15 DECEMBER 2003 KOROLEV AND MAZIN 2957
Supersaturation of Water Vapor in Clouds
ALEXEI V. K OROLEV Sky Tech Research, Inc., Richmond Hill, Ontario, Canada
ILIA P. M AZIN Center for Earth and Space Research, George Mason University, Fairfax, Virginia
(Manuscript received 18 October 2002, in ®nal form 26 June 2003)
ABSTRACT A theoretical framework is developed to estimate the supersaturation in liquid, ice, and mixed-phase clouds. An equation describing supersaturation in mixed-phase clouds in general form is considered here. The solution for this equation is obtained for the case of quasi-steady approximation, that is, when particle sizes stay constant. It is shown that the supersaturation asymptotically approaches the quasi-steady supersaturation over time. This creates a basis for the estimation of the supersaturation in clouds from the quasi-steady supersaturation calcu- lations. The quasi-steady supersaturation is a function of the vertical velocity and size distributions of liquid and ice particles, which can be obtained from in situ measurements. It is shown that, in mixed-phase clouds, the evaporating droplets maintain the water vapor pressure close to saturation over water, which enables the analytical estimation of the time of glaciation of mixed-phase clouds. The limitations of the quasi-steady ap- proximation in clouds with different phase composition are considered here. The role of phase relaxation time, as well as the effect of the characteristic time and spatial scales of turbulent ¯uctuations, are also discussed.
1. Introduction centration and its average size (Squires 1952; Mazin 1966, 1968). Questions relating to the formation of su- The phase transition of water from vapor into liquid persaturation in clouds were discussed in great details or solid phase plays a pivotal role in the formation of by Kabanov et al. (1971). Rogers (1975) studied the clouds and precipitation. The direction and rate of the supersaturation equation for liquid clouds numerically phase transition ``liquid-to-vapor'' or ``ice-to-vapor'' and considered analytical solutions for some special cas- are determined by the vapor supersaturation with respect es. Subsequent re®nements of the supersaturation equa- to liquid or ice, respectively. The early numerical mod- tion and the time of phase relaxation were made by eling by Howell (1949) and Mordy (1959) has shown Korolev (1994). Most studies of the supersaturation are that the supersaturation in a uniformly ascending cloud related to liquid clouds. Only few works consider su- parcel reaches a maximum near the cloud base during persaturation in glaciated clouds (e.g., Juisto 1971; activation of droplets and then monotonically decreases. Heyms®eld 1975). Squires (1952) derived a relationship between supply The present paper is focused on a study of the super- and depletion of water vapor in a vertically moving saturation in clouds of any phase composition: mixed, cloud parcel. The solution of the Squires equation sug- liquid, and ice. In the section 2 we consider the equation gested that, to a ®rst approximation, the supersaturation for supersaturation in a general form for a three-phase is linearly related to a vertical velocity and inversely component system consisting of water vapor, liquid, and proportional to the concentration and the average size ice particles. Section 3 presents a solution of the super- of the cloud droplets. This relationship was later used saturation equation for the case of quasi-steady approx- to determine supersaturation in convective clouds by imation, that is, when the sizes of cloud particles stay Warner (1968), Paluch and Knight (1984), and Politov- constant. Sections 4 and 5 examine supersaturation in ich and Cooper (1988). The characteristic time of phase liquid, ice, and mixed-phase clouds. In section 6 the an- transition due to water vapor depletion by cloud droplets alytical solution for the glaciating time of mixed-phase is inversely proportional to the product of droplet con- clouds for zero vertical velocity is obtained. Section 7 justi®es the possibility of estimating the in-cloud super- saturation from the quasi-steady supersaturation, which Corresponding author address: Alexei V. Korolev, Sky Tech Re- search, Inc., 28 Don Head Village Blvd., Richmond Hill, ON L4C can be calculated from in situ measurements of particle 7M6, Canada. size distributions and vertical velocity. In section 8 the E-mail: [email protected] limiting conditions for the results are discussed.
᭧ 2003 American Meteorological Society 2958 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 60
2. General equation dqw ϭ BSNrww ww, (4) Consider a vertically moving adiabatic cloud parcel dt consisting of a mixture of liquid droplets and ice par- where Bw ϭ 4wAw/a and r w is the average radius of ticles uniformly distributed in space. Assume that the cloud droplets. cloud particles move with the air and stay inside the Similarly, an equation for changes in the mixing ratio parcel at all times and no activation of new cloud drop- of ice can be written as lets and nucleation of ice particles occur. Thus, the num- ber of droplets and ice particles per unit mass of the air dq 4Ndr1 imax ϱ iif (r , , c) rdrd2 idc. stays constant. Another assumption is related to so ϭ ii i i i i dt a ͵͵ ͵ dt called regular condensation; that is, the water vapor 0 imin 0 pressure and temperature ®elds at large distance from (5) cloud particles are considered to be uniform, and all Here, Ni is the number concentration of ice particles; ri cloud particles grow or evaporate under the same con- is the characteristic size of ice particles; c is a shape ditions. factor of ice particles characterizing ``capacitance'' in The changes in the supersaturation in the cloud parcel the equation of the growth rate; i is the density of ice can be described by an equation (appendix A): particles (imin Ͻ i Ͻ imax); and f i(ri, i, c)isthe distribution of ice particles by characteristic size, den- 1 dS gL g 1 Ldq2 wwϭϪu Ϫϩ wwsity, and shape factor normalized to unity. The size of 22z Swpϩ 1 dt cRT RTa q cRTp dt an ice particle is an ambiguous parameter, and it can be de®ned in a number of different ways. The ice particle 1 LL dq size r is related to the shape factor c through the rate Ϫϩiw i. (1) i 2 of the mass growth. In the following consideration, the qcRTdt p size ri will be de®ned as half the maximum dimension Here, Sw ϭ (e Ϫ Ew)/Ew is the supersaturation with of the ice particle. For this de®nition of ri, the shape respect to water; e is the water vapor pressure; Ew(T) factor varies in the range 0 Ͻ c Յ 1, being equal to 1 is the saturated vapor pressure over liquid water at tem- for ice spheres. perature T; q , qw, and qi are the mixing ratios of water Using Jeffery's electrostatic analogy, the diffusional vapor, liquid water, and ice, respectively (mass per 1 kg growth rate of the ice particles can be written as (e.g., of dry air); R and Ra are the speci®c gas constants of Pruppacher and Klett 1997) water vapor and air, respectively; cp is the speci®c heat capacitance of the air at constant pressure; L and L driii cA S w i ϭ . (6) are the latent heat of water evaporation and ice subli- dt ri mation, respectively; g is the acceleration of gravity; uz 2 2 Ϫ1 is the vertical velocity of the cloud parcel. Here, Ai ϭ (iLi /kR T ϩ iR T/EiD) ; Ei is the sat- The rate of changes in the liquid water mixing ratio urated water vapor pressure over a ¯at ice surface at due to condensation or evaporation of cloud droplets temperature T; Si is the supersaturation over ice. can be described as The supersaturation over ice can be related to the supersaturation over water with dq 4 Ndrϱ wwwϭ f (r )rdr2 w. (2) www w e Ϫ Ei dt a ͵ dt 0 SiwϭϭS ϩ Ϫ 1, (7) Ei Here, f w(rw) is the size distribution of cloud droplets normalized to unity, rw is the droplet radius, Nw is the where (T) ϭ Ew(T)/Ei(T). Then, substituting Eqs. (6) number concentration of the cloud droplets, w and a and (7) into Eq. (5) and averaging over ri and c gives are the densities of liquid water and dry air, respectively. dq 4ANrc The mass transfer of water from vapor to liquid phase iiiiϭ (S ϩ Ϫ 1) , dt w inside clouds is mainly related to droplets having rw Ͼ a 1 m. For such droplets, the corrections for the droplet which for simplicity yields curvature and salinity can be disregarded, and the equa- tion for a droplet growth can be written as (e.g., Prup- dq i ϭ (BS ϩ B*)Nr (8) pacher and Klett 1997) dt iw i ii dr A S wwwϭ . (3) if one assumes that c is the same for all ice particles dt rw and that
2 2 Ϫ1 Here, Aw ϭ (wLw /kR T ϩ wR T/EwD) , D is the 4 cA 4 c B ϭ ii; B* ϭ i( Ϫ 1)A . coef®cient of water vapor diffusion in the air, and k is iii the coef®cient of air heat conductivity. Substituting Eq. aa (3) into Eq. (2) and averaging over r results in Substituting Eqs. (4) and (8) in Eq. (1) yields 15 DECEMBER 2003 KOROLEV AND MAZIN 2959
1 dS Here, we denote a ϭ g(LR /c R T Ϫ 1)/R T; a ϭ w au aB*Nr 0 a p a 1 ϭ 0 z Ϫ 2 iii 1/q ϩ L2 /c R T 2; a ϭ 1/q ϩ L L /c R T 2. Sw ϩ 1 dt w p 2 w i p Integrating Eqs. (3) and (6), and assuming that the size
Ϫ (aB1 www N r ϩ aBNr2 iiiw)S . (9) distributions of ice particles and droplets are monodisperse and then substituting rw and ri in Eq. (9) results in
1 dS t w 2 ϭ au0 z Ϫ aN2 ii r0 ϩ 2cAi [Sw(tЈ) ϩ Ϫ 1] dtЈ Sw ϩ 1 dt ͵ Ί 0
tt Ϫ aB N r22ϩ 2AS(tЈ) dtЈϩaBN r ϩ 2cA [S (tЈ) ϩ Ϫ 1] dtЈ S . (10) 1 ww w0 w ͵ w 2 ii i0 i ͵ ww Ά·ΊΊ00
1 Here, rw0 and ri0 are droplet and ice particle sizes at the saturation Sqsw is called quasi-steady. We will keep this time t ϭ 0. terminology in the frame of this study. Similarly, the
Equation (10) describes supersaturation in a general assumption rrw ϭ constant and i ϭ constant in the su- form in clouds consisting of ice particles and liquid persaturation equation (9), will be called quasi-steady droplets having monodisperse size distribution, and it approximation. Following Mazin (1966, 1968), the con- can be used for estimation of supersaturation in clouds stant p herein will be called the time of phase relax- with any phase composition. Some special cases, when ation. The term integral radius will be applied to Nwwr analytical solutions can be found, are considered in the and Niir following Politovich and Cooper (1988). following sections. The dependence of q on supersaturation in the co- ef®cients a1 and a 2 has been neglected in Eqs. (1), (9), and (10). The comparison of numerical modeling results 3. Quasi-steady approximation (appendix B) with the solution of Eq. (10) shows that Assuming that changes in the size of the cloud par- for most clouds in the troposphere this assumption works with high accuracy. The solution for Eq. (9) in ticles can be neglected so that rrw and i are constant, Eq. (9) yields a solution: a more general form, including a dependence of q ver- sus Sw, is given in appendix C.
Sqsw Ϫ C0 exp(Ϫt/ p) Sw ϭ , (11) 1 ϩ C0 exp(Ϫt/ p) 4. Supersaturation in single-phase clouds where a. Liquid clouds
au0 ziiiϪ b*Nr Assuming Ni ϭ 0, Eq. (10) gives the equation for Sqsw ϭ (12) bNrwwwϩ bNr iii supersaturation in liquid clouds having a monodisperse droplet size distribution (Korolev 1994): is the limiting supersaturation, and 1 dSw 1 Sw ϩ 1 dt p ϭ (13) au0 zwwwiiiiϩ bNr ϩ (b ϩ b*)Nr t is the time constant characterizing a time of asymptotical ϭ au Ϫ aB N S r2 ϩ 2cA S (tЈ) dtЈ. (14) 0 z 1 www w0 w ͵ w approach of Sw to Sqsw, that is, at t/p k 1, Sw ϭ Sqsw. Ί 0 Here, C ϭ (S Ϫ S )/(1 ϩ S ); S is the super- 0 qsw w0 w0 w0 Figure 1 shows solutions of Eq. (14) for different saturation at t ϭ 0; bw ϭ a1Bw; bi ϭ a 2Bi; b*i ϭ vertical velocities uz. In the ascending parcel, the su- a 2B*i . The limiting supersaturation S [Eq. (12)] coincides persaturation decreases after reaching a maximum (Fig. qsw 1a). The presence of the maximum depends on initial with the solution of Eq. (9), when dSw/dt ϭ 0. Therefore, supersaturation Sw(0). If Sw(0) is high enough, Sw(t) will the physical meaning of Sqsw can be interpreted as an equilibrium supersaturation; that is, when an increase monotonically decrease at all times. It can be shown (decrease) of the relative humidity due to air cooling that for t k p the supersaturation asymptotically ap- (heating) is balanced by the depletion (release) of the water vapor by cloud particles. Traditionally, after 1 Squires (1952) used the term quasi-static for Sqsw, which has a Squires (1952) and Rogers (1975), the limiting super- meaning close to quasi-steady. 2960 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 60
FIG. 2. Quasi-steady supersaturation vs Nwwr in liquid clouds for different vertical velocities (Ϯ0.1, Ϯ1, and Ϯ10msϪ1) and different
temperatures (ϩ5Њ and Ϫ30ЊC). Gray color indicates Nwwr typical for the liquid clouds in the troposphere; P ϭ 690 mb.
for the supersaturationS*w (t) for the quasi-steady ap- proximation, that is, when Nwwr ϭ const and Ni ϭ 0. As seen from Eq. (14) the quasi-steady approximation can be applied for the time period when
t r2 k 2AS(tЈ) dtЈ . (15) w0 w ͵ w ΗΗ0
Figure 1 shows that the supersaturation Sw(t) agrees well withS*w (t) when t Ͻ p. At larger times the difference between Sw(t) andS*w (t) increases since condition (15) FIG. 1. Changes of supersaturation (solid line) with time in adia- batic (a) ascending and (b) descending liquid cloud parcels containing becomes invalid. Ϫ3 Substituting N 0 in Eqs. (12) and (13) gives the liquid droplets with rw0 ϭ 5 m and Nw ϭ 200 cm . The solution i ϭ of Eq. (9) for the quasi-steady approximation (Nwwr ϭ constant and expression for quasi-steady supersaturation
Ni ϭ 0) is shown by a dashed line. The time of phase relaxation (p 3.3 s) is indicated by the vertical line on the left-hand side of (a) au0 z ϭ S ϭ (16) and (b). The initial conditions are S(0) ϭ 0; T(0) ϭ 0ЊC, P(0) ϭ 870 qsw bNr mb. www and the time of phase relaxation proaches zero as S (t) ϳ tϪ1/3 [Sedunov 1974, Eq. 1 w . (17) (4.3.26) p. 116]. In the descending parcel the supersat- p ϭ au0 zwwwϩ bNr uration is negative and decreases monotonically with time to the moment of complete evaporation of droplets For characteristic values of Nwwr 0 and uz typical for (Fig. 1b). After the droplets evaporate the supersatu- liquid clouds, bwNwwr k a 0uz. Then, neglecting a 0uz in Eq. (16), gives ration changes as Sw ϭ exp(Ϫa 0uzt) Ϫ 1. Equation (14) gives a rather accurate solution of Sw. 1 Comparison shows that the agreement between super- p ϭ . (18) bNr saturation derived from a numerical solution of a system www of differential equations Eqs. (B1)±(B7) (appendix B) An expression similar to Eq. (16) was originally de- and that of a single Eq. (14) is better than 1% for a duced by Squires (1952). The time of phase relaxation vertical displacement of a few hundred meters. For larg- in the form similar to Eq. (18) was ®rst obtained by er displacements, the deviation between Sw computed Squires (1952) and Mazin (1966). from Eq. (14) and Eqs. (B1)±(B7) (appendix B) in- Figure 2 shows Sqsw calculated from Eq. (16) for dif- creases because Eq. (14) does not take into account ferent values of Nwwr and uz. The gray color indicates changes of a 0, a1, and Aw with T and P. the range of Nwwr typical for liquid clouds in the tro- Dashed lines in Fig. 1 show the solution of Eq. (9) posphere. As seen in Fig. 2 the supersaturation varies 15 DECEMBER 2003 KOROLEV AND MAZIN 2961
FIG. 3. Time of phase relaxation vs Nwwr in liquid clouds for dif- ferent vertical velocities. T ϭ 0ЊC, P ϭ 680 mb. Gray color indicates
Nwwr typical for the liquid clouds in the troposphere.
between Ϫ0.5 Ͻ Sw Ͻ 0.5% for vertical velocities Ϫ1 Ϫ1 Ͻ uz Ͻ 1ms . Such vertical velocities would limit the characteristic range of vertical velocity ¯uctuations in stratiform clouds (Mazin et al. 1984). In convective clouds where the vertical velocities vary in a wider Ϫ1 range, for example, Ϫ10 Ͻ uz Ͻ 10 m s , the super- saturation would experience ¯uctuations of Ϫ5 Ͻ Sw Ͻ 5%, which is an order of magnitude larger than those in stratiform clouds. The supersaturation in a vertically moving parcel also depends on temperature and pressure. For ®xed Nwwr and uz, the supersaturation increases with decreasing FIG. 4. Changes of supersaturation with time (solid line) in adia- temperature and increasing air pressure (Rogers 1975). batic (a) ascending and (b) descending ice cloud parcels containing Ϫ3 Figure 3 shows the time of phase relaxation versus ice particles of ri0 ϭ 20 m and Ni ϭ 1cm . The solution of Eq. (9) for the quasi-steady approximation (Niir ϭ constant and Nw ϭ 0) Nwwrrcomputed from Eq. (17). For typical Nww, the is shown by a dashed line. In (a), line Sw ϭ 0 corresponds to saturation characteristic time of the phase relaxation in liquid Ϫ1 over water at uz ϭ 1ms . The vertical lines indicate the initial Ϫ1 clouds is a few seconds. The effect of vertical velocity values of time of phase relaxation: (a) p ϭ 168 s, (uz ϭ 0.1 m s ), 152s(u 1msϪ1, not shown); (b) 173s(u 0.1 uz becomes signi®cant at small Nwwr . However, such a p ϭ z ϭ p ϭ z ϭϪ Ϫ1 Ϫ1 ms ), p ϭ 193s(uz ϭϪ1ms , not shown). The initial conditions combination of high uz and low Nwwr is unlikely in liquid are Si(0) ϭ 0, T(0) ϭϪ5ЊC, and P(0) ϭ 870 mb. clouds. Therefore, the term a 0uz in the denominator in Eq. (17) can be neglected, and for practical calculations of p in liquid clouds, Eq. (18) is an accurate approx- 1 dSi imation. S ϩ 1 dt Equations (9)±(18) neglect the corrections of curva- i ture and salinity of droplets on the rate of droplet growth t [Eq. (4)]. Kabanov et al. (1971) showed that the effect ϭ au Ϫ aB NS r2 ϩ 2cA S (tЈ) dtЈ. (19) 0 z 3 i0 ii i0 i ͵ i of these corrections on Sqsw and p is negligible. Omitting Ί 0 these corrections does not affect our considerations and Here, a ϭ 1/q ϩ L2/c R T 2; B ϭ 4 cA / . the ®nal conclusions. 3 i p i0 i i a Figure 4 shows changes of supersaturation Si(t)inice Ϫ3 cloud with ri0 ϭ 20 m and Ni ϭ 1cm for different b. Ice clouds vertical velocities. The ascending parcel takes longer to reach maximum supersaturation in glaciated cloud than
Similarly to section 2, the equation for supersatura- in liquid clouds because of lower values of Niir com- tion with respect to ice in glaciated clouds with mono- pared to Nwwr . Dashed lines show the supersaturation disperse ice particles and Nw ϭ 0 can be written as S*i (t) resulted from the solution of Eq. (19) for the quasi- 2962 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 60
FIG. 5. Quasi-steady supersaturation vs Niir in ice clouds for dif- FIG. 6. Time of phase relaxation vs Niir in ice clouds for different ferent vertical velocities. Horizontal lines indicate saturation over vertical velocities; T ϭϪ10ЊC; P ϭ 680 mb. Gray color indicates water at Ϫ5ЊC (line 1); and at Ϫ35ЊC (line 2). P ϭ 870 mb. Gray Niir typical for the ice clouds. color indicates Niir typical for the ice clouds.
of the water vapor and prevent continuous increase of steady approximation, that is, when N r ϭ const. The ii the supersaturation, keeping the water vapor pressure difference between S (t) andS* (t) is also typically larger i i close to the saturation over water. in glaciated clouds than in liquid clouds. Figure 6 shows the time of phase relaxation calculated The supersaturation in Eq. (9) was considered with from Eq. (21). For typical N r in ice clouds the char- respect to water. Similarly, this equation can be rewritten ii acteristic time of the phase relaxation may vary from a for the supersaturation with respect to ice. The solution few minutes to a few hours. Comparing diagrams in of this equation yields the quasi-steady supersaturation Figs. 3 and 6, it can be concluded that the processes au related to changes in water vapor due to evaporation or S ϭ 0 z (20) qsi bNr condensation of ice particles in the glaciated clouds are i0 ii on average slower than those related to liquid droplets. and the time of phase relaxation In other words the ``condensational'' activity of droplets 1 in tropospheric clouds is typically higher than that of p ϭ , (21) ice particles due to on average lower values of integral au0 ziϩ bNr0 ii radius of the ice particles than those of the liquid drop- lets. The effect of u on is more pronounced in ice where bi0 ϭ a3Bi0. z p clouds compared to liquid ones. Figure 5 shows Sqsi calculated from Eq. (20) for dif- The ice particles in this section were treated as ice ferent values of Niir and uz. In situ measurements showed that the average concentration of ice particles spheres having the capacitance c ϭ 1. Because for the Ϫ3 in glaciated clouds is 2±5 cm with a characteristic size majority of ice particles c Ͻ 1, the coef®cients Bi and between 25±40 m in the temperature range Ϫ35ЊϽ B*i will be reduced. This would result in supersaturation T Ͻ 0ЊC (Korolev et al. 2003). Due to lower values of and time of phase relaxation values higher than those shown in Figs. 4 and 5. Therefore, the values of S , S Niir , the supersaturation Sqsi in glaciated clouds may be i qsi and should be considered as lower estimates. signi®cantly higher compared to Sqsw in liquid clouds. p Ϫ1 As seen from Fig. 5 for uz ϭ 1ms and for typical N r , the supersaturation S may range from a few to ii qsi 5. Supersaturation in mixed-phase clouds hundreds of percent with respect to ice. However, such a high supersaturation will never be reached in clouds Cloud droplets may stay in a metastable liquid con- at T ϾϪ40ЊC, since an increase of the supersaturation dition down to about Ϫ40ЊC. This results in a population is limited by the saturation over water. Lines 1 and 2 of cloud particles below 0ЊC that may consist of a mix- in Fig. 5 show the saturation over water for Ϫ5Њ and ture of ice particles and liquid droplets. Such clouds are Ϫ35ЊC, respectively. As soon as the saturation with re- usually called ``mixed-phase'' or ``mixed'' clouds. Due spect to water is exceeded, the activation of cloud con- to the difference of water vapor saturation over ice and densation nuclei (CCN), which always exist in the tro- liquid, the mix of ice particles and liquid droplets is posphere in large numbers, will occur. The activation condensationally unstable and may exist only for a lim- of newly formed droplets will result in rapid depletion ited time. The behavior of supersaturation in mixed- 15 DECEMBER 2003 KOROLEV AND MAZIN 2963
FIG. 8. Quasi-steady supersaturation vs Nwwr in mixed-phase clouds Ϫ1 for different Niir and two vertical velocities: 0.1 m s (dashed line) and1msϪ1 (solid line); T ϭϪ10ЊC, P ϭ 680 mb. Gray color indicates
Nwwr typical for liquid and mixed phase clouds in the troposphere.
ciation the time of phase relaxation p jumps up (Fig. 7d), and the supersaturation Sw decreases toward the value equivalent to the saturation over ice, while Si de- creases toward zero.
Dotted lines in Figs. 7a,b show solutions for Sw and Si calculated from Eq. (10). The solution of Eq. (10)
FIG. 7. Time history of (a) Sw, (b) Si, (c) LWC and IWC, and (d) gives a reasonable agreement with that of (B1)±(B7). in an ascending mixed-phase parcel; T(0) ϭϪ10ЊC, P(0) ϭ 680 Figure 8 shows quasi-steady supersaturation in Ϫ3 Ϫ1 mb, Sw(0) ϭ 0, Ni ϭ 1cm , ri(0) ϭ 10 m, LWC(0) ϭ 0.1gkg , Ϫ3 Ϫ1 mixed-phase clouds as a function of Nwwr for different Nw ϭ 200 cm , uz ϭ 0.8ms . Ϫ1 Niir for two updraft velocities 0.1 and 1 m s . The supersaturation Sw in the ascending mixed-phase cloud phase clouds is more complex compared to that in sin- may be either positive or negative. The sign of the su- gle-phase liquid or ice clouds. Since the saturating water persaturation is related to the balance between the rate vapor pressure over ice is less than that over water, there of supersaturation growth due to adiabatic cooling and are several possible scenarios for the evolution of the the rate of water vapor depletion by ice particles and three-phase colloidal system. Depending on the values depletion or supply by liquid droplets. For example, if of uz, P, T, Ni, rri, Nw, and w, the liquid and ice particles Niir is high enough then the ice particles will absorb may (a) both grow or (b) both evaporate or (c) ice par- water vapor rapidly reducing it to saturation over ice. ticles may grow while liquid droplets evaporate. The If Niir is too low, then ice particles will play no sig- last process, when the ice particles grow at the expense ni®cant role in the water vapor depletion, and the su- of evaporating droplets, is known as the Wegener±Ber- persaturation will be controlled mainly by liquid drop- geron±Findeisen (WBF) mechanism (Wegener 1911; lets. Bergeron 1935; Findeisen 1938). Figure 9 shows the time of phase relaxation in mixed-
Figure 7 shows time histories of Sw, Si, liquid water phase clouds for uz ϭ 0. In the mixed phase clouds with content (LWC), ice water content (IWC), and p in a typical Nwwr (Korolev et al. 2003), the ice particles do Ϫ1 cloud parcel ascending with uz ϭ 0.8 m s . The time not affect the time of phase relaxation until Niir reaches Ϫ3 of phase relaxation p was calculated for current values values above 100 mcm . This means that, for such of rw(t), ri(t), e(t), T(t), and P(t). During the ®rst 15 s mixed-phase clouds, the time of phase relaxation is or so, both droplets and ice particles are growing. During mainly de®ned by liquid droplets. At the ®nal stage of this period of time the ice particles deplete water vapor cloud glaciation due to the WBF mechanism, when at a higher rate compared to the liquid droplets, even- Nwwr becomes small, p is mainly de®ned by ice par- tually reducing Sw to below zero. As a result, the droplets ticles. start to evaporate, whereas the ice particles keep grow- In mixed-phase clouds, the droplets would stay in ing. After 140 s the droplets completely evaporate and equilibrium (i.e., they will neither grow nor evaporate) the cloud becomes glaciated. After the moment of gla- if Sw ϭ 0. The ice particles under this condition will 2964 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 60
FIG. 9. Time of phase relaxation vs Nwwr in mixed-phase clouds FIG. 10. Threshold vertical velocity such that, in a mixed-phase for different N r ; T 10 C, P 680 mb. Gray color indicates ii ϭϪ Њ ϭ cloud at uz Ͼ u*z , both droplets and ice particles grow. Calculations Nwwr typical for the liquid and mixed phase clouds in the troposphere. were done for temperatures Ϫ5Њ and Ϫ35ЊC; P ϭ 680 mb. Gray color
indicates Niir typical for the mixed phase and ice clouds. keep growing because S Ͼ 0. Substituting S ϭ 0in i qsw should be scaled down by a factor of c. Therefore, u* Eq. (12) yields a threshold velocity of ascent when drop- z shown in Fig. 10 should be considered an upper esti- lets remain in equilibrium: mate.
b*iiiNr An example when an ice cloud may transfer into a u*z ϭ . (22) mixed one is shown in Fig. 4a for the parcel ascending a0 Ϫ1 at uz ϭ 1ms . The relative humidity in the parcel Substituting Eq. (7) in (12) and assuming S 0 i ϭ exceeds saturation over water (curve Sw ϭ 0) between yields a condition for the equilibrium of ice particles; 140 and 210 s. Liquid droplets will be activated during that is, ice particles neither grow nor evaporate: this period and the ice cloud will turn into a mixed cloud. (1 Ϫ )bNr The study of Korolev and Isaac (2003) showed that o www uz ϭ . (23) periodic oscillations of vertical velocity may result in a a0 periodic activation and evaporation of liquid droplets in ice clouds. The most interesting part of this process is Thus, if uz Ͼ u*z , both droplets and ice particles grow; 0 that the periodic activation and evaporation of droplets if uuzzϽ uz Ͻ *, the droplets evaporate, whereas the ice 0 in ice clouds under such oscillations are not limited in particle grow; if uz Ͻ uz , both droplets and ice particles evaporate. time and may occur over an in®nitesimally long period. Figure 11 shows a diagram ofur0 versus N . The The velocityu*z may also be considered as a threshold z ww velocity for the activation of liquid droplets in ice clouds evaporation of both droplets and ice particles may occur only in downdrafts. For typical N ruthe values of 0 when uz Ͼ u*z . Basically, Eq. (22) de®nes a condition ww z for maintaining a mixed-phase in a cloud. It is worth are unrealistically high. In practice the simultaneous evaporation of droplets and ice particles may occur only noting that the threshold velocityu*z does not depend 0 during the ®nal stages of cloud glaciation, when N r on Nwwru, andz does not depend on Nii r. ww Figure 10 shows the dependence of the threshold ve- becomes small. locityu*zz versus Niiru. The velocity* does not exceed a few meters per second for typical values of Niir (Fig. 6. Glaciating time of mixed-phase clouds 10). Such vertical velocities can be generated by tur- bulence in stratiform clouds or regular convective mo- Based on Eq. (12) in the absence of a vertical velocity tions in cumulus clouds. This results in an important (uz ϭ 0), the supersaturation over water in a mixed- conclusion that liquid droplets may be easily activated phase cloud is always negative: in ice clouds under relatively small vertical velocities b*iiiNr having turbulent or regular nature keeping the cloud in Sqsw ϭϪ . (24) bNrwwwϩ bNr iii mixed-phase condition. It should be noted thatu*z was calculated for spherical ice particles, that is, c ϭ 1, As seen from Fig. 12, for the case uz ϭ 0 for typical whereas usually for natural ice particles c Ͻ 1. Because Niirrand Nww, the supersaturation in mixed-phase u*zzis linearly related to c [Eq. (22)], the actualu* would clouds is close to zero ( | Sqsw | Ͻ 1%). It means that in be lower than that calculated above, andu*z in Fig. 10 mixed-phase clouds the evaporating droplets tend to 15 DECEMBER 2003 KOROLEV AND MAZIN 2965
FIG. 11. Threshold vertical velocity such that, in a mixed-phase FIG. 12. Quasi-steady supersaturation in mixed-phase cloud for uz 0 cloud at uz Ͻ uz , both droplets and ice particles evaporate. Calcu- ϭ 0 for different Nwwr ; T ϭϪ10ЊC, P ϭ 680 mb. Gray color indicates lations were done for temperatures Ϫ5Њ and Ϫ35ЊC; P ϭ 680 mb. Niir typical for the mixed phase and ice clouds. Gray color indicates Nwwr typical for the liquid and mixed phase clouds in the troposphere.
Wi( gl) Wi(t0) ϩ Ww(t0) mi( gl) ϭϭ . (28) maintain the water vapor pressure close to saturation NNii over water. This creates a basis for the analytical esti- Integrating Eq. (25) from mi(t0)tomi(gl), and substi- mation of the time of glaciation in mixed-phase clouds. tuting Eqs. (27) and (28), yields the glaciation time The purpose of the following section is to consider the 1/3 2/3 glaciating time of mixed-phase clouds at uz ϭ 0 solely 19iwW (t0) ϩ Wi(t0) due to the WBF mechanism. gl ϭ 4cAii S 2 Ά[]N i Consider an adiabatic parcel containing a mixture of liquid droplets and ice particles. As in section 2, as- W (t ) 2/3 sume that the concentration of droplets and ice particles Ϫ i 0 . (29) stay constant, and no nucleation of ice particles occur. []Ni · The mass growth of an individual ice particle can be If Ww(t 0) k Wi(t0), then Eq. (29) will be simpli®ed written as (Mazin 1983):
2/3 dmi 1/3 ϭ 4iiiicA r S . (25) 19iwW (t0) dt gl ϭ . (30) 4cA S 2 N ii[] i Because the water vapor pressure in mixed-phase clouds Equation (29) indicates that to a ®rst approximation is close to the saturation over water, the supersaturation the time of glaciation is insensitive to the size spectra in mixed-phase clouds can be estimated as S ϭ [E (T) i w of liquid droplets and depends only on ice particles num- Ϫ E (T)]/E (T). Therefore, the ice particles are growing i i ber concentration and the initial LWC and IWC. Figure at the expense of liquid droplets only, and at the moment 13 shows the dependence of versus T for different of glaciation ( ), the mass of all ice particles will be gl gl N calculated from a numerical model (appendix B), and approximately i that derived from Eq. (29). The calculations were done Ϫ3 Wi(gl) ϭ Wi(t0) ϩ Ww(t0). (26) for Ww0 ϭ 0.1gm and P ϭ 680 mb. As seen from Fig. 13, gl has a minimum around Ϫ12ЊC, where the Here, Wi(t0) and Ww(t 0) are initial ice and liquid water difference between the saturation vapor pressure over content, respectively. Assuming that the ice particles are water and ice has a maximum. The glaciation time in- monodisperse, the individual ice particle mass at the creases toward 0ЊC and cold temperatures, where the initial moment of time t 0 ϭ 0is difference Ew(T) Ϫ Ei(T), and therefore Si, approaches W (t ) zero. m (t ) ϭ i 0 . (27) i 0 N The glaciation time derived from Eq. (29) agrees well i with that calculated from the numerical model for ice Ϫ3 Because the number concentration of the ice particles concentrations less 1 cm . The difference between gl Ni stays constant, the mass of a single ice particle at calculated from the numerical model and Eq. (29) in- the moment of glaciation will be creases with an increase of Ni (Fig. 13). For example, 2966 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 60
FIG. 14. Comparison of the supersaturation Sw(t), Sqsw(t), andS*w (t) Ϫ3 in ascending liquid clouds having the same Nwwr ϭ 1000 mcm , FIG. 13. Time of glaciation of a mixed-phase cloud vs temperature Ϫ1 Ϫ1 but different Nw and r w0; uz ϭ 1ms ; T(0) ϭ 0ЊC; P(0) ϭ 870 mb. for uz ϭ 0; LWC0 ϭ 0.1gkg ; T ϭϪ15ЊC; P ϭ 680 mb. Gray lineS*w (t) is the solution of Eq. (9) for the quasi-steady ap-
proximation (Nwwr ϭ constant and Ni ϭ 0). for N ϭ 10 cmϪ3 and N ϭ 1cmϪ3 the relative difference i i only be reached at the point where S (t) has a maximum; at T ϭϪ15ЊC reaches about 40% and 5%, respectively. w Ϫ1 Ϫ3 that is, dSw/dt ϭ 0. Politovich and Cooper (1988) stated For ice particle concentrations less than Ni ϭ 10 cm , the maximum relative error between the model and Eq. that ``the value of S may be assumed near its quasi- (29) does not exceed 1%. Therefore, Eqs. (29) and (30) steady value when the contributions to the temporal var- iations of S from frequencies Ϫ1 are negligible.'' can be used as an estimate of . The agreement between qs Ͼ p gl So far it is not clear how accurate the estimate of S Eq. (29) and the numerical model indicates that, in w from S calculated from measurements of u and mixed-phase clouds, the vapor pressure is close to sat- qsw z N r is. The difference between S and S can be found uration over liquid water, and the rate of decrease of ww w qsw by substituting the current values of r (t), N (t),2 e(t) LWC is equal to the rate of growth of IWC. w w For nonspherical particles, the glaciating time should T(t), and P(t) calculated from a numerical model (ap- be scaled up by a factor of cϪ1 according to Eq. (29). pendix B) into the expression for the quasi-steady ap- In real clouds, the glaciating time can be reduced com- proximation [Eq. (16)]: pared to Eq. (29) due to ice multiplication and freezing a (t)u S (t) ϭ 0 z of droplets, or increased due to sedimentation and ag- qsw b (t)N (t)r (t) gregation. Glaciation due to the WBF mechanism in www vertically ascending and descending mixed-phase cloud a (t)u ϭ 0 z . (31) parcels was considered in detail by Korolev and Isaac t (2003). b (t)N (t) r2 ϩ 2A (t) S (tЈ) dtЈ ww 0ww͵ w Ί 0
7. Relation between S and Sqs A remarkable property of Sqsw(t) is that it approaches S (t) asymptotically over time when u constant; that In situ measurements of supersaturation in clouds is w z ϭ a great challenge. At present, there are no aircraft in- is, struments providing measurements of the supersatura- Sqsw(t) tion. Our knowledge about the in-cloud supersaturation lim ϭ 1. (32) → S (t) i ϱ []w is based mainly on numerical modeling. Another ap- proach was used in studies by Warner (1968), Paluch A rigorous mathematical proof of Eq. (32) is not trivial and Knight (1984), and Politovich and Cooper (1988). and it goes beyond the scope of this paper. Numerical They have estimated the supersaturation in liquid clouds modeling indicates that Eq. (32) is valid both for liquid, from Eq. (16) based on in situ measurements of uz and ice and mixed clouds. Some examples are shown in Figs. Nwwr . It was considered that ``supersaturation can be 14, 15, and 16. The difference between Sqsw(t) and Sw(t) computed using a quasi-steady assumption that the con- densation rate just balances the tendency for supersat- 2 Note, that the concentration N changes with time, due to changes uration to increase during ascent'' (Paluch and Knight w of the air density a. However, the concentration nw ϭ Nw /a stays 1984). However, as seen in Fig. 1, such a balance may constant. 15 DECEMBER 2003 KOROLEV AND MAZIN 2967
FIG. 16. Comparison of the supersaturation Sw(t), Sqsw(t) andS*w (t) FIG. 15. Comparison of the supersaturation S (t), S (t), andS* (t) i qsi i in ascending mixed-phase clouds; N ϭ 200 cmϪ3, LWC(0) ϭ 0.2 g in ascending ice clouds having the same N r 10 mcmϪ3, but w iiϭ mϪ3, N ϭ 1cmϪ3, r ϭ 10 m, u ϭ 0.25 m sϪ1; (a) T(0) ϭϪ10ЊC; Ϫ1 i i0 z different Ni and r i0; uz ϭ 0.5ms ; (a) T(0) ϭϪ10ЊC; P(0) ϭ 870 P(0) ϭ 870 mb; (b) T(0) ϭϪ30ЊC, P(0) ϭ 360 mb. Thin line Sqsw(t) mb, (b) T(0) ϭϪ35ЊC, P(0) ϭ 360 mb. Thin line Sqsi(t) is the quasi- is the quasi-steady supersaturation calculated from Eq. (12) for cur- steady supersaturation calculated from Eq. (20) for current ri(t), Ni(t), rent rw(t), Nw(t), ri(t), Ni(t), e(t), T(t), and P(t). Gray lineS*w (t) is the e(t), T(t), and P(t). Gray lineS*i (t) is the solution of Eq. (9) calculated solution of Eq. (9) for the quasi-steady approximation (Nwwr ϭ con- with respect to ice for the quasi-steady approximation (Niir ϭ con- stant and Niir ϭ constant). stant and Nw ϭ 0).
[or Sqsi(t) and Si(t)] becomes less than 10% usually with- from about 200 to 800 s. In high-altitude clouds (Fig. in a characteristic time 3p depending on Nw, r w0, uz and initial S(0), T(0), and P(0). Under some conditions, 15b, H ϭ 7000 m, T ϭϪ35ЊC) the ice particles ac- commodate the water vapor excess faster than that in the 10% agreement between Sqsw(t) and Sw(t) can be low-level clouds (Fig. 15a, H ϭ 1000 m, T ϭϪ10ЊC). reached during a time period less than p. For example Ϫ3 Figure 16 shows changes of the supersaturation in in Fig. 14, for the case with Nw ϭ 50 cm , r w0 ϭ two identical mixed clouds at two altitudes. The super- 20m, and Sw(0) ϭ 0, the ratio ⌬Sw/Sw ϭ (Sqsw Ϫ Sw)/ Sw becomes less than 10% after about 3p (ϳ10 s), for saturation of mixed-phase clouds changes in a step-wise Ϫ3 Nw ϭ 1000 cm and r w0 ϭ 1 m the ratio ⌬Sw/Sw manner (Fig. 16). Before droplet evaporation the su- reaches 10% difference after 1.5p (ϳ5 s). Note that persaturation Sw is close to zero, which is in agreement Nwwr is the same for both cases. with the discussion in section 5. After complete droplet The behavior of Sqsi(t) in glaciated clouds is similar evaporation, Sqsw(t) drops to the value equivalent to the to that in liquid clouds (Fig. 15). The characteristic time saturation over ice, whereas Sw(t) gradually approaches of approach of Sqsi(t)toSi is about two orders of mag- the same value. It is worth noting that glaciation occurs nitude higher than in liquid clouds for typical Nwwr and faster in high-altitude clouds (Fig. 16b) compared to Niir . For the cases shown in Fig. 15 this time ranges that in low-level clouds (Fig. 16a). 2968 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 60
8. Discussion threshold velocity required for activation of interstitial cloud condensation nuclei CCN (Korolev 1994). For In this section we will formulate and discuss the typical N the characteristic threshold velocity is ϩ limiting conditions for the quasi-steady approximation. wwruz Ϫ1 0 Ϫ1 The time of phase relaxation is the characteristic ϳ 10 ±10 ms . Numerical modeling showed that p the rate of change of concentration during secondary timescale of changing of Sqsw . Therefore, the charac- Ϫ3 teristic time of changes of different coef®cients and activation may vary in the range ⌬Nw/⌬t ϳ 5±20 cm parameters, considered as constant, should be much sϪ1 (Korolev 1994). These estimations suggest that the changes of droplet concentration due to secondary ac- larger than p (Kabanov et al. 1971). For the following estimations it will be assumed that pressure and tem- tivation depending on the rate ⌬Nw/⌬t may result in a perature change from 1000 to 300 mb, and from 0Њ to difference between Sw and Sqsw.
Ϫ40ЊC, respectively; for liquid clouds p ϳ 1±10 s, The characteristic rate of the changes of the concen- Ϫ1 2 3 | uz | ϭ 0.1±10 m s ; for ice clouds p ϳ 10 ±10 s, tration due to collision±coalescence based on numerical Ϫ1 Ϫ4 Ϫ5 | uz | ϭ 0.1±1 m s . modeling is of the order of ⌬Nw/⌬t ϳ 10 to Ϫ10 cmϪ3 sϪ1 (Pruppacher and Klett 1997), and therefore, does not play any signi®cant role in the formation of a. Changes of pressure and temperature the Sqsw. The sedimentation may both decrease the par- The following limitation should be imposed on the ticle concentration due to their fallout of the cloud parcel changes of pressure and temperature in the frame of the and increase it due to new particles entering the same quasi-steady approximation: parcel. For spatially uniformly distributed cloud parti- ⌬PP cles the balance of the decrease and increase of the K (33) droplet concentration due to sedimentation will be close ⌬t ΗΗ p to zero. Therefore, the effect of the sedimentation is expected to be negligible and it satis®es the condition (35). However, this may not be the case near the cloud ⌬TT edges, where the spatial inhomogeneity of droplets is K . (34) high. ΗΗ⌬t p The entrainment of dry out-of-cloud air with subse- Using the hydrostatic approximation [Eq. (A9)] for quent mixing with the saturated in-cloud environment liquid clouds | dp/dt | ϳ 10 0±10 2 NmϪ2 s Ϫ1 , and for will result in a decrease of supersaturation followed by ice clouds | dp/dt | ϳ 10 0±101 NmϪ2 s Ϫ1 . The char- the droplets evaporation. The effect of the entrainment acteristic values of P/p in liquid and ice clouds are and mixing will be most signi®cant near the cloud 10 4±10 5 NmϪ2 s Ϫ1 and 10 2±103 NmϪ2 s Ϫ1 , respec- boundaries. tively. The freezing of liquid droplets will reduce the con- Ϫ1 Ϫ3 Ϫ1 In liquid clouds | dT/dt | Ͻ ␥ auz ϳ 10 ±10 Ks centration of droplets and increase the concentration of and in ice clouds | dT/dt | Ͻϳ10Ϫ2±10Ϫ3 KsϪ1. The ice particles. Unfortunately, at present, the rate of ice characteristic values of T/p in liquid and ice clouds are nucleation at different temperatures is poorly known. In 101±102 KsϪ1, and 10Ϫ1±100 KsϪ1. situ measurements in wave clouds showed that the gla- Therefore, the conditions (33) and (34) are satis®ed ciation of a downstream part of the cloud occurs within both in liquid and ice clouds. tens of seconds (e.g., Cotton and Field 2002). That sug- gests that condition (35) is satis®ed and the rate of nat- b. Changes of particle concentration ural nucleation of ice likely does not affect Sqsw in liquid and mixed-phase clouds. The changes of the cloud particle concentration in the In ice clouds, the value N / is comparable to or may quasi-steady approximation will be limited by the con- i p be much less than ⌬N /⌬t due to sedimentation, aggre- dition i gation, and ice nucleation. Therefore, Eq. (20) should ⌬NN be used carefully for estimation of the supersaturation K . (35) from S in ice clouds. ΗΗ⌬t p qsi Turbulent ¯uctuations may result in inhomogeneity In liquid and mixed clouds the characteristic values of the droplet concentration (preferential concentration), 0 3 Ϫ3 Ϫ1 which would cause perturbations of the supersaturation of Nw/p ϳ 10 ±10 cm s , in ice clouds Ni/p ϳ 10Ϫ5±10Ϫ2 cmϪ3 sϪ1. and nonuniform growth of cloud particles. Detailed The concentration of droplets inside liquid and mixed studies by Vaillancourt et al. (2001, 2002) showed that, clouds may change due to (a) secondary activation, (b) in cloud cores, the perturbation of supersaturation evaporation, (c) sedimentation, (d) collision±coales- caused by preferential concentration is relatively small, cence, and (e) droplet freezing. The secondary activa- and therefore, the effect of the preferential concentration ϩϩ tion of droplets may occur if uz Ͼ uuzz, where is the can be neglected. 15 DECEMBER 2003 KOROLEV AND MAZIN 2969
FIG. 17. Time variations of Sqsw(t) and Sw(t) in a vertically oscil- FIG. 18. Time variations of Sqsi(t) and Si(t) in a vertically oscillating Ϫ3 lating liquid cloud with (a) p K t, high-frequency vertical oscil- ice cloud with (a) p ϳ t, Nw ϭ 0.1 cm , ri0 ϭ 10 m, p ϭ 171 Ϫ1 Ϫ3 lations D ϭ 2m,uvort ϭ 1ms , t ϭ 1 s; and (b) p k t, low- s; and (b) p k t, Ni ϭ 0.01 cm , ri0 ϭ 10 m; p ϭ 1710 s. For Ϫ1 Ϫ1 frequency oscillations D ϭ 100 m, uvort ϭ 0.5ms , t ϭ 100 s. both clouds, D ϭ 100 m, uvort ϭ 0.2ms , ϭ 250 s, T(0) ϭ Ϫ3 For both clouds Nw ϭ 100 cm , rw0 ϭ 5 m; p ϭ 6.6 s, T(0) ϭ Ϫ35ЊC; P(0) ϭ 360 mb. The vertical velocity was changed to uz ϭ
0ЊC, P(0) ϭ 870 mb. The vertical velocity was changed to uz ϭ uvort uvort sin(2tuvort/D ). sin(2tuvort/D ).
c. Changes of vertical velocity and characteristic examples. Figure 17 shows temporal variations of Sw (t) timescale and Sqsw (t) in a vertically oscillating parcel in a liquid The limitation for the vertical velocity for the quasi- cloud. The vertical velocity was changed as uz ϭ uvort steady supersaturation yields sin(t), where ϭ 2uvort/D ; D and uvort are the di- ameter and tangential velocity of the vortex, respec- ⌬uu zzK . (36) tively. The characteristic time of such vertical ¯uctu- ations can be estimated as ϳ 1/. Figure 17a dem- ΗΗ⌬t p t onstrates the case with p Ͼ t (t ϭ 1s,p ϭ 6.6 s). In real clouds, vertical velocity experiences contin- As a result the variations of the supersaturation Sw are uous ¯uctuation due to turbulence, wave motions, or signi®cantly less compared to Sqsw (t). In this case convection. Early works of Sedunov (1965), Mazin Sqsw (t) cannot be used to get an estimation of Sw . The (1967), Kabanov and Mazin (1970), and Kabanov frequency of the vertical ¯uctuations in Fig. 17a is too (1970), have studied the relation between the turbulent high and unlikely in natural clouds. However, such velocity ¯uctuations and supersaturation in the cloudy ¯uctuations might occur in the trails behind airplanes. atmosphere. They showed the cloud droplets would ac- Figure 17b shows the case with p K t (t ϭ 100 commodate the vapor, and the processes inside a cloud s, p ϭ 6.6 s). Such ¯uctuations (D ϭ 100 m and uvort would be wet adiabatic if ϭ 0.5 m sϪ1 ) are typical for stratocumulus, and its
period (2D /uvort ഠ 10 min) is close to that of the ptK ; (37) gravity waves. As seen, the Sw (t) and Sqsw (t) are nearly 2 1/3 here i ϳ (L /) is the characteristic timescale of the consistent, and Sqsw (t) gives a very accurate estimate vertical turbulent ¯uctuation. If p k t, then the re- of Sw (t). sponse of the supersaturation on the vertical motion will Figure 18 shows time changes of Si(t) and Sqsi(t)in be reduced. In this case the droplets, due to the con- the vertically oscillating parcel in a glaciated cloud with densational inertia, would not have enough time to adapt p ϳ t(t ϭ 250 s, p ϭ 170 s). The values of Ni, r i, the water vapor and the processes inside such a parcel uz, and D chosen in Fig. 18 could occur in cirrus clouds. would follow the dry adiabat. The amplitudes of Si(t) and Sqsi(t) for this cloud are The characteristic timescale of turbulence t may be about the same, though the phase of the oscillations is considered as the time of a turnover of a turbulent biased. It should be noted, that though Si(t) and Sqsi(t) vortex. In this case, ⌬uz can be estimated as ⌬uz ϳ are different, frequency distribution of Sqsi(t) calculated 2uz . Substituting this estimate into Eq. (36) results in from a statistically signi®cant ensemble of Niir and uz Eq. (37). would be about the same as the distribution of the actual
The behavior of the supersaturation under the limits Si. p K t and p k t can be illustrated by the following Figure 18b demonstrates the case when p k t(t ϭ 2970 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 60
250 s, p ϭ 1710 s). The S(t) and Sqsi(t) are signi®cantly saturation Sqs(t) calculated for current values of uz(t), different as expected. rrw(t), Nw(t), i(t), and Ni(t). The characteristic time
of the approach of S(t)toSqs(t) is de®ned by the time of phase relaxation [Eqs. (13), (18), (21)]. d. Changes of particle sizes p This creates a theoretical basis for the use of Sqs for The limitations on the changes of the cloud particle estimating supersaturation in clouds from in situ sizes may be de®ned from Eq. (15). Assuming that S measurements of Nwwrr, Nii, and uz as was done by ϭ Sqsw, t ϭ p, and substituting Eq. (16) and Eq. (18) Warner (1968), Paluch and Knight (1984), and Pol- into Eq. (15), yields itovich and Cooper (1988). rN4 2 2Aa 3) In estimating S from in situ measurements, the av- wwk w0 . (38) eraging of N rr, N , and u should be done over 2 ww ii z |uzw| b the scale l Ͼ lp. The quasi-steady supersaturation Sqs Equation 37 is valid for ice clouds, if the index ``w'' is gives a good estimate of S in liquid and mixed-phase replaced with ``i.'' Differentiating r w and uz in Eq. (38) clouds because usually p Ͻ t in these clouds. In by t, and using Eqs. (3) and (18) results in | ⌬uz/⌬t | ice clouds, estimations of S should be made with K 2uz/p, which is similar to Eq. (36), Thus, the lim- caution, because in many cases p Ͼ t. itation on the changes of the particle size reduces to the 4) Solution of the equation for the quasi-steady su- limitation on the vertical velocity. persaturation suggests that for typical Nwwr and
Niir , the relative humidity in liquid and mixed phase clouds is close to the saturation over water, e. Characteristic spatial scale and the characteristic time of settling of the super-
The above consideration in sections 2±4 assumes that saturation (p ) is of the order of seconds. In ice all processes are adiabatic. In real clouds, this assump- clouds the quasi-steady supersaturation is close to tion is violated due to entrainment and mixing. The saturation over ice. However, for typical Niir in ice entrainment and mixing would generate ¯uctuations of clouds, the characteristic time of approaching of the the supersaturation, which are not related to ¯uctuation supersaturation to its quasi-steady value may range of u , N rr, and N . After each event of entrainment z ww ii from minutes to hours (p Ͼ t ). Therefore, in situ and mixing, the supersaturation will recover to its quasi- measurements of the relative humidity in ice clouds steady value during a characteristic time p. If the char- may result in any value below the saturation over acteristic spatial scale of the turbulent ¯uctuations lt K water; that is, it may be both higher or lower than lp, then the quasi-steady approximation cannot be ap- saturation over ice. plied. Here, 5) Liquid water droplets can be activated in ice clouds 1/2 3/2 with N r Ͻ 1cmϪ3 at u ϳ 1msϪ1. The vertical lppϳ (39) ii z turbulent and regular motions observed in natural is the characteristic phase scale (Mazin 1966; Kabanov clouds may maintain the cloud in mixed phase for and Mazin 1970); is the turbulent energy dissipating a long time. The condition for the activation of liquid rate. At the spatial scale lt K lp, the supersaturation droplets and maintaining the cloud in mixed phase Sqsw(t) may be signi®cantly different from the actual is determined by Eq. (22). This may be a possible Sw(t). Thus, we come to an important conclusion that explanation of the large observed fraction of mixed the estimations of supersaturation from in situ mea- phase clouds. surements of Nwwrr, Nii, and uz, the averaging should be done over the scale l Ͼ lp. In conclusion we would like to stress the impor- In stratocumulus the characteristic values ϳ10Ϫ3 tance that the integral radii should not be overlooked. m2 sϪ3 (Mazin and Khrgian 1989) and ϳ 10 s result p The values of integral radii N wwrrand N iide®ne Sqsw Ϫ2 Ϫ1 2 in lp ϳ 1 m. In convective clouds, ϳ10 ±10 m and p . Thus, the ®rst moment of the cloud particle Ϫ3 s and p ϳ 10 s result in lp ϳ 3±10 m. size distribution (Nr ), similar to the second moment (Nr 2 ) de®ning optical properties of cloud, and the 9. Conclusions third moment (Nr3 ) de®ning cloud water content, should be considered one of the main parameters in In the framework of this study the following results cloud physics. were obtained. 1) An equation for quasi-steady supersaturation and the Acknowledgments. We appreciate Prof. Roddy Rogers time of phase relaxation for the general case of (National Science Foundation) for supporting this paper. mixed-phase clouds was obtained. Many thanks to Monika Bailey (Meteorological Service 2) It is shown that the supersaturation S(t) in a uni- of Canada) for editing the text. The authors express formly vertically moving cloud parcel asymptoti- gratitude to two anonymous reviewers for useful com- cally approaches, over time, the quasi-steady super- ments that help to improve the paper. 15 DECEMBER 2003 KOROLEV AND MAZIN 2971
APPENDIX A dq dq dq ϩϩϭwi0, (A10) dt dt dt Supersaturation in a Vertically Moving Parcel and Eqs. (A1) and (A3) to derive dp/dt, dq /dt, e/Ew, In the following consideration, the supersaturation is and p, respectively, and substituting in Eq. (A8) yields de®ned as 2 e Ϫ E 1 dSww gL g 1 Ldq ww w ϭϪu Ϫϩ Sw ϭ , (A1) 22z E Swpϩ 1 dt cRT RTa q cRTp dt w where e is the water vapor pressure and E is the sat- w 1 LL dq urated water vapor pressure. The rate of change of su- Ϫϩiw i. (A11) 2 persaturation in a vertically moving adiabatic parcel can qcRTdt p be found by differentiating Eq. (A1): In the above consideration, we used the following dS 1 de e dE approximations: p ഡ pm, where p, pm are the pressures wwϭϪ . (A2) 2 of dry and moist air, respectively. The quasi-hydrostatic dt Eww dt E dt Ϫ1 approximation Eq. (A9) requires uz Ͻ 10 m s , which To ®nd de/dt, we express the water vapor pressure as satis®es stratiform and shallow convective clouds con- ditions. R e ϭ qp . (A3) Ra APPENDIX B
Here, p is the pressure of a dry air; q ϭ m /ma is the mixing ratio of water vapor, that is, the mass of water Numerical Model of Supersaturation in a Vertically vapor m per mass of the dry air ma. Differentiating Eq. Moving Cloud Parcel (A3) gives The process of cloud-phase transformation in an adi- de R dq R dp abatic parcel can be described by a system of the fol- p q . (A4) ϭ ϩ lowing equations: dt Raa dt R dt
Using the Clayperon±Clausius equation dEw/dT ϭ the pressure variation equation 2 LwEw/R T , the changes of the saturated vapor pressure dp gpu can be presented as ϭϪ z ; (B1) dt Ra T dE dE dT L E dT wwϭϭ ww. (A5) 2 the energy conservation equation dt dT dt R T dt dT gU L dq L dq The term dT/dt can be found from the energy conser- ϭϪzwwii ϩ ϩ ; (B2) vation equation for an adiabatic parcel dt cp (1 ϩ q )cdtp (1 ϩ q )cdtp dp the water mass conservation equation cdTpaϪ RT Ϫ Ldq wwiiϪ Ldq ϭ 0. (A6) p dq dq dq ϩϩϭwi0; (B3) Differentiating Eq. (A6) and substituting dT/dt in Eq. dt dt dt (A5) yields the rate of change of the liquid droplets mass dE R L E dp L2 E dq L L E dq wϭϩϩ aww ww w iww i. (A7) dq dt pc R T dt c R T22 dt c R T dt w ϭ 4 AnrS; (B4) p p p dt wwwww Substituting Eqs. (A4) and (A7) in Eq. (A1) results in the rate of change of the ice particles mass dSw 1 RdqR dp p q dqi ϭ ϩ ϭ 4c AnrS; (B5) dt Ewa R dt R a dt dt iiiii e R L E dp L2 E dq L L E dq the rate of change of droplet size Ϫϩϩaww ww w iww i. 222 Ewp pcRTdt cRTp dt cRTp dt drwww A S ϭ ; and (B6) (A8) dt rw Using the equation for quasi-hydrostatic approximation the rate of change of ice particle size
dp gp driii cA S ϭϪ uz, (A9) ϭ . (B7) dt Ra T dt ri the equation for the conservation of total water mass The explanation for the symbols is provided in ap- 2972 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 60
pendix D. For k(T), Lw(T), and Li(T), the dependence A ϭϪaB6 www N r Ϫ aBNr5 iii; (C5) on temperature was taken into account. For D(T, P) both temperature and pressure were considered. The ice par- B ϭ aB5 *iiiNr ϩ (a46ϩ a )BNrwww ticles in the above model were assumed to be spheres, ϩ (a45ϩ a )BNriiiϪ au0 z; (C6) having initial size ri(t 0) ϭ 1 m. Both droplets and ice particle were assumed to have monodisperse size dis- C ϭ au0 z Ϫ (a45ϩ a )B*iiiNr; (C7) tributions. pR L L L2 a ϭ ; a ϭ wi; a ϭ w . 4522 6 APPENDIX C ERwa cRT p cRTp In the frame-work of quasi-steady approximation (r Equation for Supersaturation w ϭ const and r i ϭ const), Eq. (C3) yields a solution In deriving Eq. (1) and Eq. (A11), the dependence of S02Ϫ St the term 1/q on the supersaturation was neglected. This S21Ϫ S exp Ϫ approximation works very well for most cloudy situa- S01Ϫ S tions in the troposphere. An accurate consideration of Sw ϭ , (C8) q results in S02Ϫ St 1 Ϫ exp Ϫ S01Ϫ S eRawa E R q ϭϭ(Sw ϩ 1) . (C1) where pR pR 2 Substituting Eq. (C1) into Eq. (1) yields B Ϯ ͙B Ϫ 4AC S ϭ and (C9) 1,2 2A 1 dS gL g wwϭϪu 11 2 z ϭϭ . (C10) Swpϩ 1 dt c R T Ra T 2 A(S21Ϫ S ) ͙B Ϫ 4AC
2 1 pR Lww dq As seen from Eq. (C8), when time t increases the Ϫϩ 2 supersaturation, S asymptotically approaches S , which Swwapϩ 1 ER cRT dt w 2 will be referred to as quasi-steady supersaturation and 1 pR L L dq denoted Sqsw: Ϫϩ iw i. (C2) 2 2 Swwapϩ 1 ER cRT dt B Ϫ ͙B Ϫ 4AC S ϭ . (C11) qsw 2A Substituting dqw/dt and dqi/dt, similar to that in sec- tion 2, results in For Nwwrr, Nii, and uz typical for tropospheric clouds, 1 dS B k AC. This allows an expansion of the expression w under the square root sign in Eqs. (C10) and (C11). S 1 dt w ϩ Then substituting Eqs. (C5)±(C7) yields gL g CauϪ c*Nr ϭϪw u S ϭ 0 ziii and (C12) 2 z qsw ഠ cRTp RTa BcNrwwwϩ cNr iiiϪ au0 z 11 1 pR Liw L ϭ . (C13) ϪϩB*Nr p ഠ 2 iii BcNrwwwϩ cNr iiiϪ au0 z Swwapϩ 1 ER cRT
Here, cw ϭ (a 4 ϩ a 6)Bw, ci ϭ (a 4 ϩ a 5)Bi ϩ a 5B*i , and 2 1 pR Lw c* ϭ (a ϩ a )B* . Comparisons show that Eq. (C12) ϪϩBNr ii4 5 S ϩ 1 ER cRT2 www and Eq. (C13) are close to Eqs. (12) and (13), respec- [wwap tively. 1 pR L L ϩϩ iw BNr S . (C3) S ϩ 1 ER cRT2 iiiw wwap ] Equation (C3) can be rewritten as dS w ϭ AS 2 Ϫ BS ϩ C. (C4) dt ww Here, the following designations have been used: 15 DECEMBER 2003 KOROLEV AND MAZIN 2973
APPENDIX D APPENDIX D List of Symbols (Continued) Symbol Description Units Symbol Description Units gLR L Latent heat for liquid water evaporation J kgϪ1 a Ϫ1 w a0 Ϫ 1 m lp Characteristic spatial phase scale m RTap cRT mi Mass of a single ice particle kg 2 1 Lw N Concentration of ice particles mϪ3 a ϩ Ð i 1 2 Ϫ3 qcRT p Nw Concentration of liquid droplets m n N / number of ice particles per unit kgϪ1 1 LL i i a wi mass or dry air a2 ϩ Ð qcRT2 Ϫ1 p nw Nw /a number of liquid droplets per unit kg 1 L2 mass or dry air a ϩ i Ð p Pressure of moist air N mϪ2 3 2 qcRT p Ϫ2 pa Pressure of dry air N m
pR ri Half of a maximum linear dimension of m a4 Ð an ice particle ERwa rw Liquid droplet radius m Ϫ1 Ϫ1 LLwi R Speci®c gas constant of moist air J kg K a Ð a 5 2 R Speci®c gas constant of water vapor J kgϪ1 KϪ1 cRTp q Ice water mixing ratio (mass of ice per 1 Ð L2 i a w Ð kg of dry air) 6 2 cRTp q Water vapor mixing ratio (mass of water Ð Ϫ1 vapor per 1 kg of dry air) 2 iiL iRT 2 Ϫ1 qw Liquid water mixing ratio (mass of liq- Ð A ϩ m s i 2 kR T Ei(T)D uid water per 1 kg of dry air) Si e/Ei Ϫ 1, supersaturation over ice Ð Ϫ1 L2 RT S e/E Ϫ 1, supersaturation over water Ð ww w 2 Ϫ1 w w Aw ϩ m s kR T2 E (T)D Sqsi Quasi-steady supersaturation with respect Ð w to ice 2 Ϫ1 bw aB1 w m s Sqsw Quasi-steady supersaturation with respect Ð
2 Ϫ1 to water bi aB2 w m s Si*(t) Solution of Eq. (9) for supersaturation Ð 2 Ϫ1 bi0 aB2 w0 m s over ice when rri ϭ constant and w ϭ
2 Ϫ1 constant bi* aB2 i* m s Sw*(t) Solution of Eq. (9) for supersaturation Ð 4 cA over water when rrϭ constant and B ii m2 sϪ1 i w i ϭ constant a T Temperature K 4 cA ii 2 Ϫ1 t Time s Bi0 m s Ϫ1 a uz Vertical velocity ms W Ice water content kg mϪ3 4 i 2 Ϫ1 W Liquid water content kg mϪ3 Bi* ii( Ϫ 1)cA m s w Ϫ1 a ␥a Dry adiabatic lapse rate Km 2 3 4 A Turbulent energy dissipating rate m s ww 2 Ϫ1 Ϫ3 Bw m s a Density of the dry air kg m a Ϫ3 i Density of an ice particle kg m c Ice particle shape factor characterizing Ð Density of liquid water kg mϪ3 capacitance 0 Ͻ c Յ 1(c ϭ 1 for w p Time of phase relaxation s spheres) Characteristic timescale of the vertical s Ϫ1 Ϫ1 t cp Speci®c heat capacity of moist air at Jkg K turbulent velocity ¯uctuations constant pressure Glaciating time of a mixed-phase cloud s 2 Ϫ1 gl ci (a4 ϩ a5)Bi ϩ a5Bi* m s E /E Ð 2 Ϫ1 w i ci* (a4 ϩ a5)Bi* m s 2 Ϫ1 cw (a4 ϩ a6)Bw m s D Coef®cient of water vapor diffusion in m2 sϪ1 the air REFERENCES e Water vapor pressure N mϪ2 Ϫ2 Ei Saturation vapor pressure above ¯at sur- Nm Bergeron, T., 1935: On the physics of clouds and precipitation. Proc. face of ice Ve AssembleÂe GeÂneÂrale de l'Union GeÂodeÂsique et Geophysique Ϫ2 Ew Saturation vapor pressure above ¯at sur- Nm Internationale, Lisbon, Portugal, International Union of Geodesy face of water and Geophysics. 156±180. Ϫ1 fw(rw) Size distribution of cloud droplets nor- m Cotton, R. J., and P. R. Field, 2002: Ice nucleation characteristics of malized on unity an isolated wave cloud. Quart. J. Roy. Meteor. Soc., 128, 2417± Ϫ1 fi(ri) Size distribution of ice particles normal- m 2439. ized on unity Findeisen, W., 1938: Kolloid-meteorologische VorgaÈnge bei Neider- g Acceleration of gravity m sϪ2 schlags-bildung. Meteor. Z., 55, 121±133. k Coef®cient of air heat conductivity J mϪ1 sϪ1 KϪ1 Heyms®eld, A. I., 1975: Cirrus uncinus generating cells and the evo- Ϫ1 Li Latent heat for ice sublimation J kg lution of cirrdorm clouds. Part III: Numerical computations of the growth of the ice phase. J. Atmos. Sci., 32, 820±830. 2974 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 60
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