Rates of phase transformations in mixed-phase clouds

Transcription

1 QUARTERLY JOURNAL OF THE ROYAL METEOROLOGICAL SOCIETY Q. J. R. Meteorol. Soc. 134: (2008) Published online 9 April 2008 in Wiley InterScience (.interscience.iley.com).230 Rates of phase transformations in mixed-phase clouds Alexei V. Korolev* Environment Canada, Toronto, Canada ABSTRACT: This paper presents a theoretical frameork describing the thermodynamics and phase transformations of a three phase component system consisting of ice particles, liquid droplets and ater vapour. The instant rates of change of ater ( q ), ice ( q i ) and vapour ( q v ) mixing ratios are described based on the quasi-steady approximation. The local thermodynamic conditions required for the equilibrium of liquid ( q = 0), ice ( q i = 0) and vapour ( q v = 0) phases are analysed. It is shon that there are four different regimes of the partitioning of ater beteen liquid, ice, and gaseous phases in mixed clouds. The WegenerBergeronFindeisen (WBF) process is identified as being relevant to to of those regimes. The efficiency of the WBF process in characterizing the capability of ice crystals to deplete the ater evaporated from liquid droplets is introduced here. It is shon that the WBF process has maximum efficiency at approximately zero vertical velocity. The analysis of the dependences of q, q i and q v on the vertical velocity, temperature, pressure and the integral radii of the cloud particles is presented. It is shon that the maximum rates of ice groth and droplet evaporation do not necessarily occur at T = 12 C here the maximum difference beteen saturation vapour pressure over ice and that over liquid is observed. Copyright 2008 Cron in the right of Canada. Published by John Wiley & Sons, Ltd. KEY WORDS mixed phase clouds; phase equilibrium; WegenerBergeronFindeisen process Received 16 July 2007; Revised 29 January 2008; Accepted 8 February Introduction Cloud droplets may stay in a metastable liquid condition don to temperatures of approximately 40 C. This results in a population of cloud particles belo 0 C that may consist of a mixture of ice particles and liquid droplets. Such clouds are usually called mixed-phase or mixed clouds. Due to the difference of saturation ater vapour pressure over ice and liquid, the mixture of ice particles and liquid droplets is condensationally unstable and may exist only for a limited time. There are three possible scenarios for the evolution of cloud particles in mixed-phase clouds. Depending on the local thermodynamic and microphysical parameters of the cloud, the liquid and ice particles may both gro, or both evaporate, or ice particles may gro hile liquid droplets evaporate (Korolev and Mazin, 2003; Korolev, 2007). The third process, here the ice particles gro at the expense of evaporating droplets, is knon as the WegenerBergeronFindeisen (WBF) process (Wegener, 1911; Bergeron, 1935; Findeisen, 1938). This paper presents a theoretical frameork for the analysis of the rates of the phase transformations in mixed-phase clouds. The approach developed is based on a so-called quasi-steady approximation, hich assumes that the sizes of cloud particles stay constant (Squires, 1952). This assumption allos for the analytical solution of the differential equation for supersaturation * Correspondence to: Alexei V. Korolev, Environment Canada, 4905 Dufferin Street, Toronto, Ontario, M3H5T4 Canada. alexei.korolev@ec.gc.ca in mixed-phase clouds (Korolev and Mazin 2003). In section 2, it is demonstrated that the use of quasi-steady supersaturation provides accurate calculations of the rates of change of liquid ( q ), ice ( q i ) and vapour ( q v ) mixing ratios. The dependences of q, q i and q v on temperature, pressure and vertical velocity are analysed in section 3. The conditions for the equilibrium of ice, liquid and ater vapour in mixed-phase clouds are considered in section 4. Four different scenarios of the phase transformation in mixed-phase clouds are discussed in section 5. In section 6, e discuss the conditions enabling the WBF process in mixed-phase clouds. It is shon that the maximum efficiency of the WBF process occurs at near-zero vertical velocity. 2. Rates of vapour, ice and liquid mass changes in mixed clouds In the folloing discussion e consider an idealized adiabatic mixed-phase cloud consisting of liquid droplets and ice particles suspended in ater vapour. The interaction beteen ice particles, liquid droplets and ater vapour occurs through processes of diffusive groth and/or evaporation. The processes related to particle-to-particle interaction, like riming, aggregation and coagulation, are not considered here. It is assumed that the spatial distribution of ice particles, liquid droplets, ater vapour and temperature is uniform. The concentrations of liquid droplets and ice particles stay constant ith time and the radiation effects are neglected. It is recognized that this idealization means that some of the quantitative results ill not Copyright 2008 Cron in the right of Canada. Published by John Wiley & Sons, Ltd.

2 596 A. V. KOROLEV be directly applicable to real cloud systems, and this ill be addressed in section 7. Hoever, such simplifications allo us to get a good theoretical understanding of mixedphase phenomena. The rate of mass change of an ensemble of liquid droplets can be described as (e.g. Squires, 1952; Korolev and Mazin, 2003) dq dt = B S N r. (1) Here q is the liquid ater mixing ratio; S = E e 1 is the supersaturation over ater; e is ater vapour pressure; E (T ) is saturation ater vapour pressure over ater at the temperature T ; N and r are the concentration and average radius of the droplets, respectively; B is a coefficient dependent upon T and P (an explanation of all the variables used in the text is provided in appendix A). Similar to the liquid droplets, the rate of mass change of ice particles can be ritten as dq i dt = B i0 S i N i r i c. (2) Here, S i = e E i 1 is the supersaturation over ice; E i (T ) is saturation ater vapour pressure over ice; B i0 is a coefficient dependent upon T and P ; N i is the concentration of ice particles; r i is the characteristic size of ice particles; c is a shape factor of ice particles characterizing capacitance in the equation of the groth rate. The size of an ice particle is an ambiguous parameter, and it can be defined in a number of different ays. The ice particle size r i is related to the shape factor c through the rate of the mass groth. In the folloing consideration, the size r i ill be defined as a half of a maximum dimension of the ice particle. For this definition of r i,theshape factor varies in the range 0 <c 1, being equal to 1 for ice spheres. For simplicity in the frame of this study, e assume c = const. This simplification does not disturb the generality of the folloing consideration. Hoever, for an ensemble of ice particles having different shapes one should consider the average value of the product r i c. The products N r and N i r i (or the first moments of particle size distributions) are usually referred to as the integral radii of the droplets and ice particles, respectively. We ill follo this convention in this ork. The supersaturation over ice can be related to the supersaturation over ater as S i = e E i E i = ξs + ξ 1, (3) here ξ(t) = E (T ). Then, substituting (3) into (2) E i (T ) yields dq i = (B i S + Bi dt )N ir i. (4) Here, B i = ξcb i0 and Bi = (ξ 1)cB i0. The supersaturation S in (1) and (4) can be approximated by the quasi-steady supersaturation S qs. Korolev and Mazin (2003) shoed that in mixed-phase clouds the quasi-steady supersaturation can be described as S qs = a 0u z bi N ir i. (5) b N r + b i N i r i In the same ork it has been demonstrated that the actual supersaturation S approaches ith time to S qs calculated for current values of N, N i, r (t), r i (t), T(t) and P(t),i.e. lim S (t) = S qs (t). (6) t For clouds ith typical integral radii N r and N i r i, the difference beteen S and S qs usually becomes less than 10% ithin a time period 3τ p,here 1 τ p = a 0 u z + b N r + (b i + bi )N, (7) ir i is the time of phase relaxation (Korolev and Mazin, 2003). In mixed-phase clouds τ p is mainly defined by the integral radius of liquid droplets N r, and typically it does not exceed a fe seconds (At the final stage of droplet evaporation, hen N r becomes small, the time of phase relaxation ill be mainly defined by the integral radii of ice particles N i r i, hich eventually results in an increase of τ p.). This time is significantly less than the characteristic lifetime of mixed-phase clouds (τ c ), i.e. τ c τ p. Therefore, the supersaturation S has enough time to relax to S qs. This justifies the use of S qs as an approximation of S in calculations of the rate of mass changes of ice and liquid. Thus, substituting (5) into (1) gives the rate of change of liquid ater mixing ratio, q = (a 0u z bi N ir i )B N r. (8) b N r + b i N i r i Similarly, substituting (5) into (4), and taking into account that Bi b i = bi B i (Appendix A) e obtain the rate of change of ice ater mass ( a 0 u z 1 ξ ) b N r B i N i r i ξ q i =. (9) b N r + b i N i r i The rate of change of the ater vapour mixing ratio ( q v ) can be found from the equation of mass conservation q v q + q i + q v = 0. (10) Substituting (8) and (9) into (10) yields = B B i (a 1 a 2 )N r N i r i a 0 u z (B N r + B i N i r i ) b N r + b i N i r i. (11) Copyright 2008 Cron in the right of Canada. Published by John Wiley & Sons, Ltd. Q. J. R. Meteorol. Soc. 134: (2008)

3 RATES OF PHASE TRANSFORMATIONS IN MIXED-PHASE CLOUDS 597 The replacement of S ith its quasi-steady approximation S qs (5) raises questions concerning the accuracy of this replacement and the factors limiting the use of (8) and (9). The accuracy of the quasi-steady approximation has been tested by comparing S, q and q i derived from (5), (8) and (9), respectively, ith that calculated from a parcel model of vertically moving adiabatic clouds. The parcel model used here is similar to that described in Korolev and Mazin (2003). Figure 1 shos the results of the comparison of S, q and q i for a set of modelled clouds, here the integral radii of droplets and ice particles ere changed in the range of 500 <N r < 5000 µm cm 3 and 0.5 <N i r i < 500 µm cm 3,respectively, and the vertical velocity varied in the range 5 <u z < 5 m/s. The initial radius of droplets varied from 2 µm to10µm. The diagrams shon in Figure 1 suggest that (5), (8) and (9) reproduce S, q and q i calculated from the parcel model ith a good accuracy. These results create a basis for the use of (8) and (9) in the folloing analysis. It as found that at the final stage of glaciation the deviation of S qs from S increases hen the droplets begin to rapidly reduce their sizes during evaporation. This deviation occurs because the limiting condition for the quasi-steady approximation (Korolev and Mazin, 2003), 2A a 0 u z b 2 r4 N 2 1, (12) is not satisfied hen the droplets become too small. The limiting condition (12) is also relevant to the initial stage of activation of droplets in the pre-existing ice cloud, hen the droplets are not yet big enough. Modelling points, here (12) as not satisfied, ere indicated by red triangles in Figure 1. The sizes of the droplets corresponding to the red triangles vary from 0.5 to 3 µm, depending on N and u z. The modelling as interrupted hen the droplets became smaller than 0.5 µm, during glaciation. Other limiting conditions applying to the rates of change of P, T, u z are discussed in detail in Korolev and Mazin (2003). It is shon that usually the conditions for P, T, u z are satisfied in mixed-phase clouds in the troposphere. The comparisons in Figure 1 excluded the points ith time t<3τ p required for adjusting S to S qs, since the initial supersaturation S (t 0 ) = 0asalays different from the quasi-steady one S qs (t 0 ) at the initial moment of time t 0. Equations (8) and (9) have a simple form and, presumably, they may be implemented in cloud models or used for parametrization in large-scale models ith relative ease. 3. Effect of thermodynamics on q, q i and q v In this section e consider the effect of u z, T and P on the rates of change of ice, liquid and ater vapour mass under conditions typical for mixed-phase clouds. Figure 2 shos the groth rate of ice and liquid in mixed-phase clouds versus u z calculated from (8) and Figure 1. Comparisons of (a) S,(b) q and (c) q i derived from Equations (5), (8) and (9), respectively, ith that calculated from a parcel model of a vertically moving adiabatic mixed-phase cloud. Modelling as conducted for 500 <N r < 5000 µm cm 3 ; 0.5 <N i r i < 500 µm cm 3, 5 <u z < 5m/s; 40 C <T < 5 C; 500 mb <P < 1000 mb. Blue circles indicate points satisfying (12). Red triangles indicate points outside the envelope of the limiting conditions for the quasi-steady approximation described by (12). This figure is available in colour online at.interscience.iley.com/qj (9) for N r = 1000 µm cm 3 and N i r i = 10 µm cm 3 at three different temperatures; 5 C, 15 C and 25 C. As follos from (8) and (9), q and q i are Copyright 2008 Cron in the right of Canada. Published by John Wiley & Sons, Ltd. Q. J. R. Meteorol. Soc. 134: (2008)

4 598 A. V. KOROLEV Figure 2. Groth rate of mixing ratio of ice q i (solid line) and liquid q (dashed line) versus u z calculated from (9) and (8), respectively. Calculations ere done for three different temperatures 5 C, 15 C and 25 C, integral radii N r = 1000 µm cm 3, N i r i = 10 µm cm 3, and pressure P = 680 mb. This figure is available in colour online at.interscience.iley.com/qj linearly related to u z. Figure 2 indicates that q is significantly more sensitive to changes of u z as compared to q i. As seen from Figure 2, q i stays nearly constant, hereas q changes signs ithin the range of u z considered in Figure 2. Formally, the loer sensitivity of q i versus u z as compared to q is explained by the difference in the coefficients B N r and B i N i r i in front of u z in (8) and (9), respectively. The difference in coefficients is mainly defined by the difference in values of N r and N i r i. Typically, in mixed-phase clouds N r N i r i and B B i. The physical explanation of the difference in sensitivity of q i and q to u z follos from the fact that the humidity in mixed-phase clouds is close to saturation ith respect to ater (Korolev and Mazin, 2003; Korolev and Isaac, 2006). It can be shon that, at temperatures 5 > T> 40 C at near ater saturation, the fluctuations of supersaturation over ater ith S < ill be equivalent to the fluctuations of supersaturation over ice ith S i S i. In mixed-phase clouds the fluctuations S < can be caused by fluctuations of vertical velocity ith u z < 5 m/s (Korolev and Mazin, 2003). Therefore, the fluctuations of ater supersaturation in mixed-phase clouds generated by fluctuations of vertical velocity ith amplitude a fe metres per second have minor effect on supersaturation over ice, and for practical purposes S i may be considered approximately constant. Thus, in mixed-phase clouds, q i is anticipated to be less sensitive to the changes of u z as compared to q. Figure 3 shos the dependence of the rate of change of the ater vapour mass q v on u z for N r = 1000 µm cm 3, N i r i = 10 µm cm 3 calculated for three different temperatures. Equation (11) indicates that q v is also a linear function of u z. In Figure 3(a) all q v curves calculated for different T appear to intersect at the same point (0, 0). Hoever, a closer look at the region of Figure 3. (a) Dependence of the ater vapour mass q v on vertical velocity u z for three different temperatures. (b) Zoomed area in the vicinity of the point ith u z = 0and q v = 0. The x- andy-axes in (b) are the same as in (a). Calculations ere done for three different temperatures 5 C, 15 C and 25 C, integral radii N r = 1000 µm cm 3, N i r i = 10 µm cm 3, and pressure P = 680 mb. This figure is available in colour online at.interscience.iley.com/qj intersection (Figure 3(b)) indicates that the q v curves intersect each other at different points. The rates q, q i and q v depend on T and P through the coefficients a 0, b, B, b i, bi, B i and ξ. Figure 4 shos q, q i and q v versus T for three different pressures P = 900 mb, 500 mb and 300 mb and to vertical velocities u z = 1m/sand1m/s. Figure 4(a) shos that the groth rate of ice has a maximum ( q imax ) at temperatures belo 12 C. The temperature corresponding to q imax is decreasing ith the decrease of P, and the amplitude of q imax is increasing ith the decrease of P. It should be noted that in mixedphase clouds the temperature corresponding to q imax is not equal to 12 C, here the maximum difference is beteen saturation pressures over ice and liquid, but may change in the range 40 <T < 12 C depending on P. For pressure 200 < P < 1000 mb the temperature for q imax varies in the range 19 <T < 12 C. Cases ith P<200 mb resulting in T< 19 C for q imax have no significance for the mixed-phase clouds formed in the troposphere. As seen from Figure 4(a), the ice groth rates q i at u z = 1m/sandu z = 1m/s practically coincide. In other ords, u z has minor effect on q i, hich is consistent ith the above discussion. Figure 4(b) shos the dependence of q on T and P. Contrary to q i, the groth rate of liquid q has a minimum q min hich, depending on P and u z,maybe located anyhere in the range 40 <T <0 C. Comparisons of Figure 4(a) and 4(b) indicate that the temperatures corresponding to the minimum of the groth rate of the liquid phase ( q min ) and the maximum groth rate of the ice mass ( q imax ) are different. Figure 4(b) also shos that q increases ith the increase of P at temperatures approximately loer than 8 C. This behaviour Copyright 2008 Cron in the right of Canada. Published by John Wiley & Sons, Ltd. Q. J. R. Meteorol. Soc. 134: (2008)

5 RATES OF PHASE TRANSFORMATIONS IN MIXED-PHASE CLOUDS 599 radii, q v becomes non-monotonic for lo vertical velocities u z < 0.1 m/s. In some papers it is speculated that the maximum rate of the phase transformation of liquid-to-ice is expected to occur at 12 C, here the maximum difference beteen saturation ater vapour pressure beteen ice and liquid is observed. In fact, the diagrams in Figure 4 suggest that the extrema of q and q i vary in a ide range of temperatures depending on P, N r, N i r i and u z, and that under the same conditions the temperatures corresponding to q min and q imax are different. 4. Points of the phase equilibrium in mixed-phase clouds The interaction beteen solid, liquid and gaseous phases of ater in mixed-phase clouds is a complex process. The rate and direction of the partitioning of the ater beteen three phases is defined by local thermodynamic (T, P, u z ) and microphysical (N r, N i r i ) characteristics of the cloud. In order to better understand the mechanisms of phase transformation in mixed-phase clouds in this section, e consider the thermodynamic conditions required for equilibrium of each of the three phases: (1) q = 0, liquid phase is in equilibrium; (2) q i = 0, ice phase is in equilibrium; (3) q v = 0, ater vapour is in equilibrium; (4) q = q i, groth rate of the liquid phase is in equilibrium ith that of the ice phase; (5) q v = q,rateofmass change of ater vapour is in balance ith that of liquid droplets; and (6) q v = q i, rate of mass change of ater vapour is in balance ith that of ice particles Case q = 0 As follos from (8), the equilibrium of liquid phase q = 0 occurs hen the vertical velocity u z becomes equal to a threshold velocity, u z = E E i E i ηn i r i. (13) Figure 4. Dependence of the groth rate of (a) ice ater mass q i, (b) liquid mass q and (c) ater vapour mass q v on temperature T,for to vertical velocities 1 m/s (solid line) and 1 m/s (dashed line) and three different pressures 900, 500 and 300 mb. Calculations ere done for integral radii N r = 1000 µm cm 3, N i r i = 10 µm cm 3.This figure is available in colour online at.interscience.iley.com/qj is opposite to that of q i (Figure 4(b)), here q i monotonically decreases ith the increase of P for the hole temperature range. Figure 4(c) shos that q v is monotonically increasing for u z = 1 m/s and decreasing for u z = 1m/s.The absolute value q v decreases ith increase of P.Calculations sho that for the considered values of integral Here, η = a 2 B i0 /a 0. A similar result as obtained in Korolev and Mazin (2003). The threshold velocity u z separates the regimes of groth and evaporation of liquid droplets in mixed-phase clouds, i.e. for u z >u z liquid droplets gro in the presence of ice, hereas for u z <u z droplets evaporate. As seen from (13), u z is alays positive. Figure 5(a) shos dependence of u z on T for different N i r i and P. As follos from Figure 5(a), u z increases ith the decrease of T. For typical values of N i r i the magnitude of the threshold velocity u z changes in the range 10 2 m/s to 10 0 m/s in the temperature interval from 5 Cto 30 C (Figure 5(a)). Such vertical velocities are common in the atmosphere, and they can be generated in clouds by turbulence or convection. For a general case, including non-adiabatic conditions, the equilibrium of liquid phase in mixed-phase clouds occurs hen e = E, hich is equivalent to S = 0. Copyright 2008 Cron in the right of Canada. Published by John Wiley & Sons, Ltd. Q. J. R. Meteorol. Soc. 134: (2008)

6 600 A. V. KOROLEV Figure 5. Dependence of the threshold vertical velocities (a) u z,(b)uo z,(c)u+ z,(d)u(i) z,(e)u z (v), and (f) u z (vi) versus temperature for different N r and N i r i. Calculations ere conducted for to pressures P = 900 mb (solid line) and P = 400 mb (dashed line). The values of u + z, u(i) z, u z (v),andu z (vi) ere calculated for N r = 1000 µm cm 3. This figure is available in colour online at.interscience.iley.com/qj 4.2. Case q i = 0 Equation (9) yields a vertical velocity u o z required for equilibrium of ice phase in a mixed-phase cloud: u o z = E i E E χn r. (14) Here, χ = a 1 B /a 0. Similar to u z, the velocity uo z separates regimes of groth and evaporation of ice particles in mixed-phase clouds, i.e. for u z <u o z ice particles sublimate in the presence of liquid droplets, hereas for u z >u o z ice particles gro. As follos from (14), uo z is alays negative. Figure 5(b) shos the dependence of u o z on T for different N i r i and P. As seen from Figure 5(b), u o z increases ith the increase of T. For typical values of N i r i, the magnitude of the velocity u o z changes in the range from 10 0 m/s to 10 3 m/s over the temperature range 5 C to 30 C (Figure 5(b)). Vertical velocities 10 <u z < 1 m/s may occur in compensating dondraughts in convective clouds. Dondraughts ith a higher velocity are unlikely to be formed in the atmosphere. The diagram in Figure 5(b) suggests that at temperatures close to 0 C, the absolute values of the vertical dondraughts required for reaching ice equilibrium do not exceed 1 m/s, and they can be generated by turbulence. Generally, the equilibrium of ice phase requires e = E i, hich is equivalent to the condition S = ξ 1in terms of the supersaturation over ater, or S i = 0 in terms of the supersaturation over ice. It should be noted that for a non-adiabatic case, the local equilibrium of ice phase in a mixed-phase cloud can be reached during mixing ith dry out-of-cloud air Case q v = 0 Equation (11) provides a velocity required for the ater vapour equilibrium: u + z = (ξ 1)(B b i b B i )N r N i r i. (15) a 0 ξ(b N r + B i N i r i ) For u z <u + z, the mass of the ater vapour is increasing, i.e. q v > 0, hereas for u z >u + z the mass of the Copyright 2008 Cron in the right of Canada. Published by John Wiley & Sons, Ltd. Q. J. R. Meteorol. Soc. 134: (2008)

7 RATES OF PHASE TRANSFORMATIONS IN MIXED-PHASE CLOUDS 601 ater vapour is decreasing, i.e. q v < 0. As follos from (15), u + z is alays positive. Figure 5(c) shos the dependence of u + z on T for different N i r i calculated for N r = 1000 µm cm 3. As seen from Figure 5(c), in mixed-phase clouds u + z is close to zero, and for typical N i r i and N r it is usually of the order of millimetres and centimetres per second. In a general case, including non-adiabatic conditions, the equilibrium of ater vapour ill be reached hen e = E v,heree v is the equilibrium vapour pressure: E v = ( 1 + S (v) ) E. (16) Here, S (v) is the supersaturation yielding the ater vapour equilibrium. In order to find S (v), substitute q v = 0 in (10), hich yields q = q i. Then, combining Equations (1), (4), (5) and q = q i, e obtain S (v) = B i N ir i B N r + B i N i r i. (17) It should be noted that the equilibrium of ater vapour in mixed-phase clouds has a dynamic nature, i.e. ater vapour released by evaporating droplets is depleted by groing ice particles Case q = q i Substituting (8) and (9) into the equality q = q i yields the vertical velocity resulting in the equilibrium beteen the rates of groth of ice and liquid phases in a mixedphase cloud: u (i) z = (a 1 + a 2 )B b i N r N i r i a 0 (B N r B i N i r i ). (18) The physical meaning of u (i) z is that at u z >u z (i) the rate of the mass groth of the liquid droplets is higher than that of the ice ( q > q i ), hereas at u z <u (i) z the mass of droplets is changing sloer than that of the ice particles ( q < q i ). As seen from Figure 5(d) the general behaviour of u (i) z is similar to that of u z (Figure 5(a)). The velocity u(i) z can be both positive, if N r N i r > B i i B, and negative, if N r N i r < B i i B. Since N r N i r i and B i B in mixed-phase clouds usually, the situation hen N r N i r < B i i B has a limited significance, and therefore this case is not shon in Figure 5(d). For typical values of N i r i and N r the magnitude of the velocity u (i) z changes in the range 10 2 m/s to 10 0 m/s in the temperature interval from 5 C to 30 C (Figure 5(d)). Such vertical velocities can be generated by turbulence or convection, and therefore parity of the mass groth of ice and liquid phases is expected to occur frequently in mixed-phase clouds. Combining (1), (4) and the equality q = q i yields the supersaturation required for maintaining equilibrium beteen ice and liquid groth rates: 4.5. Case q v = q S (i) Bi = N ir i. (19) B N r B i N i r i Substituting (8) and (11) into the equality q v = q yields the vertical velocity required to equilibrate the rate of mass changes of vapour and liquid phases: u (v) z = (2a 2 a 1 )B B i N r N i r a 0 (2B N r + B i N i r i ). (20) For vertical velocities u z >u z (v) the liquid ater mixing ratio is increasing faster than that of ater vapour ( q v < q ), hereas at u z <u z (v) the mass of liquid droplets is increasing sloer than that of the ater vapour ( q v > q ). The velocity u z (v) is alays positive, and for typical values of N i r i and N r the characteristic values of u z (v) are of the order of millimetres to tens of centimetres per second (Figure 5(e)). Therefore, it is expected that in mixed-phase clouds the rates of change of mass of ater vapour and liquid droplets are frequently equal to each other during turbulent fluctuations. In order to find the supersaturation providing the equilibrium beteen ater vapour and liquid droplets, substitute q v = q in (10), hich yields 2 q = q i. Then, combining Equations (1), (4) and 2 q = q i,e obtain S (v) Bi = N ir i. (21) 2B N r + B i N i r i 4.6. Case q v = q i Substituting (9) and (11) into the equality q v = q i yields the vertical velocity required for balancing the mass groth rates of ater vapour and ice phases: u (vi) z = (a 2 2a 1 )B B i N r N i r a 0 (B N r + 2B i N i r i ). (22) For the vertical velocity u z >u (vi) z the ice mixing ratio is changing faster than that of ater vapour ( q v < q i ). As seen from Figure 5(f) the velocity u z (vi) is alays negative, and for typical values of N i r i and N r the magnitude of u z (v) is of the order of a fraction of a millimetre to tens of centimetres per second. Therefore, it is expected that in mixed-phase clouds the condition q v = q i may frequently occur during turbulent fluctuations. In order to find the supersaturation providing the equilibrium beteen ater vapour and ice particles droplets, substitute q v = q i in (10), hich yields 2 q i = q. Then, combining Equations (1), (4) and 2 q i = q, e obtain S (vi) = 2B i N ir i B N r + 2B i N i r i. (23) Copyright 2008 Cron in the right of Canada. Published by John Wiley & Sons, Ltd. Q. J. R. Meteorol. Soc. 134: (2008)

8 602 A. V. KOROLEV In the concluding remarks to this section, it should be noted that the supersaturations S (v), S(i) and S (vi), S(v) depend on the ratio N r N i r, but not on the absolute i values of N i r i and N r. For the case N r inequality N i r i > B i B the S (i) >S (v) >S (v) >S(vi) (24) is alays true (Appendix B). Since B i B 1, the condition N r N i r > B i i B can be simplified to N r >N i r i. This type of relationship, beteen the integral radii of droplets and ice particles, is typically satisfied in mixed-phase clouds. At the final stage of glaciation, or at the initial stage of droplet activation, the droplets are small enough, hich may result in N r N i r < B i i B. In this case, the inequality (24) turns into S (v) >S (v) >S(vi) >S(i). (25) 5. Different regimes of the phase transformation in mixed-phase clouds Three points of the phase equilibria q = 0, q v = 0, q i = 0 and their associated vertical velocities u z, u+ z, u o z, discussed in the previous section, play a fundamental role in the understanding of the phase transformation in mixed-phase clouds. The analysis of Equations (13), (14) and (15) yields that the inequality u o z <u+ z <u z (26) is alays satisfied for any N i r i and N r (Appendix C). The inequalities in (26) enable the phase transformation in mixed clouds to be separated into four regimes: (1) If u z <u o z, then q v > 0, q i < 0 and q < 0. In this case, both ice particles and droplets evaporate, hereas the mass of the vapour increases. In terms of the vapour pressure, this case corresponds to the condition e<e i. (2) If u o z <u z <u + z,then q v > 0, q i > 0and q < 0. Under these conditions ice particles gro, droplets evaporate, and the ater vapour mixing ratio increases. The ater vapour pressure in this case changes in the range E i <e<e v. (3) If u + z <u z <u z,then q v < 0, q i > 0and q < 0. In this case, ice particles gro, droplets evaporate, and the ater vapour mass decreases. This case corresponds to the ater vapour pressure E v <e< E. (4) If u z >u z,then q v < 0, q i > 0and q > 0. In this case, both ice particles and liquid droplets gro, and the ater vapour mass decreases. Under this condition the ater vapour pressure ill be e>e. Figure 6. Conceptual diagram of four different scenarios of phase transformation in mixed-phase clouds: (a) u z <u o z ;(b)uo z <u z <u + z ; (c) u + z <u z <u z ;(d)u z >u z. The arros indicate the direction of mass transfer. The thickness of the arros in (b) and (c) indicates conditional rate of mass transfer. A conceptual diagram in Figure 6 illustrates the four regimes of the phase transformation in mixed-phase clouds. Depending on the thermodynamic conditions, the flo of the ater vapour may be directed either from both liquid and ice particles to the gaseous phase (Figure 6(a)), or from the gaseous phase to both liquid and ice cloud particles (Figure 6(d)), or from liquid particles to ice particles through the gaseous phase (Figure 6(b) and (c)). 6. WBF process The WBF process is defined as the process hen... ice crystals ould gain mass by vapour deposition at the expense of the liquid drops that ould lose their mass by evaporation (American Meteorological Society, 2000). The thermodynamic conditions hich are a result of the aforementioned condition occur hen the vapour tension ill adjust itself to a value in beteen the saturation values over ice and over ater (Wegener, 1911, p.81). Based on the above statements it can be concluded that the WBF process is determined by a set of conditions: q < 0, q i > 0 and E i <e<e. Thus, of the above four scenarios described in section 5, only cases (2) and (3) suit the definition of the WBF process, i.e. hen ice particles are groing at the expense of evaporating liquid droplets. These cases correspond to the velocity range u o z <u z <u z. For the vertical velocities u z >u z, both ice and liquid particles gro simultaneously. Under these conditions the liquid droplets compete ith the ice particles for the ater vapour. It can be shon that for u z >u z, liquid droplets slo don the rate of groth of ice particles q i (Appendix D). In other ords, hen u z >u z, ice Copyright 2008 Cron in the right of Canada. Published by John Wiley & Sons, Ltd. Q. J. R. Meteorol. Soc. 134: (2008)

9 RATES OF PHASE TRANSFORMATIONS IN MIXED-PHASE CLOUDS 603 crystals gro faster ithout droplets as compared to hen the droplets are present. This type of behaviour of liquid droplets in the presence of ice particles is different from that described by the WBF process, hich is hen liquid droplets enhance the groth of ice particles by providing them ith ater vapour through their evaporation (Wegener, 1911; Bergeron 1935; American Meteorological Society, 2000). For a case hen u z < u o z, both ice particles and liquid droplets simultaneously evaporate, hich, again, does not match the definition of the WBF process, since ice particle evaporate in the presence of liquid droplets (American Meteorological Society, 2000). Earlier, Reisin et al. (1996) came to a conclusion, based on the analysis of results of numerical modelling of convective clouds, that the WBF process did not play a significant role in the rain formation process and that in strong updraughts ice particles gre by deposition, but did not cause the evaporation of the drops. A similar statement can be found in Pruppacher and Klett (1997, p. 549). Although it as understood that the WBF process may play a limited role in mixedphase clouds, at that point it as not clear as to hat conditions activated the process. In the present section e demarcate three distinctly different regimes of the behaviour of the condensed ater in mixed-phase clouds and find the conditions for each of them. For adiabatic parcels, the regimes of groth and evaporation of the droplets and ice particles can be ell depicted by the ratio of the groth rates of liquid and ice q / q i. Dividing (8) by (9) yields: q q i = (a 0 u z bi N ir i )B N r ( a 0 u z 1 ξ ). (27) b N r B i N i r ξ Figure 7 shos the dependence of the ratio q / q i on the vertical velocity u z for N r = 500 µm cm 3, Figure 7. Ratio q / q i versus vertical velocity u z in a mixed-phase cloud. Grey areas and numbers indicate areas here: (1) q i < 0and q < 0, both droplets and ice particles evaporate; (2 and 3) q i > 0 and q < 0, liquid droplets evaporate, hereas ice particles gro (this condition corresponds to the WBF process); (4) q i > 0 and q > 0, both droplets and ice particles gro. T = 5 C; P = 680 mb; N r = 300 µm cm 3 ; N i r i = 10 µm cm 3. N i r i = 10 µm cm 3 at T = 5 C. Grey areas in Figure 7 indicated by numbers 1, 2, 3 and 4 correspond to four different regimes as described in section 5. The vertical dashed lines in Figure 7 sho the threshold velocities u z (Equation (11)), uo z (12) and u+ z (15) separating the four regimes of mixed phase evolution. For the vertical velocity u z, the groth rate of liquid ater q = 0 and, therefore, q / q i = 0. For the velocity u o z, the groth rate of ice q i = 0 and therefore, q / q i ±. The WBF process is enabled only in the range of vertical velocities u z >u z >u o z (areas 2 and 3 in Figure 7). During the WBF process the ice particles are groing not only due to the evaporating liquid droplets. The mass of the ater vapour deposited on the ice crystals may be partitioned beteen evaporated and pre-existing liquid droplets ithin the cloud parcel ater vapour. This groth regime corresponds to the condition 1 < q / q i < 0 (area 3 in Figure 7). In other ords, in area 3 in Figure 7 hen u + z <u z <u z, ice particles consume more ater vapour than as evaporated by the liquid droplets. When u z is approaching u z, the liquid droplets slo their evaporation don, hereas ice particles increase their groth rate. At u z = u z, liquid droplets reach equilibrium ith ater vapour and cease groing. In the case of < q / q i < 1 (area 2 in Figure 7), only a fraction of ater vapour evaporated by liquid droplets ill be depleted by groing ice crystals, and another fraction ill stay in the gaseous phase. In area 2 in Figure 7, hen u o z <u z <u + z, ice particles take up less ater vapour than that evaporated by the liquid droplets. When u z is approaching u o z, the liquid droplets evaporate faster, hereas ice particles slo don their groth, and at u z = u o z ice particles stay in equilibrium ith the ater vapour and stop groing. The genuine WBF process, hen all ater evaporated by the droplets is equal to that consumed by the ice particles, occurs hen q / q i = 1. Folloing (10), the condition q / q i = 1 is equivalent to q v = 0, hich for an adiabatic case is reached hen u z = u + z (Equation (15)). In order to characterize the effect of evaporating droplets on the groth of ice particles, e introduce the efficiency of the WBF process (ω). The WBF efficiency is defined as the fraction of evaporated liquid ater converted into ice (ω = q i / q )foru o z <u z <u + z.for the velocity range u + z <u z <u z the groth rate of ice q i becomes larger than that of liquid q, since ice gros both due to evaporating droplets and pre-existing ater vapour. Therefore, for u + z <u z <u z the WBF efficiency is defined as the fraction of evaporated liquid to the total ater vapour converted into ice, i.e. ω = q / q i. Figure 8 shos the dependence of the WBF efficiency versus u z. As seen from Figure 8, ω = 0foru z = u z and u z = u o z, and it reaches maximum (ω = 1), hen u z = u + z. The ratio q / q i is temperature-sensitive. Figure 9 shos the behaviour of q / q i for the different temperatures in the vicinity of u z = 0. As seen from Figure 9, Copyright 2008 Cron in the right of Canada. Published by John Wiley & Sons, Ltd. Q. J. R. Meteorol. Soc. 134: (2008)

10 604 A. V. KOROLEV has a maximum efficiency (ω = 1) at u z 0 and to a first approximation it is not dependent on the microphysics of mixed-phase clouds, temperature or pressure. 7. Limitations and assumptions Figure 8. Efficiency of the WBF process versus u z. The conditions are the same as in Figure 7: T = 5 C; P = 680 mb; N r = 300 µm cm 3 ; N i r i = 10 µm cm 3. Figure 9. Dependence of the ratio q / q i on vertical velocity u z for three different temperatures. Calculations ere done for integral radii N r = 1000 µm cm 3, N i r i = 10 µm cm 3 and pressure P = 680 mb. This figure is available in colour online at.interscience.iley.com/qj the dependence of q / q i on u z eakens ith the decrease of temperature. At u z = 0 the ratio q / q i becomes independent of integral radii N r and N i r i, and it is equal to ( ) q q i u z =0 = a 2 a 1. (28) The ratio a 2 /a 1 is a eak function of T and P, and it changes in the range of 1 <a 2 /a 1 < 1.08, for changes of temperature and pressure in the ranges 40 <T <0 C and 300 <P <1000 mb, respectively. Thus, for practical purposes, it can be considered, ith a high degree of accuracy, that q / q i 1 at u z = 0 regardless of N r, N i r i, T and P. This finding results in the conclusion that the WBF process The consideration of an idealized mixed-phase cloud as discussed at the beginning of section 2 resulted in the neglecting of some physical processes inherent in real clouds, such as: (1) sedimentation; (2) aggregation of ice crystals; (3) riming; (4) nucleation of ice particles; (5) activation and evaporation of liquid droplets; (6) coalescence of droplets; (7) entrainment and mixing; (8) radiation effects. The above processes affect the rates of phase transformation through the changes of N i r i (processes 14), N r (processes 5 and 6), and the humidity and temperature (processes 7 and 8). In the present ork the instant rates of the phase transformation ere considered for the diffusive stage. The characteristic time-scale of the diffusional processes is determined by the time of phase relaxation τ p (Equation (7)). Typically, in mixedphase clouds τ p is of the order of a fe seconds (Korolev and Mazin, 2003). The magnitude of the characteristic time-scales of processes related to aggregation, riming, coagulation, sedimentation, radiative heating or cooling varies from minutes to hours (Prupacher and Klett, 1997) and it is much larger than τ p. Due to a naturally lo concentration of ice nuclei, secondary ice nucleation in already pre-existing ice or mixed-phase clouds is considered to be a relatively slo process, hich is not expected to change N i r i ithin a fe τ p. Therefore, it is expected that the effect of the processes 14, 6 and 8 on the instant rates of q, q i and q v ill be negligible. There are to main issues to consider: activation and evaporation of liquid droplets, and entrainment and mixing. The entrainment and mixing ith the dry outof-cloud air changes the local relative humidity and temperature. Entrainment and mixing may result in a rapid decrease of the relative humidity folloed by the evaporation of cloud particles to compensate for the deficit in humidity. This is an essentially nonequilibrium process, hich cannot be described through the quasi-steady approximation. Since at time-scales t τ p the relative humidity in a cloud parcel relaxes to its quasi-steady value, the mixing is expected to have maximum effect on the steady state q, q i and q v at t<τ p. Entrainment and mixing is a most common phenomenon in the vicinity of the cloud interfaces, and its effect decreases hen moving aay from the cloud boundaries deep into the cloud. Therefore, it is expected that the developed theoretical frameork may not be applicable to the cloud regions in the vicinity of cloud boundaries ith intensive entrainment and mixing. Copyright 2008 Cron in the right of Canada. Published by John Wiley & Sons, Ltd. Q. J. R. Meteorol. Soc. 134: (2008)

11 RATES OF PHASE TRANSFORMATIONS IN MIXED-PHASE CLOUDS 605 The activation and evaporation of droplets imposes another limitation on the use of the quasi-steady approximation. During the initial stage of activation or the final stage of evaporation of droplets their sizes become too small, and they do not satisfy the condition (12), imposing limitations on the use of the quasi-steady approximation. The activation of the droplets may occur in pre-existing ice clouds hen updraughts u z >u z.the evaporation of droplets in a mixed-phase cloud may happen during the process of cloud glaciation during ascent or descent, hen u z <u z. The examples of the cases hich do not satisfy (12) are shon in Figure 3 ith red triangles. It should be noted that (12) as derived assuming that N r N i r i. This condition is typically satisfied in mixed-phase clouds. For ice clouds, a limiting condition can be obtained from (12) by simply replacing subscript by i. Despite the limitations discussed above, the quasisteady approximation provides a relatively accurate estimation of the instant rates of the phase transformations (Figure 1). A more accurate consideration of the phase transformations can be achieved by utilizing the complete set of differential equations describing all processes mentioned at the beginning of this section. Such elaboration may provide more accurate estimations of the threshold velocities u o z, u+ z, u z and the rates q v, q i and q. An important assumption used in this study is that the mass of ice crystals follos Maxellian groth, i.e. m t 3/2. This assumption as incorporated in (2) hen describing the mass groth rate of an ensemble of ice particles, and in (5) hich resulted from the quasisteady approximation. Even though for some cases the groth rate of ice particles as found to deviate from the Maxellian groth rate, numerous laboratory experiments indicated that for steady-state conditions for most cases, including a ide range of conditions for T and S i, and variety of ice particle habits, m t 3/2 (Pruppacher and Klett, 1997). As mentioned in section 1, the groth rate of ice particles depends on the shape factor c. Unfortunately, the shape factor of ice crystals is poorly knon, because of the lack of laboratory experiments and a great variety of ice particle habits. Even more uncertainty in c comes from cases in hich ice particles transit through the conditions ith different crystallographic groth regimes. In the frame of this study, it as assumed that c = 1 in the calculation of coefficients related to the groth rate of ice (i.e. A i, b i, bi, B i, Bi ). This is equivalent to spherical ice groth. Spheres have the fastest groth rate in comparison to other habits of ice crystals. For real ice particles the shape factor varies in the range 0 <c<1, resulting in a loer groth rate as compared to that in this paper. The calculation of u + z and u z for c<1 ill result in loer velocities than those presented in Figure 5(b) and (c). The assumption about the shape factor does not limit the generality of the developed theoretical frameork. 8. Conclusions One of the goals of this ork is to demonstrate a complexity of interaction of liquid, ice and gaseous phases in mixed-phase clouds on a diffusive level. One of the important outcomes of this ork is that the interaction beteen the three phases is not limited just by the WBF process, describing a one-directional phase transition. In fact, mixed phase has several points of equilibrium, the phase transformation has different regimes and the WBF process is one of them. The direction and the rate of the phase transformations are tightly related to the local thermodynamic and microphysical properties of a mixedphase cloud. It has been shon that for typical integral radii N i r i and N r the vertical velocities separating different regimes of the phase transformation can be generated by turbulent fluctuations. In other ords, during turbulent fluctuations the direction of partitioning of ater beteen ice, liquid, and vapour are constantly changing direction, and occasionally folloing the WBF process. During turbulent fluctuations cloud parcels are continuously passing through different points of equilibrium, therefore resulting in a continuous change of the rate and direction of partitioning of the mass beteen ice, liquid and ice phases. Since the integral radii of ice particles and droplets are continuously changing due to the diffusive groth and/or evaporation of droplets and ice particles, the vertical velocities defining the equilibrium beteen phases are also changing ith time. The folloing results ere obtained ithin the frameork of this study: 1. It is shon that mixed-phase clouds have three basic points of phase equilibrium for liquid, ice and vapour phases, hich separate the phase transformation in mixed-phase clouds into four different regimes. 2. For typical N r and N i r i the groth rate of ice particles is significantly less sensitive to the vertical velocities in mixed-phase clouds in comparison to liquid droplets. 3. The temperature corresponding to the maximum rate of the groth of ice and evaporation of liquid changes depending on P, N r, N i r i and u z, and it varies ithin a ide range. This temperature is not necessarily equal to 12 C, here the maximum difference beteen saturation ater vapour pressure beteen ice and liquid is observed. 4. Maximum efficiency of the WBF process, i.e. hen all ater vapour evaporated by the liquid droplets is deposited on the ice crystals, occurs at u z = u + z.itis shon that in mixed-phase clouds u + z 0. Acknoledgements The authors express gratitude to to anonymous revieers for useful comments and Maria Koroleva for the technical support in preparation of the manuscript. Copyright 2008 Cron in the right of Canada. Published by John Wiley & Sons, Ltd. Q. J. R. Meteorol. Soc. 134: (2008)

12 606 A. V. KOROLEV Appendix A List of Symbols Symbol Description Units a 0 g R a T ( ) L R a c p R v T 1 m 1 a 1 1 qv + L2 c p R v T 2 a 1 2 qv + L L i c p R v T 2 ( ρi L 2 A i i kr v T 2 + ρ ) 1 ir v T m E i (T )D s 1 ( ρ L 2 A kr v T 2 + ρ ) 1 R v T m E (T )D s 1 b a 1 B m 2 s 1 b i a 2 B i m 2 s 1 b i0 a 2 B i0 m 2 s 1 bi a 2 Bi m 2 s 1 B i ξcb i0 = 4πρ iξca i ρ m 2 s 1 a 4πρ B i A i i0 ρ m 2 s 1 a Bi (ξ 1)cB i0 = 4π ρ ρ i (ξ 1)cA i m 2 s 1 a 4πρ B A ρ m 2 s 1 a c ice particle shape factor characterizing capacitance 0 <c 1(c = 1 for spheres) c p specific heat capacity of moist air at constant pressure Jkg 1 K 1 D coefficient of ater vapour diffusion in the air m 2 s 1 e ater vapour pressure N m 2 E i saturation vapour pressure above Nm 2 E v flat ( surface ) of ice 1 + S (v) E equilibrium ater vapour pressure hen q v = 0 Nm 2 (Eq. 16) E saturation vapour pressure above Nm 2 flat surface of ater g acceleration of gravity m s 2 k coefficient of air heat conductivity J m 1 s 1 K 1 L i latent heat for ice J kg 1 L latent heat for liquid ater J kg 1 m cloud particle mass kg N i concentration of ice particles m 3 N concentration of liquid droplets m 3 p pressure of moist air N m 2 q i ice ater mixing ratio (mass of ice per kg of dry air) q v ater vapour mixing ratio (mass of ater vapour per kg of dry air) q liquid ater mixing ratio (mass of liquid ater per kg of dry air) q i rate of change of ice ater mixing s 1 ratio (Eq. 9) q v rate of change of ater vapour s 1 mixing ratio (Eq. 11) q rate of change of liquid ater s 1 mixing ratio (Eq. 8) r i average of half of a maximum m linear dimension of an ice particle r average liquid droplet radius m R a specific gas constant of moist air J kg 1 K 1 R v specific gas constant of ater vapour Jkg 1 K 1 Appendix A (Continued) Symbol Description Units S qs quasi-steady supersaturation (Eq. 5) S i e/e i 1, supersaturation over ice S e/e 1, supersaturation over ater S (i) supersaturation, hen q i = q (Eq. 19) S (v) supersaturation, hen q v = 0 (Eq. 17) S (v) supersaturation, hen q v = q (Eq. 21) S (vi) supersaturation, hen q v = q i (Eq. 23) t time s T temperature K u z vertical velocity m s 1 u z vertical velocity, hen q = 0 ms 1 (Eq. 13) u o z vertical velocity, hen q i = 0 ms 1 (Eq. 14) u + z vertical velocity, hen q v = 0 (Eq. 15) ms 1 u z (i) vertical velocity, hen q i = q ms 1 (Eq. 18) u z (v) vertical velocity, hen q v = q ms 1 (Eq. 20) u z (vi) vertical velocity, hen q v = q i ms 1 (Eq. 22) z altitude m η a 2 B i0 /a 0 m 3 s 1 χ a 1 B /a 0 m 3 s 1 ξ E /E i ρ a density of the dry air kg m 3 ρ i density of an ice particle kg m 3 ρ density of liquid ater kg m 3 τ p time of phase relaxation (Eq. 7) s τ c characteristic lifetime of a cloud s ω efficiency of the WBF process Appendix B Relationship beteen supersaturation S (i), S(v), S(vi) and S(v) Introduce a ne variable x = B N r B i N i r. Substituting x i into (17), (19), (21), (23) yields S (v) = 1 x + 1, S (i) = 1 x 1, S (v) = 1 2x + 1, S (vi) = 2 x + 2. (B1) (B2) (B3) (B4) It is easy to see from (B1)(B4) that for x> 1 (i.e.n r /N i r i >B i /B ), the folloing relationship Copyright 2008 Cron in the right of Canada. Published by John Wiley & Sons, Ltd. Q. J. R. Meteorol. Soc. 134: (2008)