1. Introduction

The retrieval of microphysics of precipitating snow from Doppler radar and other remote sensing measurements, as well as snow microphysical parameterizations, requires knowledge of the form of the size distribution that allows an accurate derivation of the distribution moments that are important for descriptions of microphysical processes. Moreover, characteristics of individual snowflakes such as representative dimensional relations of mass and velocity are needed. The study of Woods et al. (2007) demonstrated an important sensitivity of the precipitation redistribution on the changes in both mass and velocity dimensional relationships for snow in bulk microphysical schemes.

In Zawadzki et al. (2010, henceforth Part I), the study of the variability of the snowflake fall velocity, the environmental parameters controlling this variability, and the uncertainties related to the velocity measurement have been presented. In this work, based on hydrodynamic theory and the results presented in Part I, we derive an approximate relation between the mass and terminal velocity of snowflake.

According to observational, laboratory, and theoretical studies, mass has a major effect on the terminal velocity of particles. Empirical power-law relations obtained by Langleben (1954) express the snowflake fall speed in terms of its melted equivalent diameter representing the mass; the coefficients are dependent on the aggregate type. The dependence on the riming intensity related to the particle density has been introduced to other empirically derived formulas (Kajikawa 1998; Barthazy and Schefold 2006). Theoretical and laboratory work on the determination of the terminal velocity as a function of the particle mass, or density, has been mainly based on hydrodynamic theory using parameterized relationships between the Reynolds number and the Best (or Davies) number, the latter expressed in terms of mass and effective cross-sectional area. By inverting this procedure, the particle mass can be estimated from the observed terminal velocity (Hanesch 1999; Schefold 2004; Lee et al. 2008). The same approach is used here as a first step to estimate the mass relation from optical spectrograph measurements of terminal velocity as a function of snowflake size. In the next step, we determine the approximate average relation between the experimentally obtained velocity–size relationship and the estimated mass–size relationship. In this way we obtain a consistent parameterization of velocity–size and mass–size relationships.

The cross-sectional area included in the calculation of the mass–velocity relation based on the Best number X and Reynolds number Re (X–Re) relationship is expected to be related the snowflake effective density as suggested by Cunningham (1978). More recently, an empirical average expression relating particle size, density, and effective area has been proposed by Heymsfield et al. (2004).

Two types of uncertainties contribute to the uncertainty of the derived relations. The first type represents fluctuations in the measured data, such as the velocity or the area ratio for a given size category, used as an important starting point for the calculations. The second type of uncertainty arises from the fact that the theoretical formulas used for the derivation of the resulting relation are not well known. These two types of uncertainties are combined when investigating the uncertainties of the derived relations.

The measurements that we use are from an optical disdrometer, which can give information not only on velocity and area for each size bin of snowflake but also on snowflake particle size distribution (PSD). Measured PSDs were used to evaluate the retrieved relation through the comparison of the expected and a measured radar reflectivity time series and also to derive some average relations between PSD bulk quantities.

We begin with the description of the experimental data used as the starting point of our calculations. In section 3, the theoretical basis and the main assumptions of our method to retrieve mass–velocity relations are described; the calculation results and uncertainties estimation are done in section 4. Section 5 is devoted to the validation of the proposed relation. Some useful applications can be found in section 6. Finally, section 7 is a summary of the results and a discussion of some of the limitations.

2. Experimental data used in the study

In this study we use a large dataset of snow measurements collected by a ground-based optical disdrometer [Hydrometeor Velocity and Shape Detector (HVSD)] (Barthazy et al. 2004). Measurements were taken at the Centre for Atmospheric Research Experiments (CARE) site (80 km north of Toronto) during winter 2005/06 as part of the Canadian CloudSat–Cloud-Aerosol Lidar and Infrared Pathfinder Satellite Observations (CALIPSO) Validation Project (C3VP) described by Hudak et al. (2006). The HVSD measurements provide particle size and terminal fall speed for each size class [assuming negligible vertical air velocity at the height of the HVSD measurement]. The study in Part I gives a detailed description of the measurements and investigates the variability in the velocity–size power-law coefficients. It shows that three environmental parameters—surface temperature, echo top temperature and, in particular, the depth of the precipitation system—determine the fall velocity fairly well. These factors express the impact of different active microphysical processes and of the time of growth.

In this study, nine snowfall events have been selected from the same database as in Part I by inspecting the vertically pointing X-band radar (VertiX) records and retaining only the snow systems uniform in time (for more details, see Part I). An additional requirement for the present study was the availability of well-sampled time series of the particle size distributions measured by the HVSD. For each snow event and each size bin, the mean values of velocity and area ratio, as well as their standard deviations, were obtained from Part I (see Table 1 for the summary of the analyzed events). We consider that during each event we have the same dominant type of particles characterized by the single mass and area relations. Values of the measured velocity and area ratio that deviate by more than 2 standard deviations from the average value for a given size class by the obtained relations were discarded as outliers.

The retrieved mass–velocity relationship is evaluated by comparing the time series of the reflectivity factor calculated from the mass–size relationship applied to the size distribution measured by the HVSD to the reflectivity obtained from the collocated small X-band bistatic Doppler radar providing measurements a few meters above ground [Precipitation Occurrence Sensor System (POSS); Sheppard 1990; Sheppard and Joe 2000]. The HVSD and POSS data are averaged over 6-min periods. The tests of the sensitivity of the results to the time averaging of the HVSD and POSS data have been done. Only very small differences in the results have been obtained for different averaging periods. The total number of analyzed spectra is 805.

The temperature at the ground during these events varied between −17° and −2°C. The observed reflectivity factor varied between about −10 and 30 dBZ.

3. Theoretical basis and main assumptions

a. Functional forms of the used dimensional relationships

The form of a terminal velocity–size and mass–size relationships adopted here is the most common power law:

where D represents a reference size of snowflake.

The coefficients a_u and b_u have been experimentally derived separately for each snow event. The range of the obtained a_u, b_u values underlines the variability between the events as shown in Part I. For a given snow event, the derived a_u and b_u, are assumed to be size independent. However, analytical relations such as those developed by Mitchell and Heymsfield (2005, henceforth MH05) or Khvorostyanov and Curry (2005, henceforth KC05) describe a_u and b_u for different size ranges. These variations with size are not very important for sizes that mostly contribute to the total mass for a given particle type having the same mass and area dimensional relationships. The calculated mean mass-weighted fall speed with constant coefficients used to characterize the particle sizes, where the major part of the mass is located, is only slightly different from the calculated fall speed with size-range-dependent coefficients. The calculations performed by McFarquhar and Black (2004) for different particle types also lead to the above conclusion.

The parameters a_m, b_m in the mass–size relation (1b) are, in general, empirically derived from observations for different types of particles of different size ranges (e.g., Mitchell et al. 1990) and at different temperatures (Heymsfield et al. 2007). Our work aims at evaluating the coefficients a_m, b_m and relating them to the a_u, b_u coefficients in (1a) in the range of snowflake sizes that dominate the total snow mass or reflectivity factor.

Moreover, application of the hydrodynamic theory requires calculation of the cross-sectional area projected normally to the flow. In general, the cross-sectional area is directly related to the area ratio A_r, defined as the area of the particle image A divided by that of a circle of diameter D:

where D_eq is area equivalent diameter. Two forms have been proposed for the parameterization of A_r for the side-view particle images, depending on the snowfall event:

where a_r, b_r, a_re, and b_re are constants determined empirically. The form selected to describe each snow event is the one with a greater correlation.

For nonspherical particles with complex structures such as ice crystals or snowflakes, the definition of size D is an issue in itself. This size is in general determined from particle image recorded with two-dimensional imaging probes; because it is noncircular, there are different possibilities to define the image size. Moreover, depending on the observation method, the image may refer to the side view when the image is projected on the vertical plane parallel to the flow (e.g., Barthazy et al. 2004) or the image projected on the horizontal plane normal to the flow (e.g., Mitchell et al. 1990) In investigations reported in the literature the dimension D has been considered in various ways—for example, as the diameter of an area-equivalent circle (i.e., the diameter that corresponds to the area as the particle shadow; Locatelli and Hobbs 1974; Francis et al. 1998), the equivalent spherical volume diameter (Brandes et al. 2007), the maximum diameter of the particle image (Kajikawa 1989; Mitchell et al. 1990), the mean of the two orthogonal extensions (Brown and Francis 1995), and the width of the enclosing box (Barthazy et al. 2004). For the mass relation a new approach is proposed by Baker and Lawson (2006), who introduce a size parameter that is a combination of the maximum dimension, width, area, and perimeter. A combination of parameters is also used by Hanesch (1999) in her study of the snowflake fall speed. In addition, the equivalent melted diameter is considered as a snowflake size parameter.

In the data collected by the HVSD the observed snowflake dimensions correspond to the two-dimensional side-view pattern. As a reference, the dimension D is chosen to be the maximum side-view size—that is, the maximum of the two perpendicular extensions: height of the image (vertical dimension as the snowflake falls) and width of the image (see Part I). This definition of the snowflake reference size is used in our dimensional relationships obtained from measurements of the velocity and area ratio and retrieved for mass. The dimensions normal to the flow required for the hydrodynamic calculations have to be estimated from the side-view projection.

b. Introduction of the Reynolds and Best (Davies) numbers

The general expression for terminal fall speed is given by equating the drag force F_D with the difference between the gravitational force mg and the buoyancy force. However, the problem with applying this expression is the dependence of the drag coefficient C_D on the fall speed itself. Thus, the common method for calculating the terminal velocity is from a determined relationship between the Reynolds number Re and the Best number X related to the drag coefficient but having no dependence on fall speed. The two numbers are defined as (e.g., List and Schemenauer 1970)

taking ρ_snow − ρ_a ≈ ρ_snow, the following expression for X is obtained:

The index ⊥ describes the quantities related to the horizontal projection (perpendicular to the flow), while the index ‖ (introduced in Eq. 7a below) represents the projection on the vertical plane (parallel to the flow).

The environmental conditions are included via the kinematic viscosity and air density, ν and ρ_a, respectively; g denotes gravitational acceleration. The expression for X contains the two primary variables that determine u (i.e., the mass m and the effective particle area projected normal to the flow A_⊥, in addition to D_*, which denotes the chosen characteristic dimension of the particle). Introducing the area ratio normal to the flow, A_r⊥, and using (2), X can be expressed as

where D_⊥ denotes the maximum diameter in the direction normal to the flow that is not measurable and therefore must be estimated. The final expression for X depends on the choice of the characteristic size D_*. The symbols X_‖ and X_⊥ below describe X calculated with D_* = D (our measured maximum side-view diameter taken as reference) and D_* = D_⊥ (estimated maximum diameter normal to the flow), respectively:

The corresponding Re numbers are given by

Assuming an idealized spheroidal shape for the snowflakes and taking D_* equal to the maximum diameter projected on the flow D_⊥, Böhm (1992) modified the general expression for X (7b) by introducing the fourth root of the inverse of area ratio and the axial ratio ϕ of this assumed oblate spheroid shape of snowflake:

c. Relations between Reynolds and Best numbers

The equation relating the X to the Re numbers can be obtained empirically from direct measurements of particle velocity, cross-sectional area, and size (e.g., Knight and Heymsfield 1983; Heymsfield and Kajikawa 1987; Redder and Fukuta 1991). However, important uncertainties are associated with these measurements, mainly those of the cross-sectional area that is normal to the flow.

A generalized theoretical relationship between Re and X derived from the boundary layer theory was developed by Böhm (1989) and Mitchell (1996) on the basis of the previous work of Abraham (1970), giving

The values of the two constants δ₀ and C₀ characterizing the boundary layer shape and thickness are 5.83 and 0.6, respectively, for ice particles. For aggregates, MH05 modified slightly the relation by adding an empirical term −a₀X^b₀ with a₀ = 0.0017, b₀ = 0.8. This additional term accounts for turbulent flow around larger aggregates so that there is better agreement with the measurements presented by Heymsfield et al. (2002b). KC05 proposed an alternate method to introduce this correction.

In this study, the particle mass is derived from the Best number that is calculated from Re. Therefore, we invert the proposed relations Re = f (X) from MH05 and KC05 by fitting log(X) to an eighth-order polynomial of log(Re):

with C_l = C_MH,l for the MH05 relation, and C_l = C_KC,l for the KC05 relation. The coefficients C_l are given in Table 2. The X–Re relations given in (11) are applied for the pairs (X_‖, Re_‖) and (X_⊥, Re_⊥). The differences between the values of X retrieved using different X–Re relationships are evident for the larger Re, which describe the snowflakes that have a larger contribution to the reflectivity values, as seen in Fig. 1, which gives a good estimate of the uncertainty associated with the relationship X–Re.

d. Assumed relations between side-view snowflake projection and the section perpendicular to the flow

Snowflake size and area measured by the HVSD represent a side view of the particle. Because of aerodynamic forcing, the falling snowflakes have in general their maximum size oriented horizontally (e.g., Magono and Nakamura 1965). On average, their horizontal dimension is larger than the value measured when viewed on side projection. Since the horizontal (normal to flow) dimensions that are required for the velocity calculations cannot be measured, they must be estimated from the side projection. For this, 1) we assume that A_r⊥ ≈ A_r (i.e., the area ratio is independent of the angle of observation, normal or parallel to the flow) and 2) we use the relation from Schefold (2004) giving the ratio of the area projected normally to the flow to the area from the side view, given as a function of the canting angle α and the side projected axial ratio ϵ. The latter is the quotient of the side projected minor axis to the side projected major axis. Thus,

where ϵ_min describes the minimal value of the axis ratio evaluated from the measurements as equal to 0.3. The last relation together with the assumption A_r⊥ ≈ A_r is also used to calculate the maximum horizontal dimension D_⊥ from the maximum side dimension D taken as the reference dimension:

Equations (12) and (13) give the ratio D/D_⊥ decreasing with D on average. For smaller sizes the mean value of this ratio is around 0.85 and decreases progressively. Since the value of 0.8 is consistent with the results from the snowflake fractal analysis by Schmitt and Heymsfield (2010), the minimal value of 0.75 is imposed on this ratio and the corresponding maximal value on f_A as calculated in (12).

Our first assumption about the area ratio being independent on the projection plane seems reasonable for snow aggregates for which the value of area ratio has been shown to be related to particle density (e.g., Heymsfield et al. 2004). Moreover, this postulate agrees with the results of the recent theoretical work on the fractal dimensions of aggregates (Schmitt and Heymsfield 2010).

4. Derivation of the mass–velocity relationship

An average relation between a snowflake mass and velocity has been derived in the following steps with the first four steps calculated separately for each snow event:

• (i) Calculation of Re from the HVSD measurements of terminal velocity and area ratio from (8a) or (8b) and (13);
• (ii) Estimation of the value of X from Re using a polynomial fit (11) with constants C_MH,l for the MH05 relation, and with C_KC,l for the KC05 relation;
• (iii) Best estimation of mass for D-size snowflake from the estimated X and area ratio normal to the flow;
• (iv) Determination by regression of the coefficients in the power-law relation m–D; and
• (v) From all events, derivation by regression of an average relation between the power-law coefficients in the mass and velocity dimensional relationships.

a. Best estimation of mass for D-size snowflake and its uncertainty

Using the value of X estimated from Re, and with the help of the relations (12) and (13) and the approximation A_r⊥ ≈ A_r, the following three equations for mass are obtained from (7a), (7b), and (9):

For each size bin, using (14a)–(14c) 24 different values of mass, m_k with k = 1, … , 24, are calculated from different combinations of the X–Re relationships, smoothed and unsmoothed measured u and A_r, and two values of axial ratio of 0.7 and 0.8 put in (14c). Each value of mass is considered as the “measured” value of mass using different methods and interrelated through the common parameters. The geometric weighted mean is taken as the best estimate of the snowflake mass with reference diameter D:

The weight of each measured mass is the inverse of the square of the corresponding uncertainty (Δ logm_k)² obtained by combining the contribution of the uncertainty related to the different quantities involved. To unify the calculation of (Δ logm_k)², we take the following single logarithmic form of Eqs. (14a)–(14c):

Here C_E is a parameter describing the environmental conditions, and is different by a factor of 4/π in (14c) relative to (14a) and (14b). The uncertainty of C_E will be neglected here.

In general, the uncertainty related to the calculated X value can be decomposed into two terms. The first represents the uncertainty of the used X–Re relationships (11) for different coefficients C_l. The second describes the contribution to the uncertainty of the computed Re value:

Assuming that the uncertainty in the X–Re relation is 30% (as in Field et al. 2008), we have (Δ logX)_Cl² ≈ 0.017. The same potential bias of log(X) is assumed for X calculated via (10). The second term on the rhs of (17) is evaluated from (11) as

The term in the {·} parentheses varies slightly with the value of Re but may be approximated by its mean value of about 1.8. The uncertainty in log(Re) is different for Re_‖ and Re_⊥. Their values can be estimated from (8a) and (8b):

Combining the relations (16)–(19), the uncertainty in log(m_k) can be written as

The relations (20) and also (17) and (19) assume that for a single m_k there is no correlation between the errors in the variables used in the calculation of m_k.

The uncertainties in u and A_r can be estimated from the measurement standard deviation (Part I). Since we are interested in the medium-size and larger snowflakes that mostly contribute to the reflectivity, the estimated relative errors are equal to about 20% for both Δu/u and ΔA_r/A_r. Note that these error values are incorrect for snowflakes smaller than about 3–4 mm. The potential systematic bias in the axial ratio and in the calculated value of f_A is difficult to estimate; a ±30% range is chosen. The error in the measurement of D is neglected, in which case Δ logD ≈ 0. Table 3 contains the summary of the estimated uncertainties used in (20).

With the assumed estimates for the different terms in (20) we obtained values of (Δ logm_k)² equal to 0.0321, 0.0363, and 0.0462 for the masses estimated from (14a), (14b), and (14c), respectively. This means that the relative error in our mass estimate is between 40% and 50% and is the same for all sizes D since the relative uncertainties were taken as constant.

To evaluate the uncertainty included in the mass estimate log [m̃(D)] given by (15), the effect of the existing correlations between the many components of every couple of m_k has to be taken into account. To do this, the uncertainty of log(m_k) has to be separated into uncorrelated and fully correlated uncertainties for each combination of two values of m_k. Then the propagation error law has to be applied. To avoid these complex calculations we only estimate the maximum uncertainty for which all m_k are fully correlated between them. In this case, the total uncertainty is simply the linear sum of the weighted uncertainties of an individual m_k:

with . The value of Δ(logm̃) computed from (21) provides the average uncertainty of m_k calculated from the relations (20) and is equal to 0.193.

b. Determination by regression of the coefficients in the power-law relation m–D

Assuming that the calculated value of m̃(D) represents the best evaluation of the snowflake mass with reference diameter D, we look for a dimensional relation m–D of the power-law form given by (1b) for a snowflake of a given size D for each snow event. Taking the logarithm of (1b) yields the linear equation

The best estimate of the two coefficients is obtained by a least squares fit. Two examples are shown in Fig. 2, where the solid line shows the best-fit power law. For the nine snow events studied, the obtained coefficients a_m and b_m and their values are given in Table 1 and shown in Fig. 3. For comparison, the mean value obtained by Brandes et al. (2007) from a very large dataset is also superposed. However, because D in their study was taken as equivalent diameter, their relation has been recalculated using the average relation D_eq–D obtained from our data: D_eq = 0.69D^0.95 in CGS units (very close to the average relation in Part I: 0.71D^0.92 obtained from the larger dataset). Moreover, the sets of the parameters in the relations of mass and maximum size are also superposed after recalculation using our relation (13) between maximum side-view size and maximum horizontal dimension, taking the value for larger particles of 1/= 0.75. The relations shown are the average mass relation from Mitchell et al. (1990) for the 1986–87 field season, from Kingsmill et al. [2004; based on the data by Heymsfield et al. (2002b)], and from Matrosov and Heymsfield (2008) obtained for precipitating clouds.

For some of the cases studied, the investigation of the regression that gives the best set (a_m, b_m) shows that a better description of the m–D relationship is obtained when the analysis is performed separately for two size regimes: for D smaller than 2–3 mm, and for D larger than 2–3 mm. This separation can be important in all microphysical studies where the lower-order moments of the size distribution become important.

As shown in Fig. 3, the obtained exponents b_m through the mass power law for different snow events cluster between values of 1.8 and 2.10, with the mean value close to 1.9. The value of 2.0 corresponds to an effective snowflake density decreasing with size as D⁻¹ and is in good agreement with numerous observational studies of snow at the surface (Mitchell et al. 1990 from the overall observed particle types; Brandes et al. 2007) and aircraft observations (Heymsfield et al. 2002a). Moreover, most of the snowfall mass reaching the ground is generally associated with aggregates of ice crystals and theoretical studies (Westbrook et al. 2004) predict a value of 2 for the exponent in the mass–dimension power law for snow aggregates. Certainly, individual crystals with forms other than aggregates may be less well represented by this exponent; however, they contribute only a small fraction of the overall snowfall characteristics. On the other hand, an exponent equal to 2 probably does not describe correctly the snowflakes larger than about 1.5 cm because of the rather small number of the observed particles with these sizes. Contribution of these particles to the reflectivity is significantly reduced because of the non-Rayleigh effect. For example, for a snowflake with diameter of 2 cm, the reduction of backscattering cross section with respect to Rayleigh regime is about two orders of magnitude, as shown from the results of the T-matrix calculations in Matrosov et al. (2009).

Taking into account the general uncertainty in the mass–size relationship, in our attempt to develop a relatively simple snow parameterization we set the exponent b_m to 2 for the power-law relation representing the precipitating snow. The relation (22) becomes

with only the coefficient a_mf to be determined in the mass–size relationship (the symbol a_mf denotes a_m with fixed b_m = 2). All symbols used here are listed in Table 4.

On the other hand, the velocity power laws of the form (1a) obtained by Part I for different homogenous snow events show the relatively small variability of the exponent b_u. They showed that the snowflake velocity can be modeled with good accuracy with a fixed exponent b_u= 0.18 and a varying coefficient a_u. This fixed value of b_u is adopted here for further calculations and the corresponding value of a_u is represented by a_uf . Subsequently, all the calculations that lead to the best set of mass–size relationship based on the measured snowflake velocities and area ratios are repeated but with an imposed value of b_u = 0.18 in the fitted velocity–size relationship and b_m = 2 in the final mass–size power law. In Table 1 the obtained values of a_mf and a_uf are given. In Fig. 2 the dashed line show the result with imposed b_m = 2 using (22′), while the solid line has been obtained through regression with no restriction on the b_m value (22). The difference is within the bounds of uncertainty.

To evaluate the impact of imposing the value of 2 for b_m we calculate, from the observed PSD for the time series of snow events, the snow ice mass content (IWC) in two ways: first, from the (a_m, b_m) set that was obtained by regression with b_m varying, named IWC_am, and second, using the a_mf value obtained for b_m = 2, named IWC_amf . The root-mean-square error RMSE = 〈(IWC_am − IWC_amf)²〉^1/2 = 0.004 g m⁻³ while the root mean of fractional error RMFE = 〈[(IWC_am − IWC_amf)/IWC_am]²〉^1/2 = 0.07 for the entire length of the time series. The same calculation performed for the reflectivity factor gives an RMSE value of 0.35 dB; for reflectivity-weighted velocity (using fixed b_u = 0.18), the calculated RMSE and RMFE are 0.006 m s⁻¹ and 0.007, respectively. The particles’ radar backscatter cross section is calculated based on the Mie theory assuming the mixing rule of Maxwell–Garnett for ice–air mixture [for more details, see model 5 in Fabry and Szyrmer (1999)]. We discuss the inaccuracy of the scattering calculations arising from the Mie calculations in section 5.

The estimates of the uncertainties in the coefficient a_mf and in the value of m predicted by (22′) for each D must take into account the effect of the correlation between calculated values of m̃ resulting from the contribution to their uncertainty of the correlated components of m_k. With the assumption of a positive correlation, the magnitude of the uncertainties rises and the analytical calculations become very complex. To simplify the calculations we use the uncertainty estimate for each log m̃ given by the error bars in Fig. 2. By imposing a slope equal to 2, the uncertainty in a_mf is estimated by finding the range of intercept values for which the regression line covers most of the error bars representing Δ(logm̃). The obtained relative uncertainties in a_mf , denoted by Δa_mf/a_mf in the last column of Table 1 for each snow event, are included in the regression calculation of the approximate relation between a_mf and a_uf in (23) below and shown by error bars in Fig. 4.

c. Derivation of average mass–velocity relationship

To reduce the number of parameters describing an individual snowflake with size D, we search for an approximate relation between its mass and velocity, assuming that other factors (such as the shape of the individual crystals) introduce negligible correction to the average mass–velocity relationship. This relation is obtained for the mass and velocity exponents fixed at 2.0 and 0.18, respectively.

The atmospheric conditions have some influence on the snowflake mass calculated from (14a)–(14c), and hence on the value of a_mf derived from experimental values of a_uf . This influence is mainly through the kinematic viscosity of the air ν, which is equal to the dynamic viscosity, a function of temperature, divided by the air density. The air density variation can be neglected here because all the observations were made at the ground and to a first approximation the air density changes with height (the temperature correction is evaluated at less than 2% with respect to the average temperature of −10°C). Besides the explicit dependence of m on the magnitude of ν given in (14), the values of the calculated Re and X also change with ν. Therefore, prior to the search of the mass–velocity relationship, the influence of temperature variations on the estimated value of a_mf has to be investigated.

The snow measurements considered here were done at temperatures between −2° and −17°C. The range in dynamic viscosity corresponding to this temperature range is from 1.72 × 10⁻⁵ to 1.63 × 10⁻⁵ kg m⁻¹ s⁻¹. The correction multiplication factor that normalizes the estimated mass, and therefore a_mf , was calculated with respect to calculations done for an average value of −10°C. The obtained values of the correction factor are 1.027 and 0.983 at −2° and −17°C, respectively. This range is very small and therefore suggests that the impact of the atmospheric conditions during the measurements on the derived a_mf–a_uf relation can be neglected.

In Fig. 4 the values of a_mf derived for b_m = 2 and b_u = 0.18 are plotted as a function of the velocity coefficient a_uf obtained from the data. Each point represents one of the nine events. The analytical approximate relation between a_mf and a_uf is assumed to have the form loga_mf = α + βa_uf . Coefficients α and β have been determined by a weighted total least squares (WTLS) regression described in Krystek and Anton (2007). For each [a_uf , log(a_mf)] pair, the corresponding uncertainties are Δa_uf ≈ 0.2a_uf from Part I and Δ(loga_mf) obtained from regression (22′). These uncertainties are shown in Fig. 4. The solid line in Fig. 4 represents the least squares fitting with α = −3.02 and β = 0.0065 in CGS units, giving

The uncertainties associated with the two coefficients are valid when the effect of correlation between the values of a_mf is not taken into account. The magnitudes of a_mf obtained from (23) are shown in Table 1. The weighted average value of a_mf for the nine analyzed events is shown in Fig. 4 by a dotted line. Its value is equal to 0.0044 g cm⁻² and is very close to the average from observations of Mitchell et al. (1990) and Kingsmill et al. (2004) recalculated using (13) as in Fig. 3. The value of a_uf corresponding to this weighted average is equal to 102 cm^0.82 s⁻¹.

Using (23), snowflake velocities are calculated as a function of melted equivalent diameter, and the results are shown in Fig. 5. In the upper graph, the derived relations between snowflake velocity and melted diameter for each of the nine events are compared with the empirically obtained relations of Langleben (1954). In the lower graph we compare snowflake velocity calculated by our relation (23) for average a_mf with some other relations previously published (Locatelli and Hobbs 1974; Kajikawa 1998; Brandes et al. 2008 for velocity at −1° and −5°C and Brandes et al. 2007 for mass relation derived from observations mostly at temperatures higher than −5°).

All presented results are derived from measurements at the surface. Therefore, when applied at other pressure levels the velocity coefficients (a_u, a_uf) have to be adjusted for a change in altitude using, as a first approximation, the adjustment given in Heymsfield et al. (2007): [a_uf(p)/a_uf(1000 hpa)] = [ρ_a0/ρ_a(p)]^0.54, where ρ_a0 and ρ_a(p) are the air density at the ground level and at pressure p.

5. Evaluation

In this study the measured reflectivity is used to evaluate the procedure for the retrieval of mass/density as a function of particle size. For each snowfall event, the mass coefficient a_mf is estimated from the velocity coefficient a_uf using (23). Knowing the mass–size relationship, the backscatter cross section of each size bin is calculated using the Mie theory from model 5 described in Fabry and Szyrmer (1999). For particles smaller than about 5 mm, the scattering at X-band is in the Rayleigh regime (i.e., that the reflectivity is proportional to mass squared, independent of the particle shape; e.g., Matrosov et al. 2009). For larger particles the non-Rayleigh effect leads to the reduced backscattering that decreases with size for particles larger than about 1.5 cm (Matrosov et al. 2009).

The other question to consider is the effect of nonsphericity. According to Ishimoto (2008), using the finite difference time domain (FDTD) method for fractal-shaped snowflakes, the sensitivity to the particle shape become important for snowflakes with a diameter of the ice-mass-equivalent sphere greater than about 2.4 mm (which corresponds on the average to D = 1.2 cm). The same result has been obtained with T-matrix method in Wang et al. (2005). The inaccuracy of the Mie calculations assuming spherical form depends on the choice of mapping of particle properties into sphere parameters. By choosing in our model the area-equivalent diameter (i.e., between ice-mass-equivalent sphere diameter and maximum dimension) as diameter of the equivalent sphere, the inaccuracy of the scattering calculations arising from the Mie calculations for snowflakes not larger than 2 cm is relatively low. Moreover, as shown in Heymsfield et al. (2008), the contribution to the X-band reflectivity of the particles larger than about 1 cm (assuming the exponential size distribution with slope equal to 6 cm⁻¹ that is representative for our dataset) is negligible. Furthermore, Matrosov et al. (2009) concluded that at X-band the spherical model for low-density snowflakes provides the necessary accuracy in the reflectivity calculations if the non-Rayleigh effects are not too important.

But perhaps the most direct way of considering the effect of nonsphericity on reflectivity is by noting that observed values of differential reflectivity in precipitating snow are a few decibels, indicating that the strong effect of low density of particles decreases the influence of shape on backscattering, which confirms the results of Matrosov et al. (2009).

The expected reflectivity factor is computed from the snowflake size distributions measured by the HVSD. The time series of the calculated reflectivity are compared with the reflectivity derived from the collocated POSS. The HVSD and POSS data are averaged over 6-min intervals. The scatterplot of the reflectivity calculated for all nine events versus the POSS measured reflectivity is presented in Fig. 6. The root mean standard error is equal to 2.71 dB.

Figure 7 presents an example of the reflectivity time series. The solid line gives the POSS-measured reflectivity; the dotted–dashed line is the reflectivity calculated from the mass relationship derived for the given event of 6 January 2006. This particular event is characterized by the lowest retrieved value of a_mf (see Table 1). To provide a limit on the importance of the mass–size relation on the uncertainty in the reflectivity calculations, the dashed line in Fig. 7 shows the reflectivity calculated for the PSDs of 6 January 2006 but using the mass–size relation retrieved for the event of 9 January 2006 where the estimated a_mf was the greatest (see Table 1). The relative difference in a_mf between these two events is about 1.5. Under the Rayleigh regime, the effective backscattering cross section is proportional to the square of the mass. Then, we have Δ_mZ_e/Z_e ≈ (1 + 1.5)² − 1 for Z_e in mm⁶ m⁻³, giving the contribution of the mass uncertainty to the calculated dBZ of Δ_mZ_e,dBZ = 10 log(1 + Δ_mZ_e/Z_e) ≈ 8 dB, as can be noted in Fig. 7. For a smaller relative uncertainty in Δa_mf/a_mf we can write Δ_mZ_e/Z_e ≈ 2Δa_mf/a_mf , from which we can deduce that Δ_mZ_e,dBZ = (10/ln10)Δ_mZ_e/Z_e, equal to about 8.7 Δa_mf/a_mf valid under Rayleigh regime.

6. Applications

The retrieved relation (23) used in conjunction with the PSD information makes it possible to derive useful relations between different bulk properties of snow PSD such as equivalent reflectivity factor Z_e, snow ice water content (IWC), liquid equivalent snowfall precipitation rate S, or reflectivity- and mass-weighted velocity U_Z and U_M. The time series of the snowflake PSDs are provided for each analyzed snowfall case by the optical imager HVSD.

In Fig. 8 the reflectivity-weighted velocity at the ground level is plotted against the reflectivity as obtained from the calculations for all snow events. The solid lines are obtained through regression for the following linear relation between U_Z and Z_e in dBZ and log(a_mf):

Note that U_Z and a_mf are in CGS units. Introducing a weighted mean value for a_mf equal to 0.0044 in CGS units as shown in Fig. 4, the average U_Z–Z_e relation becomes

for precipitating low-density snow. The last term in (24) introduces the estimated sensitivity of U_Z to the error in mass–dimensional relationship. The uncertainty in U_Z resulting from the uncertainty in the value of a_mf may be estimated as ΔU_z (cm s⁻¹) ≈ 50Δa_mf/a_mf . The relation (24) when a_mf is given is expected to provide a more accurate U_Z–Z_e relation compared to (24′). For higher altitudes, the air density correction for U_Z has to be introduced as given at the end of section 4. The mean ratio of calculated reflectivity-weighted velocity to mass-weighted velocity is equal to 0.93.

The second useful relationship between two bulk quantities is the power law relating the equivalent reflectivity factor Z_e and the snowfall precipitation rate S: Z_e = χS^ω, with Z_e in mm⁶ m⁻³ and S in mm h⁻¹. In Fig. 9, the calculated snow precipitation rate S is plotted as a function of the calculated Z_e in dBZ. The solid line shows the best fit obtained by the WTLS method applied to the logarithmic form of the Z_e–S power law. The uncertainties taken into account for Z_e and S in WTLS calculations describe the contributions from the uncertainties in snowflake fall speed and mass represented by the uncertainties in a_uf and a_mf , and the correlation between them. The obtained values of the constants are χ = 494 and ω = 1.44.

Our Z_e–S relationship is in a general agreement with some of relations previously derived empirically, semiempirically, or theoretically that are drawn in Fig. 9 [Ohtake and Henmi (1970) for snowflake consisting of spatial dendrites; Fujiyoshi et al. (1990) for 1-min and 30-min average; Matrosov (1992) for snow density of 0.02 g cm⁻³; Zawadzki et al. (1993) recalculated using our average relation S–IWC: S = 3.3IWC^1.05, with IWC in g m⁻³; and seven relations retrieved by Huang et al. (2010)]. However, compared with the recent work of Matrosov et al. (2009), our value of χ is much larger for similar values of the exponent ω. This difference may be partly explained by the applied snow velocity relationship. Our measured velocity for low-density snowflakes as shown in Part I is in general lower than that obtained from the relation used in Matrosov’s work.

The increase of snowflake density (i.e., the value of a_mf in our parameterization) is related to the increasing velocity described by a_uf . For the same snowflake spectrum, this increase results in the increase of both Z_e (proportional to under Rayleigh regime) and S (proportional to the product a_mf × a_uf). To estimate the impact of these changes on the coefficients in the Z_e–S relation, we present in Fig. 10 the calculations of Z_e and S repeated 3 times for the time series of PSDs on 9 January 2006 using 1) a_mf and a_uf obtained for this event (solid circle), 2) these two coefficients retrieved for a different event, 6 January 2006 (open circle), and 3) weighted average values of these coefficients as shown in Fig. 4 (symbol “X”). The calculation results presented in Fig. 10 show that the coefficients of the power law Z_e–S for low-density snowfall do not exhibit a significant dependence on the assumed mass–velocity relationship if (23) is satisfied. Matrosov et al. (2009) show the slight dependence of the coefficient χ on the assumed mass dimensional relationship. However, they used the density independent expression for snowflake velocity. Our result suggests that for the unrimed snowfall of our study, the observed spread in the Z_e–S power law is mainly due to the PSDs differences, as shown in Matrosov et al. (2009), and to a lesser degree to the assumed mass relation.

7. Summary and discussion

Both modeling and remote sensing studies require the parameterizations of bulk quantities. Generally for snow the accuracy of these calculated variables and the relations between them are dependent on the parameterization of the PSDs used and on the mass and velocity dimensional relationships as well. The variability of snowflake velocity discussed in Part I between different snowfall events is related here to the variability in the mass dimensional relationship based on the idea that the reliable estimators of snowflake velocity are its density together with its size. Even if only light riming (or no riming) cases were used here, we see appreciable differences in (23) and all the derived relationships.

Radar observables provide an alternative verification of microphysical description of snow. Our ultimate goal is to derive the microphysical relations that are consistent with radar measurements and suitable for modeling and optimized radar data assimilation. We have limited here our interest to precipitating unrimed or lightly rimed snowflake aggregates for which we derived the mass–size–velocity relationship that is a simpler expression of the general relations of mass, area, size, and velocity developed in MH05 and KC05. In our relationship the area is eliminated: the cross-sectional area, related to the details of the crystal types composing the aggregates, is closely related to the particle mass, and therefore its direct impact on the snowflake fall speed is reduced. In this way, in our formulation the variability in the fall speed for snowflakes with the same size is only attributed to particle density (or mass).

The main goal of this work was to retrieve an average relation between unrimed or slightly rimed snowflake velocity and mass using the observational data and theoretical calculations. The retrieval is through the following five steps: 1) calculation of the Re number from HVSD measurements of terminal velocity and area ratio for each snowfall event; 2) estimation of the X number from Re; 3) retrieval of mass for each size bin from the X–Re relationship and the estimated area ratio normal to the flow; 4) determination by regression of the coefficients in the power-law relation m–D for each event; and 5) from all snow events, derivation by regression of an average relation between the power-law coefficients in the mass and velocity dimensional relationships (setting the exponents to the fixed values of 2 and 0.18). This relation is representative for the snowflake sizes dominating the snow bulk quantities such as mass content or reflectivity factor. Given all other uncertainties, the variation of the mass and velocity coefficients with particle size is not taken into account. These coefficients and the derived relation are considered to be representative for the range of sizes that dominates the radar signal and where the majority of snowfall mass is located. For very light precipitating snow with an important contribution to IWC of particles smaller than about 1–2 mm, the derived relation is not valid. If applying the relations obtained in this study, it is important to account for possible differences in the definition of snowflake reference size, taken here as maximum side-view dimension.

The evaluation of the results is made by comparing the time series of the reflectivity factor calculated for a derived mass–size relationship for an individual snowflake and applied for the size distribution measured by the HVSD, with reflectivity obtained from the zeroth moment of the spectrum measured by the collocated POSS.

Using the measurements of snow velocity and retrieved snowflake mass together with the observed PSDs, different bulk quantities have been calculated and approximate average relations between them are derived. The sensitivity to the parameter a_mf or a_uf , related by (23), is significant for calculated Z_e, S, IWC, U_Z, and U_M and for the derived expression U_Z–Z_e. However, the coefficients in the power law Z_e–S are not affected by the changes in the interdependent a_mf and a_uf and are in a generally good agreement with the relations obtained from the literature for the same meteorological conditions. Taking into account the fact that the lower size detected by the HVSD is about 0.3 mm, the observed PSDs are truncated mainly at the lower size limit and the calculated relations may be not applicable to the light snow precipitation for which the effect of this truncation can be important.

Because of this interdependence of snowflake mass and velocity, the coefficients in the mass and velocity dimensional relationships selected in the microphysics scheme have to be related to each other to ensure consistency between calculated quantities. The requirement of the self-consistency between snowflake mass and velocity may be partly satisfied by incorporating our retrieved relation (23) into microphysical calculations.

Heavy rimed snow is excluded from our analysis, and consequently the average relationships derived in this study may not be valid for this type of snow, as in the cases of reflectivity higher than about 25–30 dBZ. Our relationships may as well not be representative of other geographical conditions. The sensitivity of the proposed relations to the PSDs parameterization has to be taken into account when applying to higher levels in the atmosphere.

In Part III of our study, we will present the third important element of the bulk snow microphysics parameterization: the PSD representation. Further study will be devoted to relating the developed snow parameterization to the environmental conditions (see Part I).