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Michael, Ryan A.
Fate of volatile chemicals during accretion on wet-growing hail
h [electronic resource] /
by Ryan A. Michael.
[Tampa, Fla] :
b University of South Florida,
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Thesis (M.S.E.S.)--University of South Florida, 2008.
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ABSTRACT: Phase partitioning during freezing of hydrometeors affects the transport and distribution of volatile chemical species in convective clouds. Here, the development, evaluation, and application of a mechanistic model for the study and prediction of partitioning of volatile chemical during steady-state hailstone growth are discussed. The model estimates the fraction of a chemical species retained in a two-phase growing hailstone. It is based upon mass rate balances over water and solute for constant accretion under wet-growth conditions. Expressions for the calculation of model components, including the rates of super-cooled drop collection, shedding, evaporation, and hail growth were developed and implemented based on available cloud microphysics literature. A modified Monte Carlo simulation approach was applied to assess the impact of chemical, environmental, and hail specific input variables on the predicted retention ratio for six atmospherically relevant volatile chemical species, namely, SO, HO, NH, HNO, CHO, and HCOOH. Single input variables found to influence retention are the ice-liquid interface supercooling, the mass fraction liquid water content of the hail, and the chemical specific effective Henry's constant (and therefore pH). The fraction retained increased with increasing values of all these variables. Other single variables, such as hail diameter, shape factor, and collection efficiency were found to have negligible effect on solute retention in the growing hail particle. The mean of separate ensemble simulations of retention ratios was observed to vary between 1.0x10 and 1, whilst the overall range for fixed values of individual input variables ranged from 9.0x10 to a high of 0.3. No single variable was found to control these extremes, but rather they are due to combinations of model input variables.
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Advisor: Amy L. Stuart, Ph.D.
x Engineering Science
t USF Electronic Theses and Dissertations.
Fate of Volatile Chemicals during Accretion on WetGrowing Hail by Ryan A. Michael A thesis submitted in partial fulfillment of the requirements for the degree of Master of Science in Engineering Science Department of Civil and Environmental Engineering College of Engineering University of South Florida Major Professor: Amy L. Stuart, Ph.D. Jeffrey Cunningham, Ph.D. Jennifer M. Collins, Ph.D. Maya A. Trotz, Ph.D. Date of Approval: July 17, 2008 Keywords: atmospheric chemistry, riming, retention, ice microphysics, cloud modeling Copyright 2008 Ryan A. Michael
Acknowledgements This thesis would not have been successfully comple ted without the help of many people to whom I am grateful and because of whom my experience has been unforgettable. First, I wish to convey my thanks to my advisor, Am y Stuart, for making this feasible. From providing funds to instilling knowle dge, your support throughout my study has been paramount to my success. Thank you f or holding me to such high standards with your perceptive remarks and construc tive criticisms at every stage of my research. More importantly, thank you for your pati ence, for sticking with me when even I wanted to throw my hands up. I thank the members of my thesis committee, Maya Tr otz, Jennifer Collins, and Jeffrey Cunningham. Thank you for your meticulous r eviews. Your insightful comments were instrumental in helping to shape this work. I am forever grateful to the late Calvin Miller who did not give me life but gave me the life I have. Thank you for believing in me, you are forever in my thoughts. I would like to express my gratitude to the followi ng persons for their help throughout this process: Chris Einmo, Monica Gray, Roland Okwen, Edolla Prince, Luis Baluto Torrez, and Nicole Watson To my loving mother, Adnic West, to whom this thesi s is dedicated, I say thank you. I am most indebted to you for your inestimable help, limitless sacrifices, and unconditional love throughout my life. Without you, this would not have been possible.
i Table of Contents List of Tables .................................... ................................................... .............................. iiiList of Figures ................................... ................................................... .............................. ivABSTRACT .......................................... ................................................... ........................... v1.Introduction ...................................... ................................................... .......................... 11.1.Background ........................................ ................................................... ......... 11.2.Convective cloud systems .......................... ................................................... 11.3.Cloud hydrometeors and chemical interactions ...... ....................................... 31.4.Ice-chemical interactions ......................... ................................................... ... 51.5.Thesis organization ............................... ................................................... ...... 72.Model Development ................................. ................................................... ................. 92.1.Retention ratio ................................... ................................................... .......... 92.2.Microphysical process variables ................... ............................................... 132.3.Model parameters and assumptions .................. ........................................... 152.4.Implementation .................................... ................................................... ..... 182.5.Testing............................................ ................................................... ........... 263.Application ....................................... ................................................... ........................ 284.Results ........................................... ................................................... ........................... 294.1.Hail factors ...................................... ................................................... .......... 294.1.1.Hail diameter ..................................... ................................................ 29
ii 4.1.2.Mass fraction liquid water content of hail ........ ................................. 344.1.3.Ice-liquid interface temperature .................. ...................................... 354.1.4.Hail shape factor ................................. ............................................... 364.1.5.Efficiency of collection .......................... ........................................... 374.2.Environmental factors ............................. ................................................... .. 384.2.1.Cloud liquid water content ........................ ........................................ 384.2.2.Drop radius ....................................... ................................................. 3 94.2.3.Pressure .......................................... ................................................... 424.2.4.Air temperature ................................... ............................................... 434.3.Chemical factors .................................. ................................................... ..... 454.3.1.Chemical effective HenryÂ’s constant and pH ........ ............................ 454.3.2.The effective ice-liquid distribution coefficient ............................... 484.4.Summary of results ................................ ................................................... ... 495.Discussion and Limitations ........................ ................................................... .............. 516.Conclusions and Implications ...................... ................................................... ............ 567.List of References ................................ ................................................... .................... 58Appendices ........................................ ................................................... ............................. 71Appendix A. Retention Model Calculations .......... ............................................... 72About the Author .................................. ................................................... .............. End Page
iii List of Tables Table 1. Methods for estimation of model parameters ........ ............................................. 16Table 2. Chemical properties and thermodynamic data ................................................... 21Table 3. Model input variables and ranges ......... ................................................... ........... 23Table 4. Dependence of simulated retention fraction on input variables ......................... 50Table A. Retention model calculations ............. ................................................... ............. 72
iv List of Figures Figure 1. Hail growth and solute transfer processes ................................................. ....... 10Figure 2. The effect of individual hail input varia bles on retention fraction.. .................. 31Figure 3. The effect of individual hail input varia bles on the growth rate boundary parameter, GB. .................................... ................................................... .......... 32Figure 4. The effect of individual hail input varia bles on the wet growth boundary parameter, WB. .................................... ................................................... .......... 33Figure 5. The effect of individual environmental in put variables on the retention fraction. ......................................... ................................................... ................. 41Figure 6. The effect of individual environmental in put variables on the growth rate boundary parameter, GB. ........................... ................................................... ... 41Figure 7. The effect of individual environmental in put variables on the wet growth boundary parameter, WB. ........................... ................................................... .. 42Figure 8. The effect of individual chemical input v ariables on the retention fraction. ..... 47Figure 9. The effect of individual chemical input v ariables on the growth rate boundary parameter, GB. ........................... ................................................... ... 47Figure 10. The effect of individual chemical input variables on the wet growth boundary parameter, WB. ........................... ................................................... .. 48
v Fate of Volatile Chemicals during Accretion on WetGrowing Hail Ryan Michael ABSTRACT Phase partitioning during freezing of hydrometeors affects the transport and distribution of volatile chemical species in convec tive clouds. Here, the development, evaluation, and application of a mechanistic model for the study and prediction of partitioning of volatile chemical during steady-sta te hailstone growth are discussed. The model estimates the fraction of a chemical species retained in a two-phase growing hailstone. It is based upon mass rate balances over water and solute for constant accretion under wet-growth conditions. Expressions for the ca lculation of model components, including the rates of super-cooled drop collection shedding, evaporation, and hail growth were developed and implemented based on avai lable cloud microphysics literature A modified Monte Carlo simulation approach was app lied to assess the impact of chemical, environmental, and hail specific input variables on the predicted retention ratio for six atmospherically relevant volatile che mical species, namely, SO2, H2O2, NH3, HNO3, CH2O, and HCOOH. Single input variables found to influ ence retention are the ice-liquid interface supercooling, the mass fractio n liquid water content of the hail, and the chemical specific effective HenryÂ’s constant (a nd therefore pH). The fraction retained increased with increasing values of all these varia bles. Other single variables, such as hail
vi diameter, shape factor, and collection efficiency w ere found to have negligible effect on solute retention in the growing hail particle. The mean of separate ensemble simulations of retention ratios was observed to vary between 1. 0x10-8 and 1, whilst the overall range for fixed values of individual input variables rang ed from 9.0x10-7 to a high of 0.3. No single variable was found to control these extremes but rather they are due to combinations of model input variables.
1 1. Introduction 1.1. Background Knowledge of the upper tropospheric ozone budget is essential to our ability to understand and predict climate change. Ozone concen trations in the troposphere are regulated by catalytic cycles involving nitrogen ox ides (NOx), hydrogen oxides (HOx), and volatile organic carbon (VOC) species. In upper tropospheric regions influenced by convection, the budget of NOx and the ratio of HOx to NOy (reactive nitrogen) are not well understood, resulting in poorly understood ozo ne amounts [ Jaegl et al. 2001]. 1.2. Convective cloud systems The availability and concentration of ozone precurs ors in the troposphere are significantly affected by the action of convective cloud systems. Convective cloud systems significantly influence tropospheric chemis try and chemical deposition to the ground by moving trace gas species from the boundar y layer to the free troposphere through chemical scavenging by cloud hydrometeors. Convective processing of trace gas species is an important means of moving chemical co nstituents rapidly between the boundary layer and free troposphere, and is an effe ctive way of cleansing the atmosphere through wet deposition. It also brings into the clo ud, species that are of a different composition, concentration, and origin than the air that ascends from the boundary layer
2 [ Barth et al. 2002]. This entrained air can affect local thermo dynamics as well as chemical and microphysical processes. Convective cloud systems have been shown to influen ce the chemical characteristics of the upper troposphere to lower s tratosphere region. They contribute to the production, transportation, and redistribution of reactive chemical constituents, including water, aerosols and long-lived tracers in the upper troposphere to lower stratosphere [ Dickerson et al. 1987; Gimson 1997; Lelieveld and Crutzen, 1994; Ridley et al. 2004; Yin et al. 2005]. These convective cloud systems can offer a rapid pathway for the vertical transport of air containing reacti ve chemical species from the planetary boundary layer to the upper troposphere [ Barth et al. 2007; 2001]. Through entrainment/detrainment processes, they can facilit ate the mixing and dispersal of pollutants, and the transport of reactive species o ver significantly shorter timescales than would occur via eddy diffusion or other atmospheric mixing processes [ Dickerson et al. 1987]. At higher altitudes, increases in wind speed may result in longer atmospheric residence times of chemical species, thus increasin g the probability of their participation in photochemical reactions and other atmospheric tr ansformations [ Ridley et al. 2004; Stockwell et al., 1990; Rutledge et al. 1986]. Thus, convective cloud systems can be thought of as chemical reactors, processing atmosph eric air and its trace chemical constituents. However, the potential cleansing effect of deep con vective cloud systems on the atmospheric boundary layer is countered by negative impacts due to scavenging, dissolution, and eventual deposition of acidic spec ies. Effects of acid deposition may include reduced buffering capacity of lakes and oth er surface water systems, forest
3 deaths, reduced visibility, material deterioration, and deleterious health effects such as bronchitis and asthma [ Cosby et al. 1985; Likens et al. 1996; Dockery et al ., 1996]. Furthermore, venting of air from the atmospheric bo undary layer by convective cloud systems may influence tropospheric ozone concentrat ions due to the migration and increased residence times of chemical species that regulates its production. These processes may significantly affect the upper tropos phere ozone budget, and consequently climate change [ Barth et al. 2002; Dickerson et al 1987]. Additionally, as a result of these strong convective processes, regional air pol lution problems may be transformed to global air pollution problems due to the long range transport of pollution plumes [ Dickerson et al 1987]. Therefore, an understanding of the microph ysical processes governing the interaction of trace chemical species and condensed phase in convective cloud system is imperative to our determination of their fate, and as understanding of tropospheric ozone budget and climate change. 1.3. Cloud hydrometeors and chemical interactions Previous studies have shown that the interaction of cloud hydrometeors with trace chemical species may significantly influence the fa te of these chemicals, and subsequently impact atmospheric chemistry. Hydromet eors refer to the different forms of condensed water that constitute convective clouds, and include ice crystals, snow, graupel, and hail. Their formation is as a direct r esult of the moisture, temperatures, pressures, and airflow conditions associated with c onvective cloud systems. These cloud hydrometeors provide surfaces for chemical phase ch anges and reactions, act as condensed-phase reactors, and serve as conduits for chemical transport from the
4 atmosphere to the ground, through scavenging and pr ecipitation [ Lamb and Blumenstein 1987; Rutledge et al 1986; Santachiara et al. 1995; Snider and Huang 1998]. Interactions of volatile trace chemicals with cloud hydrometeors include absorption, condensation, diffusion, vapor deposition, or incor poration into the growing hydrometeor [ Flossmann et al, 1985; 1987 ; Pruppacher and Klett, 1997]. Once dissolved in the hydrometeor, depending on the characteristics of th e phase, the trace chemicals may dissociate or undergo further chemical reactions, t hus affecting and modifying cloud and atmospheric chemical distributions [ Barth 2000; 2001; 2007]. For example, the removal of odd hydrogen species due to interactions with cl oud hydrometeors has been found to significantly affect the oxidizing capacity of the troposphere and contribute to increased levels of sulfur species in precipitation [ Audiffren et al, 1999; Snider 1998]. Consequently, emphasis has been placed on understan ding the interaction of trace chemical species with hydrometeors in convective cl oud systems through observational and modeling studies. These include studies focusin g on acid deposition [ Barth et al. 2000; Chameides 1984; Daum et al. 1984; Kelly et al. 1985], ozone in the troposphere [ Lelieveld and Crutzen 1994; Pickering et al., 1992; Prather and Jacob 1997], and the interaction of hydrometeors with other species [ Chatfield and Crutzen 1984]. Most of the models focused on liquid phase hydrometeors, and ch emical species were distributed based on processes governing liquid-phase exchanges Consequently, little is known about how microphysical processes involving ice aff ect chemical fate.
5 1.4. Ice-chemical interactions Previous work indicates that ice-chemical interacti ons may have significant impacts on cloud and atmospheric chemistry [ Pruppacher and Klett 1997; Stuart and Jacobson 2003; 2004; 2006]. Many researchers have included limited ice-chemical interactions in cloud models [ Audiffren et al ., 1999; Chen and Lamb, 1994; Cho 1989; Rutledge et al ., 1986]. Audiffren et al. , in their two-dimensional Eulerian cloud model, utilized the formulations of Lamb and Blumenstein  and Iribarne and Barrie  in their parameterization of the entrapment of chemical species in a growing ice-phase hydrometeor. Elucidation of the m echanism by which trace chemicals and ice interact is important in order to predict c hemical fate and to find suitable parameterizations for larger scale modeling. Recent studies indicate that one microphysical proc ess, the freezing of supercooled drops via accretion, may significantly influ ence the venting of chemicals by clouds [ Yin et al., 2002, Barth et al. 2007; Cho et al., 1989]. Ice hydrometeors in convective clouds may form and grow due to distinct microphysical transformations. Three major categories of such transformations exis t, delineated by specific environmental conditions and giving rise to distinc t hydrometer types [ Pruppacher and Klett 1997]. These are non-rime freezing, dry-growth ri ming, and wet-growth riming. Non-rime freezing involves the freezing of supercoo led droplets without contact with an already frozen substrate hydrometeor. This phenomenon is normally associated with very low temperatures (< -30C), and the associated ice nucleation may be either homogeneous or heterogeneous in nature. Hydrometeor s formed via this process
6 generally retain the approximate shape of the origi nal supercooled drop [ Hobbs 1974; Pruppacher an Klett, 1997]. Conversely, riming involves the collision and colle ction of supercooled water drops by solid substrates, which may include, ice c rystals, graupel, and hail, due to the differences in velocity of the drops and substrate. Riming can be further classified into either of two broad categories: wet-growth or dry g rowth riming. The regime a hydrometeor grows in is greatly dependent on specif ic conditions of drop size, hydrometeor speed, cloud water content, and tempera tures (of air, drop, and riming substrate). Wet-growth riming results in a partiall y frozen hydrometeor, which may contain pockets of water in the hydrometer, or on t he surface of the hydrometeor, and a surface temperature of approximately 0C. Due to these conditions, drop interference and coalescence occurs, resulting in a more dense and t ransparent structure. Some liquid water may be shed from the riming hydrometeor. Conv ersely, dry-growth riming is associated with conditions of lower cloud water con tent and surface temperatures below 0C. Due to the low temperatures associated, drop fre ezing occurs independently, without coalescence, resulting in less dense, opaque hydrom eteor. Because of the varying environmental conditions and processes characterizi ng the different freezing categories, factors affecting volatile chemicals retention in t he frozen hydrometeor due to these microphysical transformations may differ significan tly. Several authors have carried out laboratory studies investigating the degree to which a volatile chemical species may be retained i n the ice phase due to the riming process. Consequently, they have characterized a re tention ratio, which gives the ratio of solute mass in the hydrometeor to that which was or iginally in the impinging droplet, i.e.,
7 the equilibrium concentration [Iribarne et al., 1983; Iribarne and Pyshnov 1990; Lamb and Blumenstein, 1987; Snider and Huang 1998]. These investigations measured the retention efficiencies of gases found in clouds inc luding O2, SO2, H2O2, HNO3, HCl, and NH3, and calculated values ranging between 0.01 and 1. Factors that varied among the studies were summarized by Stuart and Jacobson  and included temperature, droplet and substrate size, solute concentration, p H, and impact speed. To address the lack of understanding regarding the factors leading to the observed differences in experimentally derived retention rat ios, Stuart and Jacobson [2003; 2004; 2006] developed theory-based retention parameteriza tions and a mechanistic model under conditions satisfying dry growth riming and non-rim e freezing. The retention ratio was found to be highly dependent on the effective Henry Â’s constant, drop velocity, and drop size. The formation of a complete or partial ice sh ell was also found to have a significant impact on retention. Chemicals with high effective HenryÂ’s constant were found to be completely retained. For those with negligible Henr yÂ’s constant values, retention was found to be highly dependent on freezing conditions However, the microphysical processes determining volatile chemical fate during ice accretion in the wet-growth regime remain poorly understood. 1.5. Thesis organization In this study, I investigate the substrate properti es, chemical characteristics, and environmental variables affecting chemical retentio n under conditions of wet growth. More specifically, this body of research attempts t o answer the following scientific questions:
8 What chemical properties affect chemical retention in hail growing under wet-growth conditions? What specific environmental conditions contribute t o volatile chemical retention in ice hydrometeors for conditions of wet growth? What particle-scale microphysical process influence volatile chemical retention for accretion under wet growth conditions? Stuart  developed a mechanistic analytical equation for the evaluation of volatile chemical retention during steady-state acc retion on wet growing hail. The derivation is presented again in Section 2.1, by pe rmission of the author, for completeness. In this thesis, I develop the express ions for microphysical process variables (Section 2.2), and model parameters (Sect ion 2.3) necessary to apply the model. Model implementation and testing are discussed in S ections 2.4 and 2.5, respectively. It is then applied (Section 3) to understand the likel y dependence of partitioning on environmental conditions and hail characteristics a nd chemical species. Results, discussions, and conclusions are presented in Secti ons 4, 5, and 6, respectively.
9 2. Model Development This model considers a two-phase (ice and liquid wa ter) hail particle, growing at a steady-rate, in a region of sufficiently high cloud liquid water content to satisfy conditions for wet growth. Growth is facilitated by the collection of super-cooled water drops in the volume swept out by the falling hail p article. The fate of solutes, originally dissolved in the impinging drops, is determined by two coupled mass balances; a water mass rate balance, and a solute mass balance over t he growing hail particle. Expressions describing the process governing hailstone developm ent, such as impingement, evaporation, and shedding, are derived from cloud m icrophysics. 2.1 Retention ratio Wet growth is characterized by higher surface tempe ratures (~0C), higher cloud water mixing ratios, and higher rates of drop impin gent on the substrate, than the conditions associated with the dry-growth regime [ Pruppacher and Klett 1997, p. 659]. Under these conditions, impinging drops coalesce pr ior to freezing on the growing hailstone. This may lead to the presence of a liqui d water layer (skin) on the surface of the hydrometeor and liquid water in entrapped pocke ts throughout the hailstone [ Johnson and Ramussen 1992, Schumann 1938]. If the skin is thick enough, water can be shed as water droplets, due to gyration (rotation) of the h ailstone [ Garcia-Garcia and List 1992].
11 Here F is the mass rate of drop collection, G is the mass rate of hydrometeor growth, E is the mass rate of solution evaporation, and S is the mass rate of shedding, each having units of mass per time. Xd, Xh, Xe, and Xl are the solute mass fractions, e.g. gram solute per gram of solution, in the drops ( d ), evaporated solution ( e ), hydrometeor ( h ), and surface (and shed) liquid ( l ). Note that this equation assumes that the liquidtogas mass transfer (evaporation) rate for chemical s olute is proportional to that for water evaporation. This is a simplification for an open s ystem (low concentrations of water vapor and solute in the surrounding air), similar d iffusivities in air, similar Schmidt numbers, and equilibrium chemical partitioning at t he liquid-gas interface. It is noted that a water rate balance require S = F Â– E Â– G and subsequent rearrangement, results in the following: ( ) () () hl h dhlelXXG XG XFXXGXXES = ++ Eqn (2) It is recognized that el XX is a mass fraction air-water distribution coefficie nt. In terms of the more traditional HenryÂ’s constant, it is equivalent to: 1el s lvl X XH r r = Eqn (3) where, H is the dimensionless effective HenryÂ’s constant i n terms of concentration in water over concentration in air, l r is the density of the liquid solution, and s vl r is the saturation vapor density over the liquid solution.
12 The term hl XX can be determined using a solute mass balance on t he hydrometeor given by: hhllii XMXMXM =+ Eqn (4) i hhllli lX XMXMXM X =+ Eqn (5) where Mh, Ml, and Mi represent the total mass, liquid phase mass, and i ce phase mass of the hailstone, respectively. Since, Xi is the mass fraction of solute in ice, and Xl is the mass fraction of solute in the liquid solution, il XX is an effective ice-liquid interfacial distribution coefficient, which we term ke. It includes the effect of crystal growth rates, dendritic trapping, and convectively enhanced solut e mass transfer in the liquid [ Hobbs 1974, p. 600 605]. Rearrangement of equation (4) results in: ()1h e lX k X hh =+Eqn (6) where, h is the mass fraction liquid water content of the hy drometeor ( lh MM ). To solve for the retention ratio during wet growth riming, we substitute equations (3) and (6) into equation (2) and rearrange the ter ms. The retention ratio or ratio of solute mass fraction in the hydrometeor to that in the ori ginal impinging drops is then given by: ( ) () Shedding Growth effect Evaporation effect1 1 1e l e s vlGk GkES Hhh r hh r+n G= +-++ n n Eqn (7)
13 Here, G represents the mass rate of chemical accumulation in the hailstone over that in the collected liquid drops, given by hd GXFX that is, the retention fraction. 2.2 Microphysical process variables To calculate the retention ratio using Equation 7, rates of the microphysical processes involved in riming must be estimated. The se include mass rates of drop collection, water evaporation, hailstone growth, an d shedding. The rate of drop collection is estimated assuming a spherical particle of radius r moving with a velocity through a region of air of d efined liquid water content, and sweeping out a volume determined by its cross-secti onal area, r2. Therefore, the rate of impingement is a function of the fall speed of the hydrometeor and the liquid water content of the air. Thus, the mass rate of drop col lection is given by [ Pruppacher and Klett 1997, p.568 Â– 570] 2 a Frv epwr = Eqn (8) where, v is the fall speed of the hailstone relative to the drop, r is the hailstone radius, ra is the density of air, w is the mass fraction liquid water content of the cloud, and e is the collection efficiency. Evaporation is represented as a first-order rate pr ocess for mass transfer from liquid to air. Assuming a spherical hailstone, and an open system, the mass rate of solute evaporation, E, is then [ Pruppacher and Klett 1997, p. 537]:
14 4s vl vHP ErD RT p =F Eqn (9) where, s vl P is the saturation vapor pressure over liquid water is the universal gas constant for water vapor, TH is the temperature of the hailstone, D is the diffusivity of water vapor in air, and F is the ventilation coefficient defining convective enhancement of evaporation due to hail motion. An open system a ssumption is used for consistency with the simplifying assumption of proportional rat es of water and solute transfer. The mass growth rate of the hail is the sum of the ice growth rate ( Gi) and the rate of change of liquid water mass of the hydrometeor ( Gw). Thus: iw GGG =+ Eqn (10) where Gi is estimated [after Stuart and Jacobson 2006] as: ()24c ii GrbT pr =D Eqn (11) Here, ri is the density of the hailstone, and [ b(DT)c] is the intrinsic crystal interface growth velocity. The form of the interface growth v elocity equation and factors, b and c are based on experimental data and theory for growt h rates of ice in super-cooled water [ Pruppacher and Klett 1997, p. 668 Â– 674]. DT = T0 Â–Tint is the super-cooled temperature of the ice liquid interface, where T0 is the equilibrium freezing temperature of water (0rC), and Tint is the ice-liquid interface temperature. As interf ace temperature are expected to be very close to 0C during wet growth, we use a b of 0.3 and c of 2 for the classical growth regime [ Bolling and Tiller 1961]. v R
15 Considering the case of constant liquid water conte nt of the hailstone the rate of change of liquid mass of can be determined by: () wi GG h h= 1Eqn (12) The mass shedding rate of the growing hail is deter mined by water mass conservation as S = F Â– G Â– E as defined above. 2.3 Model parameters and assumptions Chemical and physical property and process paramete rs are necessary for application of the above equations. Expressions des cribing properties of water phase change, dry and moist air, and water vapor were def ined based on available literature. Properties of the ice substrate, and super-cooled d rop, such as ventilation characteristics, were similarly defined. Table 1. lists model parame ters, literature references for the estimation method, and assumptions applied.
16 Table 1. Methods for estimation of model parameters Parameter Method Reference and Assumptions Hail Temperature ( Th) Assumed equal to 0 C Latent heat of water melting ( L m ) Jacobson [2005, p. 40, Eqn. 2.55] Latent heat of water sublimation( L s ) Jacobson [ 2005, p. 40, Eqn. 2.56] Latent heat of water evaporation L s Â– L m ; Jacobson [2005, p. 40, Eqn. 2.56] Saturation vapor pressure over liquid water ( s vl P ) Jacobson [ 2005, p. 41, Eqn. 2.62] Saturation vapor density over liquid water ( s vl r ) Jacobson [2005, p. 31, Eqn. 2.25] Saturation vapor density over ice Jacobson [2005, p. 43, Eqn. 2.64] Water-air surface tension Jacobson [2005, p. 485, Eqn. 14.19] Dynamic viscosity of dry air Jacobson [2005, p. 102, Eqn. 4.54] Heat capacity of dry air at constant pressure Smith and Van Ness [2001, p. 109, Table 4.1] Partial pressure of water vapor in air ( P a ) Jacobson [2005, p. 21, Eqn. 2.27] Mass mixing ratio of water vapor in air Jacobson [2005, p. 32, Eqn. 2.31] Thermal conductivity of dry air Jacobson [2005, p. 20, Eqn. 2.5] Gas constant for moist air Jacobson [2005, p. 33, Eqn. 2.37] Gas constant for water vapor ( R v ) Jacobson [2005, p. 22, Eqn. 2.21] Molecular weight of moist air Jacobson [2005, p. 31, Eqn. 2.26] Density of moist air ( r a ) Jacobson [2005, p. 33, Eqn. 2.36] Kinematic viscosity of moist air Jacobson [2005, p. 102, Eqn. 4.55] Mean free path of moist air Jacobson [2005, p. 506, Eqn. 15.24] Effective Henry Constant (H*) Seinfeld &Pandis [1998, p. 340 Â– 350] Heat capacity of moist air at constant pressure Jacobson [2005, p. 20, Eqn. 2.5] Diffusivity of water vapor in air ( D ) Pruppacher & Klett [1997, p. 503, Eqn. 133] Heat capacity of (supercooled) water Pruppacher & Klett [1997, p. 93, Eqn. 3-16] Density of (supercooled) liquid water ( r l) Pruppacher & Klett [1997, p. 87, Eqn. 3-14] Density of ice ( r i ) Pruppacher & Klett [1997, p. 9, Eqn. 3-2] Density of hailstone ( r h) r h =[(h / r l ) + ((1h) / r i )] 1 Weighted reciprocal average of ice and water densities. Drop terminal fall velocity Jacobson [2005, p. 664, Eqn. 20.9] Hailstone fall velocity Pruppacher & Klett [1997, p. 87, Eqn. 10175 Â– 10-178], Jacobson [ 2005, p. 507, Eqn.15.26] Impact speed of drops and hailstone ( v ) Assumed equal to v h Â– v d Reynolds Number for flow around drops Jacobson [2005, p. 664, Eqn. 20.6] Reynolds Number for flow around hailstone Seinfeld &Pandis [1998, p. 463, Eqn. 8.32] Prandtl Number Jacobson [2005 p. 532, Eqn. 16.32] Schmidt Number Jacobson [2005 p. 531, Eqn. 16.25] Stokes, Nusselt, and Sherwood Numbers Stuart and Jacobson [2004, Section 2.3] Ventilation Coefficient ( F ) Pruppacher & Klett [1997, p. 537, Eqn. 1352] Critical liquid water content ( w c ) Stuart and Jacobson [2004, Eqn. 14]
17 Calculations of the parameters listed in Table 1. w ere based on several assumptions. It was assumed that the air is saturat ed with respect to water. Temperatures of the air, hail ice, and super-cooled drop were as sumed equal, whilst, hail water temperature was assumed equal to the equilibrium fr eezing temperature. Saturation vapor densities and critical water limit for wet growth, as well as, the temperature dependence of the chemical specific HenryÂ’s Law constant, and dissociation constants describing pH dependence, were calculated using the equilibrium f reezing temperature. Here, I used an average hail density for simplification which compa red well to parameterizations developed by Heymsfield and Pflaum  and Macklin  for riming, based upon drop radius, a the temperature of the ice substrate, Ts, and the impact speed of the drops, Uimp, (of the form, Y= -aUimp/Ts). Drop fall velocity was calculated as discussed i n Jacobson [2005, p. 661 664]. Hail fall velocity was deter mined accounting for the inertial effect of the particle given by the empiri cal drag coefficient, CD as follows in Seinfeld and Pandis [1998, p. 462 468], with an initial hail fall sp eed based on parameterizations discussed in Pruppacher and Klett [1997, p. 441 Â– 444]. Symbols are provided in Table 1. for parameters used elsewhere in the text.
18 2.4 Implementation Model calculations were performed using Microsoft E xcel spreadsheets. Statistical ensemble modeling runs were performed u sing OracleÂ’s statistical and risk analysis software, Crystal Ball, to assess impact of variation in the input parame ters on the modeled retention fraction. The basis of the en semble runs is the generation of random numbers, bounded by the range previously def ined for system variables of interest. That is, to assess the impact of a partic ular variable, it is assigned a fixed possible range, and all other system variables are assigned random values, by a random number generator, bounded by a predefined range det ermined by the model conditions. By choosing regular intervals within its range for the controlled variable whilst simultaneously randomly varying the values assigned to the other parameters, any statistical dependence between the manipulated vari able and system parameters is established. Such probabilistic models, involving t he element of chance, are called Monte Carlo simulations. Initial simulations were performed to determine app ropriate ranges for trace chemical parameters. Additional input parameters in cluded those controlling the variability of hail and environmental factors. The range of values assigned to the input parameters were based on literature values for cond itions applicable to wet growth, and each was assumed to confirm to a uniform distributi on. Chemical input variables are the effective HenryÂ’s constant and the effective iceliquid distribution coefficient. Although the equil ibrium ice-liquid distribution coefficient, D k is chemical specific, the effective ice-liquid di stribution coefficient, e k is
19 a strong function of the kinetics of freezing and i s less dependent on the specific chemical. The equilibrium ice-liquid distribution c oefficient is defined as the ratio of the solute concentration directly adjacent to the inter face in the solid, Cs(i) to the solute concentration directly adjacent to the interface in the liquid, Cl(i) as discussed by Hobbs [1974, p. 600] () () s D l Ci k Ci = Eqn (13) Following the discussion of Hobbs , the equilibrium ice-liquid distribution coefficient describes the extent at which solute mo lecules are incorporated into the growing ice phase and is a direct measure of the di stortion imposed by solute molecules on the molecular arrangement in the solid. For ioni c solutes in water, D k is always very much less than unity Hobbs . However, under steady-state conditions, as the solute concentration builds up in the liquid phase and dif fuse away from the interface, the width of this liquid layer next to the interface may chan ge thus affecting the localized equilibrium. Thus, it will depend on the rate of fr eezing, the equilibrium ice-liquid distribution coefficient, and the diffusivity of th e solute molecules. Therefore, an effective iceliquid distribution coefficient is d efined by Hobbs : s e C k C = Eqn (14) Here, C is the concentration of the bulk solution at a poi nt far removed from the interface. Thus, e k considers other processes affecting water-to-ice m ass transfer such as
20 crystal growth rates, dendritic trapping, and conve ctively enhanced mass transfer in the liquid phase. However, based on experimental data p resented by Hobbs , and other sources cited therein, there is little variation in the derived e k for varying chemical species, with the ranges presented having the same order of magnitude. Hence, I assume the same range of values, based on the measured eff ective ice-liquid distribution coefficients, for all species considered. To determine the range of effective HenryÂ’s constan ts to consider, a second spreadsheet was used to calculate the pH dependent effective HenryÂ’s constant, H*, accounting for dissociation, for each atmospherical ly relevant chemical species considered. Calculations were based on formulations presented in Seinfeld and Pandis [1998, p. 340 Â– 385] using tabulated values of Henr yÂ’s constants, aqueous equilibrium constants, and reaction enthalpies given in Table 2 Distributions of H* for each chemical of interest were obtained with ensemble simulations in which the pH was allowed to vary randomly with a uniform distribution from 2 to 8, a nd water temperature was assumed equal to the equilibrium freezing temperature. The resulting overall species maximum and minimum effective HenryÂ’s constant, derived fro m these simulations, was then used to define the range for subsequent calculations of retention. The results are discussed and presented in Section 4.
21 Table 2. Chemical properties and thermodynamic data Â§ Chemical properties were taken from Seinfeld and Pandis [1998, Chap. 6, p. 341 Â– 391], for values observed at 298K. Â† DHC Â– dimensionless HenryÂ’s constant, was calculate d at the equilibrium freezing temperature based on the temperature dependence of the HenryÂ’s constant, and other dissociation constants, given by Seinfeld and Pandis [1998, p. 342, Eqn. 6.5]. Chemical Â§HenryÂ’s Law Coefficient, M/atm. Enthalpy of Dissolution of Henry Law Coefficient, kcal/mole 1st dissociation constant, M 2nd dissociation constant, M Â†DHC range observed Sulfur Dioxide, SO 2 1.23 -6.25 1.310 2 6.610 8 6.6 Â– 8.5 Hydrogen Peroxide, H 2 O 2 74500 -14.5 2.210 12 7.2 Â– 8.6 Ammonia, NH 3 62 -8.17 1.710 5 5.8 Â– 10.5 Nitric Acid, HNO 3 210000 15.4 11.2 Â– 18.5 Formaldehyde, CH 2 O 2.5 -12.8 2.5310 +3 5.4 Â– 12.3 Formic Acid, HCOOH 3600 -11.4 1.810 4 2.5 Â– 9.4
22 This model considers hail growing in the wet-growth regime. Thus, only conditions of cloud liquid water content greater th an the Schumann-Ludlam limit calculated critical water content limit, Wc, for a given set of environmental conditions were considered. The Schumann-Ludlam limit, which c onsiders a heat balance on the riming substrate, is given by [ Stuart and Jacobson, 2004; after Macklin & Payne, 1967, and Young, 1993]: () ()()2sat c aoasia mwoaf WNukTTShDL vrLcTTrr e =-+n -n Eqn (15) Here, f is the shape factor of the substrate, e is the efficiency of collection, v is the impact speed, r is the hail radius, Nu and Sh are the Nusselt and Sherwood numbers, respectively, ka is the thermal conductivity of air, D is the diffusivity of water vapor, cw is the heat capacity of water, and, Lm and Ls are the latent heats of fusion and sublimation of water vapor, respectively. Essentially, the rate pe r unit area at which heat is being dissipated to the environment by convection and eva poration is compared to the rate at which it is being added due to freezing of the drop lets. Hence, for a given ambient temperature, air speed, and particle size, there ex ists a critical liquid water concentration for which all the accreted drops may be just frozen Exceeding this critical liquid water concentration results in excess water remaining unf rozen on the hail, and growth occurs in the wet regime. For the purposes of model implem entation, a wet growth boundary parameter, WB was calculated for all sets of input conditions c onsidered. If WB was positive (w >Wc, where w is the cloud liquid water content as given in Tabl e 3.), the
23 results were considered in our analysis. If WB was negative, they were only retained to understand the implications of the constraint on th e overall results. Table 3. Model input variables and ranges Name and Symbol Units Range Reference and assumptio ns Hailstone diameter (Dh) mm 1 Â– 50 Pruppacher and Klett [1997, p.71] Hailstone liquid water content ( h ) [-] 10 4 Â– 0.5 Maximum observed water fraction, Lesins and List  Ice interface supercooling ( D T) C 10 4 Â– 10 For classical growth regime, Pruppacher and Klett [1997, p. 668] Hailstone shape factor ( f ) [-] 3.14 Â– 4 Macklin and Payne , Jayaranthe  Collection efficiency ( e ) [-] 0.5 Â– 1 Assumed close to 1, Lin et. al.  Cloud liquid water content ( w ) gm 3 2 Â– 5 Pruppacher and Klett [1997, p. 23] Drop radius (a) m m 5 Â– 100 Jacobson, [2005. Tab 13.1, p. 447] Atmospheric pressure (P) mb 200 Â– 1013 Tropospheric pressures, Jacobson [2005, App. B.1]. Air temperature (Ta) C -30 Â– 0 Observed wet-growth regime limits, Pruppacher and Klett [1997, p. 682] Effective HenryÂ’s constant (H ) [-] 10 2.5 Â– 10 18.5 Calculated for pH range, Seinfeld and Pandis [1998, p. 340-385] pH 2 Â– 8 Approximate range observed in experimental retention studies Effective Ice-liquid distribution coefficient(k e ) [-] 10 5 Â– 10 3 Experimental data and theory, Hobbs [1974, p. 600-606] Air temperature range was assigned based on limits to wet growth for maximum hail radius, maximum liquid water content, and mini mum pressure as discussed in Pruppacher and Klett [ 1997, p. 682]. The pH occurring in the troposphe re depends on the types and concentrations of dissolved chemical species present. Ranges used were based on typical midrange tropospheric pH variation as discussed in Seinfeld and Pandis
24 [1998, p. 345]. Cloud liquid water content range re presents values occurring in deep convective clouds with high updrafts, as discussed in Pruppacher and Klett [1997, p. 23]. The range assigned to mass fraction liquid water co ntent of the hail particle considers the higher liquid mass associated with the wet-growth r egime. For higher temperatures and liquid water contents, the ice fraction assumes a c onstant minimum of 0.5 [ Lesins and List 1986]. Pruppacher and Klett [1997, p. 668], and other sources cited therein, discusses the dependence of ice growth rate on bath supercooling with parameterizations covering the range 0.5C to 10C. Here, the range assigned for DT accounts for lower velocities due to higher temperatures associated wi th the growth regime. The distribution coefficient for solute in ice is discussed in text. The collection efficiency is an assumed value, chosen between 0 and 1, but greater than 0.5 based on higher liquid water concentrations associated with the wet growth regim e as discussed by Lesins and List . The range given for the shape factor consid ers that the substrate assumes geometry somewhere between a cylinder and a sphere [ Macklin and Payne 1967] An additional constraint on the model was also nece ssary to ensure consistency between the ice growth rate and the mass available for growth. The intrinsic growth rate of the ice phase depends predominantly on the ice-l iquid interface temperature, DT (Equation 11) which is represented as an input vari able as there is no way to determine it within the scope of this model. Consequently, some combinations of model input parameters may result in all the liquid mass on the hail freezing, thus violating the wetgrowth concept. Here I defined an allowable growth rate by considering the amount of water mass present on the hail after accounting for evaporation (F Â– E) A growth rate boundary parameter, GB was then calculated for all sets of input conditi ons by
25 comparing the hail growth rate (see Equation 10) wi th the calculated allowable growth rate. If GB was positive, [ (F-E) >G ], the results were considered in our analysis. If GB was negative, they were only retained to understand the implications of the constraints on the overall results. It must be noted that the GB and WB constraints discussed above were applied simultaneously.
26 2.5 Testing The validity of the data obtained from derived mode l parameters was assessed by comparison to published data. The temperature depen dence of chemical specific HenryÂ’s constants was compared to data discussed in Seinfeld and Pandis [1998, p. 340 Â– 350] and other sources cited therein, and was found to b e in good agreement. Similarly, Reynolds number averaged drop and hail settling vel ocities were found to compare well with data given by Jacobson [2005, p. 507] and, Pruppacher & Klett . The model was checked for conservation of water and solute mass mass balance analysis. A mass balance test on hail water mass wa s conducted by considering the fundamental model equation describing the water rat e balance around the growing hail particle discussed in Section 2., given as, S = F Â– G Â– E Since the rates of impingement, growth, and evaporation were independently derived, the mass balance consisted of equating the sum of these processes, with the mass rate of drop shedding. Perfect conservation of mass was observed for all variation s of model parameters that met the model constraints. A solute mass balance required tracking a defined m ass of solute through the model logic and ensuring conservation of mass. An i nitial concentration of trace chemical was defined in the air phase, Ca. Its concentration in each medium was subsequently derived from the original model equations describin g solute mass fraction expressions as given in Section 2.1. Thus, the mass fraction of so lute in the drops, hail ice, shed liquid and evaporated solution, were calculated from the f ollowing expressions:
27 da w H XC r = Eqn (16) hd XX =G Eqn (17) ()() 1h l eX X k hh = +Eqn (18) *1l el s vl XX H r r = Eqn (19) Multiplying these solute mass fraction expressions with the appropriate water mass rates gives the solute mass accumulation rate in each com partment. Perfect conservation of solute mass was observed for all variations of mode l parameters that meet the model constraints.
28 3. Application To investigate the dependence of retention on envir onmental, microphysical, and chemical factors, retention ratios were calculated for a range of chemicals of atmospheric interest using an ensemble modeling approach. Six t race chemicals were considered, namely, SO2, H2O2, NH3, HNO3, CH2O, and HCOOH. Appropriate ranges for H* were first calculated for these species as discussed in Section 2.4 using a 100 member ensemble. With the Effective HenryÂ’s Constant range defined, a modified Monte Carlo ensemble modeling approach was then used to determi ne the dependence of retention on input variables. In this approach, a series of ense mble simulations was run for each input variable previously defined. The focus variable of each series was held constant at discrete values uniformly spaced over the range lis ted in Table 3. For each of those values, a 100-member ensemble was assembled by allo wing all other variables to vary randomly over a continuous uniform distribution def ined by the range of each variable as listed in Table 3. Only those results meeting the m odel constraints were retained in the analysis. Resulting output distributions of retenti on ratios and constraint conditions are presented and discussed in Section 4.
29 4. Results The results generated from modified Monte Carlo en semble runs were categorized and presented by type as, hail factors (mass fracti on liquid water content of hail, h, hail diameter, Dh, hail shape factor, f ice-liquid interface temperature, DT and hail collection efficiency, e), chemical factors (effective HenryÂ’s constant, H*, and effective ice-liquid distribution coefficient, ke), and environmental factors (air temperature, Ta, pressure, P cloud liquid water content, w, and drop radius, a ). 4.1 Hail factors 4.1.1. Hail diameter Result of the dependence of simulated retention on hail diameter is shown in Figure 2(a). The mean simulated retention varied be tween 0.068 and 0.15 for distinct hail diameters, with an overall distribution range of 5. 110-6 to 0.88. No clear trend is observed between the mean or other distribution par ameters of the retention ratio and hail diameter. Although no clear dependence of retention on hail diameter can be ascertained, a trend was observed in the number of ensemble memb ers within model constraints (see bold numbers in Figure 2a). The number of valid run s was observed to increase with increases in diameter. From Equation 8, an increase in hail diameter is ex pected to result in an increase the drop collection rate, F by increasing the swept volume of drops collected Hail fall
30 velocity (and hence impact velocity) also increases with hail diameter, also increasing F From Equation 9, it is expected that increasing hai l diameter will result in an increase the evaporation rate, E through increased surface area and ventilation, f Since the particle Reynolds number is proportional to its radius and f all speed, greater ventilation is expected with increases in diameter, which further enhances vapor and energy transfer processes. The hail diameter also affects the criti cal liquid water content, which indirectly affects retention. Finally, hail diameter has effec ts on shedding due to effects on hail motion, but this is not captured in this model. Her e, shedding, S will increase if F increases or E or G (mass growth rate) decrease. Hence, overall a compl icated relationship between hail diameter and retention is expected, due to the counteracting effects of F S and G on retention (see Equation 7). The lack of observe d dependence on hail diameter indicates that no one effect dominate s. Additionally, the large range indicates that other parameters or combinations of parameters are more important to controlling retention. Results of the dependence of the constraint paramet ers on hail diameter are shown in Figure 3(a) and 4(a). It was observed that as ha il diameter increases, the mean of the growth rate boundary parameter, GB increases slightly (i.e. becomes more positive), with fewer member runs outside the boundary (less negati ve values). However, larger variability in GB is observed as the hail diameter increases, indica ting that as hail diameter increases, the influence of other input pa rameters of the growth rate of the hail becomes more pronounced. As previously mentioned, h ail diameter is expected to have a direct correlation with drop collection rate and wa ter evaporation rate. For the wet growth boundary parameter, WB shown by Figure 4(a), a trend of increasing mean WB
31 parameter with increasing hail diameter is observed Equation 15 indicates that as hail diameter increases the calculated critical water co ntent for wet growth will decrease, thus increasing the WB parameter. As hail diameter increases, an increase in the number of valid simulation runs is also observed, as well as, a decrease in the variability of the WB parameter. This indicates that as hail diameter inc reases, the influence of the other input parameters on the growth regime decreases. nr n rrrr !" nn #"$% n %& !" !" Figure 2. The effect of individual hail input varia bles on retention fraction. The box plots character ize the ensembl e distribution of simulated results with the abscissa held constant and other parameters varied randomly The italicized value above each box plot provides the number of en semble member runs that met model constraints.
32 nnnnrn r (#"$% rrrrr)*$+# !", n n)*$+# %& r nrrr)*$+# !" !" Figure 3 The effect of individual hail input variables on th e growth rate boundary parameter, GB. The box plots characterize the ensemble distribution of simulated results with the abscissa held constant and other parameters varied randomly. The italicized value above each box plot provides the number of ensemble member runs that met model constraints.
33 r r r #"$% rr r n r rr -*$r n r r r n-*$r %& r rrrrr-*$r !" !" !" Figure 4. The effect of individual hail input variables on th e wet growth boundary parameter, WB. The box plots characterize the ensemble distribution of simulated results with the abscissa held constant and other parameters varied randomly. The italicized value abo ve each box plot provides the number of ensemble me mber runs that met model constraints.
34 4.1.2. Mass fraction liquid water content of hail The effect of hail liquid water content, h, on retention is shown in Figure 2(b). Mean simulated retention varied from 7.510-3 to 0.27 with increasing h, exhibiting a strong dependence of retention on the mass fraction liquid water content of the hail. The overall range observed ranged from 6.010-8 to 0.99, with no obvious trend in variability, or number of valid runs with changes in h. From Equation 7, a direct dependence of the retenti on ratio on h is observed in the numerator. This is because as h increases, with comparatively negligible partition ing to ice (significantly low ke), more solute can be stored in the liquid ( Xl) (see Equation 6). However, the retention ratio is also indirectly inf luenced by h through its effects on hail growth, G, and shedding, S It is expected that increasing h will result in an increase in the growth rate of the hail as indicated by Equatio ns10 and 12. Shedding is determined by conservation of water mass, so as growth rate incre ases, shedding decreases (with E constant) with the resultant opposite effect on the retention ratio. Since an increasing trend of retention with increases in h is observed, it can be surmised that the direct (numerator) effect and/or shedding dominate the gro wth effect. The effect of h on the constraint parameters is shown in Figure 3( b) and 4(b). No definite trend is observed between the mean and val ues of GB or WB and hail water fraction. As h increases, an increase in the variability of the GB parameter was observed, indicating increasing influence of other input para meters on the growth rate. Additionally, for WB extremely high negative values were observed for some
35 combination of the input parameters, suggesting tha t there were combinations of input parameters that strongly affect the growth regime. 4.1.3. Ice-liquid interface temperature Figure 2(d) gives the results of simulated retenti on on the ice-liquid interface temperature, DT Mean simulated retention increased fom 9.010-7 to 0.30 with increasing DT indicating a dependence of retention on DT The overall range varied between 1.110-4 and 0.99 with increasing variability as DT increased. The number of valid runs converely decreased with increasing DT (closer to zero) From Equation 11, it is expected that as DT increases, the intrinsic growth rate of the hail ice will increase, thus increasing the ove rall hail growth rate, G From Equation 7, is is expected that the quantity termed the growth effect will increase with increasing G leading to a subsequent decrease in retention. Ho wever, this is countered by the indirect effect of shedding. Here, an increase in G leads to a decrease in the shedding term, which has a greater influence, and results in an general increase in the observed retention. The effect of DT on the growth rate boundary prameters is shown in Figures 3(d) and 4(d). There is no significant dependence observ ed between the mean or other distribution parameter of the GB parameter and DT However, a definite decrease in the number of valid model runs is observed as DT increases via its direct effect on the intrinsic ice growth rate, subsequently affecting the model constraint directly. For WB
36 no definite trend exists in the mean values, but th ere is an observed decrease in the variability of WB as DT increased. 4.1.4. Hail shape factor The dependence of simulated retention ratio on the shape factor, f is shown in Figure 2(c). The mean retention varies from 0.91 to 0.19, with an overall distribution range of 1.310-6 to 0.95. There is no definite relationship observe d between the mean or other distribution parameter of the modeled retenti on ratio and shape factor. Since the shape factor only appears in the wet grow th boundary constraint, Equation 15, it can only influence retention throug h that constraint. From Equation 15, it is observed that the shape factor influences the h eat balance on the riming substrate by determinig the enhancement of energy transfer due t o the curvature of the interface [ Macklin 1964] Figure 3(c) gives the effect of the hail shape fact or on the growth rate boundary constraint. There is no relationship observed betwe en the distribution parameters of the growth rate boundary and the the shape factor. A d irect relationship between the shape factor and the growth boundary is not expected. Ho wever, large variations in the distribution of the GB parameter is observed, due to combination of effe cts of the other input parameters. The effects of the shape factor on the wet growth b oundary parameter is given by Figure 4(c). The was no trend observed between the shape factor and the distribution parameters describing the wet growth boundary. From Equation 15 it is expected that as the particle transitions from a cylinder to a spher e, the critical liquid water required for it
37 to remain in the wet growth regime will increase, t hus influencing the GB However, it is also affected by the ambient temperature, particle size, and the impact speed. Thus, no direct correlation is observed. High variability in the GB simulations is observed with higher negative values. There was no variation in t he number of valid model runs, however. 4.1.5. Efficiency of collection Figure 2(e) characterizes the effect of the collec tion efficiency, e of the hail on the simulated retention ratio. Mean retention range d from 0.10 to 0.17, and the overall distribution indicated possible values ranging from 2.810-6 to 0.97. No clear trend is observed in the mean or variability of the simulate d retention ratio. From Equation 8 it is expected that, as e increases, the mass rate of drop collection increases. As given by Equation 7, the e ffect of an increase in drop collection depends on the relative rates of shedding, evaporat ion and ice growth. Since no trend is observed, no one effect appears to dominate. The influence of the collection efficiency on the g rowth rate boundary is shown by Figure 3(e). There is no trend observed between the distribution parameters of the simulated growth rate boundary parameter and the c ollection efficiency. Also no trend is observed in the variability of the GB parameter with changes in collection effeciency, although high overall vairability in the data. This indicates that there is no direct effect of the collection efficiency on the constraint. Figure 4(e) gives the relationship between the coll ection efficiency and the wet growth boundary parameter. There is no definite tre nd observed between the collection
38 efficiency and the distribution parameters characte rizing the growth regime boundary. Additionally, there is no trend observed in the va riability of the GB parameter distribution. However, greater negative values is o bserved in the distribution. The collection effeciency is not expected to significan tly impact the growth regime of the particle. 4.2 Environmental factors 4.2.1. Cloud liquid water content The dependence of the simulated retention ratio on the cloud liquid water content, w, is shown in Figure 5(a). The mean of the simulate d retention ratio varied between 0.11 and 0.21. The overall distribution ranges from 1.1 10-6 to 0.96. There is no relationship observed between w and the distribution parameters describing the mod eled retention ratio. The direct effect of an increase in the cloud liqui d water content is an increase in the rate of drop collection on the hail, as given b y Equation 8. Following Equation 7, this should result in a decrease in the growth and evapo ration effect, and a corresponding increase in retention. However, due to the countera cting effects of drop shedding, a definite trend is not observed. The dependence of the growth rate boundary on cloud liquid water content is given by Figure 6(a). There is a perceptible trend observed between the mean of the GB parameter and w, with the GB increasing with increases in w. However, there is no trend observed in the variability of GB or the number of valid model runs.
39 Figure 7(a) shows the relationship between the clou d liquid water content and the wet growth boundary parameter. The mean of the WB parameter is observed to increase slightly with increasing cloud liquid water. It is expected that parameter to increase with increasing cloud liquid water, since it is defined by comparing the calculated critical liquid water content of the hail to the actual clou d liquid water content. However, I do acknowledge that other parameters, such as the part icle size, ambient temperature, and impact speed play important role in the calculated critical water content. The distribution of the GB parameter showed no obvious trend in variation, bu t was negatively skewed. It is observed that as the cloud liquid water increase s, the number of valid model runs increases. This is expected, since as mention previ ously, the boundary is based on the comparison of the calculated critical liquid water content with the observed cloud liquid water. 4.2.2. Drop radius The dependence of the simulated retention ratio on drop radius is shown by Figure 5(b). The mean of the simulated retention ra tio is observed to vary from 0.12 to 0.15. The overall range of variability of retention ratio was between 4.910-5 and 1.1. There was no significant trend observed in mean or variability of the simulated retention ratio with changes in drop radius. It is expected that as drop radius increases, it ma y lead to a decrease in the impact speed, due to the decrease in relative velocities o f drop and hail particle, and a subsequent decrease in the drop collection term in Equation 8. The effect of a decrease in drop
40 collection depends on the relative rates of the oth er model parameters, E S, and G as discussed previously. The effect of drop radius on the growth rate bounda ry parameter is characterized by Figure 6(b). No trend is observed between the me an or other distribution parameter of the GB parameter and the drop radius. Based on the defini tion of the growth rate boundary, there is no indication of a direct relati onship between the drop radius and the growth boundary parameter. Similarly, though there was some amount of variability in the GB parameter, no trend was observed in the variabilit y. Additionally, no trend was observed between drop radius and the number of mode l runs meeting the model constraint. Figure 7(b) gives the relationship between the wet growth boundary parameter and the drop radius. There is no trend observed bet ween the mean or other distribution parameter of the WB parameter and the drop radius. From Equation 13, the drop radius is expected to impact the critical liquid water conten t required for wet growth by affecting the rate of heat dissipation of the freezing drop. This may result in greater water mass on the hail and consequently a decrease in the critica l water content required. However, no trend was observed indicating a relationship betwee n drop radius and growth regime observed. Additionally, no trend was observed in th e number of valid model runs.
41 r r r)*$+# %.$+r nr r r r "# r r rr)*$+# /##0 r r r1 2"% !" !" rr1 3" % n nr 4"# n rr /##0 r r %.$+r !" !" Figure 6. The effect of individual environmental in put variables on the growth rate boundary parameter GB. The box plots characterize the ensemble distribution of si mulated results with the abscissa held constant and other parameters varied randomly. The italicized value above each bo x plot provides the number of ensemble member runs that met model constraints. Figure 5. The effect of individual environmental in put v ariables on the retention fraction. The box plots c haracterize the ensemble distribution of simulated results with the abscissa held constant and other parameters va ried randomly. The italicized value above each box plot provides t he number of ensemble member runs that met model co nstraints.
42 4.2.3. Pressure Figure 5(c) characterizes the effect of pressure on the modeled retention ratio. The mean simulated retention ratio varied from 0.11 to 0.15, whilst the overall distribution had a minimum value of 1.310-8 and a maximum value 0.88. There was no clear relationship observed between the mean retention an d pressure, though the variability appears to decrease as pressure increases. The numb er of valid model runs increased gradually, as the pressure increased. As pressure decreases, an increase in the diffusivi ty of water vapor is expected, leading to a subsequent increase in solution evapor ation as given by Equation 9, and a rrr r r1 2"% rr r rr-*$r /##0 r r r $r %.5$+r6 nrn r "# !" !" Figure 7. The effect of individual environmental input variab les on the wet growth boundary parameter, WB. The b ox plots characterize the ensemble distribution of sim ulated results with the abscissa held constant and o ther parameters varied randomly. The italicized value above each bo x plot provides the number of ensemble member runs that met model constraints.
43 decrease in retention. However, it, may also direct ly lead to a decrease in the shedding term and subsequently an increase in the modeled re tention ratio. No one effect appears to dominate. The effect of pressure on the growth rate parameter is shown by Figure6(c). There is no trend observed between the distribution param eters describing the growth rate parameter and pressure. Greater variability, as wel l as, an increase in the number of valid model runs was observed. Figure 7(c) characterizes the effect of pressure on the wet growth parameter. There is no clear trend observed between the mean o r other distribution parameter defining the wet growth boundary and pressure. The variability of the WB parameter appears negatively skewed, with extremely large neg ative values, but there appears to be no clear trend in the variability. However, the num ber of valid model runs showed an increase as pressure decreased. 4.2.4. Air temperature The dependence of the simulated retention ratio on the temperature of air is shown in Figure 5(d). Whilst the mean retention rat io varied between 0.12 and 0.16, the overall range observed varied from 5.610-6 to 0.93. There was no apparent individual effect on the mean retention fraction due to variat ion in air temperature. An increase in the number of valid model runs with increasing temp erature is observed, Some anticipated direct effects of temperature on t he retention ratio include its effect on solution evaporation, by affecting both t he diffusivity and the solution saturated vapor density. From Equation 9, it is expected that that these two terms would generate
44 opposite effects on solute evaporation. Hence, the relationship between retention and temperature is expected to be complex. Figure 6(d) shows the relationship between temperat ure and the growth rate boundary. There is no trend observed between the gr owth rate boundary distribution parameters and temperature. Based on the definition of the GB parameter, it is expected that, as temperature increases, it may increase the mass rate of solution evaporation, thus leading to a decrease in the growth boundary. Addit ionally, no trends in the variability of the distribution of the GB parameter with increases in temperature are observ ed. There is also no observed effect of temperature on the numbe r of valid model runs. Figure 7(d) characterizes the relationship between the wet growth boundary parameter and temperature. There is a relationship observed between the growth boundary parameter and temperature, with the WB parameter increasing as temperature increases. From Equation 13, a complex relationship between temperature and the wet growth boundary parameter is expected. A definite i ncrease in the number of valid model simulations is observed as temperature increased, a s well as a decrease in the variability of the WB parameter distribution indicating a relat ionship between the model constraint and temperature.
45 4.3 Chemical factors 4.3.1. Chemical effective HenryÂ’s constant and pH Figure 8(a) characterizes the dependence of the sim ulated retention ratio on the effective HenryÂ’s constant, H*. The mean retention varied between 1.610-3 and 0.17, whilst the distribution parameters show retention v arying between 1.410-4 and 0.79. A trend is observed between H* and the mean of the simulated retention ratio. An increase in retention is observed as the HenryÂ’s constant in creases. There is no observed trend in the variation of valid model simulations with chang es in the effective HenryÂ’s constant. Additionally, no trend in the variation of the rete ntion ratio distribution is observed, as H* is increased. From Equation 7, it is expected that as the effecti ve HenryÂ’s constant increases, the evaporation term decreases, that is, the evapor ate-to-liquid solution chemical mass rate ratio decreases, thus resulting in an increase in retention. However, it is also recognized that drop shedding, which also depends o n the solute evaporation term, plays a large role in determining the retention fraction, where, as the evaporation term decreases, it leads to an increase in the shedding term, and a subsequent reduction in retention would result. As an increasing trend is o bserved, it is expected that the direct effect of evaporation dominate that of shedding. It must be noted that the effective HenryÂ’s constant is significantly dependent on pH. The range used for effective HenryÂ’s constant model simulations were derived based on th e dependence of the effective HenryÂ’s constant on pH as shown by Figure 8(c). As the figure shows, the effective HenryÂ’s constant for acidic species (HNO3, SO2) increases by orders of magnitude as pH
46 increases from 4 to 7, whilst, the opposite effect is seen for basic species (NH3). Hence, since retention is affected by H* it is also significantly affected by pH. Figure 9(a) shows the effect of the effective Henry Â’s constant on the growth boundary parameter. There is no trend observed by t he distribution parameters characterizing the model constraints and H*. Additionally, no trend was observed in the variation of the growth boundary. There was no effe ct of H* on the number of valid model runs observed. The effect of the effective HenryÂ’s constant on the wet growth boundary parameter is shown by Figure 10(a). There is no tre nd observed in the mean of the wet growth boundary or any of the other distribution pa rameters describing the GB parameter with variations of H*. Large negative values were observed in the minimu m distribution parameter but there was no definite trend in the va riation observed.
47 r7$8" 9 9 :r :9r %9 %99 1;&<##57$86 1 1 1 1 1 r 1;#0 !" !" n r r )*$+#1;&<##57$ 86 rn r r 11111r)*$+#1;#0 !" !" Figure 9. The effect of individual chemical input v ariables on the growth rate boundary parameter, GB. The bo x plots characterize the ensemble distribution of simulated results with the abscissa held constant and other parameters varied randomly. The italicized value above each box plot provides the number of ensemble member runs that me t model constraints. Figure 8. The effect of individual chemical input v ariables on the retention fraction The box plots characterize the ensemble distribution of simulated results with the abscissa held constant and other parameters varied randomly. The italicized value above each box plot provides the n umber of ensemble member runs that met model constr aints.
48 4.3.2. The effective ice-liquid distribution coefficient The effect of the effective ice-liquid distribution on the simulated retention ratio is shown in Figure 9(b). The mean simulated retention varied between 0.087 and 0.14, whilst the minimum and maximum distribution paramet ers varied between 5.210-6 and 0.95. There is no trend observed between the mean o r other distribution parameters characterizing the simulated retention ratio and th e effective ice-liquid distribution coefficient. nnr -*$r 1;&<##57$ 86 r 11111r-*$r 1;#0 !" !" Figure 10. The effect of individual chemical input variables on the wet growth boundary parameter, WB The box plots characterize the ensemble distribution of sim ulated re sults with the abscissa held constant and other par ameters varied randomly. The italicized value above each bo x plot provides the number of ensemble member runs that met model constraints.
49 From Equation 7 it is expected that as ke increases, both the numerator and denominator will increase. However, due to the magn itude of ke, (~10-4), it has little influence. The effect of ke on the model constraints is characterized by Figure 10(b).There was no trend observed by the distribution parameter s characterizing the model constraints and ke. There was no trend observed between the number of valid model parameters observed and variation in effective ice-liquid dist ribution coefficient. The effect of the ice-liquid distribution coefficie nt on the wet growth boundary parameter is given by Figure 11(b). There was no tr end observed in the mean of the WB parameter and ke. Similarly, no trend was observed in the variability in the distribution parameters, however, greater negative (minimum) val ues were observed. 4.4 Summary of results Table 3 provides a summary of the parameters invest igated and their observed effect on the retention ratio as given by the simul ations conducted. The iceliquid interface temperature, D T, hail liquid water fraction, and the chemicalÂ’s e ffective HenryÂ’ constant, were found to individually affect retenti on, with retention increasing as each of the parameters were increased. The cloud liquid wat er content, w and collection efficiency, e showed possible inverse relationship with the ret ention ratio. All the other parameters do not alone appear to have a clear rela tionship with the retention ratio. Maximum values of retention observed were clearly n ot resulting from any particular model parameter, but rather from a combination of m odel parameters. Finally, retention ratios greater than 1.0 were observed for certain c ombinations of conditions.
50 Table 4. Dependence of simulated retention fraction on input variables Parameter Effect Description Range of Ensemble Means Overall Range Interface supercooling, D T Large, direct, monotonic 9.010-8 Â– 0.30 1.110-8 Â– 0.99 Mass fraction hail liquid water content, h Large, direct 7.510-3 Â– 0.27 6.010-8 Â– 0.99 ChemicalÂ’s effective HenryÂ’s constant, H* Large, direct, levels off 1.610-3 Â– 0.17 1.410-4 Â– 0.79 Cloud liquid water content, w Very small, non-monotonic 0.11 Â– 0.21 1.110-6 Â– 0.96 Hail shape factor, f Very small, non-monotonic 0.091 Â– 0.19 1.310-6 Â– 0.95 Hail diameter, Dh Very small, non-monotonic 0.068 Â– 0.15 5.110-6 Â– 0.88 Collection efficiency, e Very small, non-monotonic 0.10 Â– 0.17 2.810-6 Â– 0.97 Effective ice-liquid distribution coefficient k e None 0.087 Â– 0.14 6.210-6 Â– 0.95 Air temperature, T a None 0.12 Â– 0.16 5.610 6 Â– 0.93 Atmospheric pressure, P None 0.11 Â– 0.15 1.310 8 Â– 0.88 Drop radius, a none 0.12 Â– 0.15 4.910 5 Â– 0.92
51 5. Discussion and Limitations The model presented explores the partitioning of vo latile chemical species during rime freezing under wet growth conditions. In an at tempt to understand and predict the retention of atmospherically relevant gases, I inve stigated the environmental factors, and hail and chemical properties influencing this proce ss. The ice-liquid interface supercooling was found to be the most important forcing variable for solute retention during wet growth of hail. Experimental studies have found a direct relationship between retention and supercool ing under mixed wet and dry growth conditions [ Lamb and Blumenstein 1987; Iribarne et al 1990; Snider et al. 1992]. In this study, retention was found to increase as supe rcooling increased (lower interface temperatures) The ice-liquid interface temperature determines the intrinsic growth rate of the hail ice, but more importantly, it is influence d by the generated heat of freezing released by impinging drops, and its dissipation. T his model assumes that the hailstone is growing in a cloud containing super-cooled water dr oplets with temperature equal to that of air. Since both the freezing of deposited water droplets and the condensation of water molecules are always accompanied by the release of latent heat, it is evident that through the period of growth, the temperature of the hail w ill be greater than that of the surrounding atmosphere [ List, 1963(a); 1963(b)]. At steady-state growth, the temp erature of the interface is such that the rate of heat libe ration due to the deposition and freezing of water equals the rate of heat dissipation from t he ice-liquid interface. Therefore,
52 processes that facilitate the removal or transfer o f the heat of freezing released, contribute to the increased growth of the hail. These include the movement of the hail particle through the air, which increases ventilation proces ses and the increase in the available surface area for heat transfer, as well as, lower a mbient temperatures. For the growth rate of ice in supercooled water, we used the parameteri zations of Bolling and Tiller  for the intrinsic crystal growth rate. Since the gr owth rate of ice is heat dissipation limited, an energy balance on the growing hail woul d be a better representation of the hail growth process. Results from our model simulation find that retenti on of volatile solutes in hail was also significantly impacted by the mass fractio n hail liquid water content of the growing hail particle. For conditions of high hail liquid mass fraction, high retention ratios were generally predicted by the model. There fore, conditions contributing to higher hail liquid mass fraction may result in higher degr ees of retention being observed. With higher liquid water content, more solute can be ret ained in the liquid water portion of the hail. Additionally, an overall hail particle can of ten contain layers formed during alternating wet and dry growth [ Pruppacher and Klett 1997, p. 73]. Since much of the solute is retained in the liquid during wet growth, it is expected that rate of formation of a surface ice layer during the transition to dry grow th will be important to final retention. For dry growth conditions, a surface layer of ice w as found to be important to trapping solute at higher concentration than would be expect ed from ice-air equilibrium solute partitioning [ Stuart and Jacobson 2003, 2004, 2006]. However, in this model, the ha il liquid mass fraction is explicitly set. In reality, it should be dependent on conditions such as temperature, impingement rates, and shedding rat es. Shedding has been determined by
53 a mass balance on the water mass on the hail. Howev er, this parameter has proven to be very influential in the determination of the effect of many other model parameters on the fraction retained. Thus, to elucidate the direct an d indirect effects of parameters such as the hail mass fraction, and hail diameter on retent ion, an explicit representation of this process is required in future work, where drop shed ding is a function of drop impingement, hail motion, and properties of the wat er phase. Chemical HenryÂ’s constant was found to be the third important determinant of retention fractions, with higher fractions observed for higher effective HenryÂ’s constants (more soluble, less volatile chemicals). This is co nsistent with previous findings for dry growth conditions and experiment studies discussed therein [ Stuart and Jacobson 2003, 2004]. However, under wet growth conditions, the ch emical HenryÂ’s constant (and pH) are likely less important than for dry growth condi tions, with only a low mean value of retention (0.2) simulated for the highest considere d effective HenryÂ’s constant. Hence, under wet growth conditions, the chemical identity is not expected to be as important to determining partitioning as for dry growth. The ran ge generated for use in the modified Monte Carlo simulations was based on observed varia tion of H* with changes in pH (over the range 4 Â– 7). There was no direct relationship observed between t he effective ice-liquid distribution coefficient and the retention ratio. D istribution coefficients for chemical species in ice occur over the range 10-3 Â– 10-5 [ Hobbs 1974]. Subsequently, model simulations indicate the effects of the ice-liquid distribution coefficient may not be influential to solute retention. This may be consid ered advantageous since the effective ice-liquid distribution coefficient is poorly chara cterized.
54 Overall, the issue of co-dependence and indirect ef fects on retention through independently varied hail parameters decreases our confidence in findings for the environmental variables (cloud liquid water content air temperature, atmospheric pressure, and drop radius). The environmental varia bles may affect the hail liquid water content and/or the interface supercooling temperatu re, which were found to significantly force retention. Hence, to better understand the im pacts of environmental variables, it will be important to calculate these two hail varia bles within the model, rather than set them independently as input variables. This will re quire calculation of the shedding rate independent of the water mass balance and a heat ba lance calculations. For the other hailrelated parameters (collection efficiency, hail sha pe factor, and hail diameter) and for the effective ice-liquid chemical distribution coeffici ent, no clear effect was observed on retention. Hence, it is less important to represent their dependence on environmental conditions or consider their effects on retention. As collection efficiency, hail shape factor, and the ice-liquid distribution coefficient are poorly understood themselves and would be difficult to calculate from physical (nonempirical) principles, this result is helpful to future microand cloud-scale modeling. Also, the very small impact of hail diameter on retention is important to the applicabi lity of this model. Since hail diameter has no effect on retention, the assumption of a con stant value is appropriate. Other model variables have no discernable effect alone on the r etention fraction. These include hail diameter and cloud liquid water content. However, s ince many of these parameters will determine the hail liquid mass fraction, and the ef fective ice-liquid distribution parameters, in reality the full influence of these parameters on the fraction retained is not captured by the model.
55 Finally, despite high predicted maximum values for all ensemble simulations (of 0.9 1), the means for all ensembles were much low er, with the highest mean predicted at 0.3. Hence, no single variable was found to be resp onsible for simulated maximum values of retention. Rather, combinations of favorable inp ut conditions were needed to generate retention fractions greater than 0.3. Further inves tigation of variable combinations that lead to the high observed values is needed.
56 6. Conclusions and Implications This investigation developed and explored the proce ss of wet growth riming, and its effect on the retention of trace atmospheric ga ses in growing hailstones. From cloud microphysics literature, expressions representing t he important process occurring at the particle scale, such as solution evaporation and dr op collection, were developed. Model parameters and calculated variables were checked an d found to be consistent with previously established experimental and theoretical values. The model was checked for conservation of water and solute mass, and was cons istent for all conditions satisfying model constraints. Results generated from model simulations indicate t hat the most important forcing variable for solute retention during wet growth is the ice-liquid interface supercooling. The modeled retention faction for wet growing hail was also found to be largely dependent on the mass fraction of liquid water pres ent on the hail particle. The chemicalÂ’s HenryÂ’s constant was found to have signi ficant impacts on retention, largely influencing the mass of chemical in the evaporated solution. Results also indicate that the effective ice-liquid distribution coefficient does not significantly affect retention. Shedding was observed to be an important process af fecting the retention ratio and needs to be explicitly represented in future work. Direct effects of hail and drop sizes on the retention ratio were not observed.
57 This body of work provides a valuable insight into the hail properties, chemical properties, and environmental conditions important to predicting the fate of volatile trace gases, of atmospheric interest, due to their intera ction with the growing ice phase. It is hoped that the insights gained can be used to devel op better parameterizations of retention for cloud modeling studies.
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72 Appendix A. Retention Model Calculations Table A. Retention model calculations 5 Input Conditions Symbol Units Input Input Formula Range Reference Air temperature Tair C -2.00E+00 -2.00E+00 input pa rameter (15, -49) C 0 to -30C Tair K 2.71E+02 2.71E+02 Pressure P mb 8.00E+02 8.00E+02 input parameter (265, 1013) mb 1013 mb to 200 mb P atm 7.90E-01 7.90E-01 pH 4.00E+00 4.00E+00 Try 4 to 7 H+ concentration 1.00E-04 1.00E-04 Input trace chemical characteristics HNO3 HNO3 NH3 Table 6.2 Trace chemical molecular weight MW g/mol 6.30E+01 1 .70E+01 Table 6.A.1 Henry's law coefficient (@298 K) H M/atm 2.10E+05 6 .20E+01 Table 6.A.1 S&P, pg 341 1st equilibrium constant at 298K K 1 M 1.54E+01 1.70E-05 Table 6.3 S&P, Pg 391 2nd equilibrium constant at 298K K 2 M 0.00E+00 0.00E+00 Table 6.4 S&P, Pg 391 Enthalpy of dissolution for Henry's Law Coefficient H kcal/mol -8.17E+00 Table 6.4 S&P, pg 342 Enthaply of 1st Equilibrium DH_1eq kcal/mol -1.73E+ 01 8.65E+00 =Ha(T1)exp[Ha/R(1/T11/T2)] S&P, pg 345 Enthalpy of 2nd Equilibriium DH_2eq kcal/mol 0.00E+ 00 0.00E+00 =K298exp[-H/R(T-1-2981)] S&P, pg 345 Henry's law coefficient @ freezing temp H A M/atm 2.10E+05 2.18E+02 Eq 6.5 1st equilibrium constant at eq. fr eezing temp. K 1 M 2.19E+02 4.50E-06 input parameter temperature depenendence of K from Appendix 6 2nd equilibrium constant at eq. freezing temp. K 2 M 0.00E+00 0.00E+00 Water equilibrium constant @ 298K K w M 1.00E-14 1.00E-14 (1+Ka1/[H+] + Ka1Ka2/[H+]2) Enthalphy of water equilibrium D Hw kcal/mol 1.34E+01 1.34E+01 HA(1+Ka1/[H+] + Ka1Ka2/[H+]2) Water equilibrium constant @To K w M 1.29E-15 1.29E-15 K H *RT o Dissociation factor 2.19E+06 3.50E+05
73 Appendix A. Continued Table A. Continued Input Conditions Symbol Units Input Input Formula Range Reference Overall Henry's constant K H M/atm 4.60E+11 7.61E+07 as follows form S&P, Eq 6.24, p346 Dimensionless Henry's Constant H* Cwtr/Cair 1.03E+ 13 1.71E+09 input parameter 104 1016 Effective ice-liq distribution coefficient k eff [-] 1.00E-03 1.00E-03 input parameter 10-3 10-5 Hobbs, 1974, pp 604 Input Drop characteristics input parameter Cloud liquid water content g/m3 3.00E+00 3.00E+00 (0.3, 5) g/cm3 Borovikov et al., 1963, Pruppacher & Klett, g/cm3 3.00E-06 3.00E-06 Drop radius (mean volume) a m 5.00E+00 5.00E+00 (5, 100) mm Jacobson, 2005. Tab 13.1, p. 447 a mm 5.00E-03 5.00E-03 a cm 5.00E-04 5.00E-04 Mean volume diameter Dd mm 1.00E+01 1.00E+01 input parameter Dd cm 1.00E-03 1.00E-03 input parameter Input Hail Characteristics input parameter (Mass fraction) liquid water content of hail 5.00E-01 5.00E-01 (1e-4,0.5) Lesins and List, 1986 Ice substrate temperature Tice C -2.00E+00 -2.00E+0 0 Varies between 0 and -20 Hail diameter Dh mm 2.00E+00 2.00E+00 1 to 50 mm Prupaccher & Klett, 1997 mm 2.00E+03 2.00E+03 input parameter: for cylinder=pi; for sphere = 4 cm 2.00E-01 2.00E-01 input paramter Hail radius r cm 1.00E-01 1.00E-01 input parameter Shape factor in equation F 4.00E+00 4.00E+00 (3.14,4) Macklin and Payne, 1967; Jayaranthe, 1993 Collection efficiency E 1.00E+00 1.00E+00 1.00 Lin et al., 1983
74 Appendix A. Continued Table A. Continued Input Conditions Symbol Units Input Input Formula Range Reference Super-cooled delta temp of iceliq interface T int C 2.00E+00 2.00E+00 vt drop, calculated below (0, Tair) Hobbs, 1974, pp 604 605, Pruppacher and Kett, 1998, vt hail, calculated below Ventilation Characteristics =vt hail vt drop Air velocity over drop u_d cm/s 3.25E-01 3.25E-01 Jacobson Eq 20.9 pg. 664 Air velocity over hiail u_a-s cm/s 6.91E+02 6.91E+0 2 Jacobson Eq 20.9 pg. 664 Impact speed u_i cm/s 6.91E+02 6.91E+02 constant constant Constants constant Universal gas constant R Latm/mol K 8.21E-02 8.21E02 constant (hPa=mb) Jac. Appendix Table A10, pg 712 Universal gas constant R cal/mol/K 1.99E+00 1.99E+0 0 constant Jac. Appendix Table A10, pg 712 Universal gas constant R g/cm2/s2/mol/ K 8.31E+07 8.31E+07 constant Jac. Appendix Table A10, pg 712 Universal gas constant R cm3mb/mol/K 8.31E+04 8.31E+04 constant Jac. Appendix Table A10, pg 712 Gas constant for water vapor Rv cm3mb/g/K 4.61E+03 4.61E+03 constant Jac. Appendix Table A10, pg 712 Gas constant for dry air Rdry cm3mb/g/K 2.87E+03 2. 87E+03 constant Jac. Appendix Table A10, pg 712 Gas constant for dry air Rdry J/g/K 2.87E-01 2.87E01 constant Jac. Appendix Table A10, pg 712 Avogadros number A molec./mol 6.02E+23 6.02E+23 con stant Jac. Appendix Table A10, pg 711 Boltzmann constant k g cm2/s2/K 1.38E-16 1.38E-16 Jac. Appendix Table A10, pg 711 Gravitational acceleration g m/s2 9.83E+00 9.83E+00 -Jac. Appendix Table A10, pg 711 g cm/s2 9.83E+02 9.83E+02 Jac. Appendix Table A10, pg 711 Properties of H2O phase change Eq 2.55, f(Tinf) Equilibrium freezing temperature To C 0.00E+00 0.00 E+00 Eq 2.56, f(Tinf) Equilibrium freezing temperature To K 2.73E+02 2.73 E+02 L s -L m Latent heat of fusion at To Lm cal/g 7.97E+01 7.97E +01 Eq 2.62, f(Tair) Jac, p. 40 (2nd ed) Latent heat of ice sublimation at To Ls cal/g 6.77E +02 6.77E+02 Jac, p. 40 (2nd ed) Latent heat of water evaporation at To Lv cal/g 5.9 8E+02 5.98E+02 n/V=P/RvTa
75 Appendix A. Continued Table A. Continued Input Conditions Symbol Units Input Input Formula Range Reference Saturation vapor pressure over liquid at air temperature Psat_a mbar 5.28E+00 5.28E+00 n/V = P/RvTw Jac, p 41 Saturation vapor pressure over liquid at To Psat_w mbar 6.11E+00 6.11E+00 " Saturation vapor density over liquid at Tair r air g/cm3 4.22E-06 4.22E-06 Eq 2.64 /R v (T inf+a ) assume atmosphere is saturated w/ respect to water Saturation vapor density over liquid at To r sat,w g/cm3 4.85E-06 4.85E-06 Eq 14.19, f(Tair) Ice sat vapor density at Tair r sat,i g/cm3 4.14E-06 4.14E-06 Ice sat vapor density at To r sat,i g/cm3 4.85E-06 4.85E-06 Jac, p 43 Water-air surface tension s w/a dyn/cm 7.59E+01 7.59E+01 Eq 4.54, f(Tair) Jac, p 485 Table 4.1, f(Tair) Properties of dry air Eq 2.5, f(Tair) Dynamic viscoscity of dry air h drya g/cm/s 1.71E-04 1.71E-04 Jac, p 102 Heat capacity of dry air at constant pressure Cp,da cal/g/C 2.39E-01 2.39E-01 Smith and Vanness, p 109 Thermal conductivity of dry air ka cal/cm/s/C 5.65E -05 5.65E-05 calculated above Jac, p. 20 p v/ (p a -p v ) Properties of moist air R dry (1+0.0608q v ) Density of water vapor in air rair g/cm3 air 4.22E06 4.22E-06 =R/R m Mass mixing ratio of water vapor in air w v g/g 4.13E-03 4.13E-03 =P a /R m T Eq 2.31 Gas constant for moist air Rm cm3mb/g/K 2.88E+03 2. 88E+03 Eq 2.37 Molecular weight of moist air Ma g/mol 2.89E+01 2.8 9E+01 ha/p a Eq 2.26 Density of moist air ra g/cm3 1.03E-03 1.03E-03 Eq 2.36 Dynamic viscoscity of moist air ha g/cm/s 1.71E-04 1.71E-04 calculated above Kinematic viscoscity of moist air na cm2/s 1.67E-01 1.67E-01 r v /R v T a definition Mean free path of moist air la cm 7.47E-06 7.47E-06 Eq 8.6
76 Appendix A. Continued Table A. Continued Input Conditions Symbol Units Input Input Formula Range Reference Heat capacity of moist air at constant pressure Cp,ma cal/g/C 2.40E-01 2.40E-01 Partial pressure of water vapour in air Pair mbar 5 .28E+00 5.28E+00 = 0.211(T/To)1.94(Po/P) Eq 2.27 Properties of water vapor Diffusivity of water vapor in air Dv cm2/s 2.63E-01 2.63E-01 calculated above Eq. 13-3 Properties of liquid water in supercooled drop = (n=0, n=6 a n Tn Supercooled drop temperature Tw_s C -2.00E+00 -2.00 E+00 =(n=0, n=4 anTn) Supercooled drop temperature Tw K 2.71E+02 2.71E+02 Density of supercooledwater rw g/cm3 1.00E+00 1.00E +00 Eq 3-14 Heat capacity of water cw cal/g/C 1.01E+00 1.01E+00 calculated above Eq 3-16 calculated above Properties of hail liquid water (during freezing (a t 0C)) = (n=0, n=6 a n Tn Water temp. Tw C 0.00E+00 0.00E+00 calculated above Water temp. Tw K 2.73E+02 2.73E+02 Density of water rw g/cm3 1.00E+00 1.00E+00 P&K p. 87 Saturation vapor density above liquid rsat,w g/cm3 4.85E-06 4.85E-06 calculated above Properties of ice substrate temperature of ice substrate Td C -2.00E+00 -2.00E+ 00 temperature of ice substrate K 2.71E+02 2.71E+02 input paramter kv (DT)c Properties of Bulk Hail Density of Hail (Calculated) n h g/cm3 9.57E-01 9.57E-01 0.1 (Heymsfield) to 1 g/cm3 Intrinsic ice growth velocity cm/s 1.20E+00 1.20E+0 0
77 Appendix A. Continued Table A. Continued Input Conditions Symbol Units Input Input Formula Range Reference Caculation of hail density problematic, need to explore Y = -av imp /Ts u m/sC 1.73E+01 1.73E+01 Bulk density of rimed ice g/cm3 1.05E+00 1.05E+00 Heymsfield parameterization P&K, p 661; Bulk density of rimed ice (for compare only) g/cm3 1.18E+00 1.18E+00 0.11(-B)^0.76 P&K, p 661; Macklin and Payne, 1962( p 41) Density of pure ice g/cm3 9.17E-01 9.17E-01 Eq 3-2, f(Tice) P&K, p79 Denisty of hail (calculated) g/cm3 9.57E-01 9.57E-01 ((h/rwtr + (1 -h)/rice)^-1 weighted reciprical average Terminal Fall Velocitiy Calculations For Hail Partice Schmidt Number Sc 6.32E-01 6.32E-01 a/Dp eq 16.25 Knudsen Number Kn 7.47E-05 7.47E-05 a /r i eq 15.23 Bond Number N Bo 4.95E-01 4.95E-01 eq 20.8 Physical property number N P 5.75E+11 5.75E+11 eq 20.8 X for polynomial fit X 1.28E+01 1.28E+01 eq 20.7 Y for polynomial fit Y 4.10E+00 4.10E+00 Cunningham Slip flow Correction factor G 1.00E+00 1.00E+00 =1+Kn[A+Bexp(-C/Kn)] eq 15 .30 Initial Termial Fall Speed (slip flow) Vest cm/s 1.22E+04 1.22E+04 eq 20.4 Initial Reynolds Number (slip Flow) Reest 1.47E+04 1.47E+04 eq 20.5 Reynolds Numbers Regimes Reynods NumberSlip Flow 1.47E+04 1.47E+04 eq 20.6 Reynolds Nnmber Continiuum (sphere) 7.47E+02 7.47E+02 eq 20.6 Reynolds Number Continuum (nonspherical) 8.30E+02 8.30E+02 eq 20.6 Final Reynolds Number (Hail particle) Refinal 8.30E+02 8.30E+02 eq 20.6 Final Fall Speed ( Hail particle) Vfinal cm/s 6.91E+02 6.91E+02 eq 29.9
78 Appendix A. Continued Table A. Continued Input Conditions Symbol Units Input Input Formula Range Reference For Drop Schmidt Number Sc 6.32E-01 6.32E-01 eq 16.25 Knudsen Number Kn 1.49E-02 1.49E-02 eq 15.23 Bond Number N Bo 1.29E-05 1.29E-05 eq 20.8 Physical property number N P 5.51E+11 5.51E+11 eq 20.8 X for polynomial fit X -3.08E+00 -3.08E+00 eq 20.7 Y for polynomial fit Y -6.46E+00 -6.46E+00 Cunningham Slip flow Correction factor G 1.02E+00 1.02E+00 eq 15 .30 Initial Termial Fall Speed (slip flow) Vest cm/s 3.25E-01 3.25E-01 eq 20.4 Initial Reynolds Number (slip Flow) Reest 1.95E-03 1.95E-03 eq 20.5 Reynolds Numbers Regimes Reynods NumberSlip Flow 1.95E-03 1.95E-03 eq 20.6 Reynolds Nnmber Continiuum (sphere) 1.86E-03 1.86E-03 eq 20.6 Reynolds Number Continuum (nonspherical) 2.74E-161 2.74E-161 eq 20.6 Final Reynolds Number (drop) Refinal 1.95E-03 1.95E-03 eq 20.6 Final Fall Speed (drop) Vfinal cm/s 3.25E-01 3.25E-01 eq 29.9 Turbulent enhancement to heat and water vapor trans fer from hail Gas Phase Prandtl number (molecular mom./heat transfer) Pr 7.25E-01 7.25E-01 a Cp/k a Eq 16.32 Schmidt number (water vapor) Sc v 6.32E-01 6.32E-01 a/D v Eq 16.25 Stokes Number w/o Cd Ns 2.25E+00 2.25E+00 r w UD d 2 / 9m a D s Eq 4-11 24/C D Re for drop 1.00E+00 1.00E+00 Eq 8.32
79 Appendix A. Continued Table A. Continued Input Conditions Symbol Units Input Input Formula Range Reference Stokes Number w Cd Ns 2.25E+00 2.25E+00 =Ns 24/C D Re after P&K, p 573 and S&P, p. 487,465 Nusselt No smooth cylinder (Avila) Nu_o 1.30E+01 1.30E+01 c RemPrn Incopera and Dewitt, 1996, p.345, Eqn 7.47 Nusselt No smooth cylinder(Incopera) 1.47E+01 1.47E+01 for smooth cylinder Incopera and Dewitt, 1996, p.345, Eqn 7.47 Nusselt No (heat) riming cylinder Nu 2.55E+01 2.5 5E+01 Eqn 9 Sherwood No (water vapor) smooth Sh_o_v 1.24E+01 1.24E+01 like Nu Eqn 5 Sherwood No (water vapor)riming Sh_v 2.43E+01 2.4 3E+01 Eqn 9 Ventilation coeff (water vapor) f 1.21E+01 1.21E+ 01 Sh/2 Determination of Schumann-Ludlam limit for wet vs d ry growth Efficiency of collection E 1.00E+00 1.00E+00 same as Lin et al., 1 Shape factor in equation F 4.00E+00 4.00E+00 For clyindr = pi ; For a sphere = 4 Critical liquid water content W_c g/cm3 2.09E-06 2. 09E-06 heat balance, Ts=0 Critical liquid water content W_c g/m3 2.09E+00 2.0 9E+00 Growth regime Wet Wet =r2v* eff collection =4rDF(Psat/RTh Pa/RTa) Water mass rates Mass rate of drop accretion F g/s 6.51E-05 6.51E-05 Mass Rate of water Evaporation E g/s 2.53E-06 2.53E -06 Intrinsic ice growth g/s 1.44E-01 1.44E-01 =4r2Kv(tint)cr ice Allowable ice growth g/s 3.13E-05 3.13E-05 =/(1-)*Gm Mass rate of ice crystal growth G m g/s 3.13E-05 3.13E-05 =Gm + Lw Eq 6.17 (Stuart 2000, PhD Dissertation, p135) Mass rate of change of liquid water in hail L w g/s 3.13E-05 3.13E-05 assumes constant liquid water content in hail Mass Growth rate G g/s 6.26E-05 6.26E-05 derived from initial model equation. Mass Growth check 6.26E-05 6.26E-05 Mass Rate of Drop Shedding S g/s 0.00E+00 0.00E+00 F-E-G assumes constant liquid water content in hail
80 Appendix A. Continued Table A. Continued Input Conditions Symbol Units Input Input Formula Range Reference Calculation of Retention keff 1.00E-03 1.00E-03 1-keff 9.99E-01 9.99E-01 h 5.00E-01 5.00E-01 1-h 5.00E-01 5.00E-01 1/H* Cair/Cwtr 9.69E-14 5.86E-10 rL g/cm3 1.00E+00 1.00E+00 rVLsat g/cm3 4.85E-06 4.85E-06 rL / rlVsat 2.06E+05 2.06E+05 n G/F 9.61E-01 9.61E-01 m E/F 3.88E-02 3.88E-02 numerator h + keff (1-h) 5.01E-01 5.01E-01 denominator 1-n-m (shedding effect) -1.04E-16 -1.04E-16 n(h+keff(1-h)) (growth effect) 4.81E-01 4.81E-01 m(1/H*rL / rvl) (evaporation effect) 7.75E-10 4.69E-06 total denominator 4.81E-01 4.81E-01 Retention ratio r 1.04E+00 1.04E+00 =h+ke(1-h) / 1-[1-h-ke(1-)]n-[(1-1/H*(rL / rvl sat)]m Mass Balance Testing Water Mass Balance Check g/s 0.00E+00 0.00E+00 Should be zero Solute mass balance Average mixing ratio of chemical in air pptv 5.00E+01 5.00E+01 Seinfeld & Pandis, pg 61 Ca g/cm3 1.12E-13 3.02E-14 =pptv/1e12*P/RT*MW
81 Appendix A. Continued Table A. Continued Input Conditions Symbol Units Input Input Formula Range Reference solute mass fraction_drop Xd g_chem/g_wtr 1.15E+00 5.15E-05 =Ca*H/r wtr solute mass fraction_hail Xh g_chem/g_wtr 1.20E+00 5.36E-05 =GX d solute mass fraction_shed liquid Xl g_chem/g_wtr 2.40E+00 1.07E-04 =X h /(h+keff(1-h) solute mass fraction_evap sol'n Xe g_chem/g_wtr 4.79E-08 1.29E-08 =Xl*(1/H)*(rl/rvlsat) solute mass in via accretion Xd*F g_chem/s 7.52E-05 3.35E-09 solute mass out via shedding Xl*S g_chem/s 0.00E+00 0.00E+00 solute mass out via evaporation Xe*E g_chem/s 1.21E -13 3.27E-14 solute mass accumulation in hail Xh*G g_chem/s 7.52 E-05 3.35E-09 solute mass balance check check g_chem/s 0.00E+00 0 .00E+00 should be zero retention check 1.04E+00 1.04E+00
About the Author Ryan Algernon Michael was born in New Amsterdam, G uyana. He completed his undergraduate degree at the University of Guyana du ring the fall of 2005, majoring in chemistry, with biology minor. He graduated with ho nors, and as awarded the DeanÂ’s Award for the Best Graduating Chemistry Student, as well as the University of Guyana Award for the Best Graduating Natural Science Stude nt. He worked briefly at Berger Paints, Barbados Ltd, in the Quality Control Depart ment for paint production. Ryan received a Master of Engineering Sciences degree fr om the University of South Florida in the fall of 2008.