-supersaturated layer underneath the liquid DE-AC02-05CH11231. The author is grateful to Alan cloud base. As discussed in section 4,theLCLisir- Betts, Ronald Cohen, Mark Lawrence, and an anonymous relevant when its temperature is below about 2388C. reviewer for their helpful feedback. At those temperatures, the LFL lies below the LCL, so rising parcels will form ice clouds before they reach the LCL. Therefore, the depth of the potentially ice- APPENDIX supersaturated layer is min(zLCL, zLFL) 2 zLDL.Thisis plotted in Fig. 5b, where we see that the maximum Empirical Saturation Vapor Pressure depth of the supersaturated layer is about 400 m. We In Fig. 1a, the equations for saturation vapor pressure can summarize Fig. 5b as follows. For any surface air ,s over solid water p*y from Sonntag (1990), Murphy and temperature, there is a sufficiently low relative hu- Koop (2005), and Wagner et al. (2011) are as follows. 8 midity(i.e.,arelativehumiditybelowthe0CLCL The equation from Sonntag (1990) is isotherm in Fig. 5b) that, in the absence of ice nuclei, will generate an ice-supersaturated layer just below the ,s 21 p*y 5 100 exp[24:7219 2 6024:5282T 10:010 613 868T cloud base. For surface air temperatures below 08C, a 2 : 2 2 : subcloud ice-supersaturated layer is a possibility for all 0 000 013 198 825T 0 493 825 77 log(T)], surface air relative humidities. The maximum depth of (A1) that layer occurs when the LCL has a temperature ,s of 2388C; at this temperature, the depth of the ice- where T is in kelvins and p*y is in pascals. The equation supersaturated layer is about 400 m. For colder LCLs, from Murphy and Koop (2005) is the depth of the supersaturated layer decreases to p ,s 5 exp[9:550 426 2 5723:265T21 1 3:530 68 log(T) about 300 m at the coldest surface air temperatures *y observed on Earth. 2 0:007 283 32T], (A2)

,s where T is in kelvins and p*y is in pascals. The equation 7. Summary from Wagner et al. (2011) is Expressions for the lifting condensation level (LCL) ,s 21 b b b * 5 u u 1 1 u 2 1 u 3 and the lifting deposition level (LDL) have been p y pt exp[ (a1 a2 a3 )], (A3)

Unauthenticated | Downloaded 10/02/21 05:47 AM UTC 3900 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 74 where pt 5 611:6573 Pa, u 5 T/Tt, Tt 5 273:16 K, Davis, W. M., 1889: Some American contributions to meteorology. 2 2 J. Franklin Inst., 127, 176–191, doi:10.1016/0016-0032(89)90145-2. a1 520:212 144 006 3 10 , a2 5 0:273 203 819 3 10 , Emanuel, K. A., and M. Zivkovic ´-Rothman, 1999: Develop- a 520:610 598 130 3 101, b 5 0:333 333 333 3 1022, 3 1 ment and evaluation of a convection scheme for use in 5 : 3 1 5 : 3 1 b2 0 120 666 667 10 , and b3 0 170 333 333 10 . climate models. J. Atmos. Sci., 56, 1766–1782, doi:10.1175/ In Fig. 1b, the equations for saturation vapor pressure 1520-0469(1999)056,1766:DAEOAC.2.0.CO;2. ,l over liquid water p*y from Sonntag and Heinze (1982), Espy, J. P., 1836: Essays on meteorology. No. IV: North east Murphy and Koop (2005),andWagner and Pruß (2002) are storms, volcanoes, and columnar clouds. J. Franklin Inst., 22, as follows. 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