Relationship between the Scavenging Coefficient for Pollutants in Precipitation and the Radar Reflectivity Factor. Part I: Derivation

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1 OCTOBER 1999 JYLHÄ 1421 Relationship between the Scavenging Coefficient for Pollutants in Precipitation and the Radar Reflectivity Factor. Part I: Derivation KIRSTI JYLHÄ* Department of Meteorology, University of Helsinki, Helsinki, Finland (Manuscript received 30 March 1998, in final form 3 November 1998) ABSTRACT The relation between the scavenging coefficient (s 1 ) for air pollutants in precipitation and the radar reflectivity factor Z (mm 6 m 3 ) is based on the fact that they are both functions of the hydrometeor size distribution. In this paper, which combines the fields of air pollution physics, cloud physics, and radar meteorology, Z relationships are derived analytically for below-cloud gaseous and particulate pollutants and, with certain restrictions, for pollutants incorporated into cloud droplets. For the types of precipitation and pollutant considered, it can be shown that az b, where the coefficient a has an order of magnitude of for submicron aerosol particles, for highly soluble gases, and 10 5 for pollutants in cloud droplets. In stratiform rain the exponent b ranges between about 0.4 and 0.6, so that an increase of Z by a factor of 10 approximately corresponds to a twofold to fourfold increase in. In snowfall, mainly due to the diversity of solid hydrometeors, the value of b may vary more considerably but probably is somewhat smaller than in rain. Because weather radar estimates the spatial distribution of Z essentially in real time, Z relationships can be used to monitor and nowcast those areas most significantly exposed to wet deposition. 1. Introduction Pollutants emitted into the atmosphere or formed there by chemical reactions or radioactive decay may be deposited to the ground by wet and dry removal mechanisms. The wet removal of pollutants includes incloud and below-cloud processes whereby pollutants become attached to cloud droplets, ice crystals, or raindrops, followed by the removal of the hydrometeors as rain or snow. In in-cloud precipitation scavenging, the pollution enters cloud droplets or ice crystals in their formation and growth phases, for example, via nucleation scavenging. When cloud particles that contain pollution collide and merge with each other, precipitation forms which, in falling to the ground, removes the pollution from the air. In below-cloud scavenging, on the other hand, pollutants adhere directly to falling precipitation. For aerosol particles and highly soluble or reactive gases that are irreversibly captured by falling precipitation, below-cloud scavenging can be considered to be * Current affiliation: Finnish Meteorological Institute, Helsinki, Finland. Corresponding author address: Dr. Kirsti Jylhä, Finnish Meteorological Institute, P.O. Box 503 (Vuorikatu 19), Helsinki FIN-00101, Finland. kirsti.jylha@fmi.fi a first-order decay process. In this case the concentration of the pollutants in the air diminishes exponentially, with a rate constant commonly known as the scavenging coefficient (s 1 ). On the other hand, if prior to the onset of precipitation practically all the pollutants in a cloudy air layer are incorporated into cloud particles, the problem concerning the cleansing effect of in-cloud scavenging reduces to the question of removal of the contaminated cloud particles by falling precipitation and also can be approximated with the aid of the scavenging coefficient (Scott 1982; Chang 1984). It has been shown both theoretically (e.g., Engelmann 1968; Scott 1982; Chang 1984, 1986; Asman 1995) and empirically (e.g., McMahon and Denison 1979, Jylhä 1991; Sparmacher et al. 1993; Okita et al. 1996) that the scavenging coefficient has an approximately power-law dependence on the precipitation rate R (mm h 1 ). On the other hand, a large number of experimental power-law relationships between R and the radar reflectivity factor Z for rain and snow have been published, the most frequently quoted relations being those by Marshall and Palmer (1948), Gunn and Marshall (1958), and Sekhon and Srivastava (1970, 1971). Combined together, the R and Z R relationships suggest that there is also a power-law dependence between and Z. To the knowledge of the author, however, very few studies have been published concerning the relationship between and Z. In their paper about the potential role of differential reflectivity in estimating scavenging of 1999 American Meteorological Society

2 1422 JOURNAL OF APPLIED METEOROLOGY VOLUME 38 aerosols, Seliga et al. (1989) additionally presented empirical power-law dependencies between and Z for four different aerosol sizes. Tzivion et al. (1989), in turn, have studied theoretically the simultaneous impacts of the raindrop spectrum evolution on the radar reflectivity factor and on the particulate below-cloud scavenging coefficient. In the current paper, the theoretical basis of the Z relation is presented, and factors affecting its form are discussed. A few simplifying assumptions on the characteristics of precipitation and pollutants will be made, and power-law dependencies then will be derived for below-cloud submicron aerosol particles and highly soluble gases and for pollutants in cloud droplets. A discussion of their use in producing estimates of wet deposition can be found in an accompanying paper (Jylhä 1999, hereinafter referred to as Part II), in which an example related to the Chernobyl accident also will be given. 2. Scavenging coefficient a. Below-cloud scavenging The parameterization of the below-cloud scavenging of aerosol particles is based on the likelihood of these particles being collected by falling hydrometeors, while the scavenging of gaseous pollutants may be treated in the same manner as the mass transfer of water vapor to the hydrometeors. For aerosol particles, for highly water-soluble gases, and for gases having a high affinity for adsorption on ice crystal surfaces, below-cloud scavenging by nonevaporating hydrometeors can be regarded as an irreversible process. In this case the dilution rate of the concentration in the air is linearly proportional to the scavenging coefficient, given for gases (subscript g) and for aerosol particles (subscript p) by and D max g D min 4 Cd f N(D) dd (1a) g D max t D min (d) A(V )E (d, D)N(D) dd, (1b) p t p respectively (see, e.g., Chang 1984; Seinfeld 1986, 621). The equivalent diameter of the collected particle is denoted by d; and D is the equivalent diameter of a collector hydrometeor, defined as the diameter of a sphere having the same volume as the actual water drop or, in the case of solid hydrometeors, as the diameter of the water drop to which a snow particle melts. The product N(D)dD gives the number of the hydrometeors per unit volume of air that have diameters in the range D to D dd, while D min and D max are the smallest and largest diameter of the falling hydrometeors, respectively. The other quantities in (1) are the shape factor C of a hydrometeor; the molecular diffusivity d g of the gas in the air; the ventilation coefficient f of the gas for the falling TABLE 1. Parameter values (in SI units) used in (8) for scavenging coefficient. Parameter Rain Snow Gaseous diffusivity d g (m 2 s 1 ) Kinematic viscosity of air (m 2 s 1 ) Parameter of the shape factor D m D relationship: coefficient exponent V D relationship: coefficient * exponent * A D relationship: coefficient exponent Ventilation coefficient: constant 1 coefficient 2 Average collection efficiency n Submicron aerosol particles (n p) Cloud particles (n cl) Minimum equivalent diameter D min (m) Maximum equivalent diameter D max (m) Intercept N 0 (m 4 ) Slope / Eq. (13) * The alternative values for snow are labeled 1 and , , Eqs. (14a, 14c) Eqs. (14b, 14c) hydrometeor; the hydrometeor s cross-sectional area A projected normal to the fall direction; the fall speeds V t and t of the hydrometeor and the aerosol particles, respectively; and the collection efficiency E p (d, D) ofthe particles by the hydrometeor. (A list of symbols is provided in appendix A.) Note that in (1b) the aerosol particles are assumed to have negligibly small dimensions in comparison with the hydrometeors. On the other hand, (1a) is valid only when the hydrometeor is far from saturated with the gas. Hence, (1a) is applicable, for example, for nitric acid (HNO 3 ), hydrochloric acid (HCl), and ammonia (NH 3 ), but not for sulfur dioxide (SO 2 ) (see, e.g., Chang 1984; Asman 1995). The aim now is to carry out the integrations in (1). In order to do this analytically, it is assumed here that the parameters involved are either constants or simple functions of D, so that (1) can be integrated with the aid of incomplete gamma functions. The remainder of this section deals with these assumptions, summarizes them in Table 1, and finally gives the results of the integrations. First, although the size distribution of precipitation particles may vary considerably in time and space due to various physical, dynamical, and kinematical processes, on average it may be approximated by an inverse exponential distribution: N(D) N 0 exp( D). (2) The intercept N 0 (m 4 ) and the slope (m 1 ) will be discussed later in section 3. Alternatively referred to as the capacitance, the shape factor is a function only of the hydrometeor geometry (Pruppacher and Klett 1978, ). It may be expressed as

3 OCTOBER 1999 JYLHÄ 1423 C D m /2, (3) where D m is the maximum dimension of the hydrometeor and is a proportionality factor (Chang 1984; Miller 1990). In rainfall, ignoring the fact that falling raindrops are seldom perfect spheres, the maximum dimension D m D and 1. Hence, C D/2 and the term 4 Cd g f in (1a) appears to be equal to a commonly used term in rain scavenging parameterizations, namely, to D 2 k g, where k g 2d g f /D is the gaseous mass transfer coefficient to a falling sphere (e.g., Seinfeld 1986, 630). In snowfall, because of the variety of shapes of solid hydrometeors, and D m are more difficult to determine. According to Houghton (1985, 280), possibly more than half of the total snowfall in midlatitudes is in the form of ice-crystal aggregates, commonly called snowflakes. If these aggregates of individual ice crystals have roughly spherical or disklike shapes, it can be assumed that ranges from about 0.5 to 1.0, with an average value of On the other hand, D m has been found to be related to the mass M of the ice particles by a powerlaw equation, the form of which depends on the shape and density of the ice particles (e.g., Mitchell 1996). Noting that M in turn is proportional to the third power of the melted-drop equivalent diameter D, D m can be given as D m D, (4) with the constants and depending on the types of snow particles. Empirical M D m relationships reviewed by Mitchell (1996) imply that 70 and 1.4 for ice-crystal aggregates. Note that SI units are used in (4) and that (4) is valid for rain, too, when one sets 1. By introducing f in (1a), one can take into account the fact that when hydrometeors are falling relative to surrounding air containing gaseous pollutants, the diffusion of the gases to the hydrometeors is altered. The ventilation coefficient may be approximated by f 1 2 Re 1/2 Sc 1/3, (5) where Re L * V t / is the Reynolds number, the ratio of the inertial to the frictional force, for a precipitation particle falling in the air; and Sc /d g is the Schmidt number, being the kinematic viscosity of the air. The characteristic length L * is defined as the ratio of the total surface area of the hydrometeor to the perimeter of its area projected in the flow direction (Pruppacher and Klett 1978, 451). For a raindrop, L * D, but for solid hydrometeors with irregular shapes it is more difficult to estimate. Following Chang (1984), however, L * also is approximated here by D in the case of snowfall. The constants in (5) are set as follows: for rain, and ; and for snow, and (see Pruppacher and Klett 1978, p. 443, p. 453). The molecular diffusivity of a gaseous pollutant in air, found both in (1a) and in the formula for Sc in (5), increases with increasing temperature and with decreasing air pressure and molecular weight of the gas (e.g., Asman 1995). It is assumed here to have a constant value of m 2 s 1 in rainfall and m 2 s 1 in snowfall. These values are close, for example, to those of HNO 3 at an air pressure of 950 hpa and temperatures of 10 and 10 C, respectively. At the above-mentioned air pressure, is m 2 s 1 at 10 C and m 2 s 1 at 10 C (Houghton 1985, p. 412). The vertical velocity of hydrometeors relative to air containing gaseous pollutants affects g through the expression for Re in (5), but the influence is minor, as will be shown in section 4. Of more importance is the fact that the fall speed partly determines the smallest size D min of those hydrometeors that can reach the ground and thus are able to cleanse the air and to contribute to wet deposition. On the other hand, the fall speed relative to the ground often is not equal to its terminal settling velocity but also is affected by vertical air motions. With increasing upward air velocity, the fall speed decreases, D min increases, and g decreases (see Asman 1995). In the current paper, however, the influence of vertical air motions on g (and on p ) through D min is neglected. For below-cloud scavenging of aerosol particles, the emphasis in this paper is placed on particles in an aerodynamic diameter range of m. This size range was selected on the basis of measured activity size distributions of particle-bound radionuclides in Finland after the accident at the Chernobyl nuclear power station (Kauppinen et al. 1986). A substantial part of other anthropogenic aerosol particles also is found concentrated in that size range (Seinfeld 1986, p. 24). The terminal settling velocities of submicron particles are negligible when compared with typical vertical air velocities in stratiform rain and, especially, convective precipitation. Because of their small inertia, aerosol particles practically follow air motion, while precipitation particles respond far less rapidly to it. On a sufficiently long timescale, however, steady-state conditions may be assumed, and time-averaged relative velocity V t t in (1b) between precipitation and aerosol particles can be considered to be independent of vertical air motion. In a reference frame moving with the vertical air motion, one can thus write t 0 and assume the fall speed V t (m s 1 ) of a precipitation particle to be equal to its terminal settling velocity. Since the latter may be approximated with the aid of a power-law dependence on D (m), one can write V t D. (6) The V t D relationship used here for raindrops, with 390 and 0.67, originally was proposed by Atlas and Ulbrich (1977). It is based on observations by Gunn and Kinzer (1949) for V t in stagnant air at sea level and fits the data in the size range m D m. The dependence of the V t D relationship on air density and, therefore, on height has been ignored here.

4 1424 JOURNAL OF APPLIED METEOROLOGY VOLUME 38 On the basis of the results of Foote and du Toit (1969), this neglect results in an underestimation in V t of 5% at 1.5 km and of about 20% at 5 km in a standard atmosphere. To select the values of and to be used in this paper for the case of snowfall, the empirical V t M relationships of Locatelli and Hobbs (1974) were converted into dependencies [(6)] between V t and D. Because and appeared to vary considerably for different kinds of snowflakes, it was decided to apply two alternative V t D relationships, one with 5 and 0.20 and the other with 25 and Actually, Mosimann et al. (1993) have derived empirical formulas for different snow-crystal shapes that give the terminal fall speed for a given crystal shape as a function of both the crystal maximum dimension and the degree of snowcrystal riming. Although methods have been found for estimating the riming degree indirectly by using vertically pointing Doppler radars (Mosimann 1995) or even conventional weather radars (e.g., Huggel et al. 1996), it was decided here to ignore the analytical dependence of V t on the riming in order to keep the integration of (1) and the resulting Z relationship as simple as possible. Instead, the influence of riming is assumed roughly to enter through the above-mentioned wide ranges of and. In (1b), the cross-sectional area of the hydrometeor, projected normal to the fall direction, is parameterized with the aid of a power-law dependence on D (m): A D. (7) For raindrops, ignoring their often oblate shapes, /4 and 2. By eliminating the maximum dimension of ice particles from empirical M D m and A D m relationships for various ice particle types, reviewed by Mitchell (1996), and by using the dependence between M and D, it can be shown that, in solid-phase precipitation, ranges between about 2.0 and 2.9. The values used in this paper are 300 and 2.6, characteristic for aggregates. On the basis of theoretical calculations (e.g., Grover et al. 1977; Wang et al. 1978; Martin et al. 1980; Miller and Wang 1989) and empirical results (e.g., Wang and Pruppacher 1977; Lai et al. 1978; Murakami et al. 1985a,b; Sauter and Wang 1989; Mitra et al. 1990a; Bell and Saunders 1991; Byrne and Jennings 1993), the collection efficiency for aerosol particles in falling precipitation may either increase or decrease with increasing hydrometeor diameter or remain almost constant, depending on the aerosol diameter and on the form, shape, and size of the collector hydrometeor. It therefore should be computed over the range of aerosol and hydrometeor sizes and densities and ambient meteorological conditions of interest. Following the methods of several authors (e.g., Chang 1986; Tremblay and Leighton 1986; Berge 1993), however, E p (d, D) is approximated here by an average value p (d), which only depends on the aerosol particle size and on the phase of precipitation (i.e., liquid or solid). In appendix B, theoretical and empirical studies of the collection efficiency have been reviewed, with a special emphasis on aerosol particles in the size range of m. On the basis of the review, values of 0.02 and 0.01 are assumed here for p (0.3 d 0.9 m) in rain and in snow, respectively. Substitution now of the above expressions for C, f, A, V t, E, and N(D) into (1), followed by an integration from D min to D max, yields g 2 dg N0 [ ] 1/2 1G(1 ) 2 G( ) 1 1/6 1/ dg (8a) and G(1 ) p(d) p(d)n 0. (8b) 1 The function G in (8) is a doubly truncated gamma function: D max x 1 t G(x) t e dt D min (x)[g( D max, x) g( D min, x)], (9) where denotes the complete gamma function [for n, an integral (n 1) n!], while values for the ratio g between the incomplete and complete gamma functions can be found, for example, from a computer-compiled mathematical table. The parameter values in (8) have been discussed above and are summarized in Table 1. In (9), the minimum diameter of raindrops reaching the ground is fixed at a value of m, which is the conventional boundary between cloud and rain hydrometeors. The potential influence of vertical air motion and evaporation on D min cannot be taken into account here (see Asman 1995). The large-size end of a raindrop spectrum is limited by the fact that large drops are unstable and tend to break up into smaller ones. In very heavy rain the maximum diameter is found to be about m (Pruppacher and Klett 1978, p. 23), the value used in this paper for the D max of raindrops. Aggregates of dendritic ice crystals, on the other hand, may have a maximum dimension D m as large as m (Pruppacher and Klett 1978, p. 45). On the basis of (4), this value corresponds to a water equivalent diameter of m. In intense, wet snowfall the diameters of the melted hydrometeors may be even larger, about m, while in moderate snowfall, diameters less than m are not uncommon, as indicated by observations of Gunn and Marshall (1958). Consequently, in the current paper the maximum and minimum equivalent diameters of snow particles are set equal to and m, respectively.

5 OCTOBER 1999 JYLHÄ 1425 b. In-cloud scavenging Following the theoretical parameterizations of incloud scavenging by Scott (1982) and Chang (1984), it is presumed here that all the pollutants in a cloudy air layer are contained within cloud droplets, which are then captured by falling precipitation. In this case, the removal of the pollutants from the atmosphere can be approximated with the aid of a scavenging coefficient cl (d) similar to that in (8b), except that d now refers to cloud droplet diameter and p is replaced by an average collection efficiency cl between cloud and precipitation particles. A discussion of cl is given in appendix C. Based on that discussion, values of 0.6 and 0.1 are assumed as feasible approximations for cl below and above the melting layer, respectively. Note that, as in the case of aerosol particles (section 2a), the dimensions and fall speeds of cloud droplets are ignored here, although in fact the size spectrum of cloud droplets may merge with that of the precipitation droplets, making a division between them arbitrary. Moreover, pollutants in a cloudy air layer may be only partly contained within the cloud droplets. In such conditions, the use of cl alone in appraisals of in-cloud scavenging results in overestimates of wet deposition onto the ground. As will be shown in Part II, this problem can be diminished to some degree by introducing another scavenging parameter, referred to by Schumann (1991) as the total in-cloud scavenging efficiency, and using it jointly with cl. 3. Radar reflectivity factor As revealed by (8), is linearly proportional to the intercept N 0 and, since the exponents,, and in (8) are positive (Table 1), even more strongly dependent on the slope of the hydrometeor size distribution. Measurements of precipitation by surface hydrometeor sampling devices (e.g., disdrometers, replica methods), optical array probes, bistatic continuous-wave Doppler radars, or polarimetric weather radars would yield direct estimates of these parameters (see, e.g., Sheppard and Joe 1994; Sauvageot 1994). By making a few additional assumptions, one then could derive the scavenging coefficient from (8). For example, results by Seliga et al. (1989) indicate excellent agreement between values of p from disdrometer observations and polarimetric radar measurements of differential reflectivity. The current paper, however, deals with a common situation in which no direct measurements of hydrometeor size distributions are available, only estimates of the radar reflectivity factor. By definition, Z is equal to the sixth-order moment of the hydrometeor size distribution: D max 6 Z DN(D) dd. (10) D min It usually is expressed in units of mm 6 m 3 (1 mm 6 m m 3 ). By substituting the two-parameter exponential size distribution (7) into (10) and then integrating, we find N0 (7) Z. (11) 7 Strictly speaking, when the integration in (10) is made from D min 0toD max, there should be a doubly truncated gamma function G(7) in (11), instead of (7). Because of the dependence [(9)] of the function G on the slope, there then would be no simple relationship between and Z in (11). It can be shown, however, for example, by using the Marshall and Palmer (1948) drop size distribution, that at rainfall rates lower than 10 mm h 1 the difference is at most 1% and at rainfall rates up to about 50 mm h 1 the difference is less than 10%. These rainfall rates correspond to radar reflectivity factors (expressed in decibels with respect to the value Z 1mm 6 m 3 ) of less than 40 dbz and about 50 dbz, respectively. Consequently, in rainfall (apart from very heavy rain) the error caused by the use of (7) in (11) is negligible. The intercept N 0 often varies temporally and spatially even in so-called stratiform rain. Assuming, for the sake of simplicity, the above-mentioned Marshall Palmer size distribution, N 0 can be taken to be equal to a constant value of m 4. The slope hence can be solved directly from (11) and substituted into (8) and (9) to calculate the scavenging coefficient of rain as a function of Z. In snow, on the other hand, the value of N 0 is even more variable than in stratiform rain (Pruppacher and Klett 1978). Furthermore, the difference between G(7) and (7) is larger for snow than for rain. For example, at a snowfall rate of 2 mm h 1 of melted water, corresponding to about 40 dbz, the difference is as large as 50% on the basis of the Sekhon and Srivastava (1970) snow size spectrum and the extreme values of the melted diameters shown in Table 1. Because the deviation between G(7) and (7) cannot be ignored in the case of snow, it was decided not to use (11) in the derivation of the Z relationship for snow. Instead, another approach was selected. According to Sekhon and Srivastava (1970), both N 0 and decrease with increasing ground-level snowfall rate R (mm h 1 ): N R 0.94 (12a) and R 0.45, (12b) while Z is related to R by Z 1780R (12c) Combining (14a) and (14b) with (14c) and substituting the resulting N 0 Z and Z relationships into (8) and (9), the snow scavenging coefficient can be obtained as a function of Z. To be precise, the quantity measured by radar is Z e,

6 1426 JOURNAL OF APPLIED METEOROLOGY VOLUME 38 FIG. 1. Scavenging coefficient vs radar reflectivity factor in (a) rain and (b) snow for below-cloud highly soluble gases (solid line) and submicron aerosol particles (dotted line) and for pollutants in contaminated cloud droplets (dashed line). In accordance with the alternative dependencies of the fall speed on ice particle size (Table 1), two curves, labeled 1 and 2, are shown in (b) for each type of pollutant. The right-hand vertical axes indicate the 0.5-folding time 0.5. the equivalent radar reflectivity factor, which by definition is not the same as the radar reflectivity factor Z (see, e.g., Sauvageot 1992, p. 112). However, if the resolution volume of a radar pulse is filled uniformly with spherical raindrops that are small compared with the radar wavelength so that they fall in the Rayleigh scattering region, then Z e is equal to Z. With radar using wavelengths of at least 5 cm, the condition for Rayleigh scattering usually is well satisfied in rainfall (e.g., Sauvageot 1992, 95 97). Thus in rainfall, radar measurements can be used to estimate Z directly. With radar wavelengths of 5 cm or longer, dry ice particles and snowflakes are also, in general, Rayleigh scatterers (e.g., Sauvageot 1992, 97 99). The dielectrical properties of ice particles, however, differ from those of liquid hydrometeors. Thus if a radar system is calibrated to measure the equivalent reflectivity factor Z e of water drops, as is the normal practice, then in the case of snowfall a correction Z 4.5Z e must be made to obtain the radar reflectivity factor Z of ice particles (for details, see Smith 1984). 4. Relationships between and Z By eliminating from (8) and (9) with the aid of (11) in the case of rain and with the aid of (12) in the case of snow, it can be shown that b b g a g1 g1 Z g1 a g2 g2 g2 Z (13a) and b n (d) a n n Z n, n p, cl. (13b) The subscript g refers to highly soluble below-cloud gases and the subscript n refers to below-cloud aerosol particles (n p) or to pollutants in contaminated cloud hydrometeors (n cl). The coefficients a m (m g1, g2, n) are linearly proportional to either d g or n (d) (appendix D), while the exponents b m increase linearly with the exponents of the D m D, V t D, and A D relationships. The factors m, defined as the ratios between the doubly truncated and complete gamma functions, are affected by the exponents of these relationships, too, but they also vary with Z. Analytical forms of m are shown in appendix D, which also presents the values of a m and b m for the parameter values summarized in Table 1. By using the Z dependencies in (13), as detailed in appendix D, Fig. 1 is produced; it shows on a logarithmic scale the scavenging coefficients for pollutants in rain and snow as functions of the radar reflectivity factor in decibels. Corresponding to the alternative dependencies of the fall speed V t on the ice particle size (Table 1) in the case of snow, two curves are shown for each type of pollutant (Fig. 1b). For below-cloud gases, however, the curves practically coincide, illustrating the weak dependence of g on V t. For rain with Z 10 dbz, the curves in Fig. 1a are essentially straight lines. This shape implies that in moderate to heavy rain simple power-law dependences between and Z are feasible for both gases and particles, in spite of the more complicated formulas in (13). In the case of snow, too, the curves are close to straight lines at Z 35 dbz for n (n p, cl) and at Z 10 dbz for g (Fig. 1b). By applying linear regression analysis in the above-mentioned domains of Z, one can derive average power-law dependences of the form az b, where the coefficient a and the exponent b are independent of Z. At the residual low or high values of Z, however, the curves in Fig. 1 bend slightly downward, indicating that for them a power-law dependence can approximate only roughly the relation between and Z. Because it is desirable for practical purposes to make use of such simple dependences, linear regression anal-

7 OCTOBER 1999 JYLHÄ 1427 TABLE 2. Approximate theoretical relationships between scavenging coefficient (s 1 ) and radar reflectivity factor Z (mm 6 m 3 )in rain and snow. Relationship az b (s 1 ) Precipitation type a b Range (dbz) Irreversibly soluble gases below cloud base Rain Snow Submicron aerosol particles below cloud base: d m Rain Snow Pollutants in cloud hydrometeors Rain Snow TABLE 3. Relationships between semiempirical scavenging coefficient (s 1 ) of below-cloud aerosol particles having a diameter d ( m) and measured radar reflectivity factor Z (mm 6 m 3 ) in rain (after Seliga et al. 1989). Here Corr is the correlation coefficient for the regression equation that relates the logarithms of and Z. Relationship az b (s 1 ) d ( m) a b Corr one or two orders of magnitude higher than those suggested in Tables 3 4. On the other hand, the value of a in the layer-averaged Z relationships in Table 4, , is close to those in Table 2 for contaminated cloud droplets. This similarity probably is caused by the dominance of in-cloud scavenging over below-cloud scavenging in the precipitation events under consideration. ysis also was carried out for these nonlinear parts of the curves but separately from the dominating linear parts. In the regression analysis, each pair of curves of in Fig. 1b, denoted by (1) and (2), was combined into a single dataset. The results shown in Table 2 suggest that for rain a has an order of magnitude of about 10 7 and 10 6 for below-cloud submicron aerosol particles and highly soluble gases, respectively, and 10 5 for pollutants incorporated into cloud particles, while b is about for gases and about for aerosol and cloud particles. In snow, a appears to be approximately equal to or higher than a in rain but b has lower values than it does in rain: about for gases and for particles. The theoretical relationships in Table 2 can be compared to those derived semiempirically by Seliga et al. (1989) for aerosol particles of diameter m(table 3). The relationships are based on the one hand on disdrometer measurements of raindrop size distributions and the tabulated numerical collision efficiencies of Beard (1974) in various raindrop size categories and on the other hand on measurements of the radar reflectivity factor. Combination of the classic Z R relationships Z 200R 1.6 from Marshall and Palmer (1948) for rain and Z 1780R 2.21 from Sekhon and Srivastava (1970) for snow with empirical R relationships also yields alternative forms for the Z dependences. Those forms based on results from Jylhä (1991), Sparmacher et al. (1993), and Okita et al. (1996) are shown in Table 4. A comparison of Tables 2 4 suggests that the relationships in Table 2 at least occasionally may overestimate below-cloud scavenging of submicron aerosol particles. While the theoretical and empirical values of b are quite well consistent with each other, for submicron aerosol particles the coefficients a in Table 2 are 5. Discussion The theoretical Z relationships derived in this paper are not universal since they depend on the properties of the precipitation and pollutants. The pollutants were characterized by fixed values for the gaseous diffusivity of below-cloud highly soluble gases and for the average collection efficiency of below-cloud aerosol particles and contaminated cloud particles. Ignoring the dependence of the collection efficiency on the size of precipitation particles, the scavenging coefficients are directly proportional to these two parameters [see (8)]. The relationships therefore can be modified easily to take into account their different values. On the other hand, only two types of precipitation particles were considered: liq- TABLE 4. Semiempirical Z relationships, based on empirical R relationships from various authors and on the Z R relationships from Marshall and Palmer (1948) for rain and from Sekhon and Srivastava (1970) for snow. Relationship az b (s 1 ) Type a b Below-cloud aerosol particles in rain d 0.46 m d 0.98 m Below-cloud aerosol particles in snow d 0.46 m d 0.98 m Author of the R relation Sparmacher et al. (1993) Sparmacher et al. (1993) Layer-averaged value for radioactive aerosols: mainly rain Jylhä (1991) Layer-averaged value for sulfate aerosols: rain, snow, and graupel Okita et al. (1996)

8 1428 JOURNAL OF APPLIED METEOROLOGY VOLUME 38 uid drops in stratiform rain and dry snowflakes. For convective rain and wet snow the Z relationships in Table 2 may be inappropriate and further investigation is needed, whereas for hail the present method presumably is not applicable at all. Even for the types of precipitation and pollutants considered, the Z relationships presented here are not exact but contain several sources of uncertainty. These sources mainly are related to the hydrometeor size distribution, the fall speed, the average collection efficiency, and the shape of the hydrometeors. First, the Marshall Palmer raindrop distribution, which N 0 was assumed to obey in the case of rain, was given originally for 1 R 25 mm h 1, corresponding to 23 Z 45 dbz, and tends to overestimate the number of drops that are smaller than about 1.5 mm in diameter. The Sekhon Srivastava size distribution, upon which the Z relationships for snow in Table 2 partly rest, also is based on a limited range of snowfall rates: 0.2 R 2.5 mm h 1, corresponding to 17 Z 41 dbz. Thus the curves in Fig. 1 are probably at their most accurate at intermediate values of Z. Despite the nonzero lower integration limit in (1), at low values of Z (and R) the scavenging coefficients may well be overestimated. Furthermore, it is quite possible that the hydrometeor size spectrum is not exponential at all but is described better, for example, by a gamma (Ulbrich 1983) or a lognormal distribution (Feingold and Levin 1986). The size distribution of precipitation particles also may evolve during their fall because of processes such as evaporation, coalescence, breakup, and sedimentation. Model calculations by Tzivion et al. (1989) indicate that these processes have a different influence on the vertical distributions Z and p. In the examples by Tzivion et al. for cases with no evaporation, a change of 3 4 dbz during a fall distance of 2 km was accompanied by a relative change in p of only 20 30%. Compared with the large uncertainty in the values of the average collection efficiency for submicron particles (appendix B), a change of that magnitude is insignificant. For contaminated cloud droplets, however, the degree of uncertainty in the cl Z relationships due to N(D) may exceed that due to cl (d). The fact that the dependence of the collection efficiency E n (d, D) ond has been ignored [E n (d, D) n (d)] naturally causes some error in the n Z relationships. With the aid of a numerical integration of (1b) this error could be overcome but only if E n (d, D) can be determined accurately. For pollutants in cloud droplets, the cl Z relationships also may suffer because the dimensions and fall speeds of cloud droplets were neglected (section 2a). The oblate shape of large raindrops and particularly the diversity of shapes of solid hydrometeors also affect the Z relationships. Those relationships derived in this paper for snowfall presumably are best applicable for ice-crystal aggregates. Depending on their composition and degree of riming, there is still some uncertainty in their V t D dependences (Table 1) and consequently also in the values of n,as indicated by the alternative curves in Fig. 1b. For below-cloud irreversibly scavenged gases, such as HNO 3, there is a pronounced difference between Figs. 1a and 1b. While g in rain decreases rapidly with decreasing logarithm of Z in Fig. 1a, Fig. 1b suggests that g is nearly independent of Z in the case of snow. Thus, in estimating wet deposition the duration of snowfall might be a more relevant quantity to consider than is the exact value of Z. It is, however, quite unexpected and not very plausible that very weak snowfall, having Z near a typical minimum value detectable by weather radar, would remove below-cloud gases from the atmosphere with nearly the same efficiency as do moderate and heavy snow. Because, as mentioned above, the Sekhon Srivastava snow size spectrum used to produce Fig. 1b is based on reflectivity factors higher than 17 dbz, it indeed is possible that the scavenging coefficients of snow with a low Z shown in Fig. 1b are artificially high. To conclude, the most important sources of error in the Z relationships presented here are probably the hydrometeor size distribution and the collection efficiency of submicron aerosol particles. On the other hand, in an emergency situation there may be very little information on the properties and dispersion heights of the harmful or toxic emissions under consideration and on the type of precipitation. In this case only a qualitative interpretation can be obtained from the Z relationships, but even so they still may be very useful. As will be discussed more closely in Part II, wet deposition fluxes of pollutants are directly proportional to, and the time intervals required for precipitation scavenging to reduce the concentrations in the air by a certain percentage also depend on it (see the 0.5-folding times 0.5 ln2/ on the right-hand vertical axes in Fig. 1). Consequently, since weather radar estimates the spatial distribution of Z essentially in real time and has a power-law dependence on it, radar measurements can be used to produce estimates of those areas most significantly exposed to wet deposition. As a rule of thumb, in rain Z b, where b , so that an increase of 10 dbz in the radar reflectivity approximately corresponds to an increase in by a factor of In snow, the dependence of on Z is probably somewhat weaker than in rain (Table 2). At equally low values of Z, snowfall scavenges pollutants more effectively than does rainfall, and vice versa at high values of Z (Fig. 1). 6. Conclusions The relationships derived in this paper between the below-cloud scavenging coefficient (in s 1 ) of air pollutants and the radar reflectivity factor (in mm 6 m 3 ) are based on the fact that they are both functions of the hydrometeor size spectrum. By assuming an exponential form for this spectrum, neglecting the dependence of

9 OCTOBER 1999 JYLHÄ 1429 an aerosol particle collection efficiency on the hydrometeor size, and regarding irreversible scavenging of gaseous pollutants only, it was found that, in rainfall with a radar reflectivity factor higher than 10 dbz, Z 0.4 for highly soluble gases and Z 0.5 for submicron aerosol particles. At lower values of radar reflectivity, the Z relationships probably have somewhat higher exponents than the above. Hence, an increase in the radar reflectivity factor of 10 dbz approximately corresponds to a twofold to fourfold increase in. In snowfall, the variety of types and shapes of solid hydrometeors adds to the uncertainty in the Z relationships. In this paper, the emphasis is placed on icecrystal aggregates. For 10 Z 35 dbz, the relationships derived here are Z 0.1 for belowcloud gases with a high affinity for adsorption on ice crystal surfaces and Z 0.4 for below-cloud submicron particles. Although the scavenging coefficient primarily describes exponential dilution of below-cloud pollutants by falling precipitation, it also may be used for approximate estimates of in-cloud scavenging. Provided that practically all the pollutants within the cloudy air mass are incorporated into cloud particles, their removal from the atmosphere can be assessed with the aid of a scavenging coefficient for contaminated cloud droplets. For 10 Z 35 dbz, cl Z b, where b Because weather radar is capable of estimating the spatial distribution of Z essentially in real time and because is correlated with Z, radar data can be extremely useful in the case of an accidental release of hazardous materials into the atmosphere. This usefulness is illustrated in Part II with the aid of an example based on data from the time of the Chernobyl accident. Acknowledgments. Thanks are due to Timo Puhakka for his comments on this manuscript and to Robin King for reviewing and correcting its language. The work was supported in part by the Jenny and Antti Wihuri Foundation; the Vilho, Yrjö, and Kalle Väisälä Foundation; and the Kone Foundation. APPENDIX A List of Symbols A Cross-sectional area of the hydrometeor (m 2 ) a Coefficient in the power-law Z relationship a g1, a g2 Coefficients in the complete g Z relationship a cl, a p Coefficients in the complete cl Z or p Z relationship b Exponent in the power-law Z relationship b g1, b g2 Exponents in the complete g Z relationship b cl, b p C D D m D min D max d Exponents in the complete cl Z or p Z relationship Shape factor of a hydrometeor (m) Equivalent diameter of a hydrometeor (m) Maximum dimension of a hydrometeor (m) Minimum equivalent diameter of a hydrometeor (m) Maximum equivalent diameter of a hydrometeor (m) Equivalent diameter of a collected particle (m) d g Gaseous diffusivity (m 2 s 1 ) E p (d, D) Collection efficiency of particles by a hydrometeor f Ventilation coefficient G Doubly truncated gamma function g Ratio between incomplete and complete gamma function k g Gaseous mass transfer coefficient to a falling sphere (m s 1 ) L * Characteristic length of a hydrometeor (m) M Mass of an ice crystal (g) N(D) Hydrometeor size distribution (m 4 ) N 0 Intercept of the hydrometeor size distribution (m 4 ) R Precipitation rate (mm h 1 ) Re Reynolds number Sc Schmidt number V t Fall speed of a hydrometeor (m s 1 ) t Fall speed of an aerosol particle (m s 1 ) Z Radar reflectivity factor (mm 6 m 3 ) Z e Equivalent radar reflectivity factor (mm 6 m 3 ) Coefficient in the D m D relationship Exponent in the D m D relationship Coefficient in the V D relationship Exponent in the V D relationship p Average collection efficiency for submicron aerosol particles cl Average collection efficiency for cloud particles Complete gamma function Coefficient in an A D relationship Exponent in an A D relationship 1, 2 Constants for f g1, g2, Ratios between G and p, cl Scavenging coefficient (s 1 ) cl Scavenging coefficient for contaminated cloud droplets (s 1 ) g Scavenging coefficient for below-cloud highly soluble gases (s 1 ) p Scavenging coefficient for below-cloud aerosol particles (s 1 ) Slope of the hydrometeor size distribution (m 1 ) Kinematic viscosity of air (m 2 s 1 ) Proportionality factor between C and D m

10 1430 JOURNAL OF APPLIED METEOROLOGY VOLUME 38 APPENDIX B The Collection Efficiency for Aerosol Particles The collection efficiency E p (d, D) for aerosol particles in falling precipitation can be taken as equal to the collision efficiency, assuming an adhesion efficiency of unity. Potential collision mechanisms between aerosol particles and hydrometeors are inertial impaction, interception, Brownian diffusion, thermophoresis, diffusiophoresis, electrical attraction, and turbulence, the capability of each of which depends on the particle and hydrometeor properties and on ambient conditions [see Pruppacher and Klett (1978) for a review]. Aerosol particles in the size range m fall into the so-called Greenfield scavenging gap, the region where Brownian diffusion and the inertial impaction of particles for precipitation-sized water drops become insignificant. According to theoretical calculations (Grover et al. 1977; Wang et al. 1978; McGann and Jennings 1991), in rainfall with nearly saturated conditions (RH 95%) the collection efficiency of particles in this size domain ranges from about 10 4 to Some experimental studies to measure the collection efficiency (e.g., Lai et al. 1978; Radke et al. 1980; Barlow and Latham 1983; Byrne and Jennings 1993) or the below-cloud scavenging coefficient (Schumann 1989; Volken and Schumann 1993; Sparmacher et al. 1993, Fig. 20) suggest, however, that the collection efficiency of submicron particles by rain could be at least one or two orders of magnitude larger than the theories predict. Although some possible contributing factors have been proposed (e.g., Radke et al. 1980; McGann and Jennings 1991; Volken and Schumann 1993), this discrepancy between theoretical and experimental results has not been resolved yet. As a compromise, in this paper a value of 0.02 is used for the average collection efficiency p (0.3 d 0.9 m) in rain. On the one hand it exceeds the abovementioned theoretical estimates but on the other hand it lies slightly below the experimental results obtained by Radke et al. (1980) and subsequently used by several authors (e.g., Hegg et al. 1984; Chang 1986; Tremblay and Leighton 1986; Berge 1993). It is, however, close to the values found by Murakami et al. (1983) in field measurements for the collection efficiency of rain for sulfate and nitrate particles. Of special interest in the companion paper (Part II) are particle-bound radionuclides from Chernobyl; because their activity size distributions were in good agreement with the concentration distributions of sulfate, nitrate, and ammonium (Baltensperger et al. 1987), the selection p (0.3 d 0.9 m) 0.02 was settled upon. For snowfall, Martin et al. (1980) and Miller and Wang (1989) have determined theoretically E p (d, D) of aerosol particle collection by planar and columnar ice crystals, respectively. By comparing the results of these theoretical calculations with each other and with those derived by Wang et al. (1978) for water drops, Wang and Lin (1995) have found that for equal geometrical sweep-out volumes per unit time, hexagonal ice plates have a one to two orders of magnitude higher efficiency in removing aerosol particles in the size range between 0.01 and 1.0 m than do columnar ice crystals and water droplets. However, because the theoretical results for the solid-phase scavenging of aerosol particles apply to ice crystals with simple shapes, they may not be applicable directly to aggregates of ice crystals that may vary greatly in shape and size and dominate medium and heavy snowfall. On the other hand, according to theoretical considerations by Miller (1990), variations in relative humidity in the air (with respect to ice) are much more significant than variations in crystal shape for the scavenging efficiency: because of the phoretic forces, a decrease in RH from 99% to 95% results in an increase in the scavenging rate of up to an order of magnitude, while a 50% change in the capacitance of the hydrometeor (see section 2a) changes the scavenging rate by a factor of 2 only. Experimental studies of the efficiency with which aerosol particles are collected by snowfall have been carried out, for example, by Murakami et al. (1983, 1985a,b), Sauter and Wang (1989), Mitra et al. (1990a), and Sparmacher et al. (1993). Using uncharged thin circular disks as rough models of snowflakes, Sparmacher et al. measured E p (d, D) of these disks for aerosol particles in laboratory conditions where phoretic and electrostatic effects were not taken into account. For aerosol particles with 0.3 d 0.9 m, the measured values increased from about 10 4 to about 10 2 with increasing particle diameter. The results were well consistent with the observations by Murakami et al. (1985a,b) and Sauter and Wang (1989), who experimentally studied the collection efficiency of natural snow crystals with simple shapes. On the other hand, in addition to the abovementioned results from laboratory experiments, Sparmacher et al. also directly measured below-cloud scavenging coefficients p by natural snowfall in outdoor experiments. For submicron particles, the measured values of p were about an order of magnitude higher than those calculated from the collection efficiency data of the laboratory experiments. Furthermore, Mitra et al. (1990a) observed that natural as well as laboratorygrown snowflakes collect aerosol particles considerably more efficiently than do single snow crystals. An explanation offered by Mitra et al. was the open mesh structure of snowflakes: the air that contains aerosol particles not only flows around a porous snowflake but also through it. It has become evident that due to the variety of shapes, sizes, and densities of snow particles and because of the importance of phoretic forces, it is possibly even more difficult to select an appropriate value for p (0.3 d 0.9 m) for the purpose of snowfall events than for rainfall cases. On the basis of the discussion above, a value of unity is a clear overestimation; instead, 0.01 is chosen here. This selection is supported by the fact that according to the field measurements and lab-