Mass of different snow crystal shapes derived from fall speed measurements

Meteorological forecast and climate models require good knowledge of the microphysical properties of hydrometeors and the atmospheric snow and ice crystals in clouds. For instance, their size, cross-sectional area, shape, mass, and fall speed. Especially shape is an important parameter in that it strongly affects the scattering properties of ice particles, and consequently their response to remote sensing techniques. The fall speed and mass of ice particles are other important parameters both for 5 numerical forecast models and for the representation of snow and ice clouds in climate models. In the case of fall speed, it is responsible for the rate of removal of ice from these models. The particle mass is a key quantity that connects the cloud microphysical properties to radiative properties. Using an empirical relationship between the dimensionless Reynolds and Best numbers, fall speed and mass can be derived from each other if particle size and cross-sectional area are also known. In this study, ground-based in-situ measurements of snow particle microphysical properties are used to analyse mass as a 10 function of shape and the other properties particle size, cross-sectional area, and fall speed. The measurements for this study were done in Kiruna, Sweden during snowfall seasons of 2014 to 2019 and using the ground-based in-situ instrument Dual Ice Crystal Imager (D-ICI), which takes high-resolution sideand top-view images of natural hydrometeors. From these images, particle size (maximum dimension), cross-sectional area, and fall speed of individual particles are determined. The particles are shape classified according to the scheme presented in our previous study, in which particles sort into 15 different shape 15 groups depending on their shape and morphology. Particle masses of individual ice particles are estimated from measured particle size, cross-sectional area, and fall speed. The selected dataset covers sizes from about 0.1 mm to 3.2 mm, fall speeds from 0.1 m s−1 to 1.6 m s−1, and masses from 0.2 μg to 450 μg. In our previous study, the fall speed relationships between particle size and cross-sectional area were studied. In this study, the same dataset is used to determine the particle mass, and consequently, the mass relationships between particle size, cross-sectional area, and fall speed are studied for these 15 shape 20 groups. Furthermore, the mass relationships presented in this study are compared with the previous studies. For certain crystal habits, in particular columnar shapes, the maximum dimension is unsuitable for determining Reynolds number. Using a selection of columns, for which the simple geometry allows the verification of an empirical Best number to Reynolds number relationship, we show that Reynolds number and fall speed are more closely related to the diameter of the


Mass derivation
The motion of hydrometeors when free-falling through the atmosphere establishes an equilibrium between two forces; the gravity and the aerodynamic drag. The resulting particle settling speed is called fall speed v. Thus, the fall speed is governed by the physical properties of the hydrometeors, including their mass and projected area, and it involves aerodynamic principles and environmental conditions. The gravitational force is proportional to the particle mass m, while the frictional or drag force 90 is proportional to both the particle projected area, i.e. the cross-sectional area A, and the square of its fall speed v. The force balance yields where g is the gravitational acceleration, ρ a the air density, and C D the drag coefficient. To determine v from the particle properties m and A using this equation, the drag coefficient C D has to be known as well. However, C D depends on maximum 95 dimension, shape, and on v itself. To circumvent these interdependencies, one can first determine the Best number X = C D ·Re 2 by rearranging Eq. 1 together with the Reynolds number where η is the dynamic viscosity of air, to get an expression that does not depend on fall speed v: (3) 100 Thus, X can be calculated from the particle properties D max , A, and m. If the relationship between Re and X is known, one can determine Re from X. In these circumstances, Eq. 2 provides the fallspeed, v. Böhm (1989) provides a relationship between Re and X for snow particles, which is shown here in the form given by Mitchell (1996) Re = δ 2 0 4 ·   1 + 4 · X 1/2 δ 2 0 · C 1/2 0 where δ 0 and C 0 are unit-less constants, and uses it together with the approach described above to determine v from the particle 105 properties D max , A, and m.
In a similar approach, one can determine particle mass if D max , A, and v are known. For this, Re is determined from v and D max using Eq. 2. Then, X is determined from Eq. 4 solved for X Finally, m is added to the dataset using Eq. 3 110 m = X · A · η 2 2 · g · ρ a · D 2 max , where the atmospheric conditions can be accounted for each particle by adapting η and ρ a to the measured temperature and pressure.
Instead of using Eq. 4 or Eq. 5 with one set of δ 0 and C 0 for all particles regardless of their shape, as proposed by Böhm (1989), Heymsfield and Westbrook (2010) suggested using a modified Best number X * , replacing X in Eq. 4 or Eq. 5, to 115 correct for effects due to open-geometry shapes. They proposed X * = X · A 1/2 r , where A r = A π 4 ·D 2 max is the area ratio, which is close to 1 for compact shapes and smaller the more open the geometry is. Heymsfield and Westbrook (2010) showed that by using this approach they could reduce errors of determined fall speeds associated to open-geometry particles with low area ratios. Using our data for simple thick columns in shape group (3), we could confirm that their approach is better than the approach by Böhm (1989) without modifying X (see Appendix C). Therefore, here, we use the modified Best number X * . 120 Consequently, Eq. 6 is modified to Note, that then the Best number determined from Eq. 5 is the modified Best number X * . In Eq. 5, we use δ 0 = 8.0 and C 0 = 0.35 from Heymsfield and Westbrook (2010).

Fitting relationships to data 125
Once mass is calculated, we can parameterize the relationships mass vs maximum dimension, m(D max ), mass vs crosssectional area, m(A), and fall speed vs mass, v(m), by fitting the following power laws to our data: 130 v(m) = a m · m 1 µg bm , which represent straight lines on logarithmic plots. Hence, linear least-squares fits to the logarithm of the data yield the param- As seen in Vázquez-Martín et al. (2021), using binned data instead of individual data reduces the data spread so that fitfunctions based on binned data are more robust than fit-functions based on individual data. Therefore, also here the data are first binned into a suitable number of bins before fitting Eq. 8-Eq. 10 to the data. Ten mass bins (for m vs D max and m vs A relationships) and ten fall speed bins (for v vs m) are used, respectively. The bins are spaced such that each bin contains 140 as close to the same number of particles as possible. As a consequence, the bin widths are variable and specific to each shape group, and thereby avoid the problem of individual bins having a disproportional effect on the fit. The binned data consist of the median values for each bin. Then, the m vs D max , m vs A, and v vs m relationships are fitted to the median masses vs median maximum dimensions, median masses vs median cross-sectional areas, and median fall speeds vs median masses, respectively. -Martín et al. (2021) found that about 40 particles in a shape group (currently the lowest number in our dataset is 37) 145 is the limit where binning can still be used. The advantages of binning become prominent only at larger numbers of particles.

Analytical derivation of relationships
These relationships may be useful for parameterizations in models and retrievals and are readily comparable to other studies.
In case a suitable dataset is not available, an alternative to fitting these relationships to measured data, is to derive particle mass analytically from previously determined parameterizations of cross-sectional area vs maximum dimension (A vs D max ), fall 150 speed vs maximum dimension (v vs D max ) and fall speed vs cross-sectional area (v vs A) given by power laws For each relationship, the inverse is also shown as the corresponding parameters are convenient for some of the derivations.
The parameter a corresponds to the cross-sectional area at D max = 1 mm, a corresponds to the maximum dimension at A = 1 mm 2 , a D to the fall speed at D max = 1 mm, a D to the maximum dimension at v = 1 m s −1 , a A to the fall speed at A = 1 mm 2 , and a A to the cross-sectional area at v = 1 m s −1 . The parameters b, b , b D , b D , b A , and b A are the exponents in the power laws.

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The resulting power laws are m(D max ) = π 1/2 · η 2 · γ 4 · g · ρ a · a 1 mm 2 v(m) = 1 m s −1 · 4 · g · ρ a · a D · 1 µg The derivation of these power laws is shown in Appendix B (Eq. B3-Eq. B6). There, also the X vs Re relationship is expressed as power law instead of using Eq. 5. This can be done by approximating Eq. 5 piece-wise in several regions of X with power laws Eq. B1 (with coefficient γ and exponent δ), as done by (Mitchell, 1996). Note, that both methods for deriving the relationships given by Eq. 8-Eq. 10, i.e., either the method described in Sect. 3.1 with fitting detailed in Sect. 3.2 or the alternative derivation from existing relationships described in this section, are equivalent if they are based on the same dataset.

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The two methods will yield the same relationships if both use the same power law approximations of X vs Re and the same atmospheric conditions (given as constant η and ρ a for the whole dataset). Thus, in this study, we have chosen to fit Eq. 8-Eq. 10 directly to our data (Sect. 3.2). This allows using environmental conditions individually for each particle and avoids the need to consider error propagation when deriving new relationships from existing ones.
4 Results and discussions 180 4.1 Results from fitting and correlations The particle masses have been determined from measured D max , A, and v with the method described in Sect. 3.1. The m vs D max , m vs A, and v vs m relationships given by Eq. 8-Eq. 10 are then fitted to the resulting data, now consisting of D max , A, v, and m, for the 15 shape groups using the fitting method based on binned data described in Sect. 3.2. Figure 1 and Table 1 show the results. For simplicity, we use short names included in Table 1 for the shape groups from here on, and Fig. 1 shows 185 their full names. The large spread in the data represented as individual points is apparent in Figs. A1-A3 in Appendix A.
When fitting m vs D max , m vs A, and v vs m relationships to the binned data, we note that, in general, there is a high correlation (0.9 R 2 1) for most shape groups. In the following, we call the correlation coefficients R 2 D , R 2 A , and R 2 m to indicate to which of the three relationships they belong to. For the m vs D max relationship, the only exceptions to high correlations are shape groups (1) Needles, (2) Crossed needles, and (3) Thick columns, as well as (6) Stellar  0.5, have a lower correlation. Only in the case of (10), it is uncertain if the fit function is representative of the measured data, as judged by the low R 2 A . In most shape groups, the coefficients R 2 D and R 2 A are similar. Only the four groups (1), (2), (3), and (10), mentioned above with lower correlation in one of the relationships, have a distinct difference between R 2 D and R 2 A . Of these, the three shape groups (1)-(3) have a better correlation for m vs A than for m vs 195 D max , which is consistent with a better v vs A correlation than v vs D max for the same groups (Vázquez-Martín et al., 2021), given that we have derived m using measured v here.
For the v vs m relationship, all values of R 2 m are 0.85 or higher. These high values indicate that v is better correlated to m than to D max or A (see the generally lower R 2 values reported in Vázquez-Martín et al., 2021). The generally very high correlations are partly also a consequence of m being derived from v, rather than being an independent measurement.  Fig. 1b show the m vs D max and m vs A relationships including, for reference, the mass of liquid water spheres symbolizing rain or fog droplets given by the power laws m = π 6 · ρ w · D 3 max and m = 4·ρw 3· √ π · A 3 / 2 , respectively, where ρ w = 1 g cm −3 is the density of liquid water. The mass of spheres is proportional to D 3 max and to A 3 / 2 . Thus, comparing to Eq. 8 and Eq. 9, one can see that the exponentsb D = 3 andb A = 1.5 for spheres. The values ofã D andã A for spheres are 524 µg, the 205 mass of a droplet with 1 mm diameter, and 752 µg, the mass of a droplet with a cross-sectional area A = 1 mm 2 , respectively.

Slopesb D andb A
The exponentb D for shape groups (12) Graupel and (15) Spherical is close to the value of 3 for spheres, 2.74 and 2.84, respectively. For the same groups,b A is close to the value of 1.5 for spheres, 1.34 and 1.43 for shape groups (12) Graupel and (15) Spherical, respectively. For these shape groups, this is expected due to their spherical or roundish morphology. These 210 exponent values, corresponding to the slopes in Fig. 1a) and b) are among the highest values for all shape groups. Shape groups (6) Stellar and (11) Spatial stellar are the only other shape groups that have similarly steep m vs D max and m vs A relationships. These two groups do not have a roundish morphology that could explain this. However, a slope similar to spherical particles may indicate that the morphology remains similar in these groups independent of size, i.e., ice particles scale equally in all three dimensions. An example for this would be hexagonal plates or columns that all have the same aspect 215 ratio. For pristine stellar particles one may not expect such a steep slope similar to spherical particles, but rather a decreasing area ratio with increasing size. Shape group (6), however, contains other shapes besides pristine stellar particles, such as rimed stellar and split stellar crystals. A particular mix of shapes may cause an apparently steep slope. Indeed, the area ratio in this shape group is approximately constant (Vázquez-Martín et al., 2021). Our dataset does not contain a sufficient number of stellar particles yet to analyse this further, by for example regrouping particle shapes. Additionally, the low number of particles in this 220 group also results in a relatively high uncertainty (b D = 2.61 ± 0.59 andb A = 1.34 ± 0.29).
For most other shape groups, the exponentb D varies between 1.2 and 2, and all otherb A values range between 0.8 and 1.2. Three shape groups, (1) Needles, (2) Crossed Needles, and (3) Thick columns, stand out with the lowest exponentsb D of approximately 0.8 or lower. These can easily be seen in Fig. 1a) as the lines with the most shallow slopes. For these groups, this is understandable due to their morphology. We have seen in Vázquez-Martín et al. (2021) that an increase in 225 D max (needle length) is directly proportional to A, indicating that the diameter of these needle-shaped particles (needle width) remains similar, when D max and consequently also A are growing. Thus, D max is approximately proportional to A, and predictably, bothb D andb A are close to 1 for these three shape groups. Vázquez-Martín et al. (2021), observing the very poor correlation between D max and measured fall speed for these shape groups, argued that D max is not suitable to determine the Reynolds number. Therefore, a more suitable characteristic length than D max should be used to determine Reynolds number 230 and derive mass from it. Otherwise, the derived mass, and consequentlyb D , are likely not useful. Jayaweera (1971) suggested a characteristic length for hexagonal crystals, for which the dimensions of the basal facet and the aspect ratio are known. We have tested this for a small subset of relatively simple columns from shape group (3). Indeed, this charecteristic length, which can easily be determined from the geometric dimensions on the image, is better suited to calculate Re. If it is used instead of D max in our method to determine m (see Sect. 3.1 and Sect. 3.2), thenb D is now 2.4 (for this subset of columns). Furthermore, the 235 mass can be estimated directly from the geometric dimensions, which allows a closure study to confirm the Re-X relationship, which is central in our method (see Sect. 3.1). The details of this closure study are reported in Appendix C.
Unfortunately, for many particles the dimensions of the basal facet and the aspect ratio are not readily available (or are not defined in case of more complex particles). Consequently, the characteristic length cannot be used and one may want to use D max instead. However, as the closure study shows (see Appendix C), for shape groups (1)-(3) D max should not be used as a 240 representative size parameter instead of characteristic length. Thus, mass determined using D max and relationships based on this mass should not be used.
The ratio between the exponentsb D andb A is equal to the exponent b, as can be seen from Eq. 8, Eq. 9, and Eq. 11a. = b, and range between 1.7 to 2.
The exceptions which have ratios much below the line are the two shape groups with the lowest R 2 D , groups (1) Needles and

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(2) Crossed needles. The ratios for shape groups (3) Thick columns, (9) Side planes, and (13) Ice particles are found slightly below the line. Of these groups, (3) and (13) are among the groups showing more uncertainty in the determined relationship, as indicated in Fig. 1a) by the larger confidence regions around the fits. For group (9) Side planes, the uncertainty is smaller and can not explain the lower ratio.
Intuitively,b D , the exponent of the m-D max relationship, should be larger than b, the exponent of A-D max , as confirmed 250 by literature, such as by Mitchell (1996). For some shape groups, however, b is larger thanb D . Not surprisingly, groups (1)- (3) that were noticed earlier for the lowestb D values are among these groups, as well as groups (9) and (10). The latter two were noticed in Vázquez-Martín et al. (2021) with very poor correlations between D max or A and fall speed. This problem likely indicates that for these shape groups D max is not suitable as size parameter to calculate Re. For simple thick columns, this is demonstrated in Appendix C. While suitable substitutes exist for regular shapes, such as the characteristic length suggested by 255 Jayaweera (1971), for an arbitrary shape our current image analysis methods cannot determine a similar quantity. Thus, the modified X * approach according to Heymsfield and Westbrook (2010) remains the best alternative for our study, it lessens the problem considerably for groups (1)-(3).

Coefficientsã D andã A
All relationships but those of shape group (15) Spherical form a cluster of lines located in a smaller region in both Figs. 1a) 260 and b). The only relationship found outside this cluster is that of shape group (15), which, if extrapolated towards larger sizes or cross-sectional areas, predicts larger masses than any relationship of the other shape groups. The fit coefficientsã D andã A reflect this since they predict the mass at the unit reference of 1 mm forã D and 1 mm 2 forã A . These values are much larger for spheres,ã D = 260 µg andã A = 404 µg, respectively, than for any other shape group. The second-largest values are for shape group (12)  As can be seen in Figs. 1a) and b), the power laws for (15) are close to the reference lines for liquid droplets, however, 270 predicting somewhat lower masses. These differences may be due to several reasons. While shape group (15) Spherical may contain liquid droplets, it also contains ice particles that have a lower bulk density ρ ice compared to the bulk density of liquid water ρ w . Also, the small frozen rain droplets that shape group (15) contains, are not perfectly spherical, which leads to overestimating mass if assuming a spherical shape. Furthermore, sizing errors cause an apparent error in fall speeds. Overestimating the size leads effectively to too low v, which in turn yields too low derived m. i.e. groups (1-3) and (6). Rather than the slopes, different speeds at any given mass distinguish the different shapes. This can 280 be seen with the values of a m , representing the fall speed predicted by the relationships at the mass given by the reference unit of 1 µg. However, 1 µg is below the masses usually encountered for most shape groups. Therefore, it is more instructive to evaluate predicted fall speeds closer to the median of masses in the dataset. At a mass of for example 3 µg, the fall speeds vary between 0.14 m s −1 and 0.53 m s −1 as seen in Fig. 1c). The highest four fall speeds at this mass correspond to shape groups

Comparison with previous studies
The mass vs particle size (m vs D) and fall speed vs mass (v vs m) relationships of the common shapes plates, dendrites, graupel, and spheres, i.e. for our shape groups (5) Heymsfield and Kajikawa (1987) H87, Kajikawa (1989) K89, Mitchell (1996) M96, andErfani andMitchell (2017) E17 and are listed in Table 2. For comparison with v vs m (see Fig. 3d) of our shape groups (12) Graupel and (15) Spherical, parameterizations of measurements by L74 (see also Fit to all data Fit to shape groups 68% Confidence region confidence region for the fits is also shown. a) The m vs Dmax relationships. For comparison, the mass of spheres, corresponding to rain or fog droplets, given by m = π 6 · ρw · D 3 max , where the density ρw = 1 g cm −3 , is shown as a grey dashed line. b) The m vs A relationships. For comparison, the mass of spheres given by m = 4·ρw 3· √ π · A 3/2 is shown as a grey dashed line. c) The v vs m relationships.
Eq. 2 using Re calculated from the Re vs X relationship Eq. 4 and X given by Eq. 3 for spherical particles having a density ρ w = 1 g cm −3 is added to the v vs m relationships in Fig. 3. This line will be referred to as [Re-X]. = b (derived from Eq. 8, Eq. 9, and Eq. 11a).
Depending on the study, the particle size D was defined somewhat differently. For VM21, as well as for H87, E17, K89, and M96, D corresponds to D max . For L74, D is the diameter of an estimated circle that has the same cross-sectional area as the 300 imaged particle.

Plates
We note that for plates (Fig. 3a), the m and D relationship for crystal with sector-like branches (P1b) reported by M96 is most similar to VM21. It is similar also with respect to its slope given by the value ofb D , all other relationships are steeper with larger values ofb D . Reasonably close to VM21 is also the relationship for hexagonal plate by M96, which, however, is 305 heavier at larger sizes than about 1 mm. For those larger sizes hexagonal plate by M96 is similar to hexagonal plate by H87, the latter having the steepest slope (withb D even larger than 3). The relationship for P1b by H87 predicts the lightest particles below about 2 mm. At about 2 mm, it is also similar to VM21 and P1b by M96. Thick plates by H87 are heavier at larger sizes, but similar to VM21 at about 0.2 mm. Our relationship VM21 for shape group (5) has a lower slopeb D than any of the other relationships from previous studies. Chen (1992) approximated hexagonal plates with spheroids and found a theoretical 310 lower limit of 2 forb D of plates, which the value 1.76 of VM21 seems to violate. While the selected previous studies with b D values larger than 2 looked at particular shapes, VM21's shape group (5) represents a mixture of plate-like shapes such as rimed plates, split plates, and double plates. Two of the shapes are represented with more than 40 particles, namely rimed plates (R1c) and double plates (P1o, see Vázquez-Martín et al., 2020), sufficient to determine their own relationships. As can be seen in Table 2

Dendrites
For stellar particles (Fig. 3b), several m and D relationships are fairly close to VM21, for example the two relationships for 320 P2a from H87, which has a similar slope, and K89, which has a lower slope than VM21. Relationships for another stellar particle type, P1d, are still relatively close to VM21. For example the one by M96 crosses VM21 somewhat above 1 mm and is higher for smaller particles, whereas the one by H87 is about a factor of 2 below VM21. This P1d relationship by H87 may be compared to the rimed stellar (R1d) of the same study H87. These two curves differ by about a factor of two, with the rimed stellar having the larger mass and being very close to VM21. Riming of stellar particles adds mass without 325 increasing their size noticeably (Erfani and Mitchell, 2017) as seen in E17, which explains the difference seen between the two mentioned relationships of H87. A similar difference is seen between the two relationships by E17 from a dataset used to study effects of riming. However, the two relationships by K89, which also feature unrimed and rimed stellar particles, respectively, do not show a significant difference. Particles included in shape group (6) Stellar of VM21 include cases of light riming. Distinguishing between unrimed and rimed stellar particles in the data of group (6) resulted in two relationships (not 330 shown) that are both, within uncertainties, identical to the one produced from all data in shape group (6).

Graupel and spheres
L74 reported three m vs D relationships for lump graupel (R4b) corresponding to three different particle densities with larger masses predicted by the relationships for higher densities. Our relationship for graupel is between L74's low and medium density relationships (Fig. 3c). It is well approximated by the mass of spherical particles with a density of 0.12 g cm −3 (not 335 shown in Fig. 3c), which is at the lower end of the density range reported by L74 for their medium density relationship (>0.10 to 0.25 g cm −3 ). The relationship by H87 for lump graupel (R4b) is similar to L74's medium density. The relationship by E17 agrees also with VM21, but only around 1 mm, as their relationship has a much lower slope (b D = 2.16) than all other relationships for graupel (2.7 to 3.1). The mass of liquid water spheres m = π 6 · ρ w · D 3 that was shown on Fig. 1a) is added also to Fig. 3c) as reference. Its comparison with VM21's line for shape group (15) Spherical is discussed in Sect. 4.2.2.

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The v vs m relationships from L74 (Fig. 3d) come, within their ranges, close to our relationship for shape group (12). In general, at a certain particle mass, the size and cross-sectional area, and thus the drag force, decrease with increasing graupel particle density. This can be seen, to some extent, for the three lines by L74. However, their lines have different slopes in a way that makes them intersect with each other. Their slopes are more shallow than the relationship of VM21, consequently they also cross that line. The slope for graupel of VM21 is more similar to that of the relationships related to spherical particles than the lines for graupel by L74. Consequently it approaches spherical particles, which represent an upper limit in speed, at a lower mass than the lines by L74.
The lines for spherical particles of G49 and [Re-X] are very close to each other, thus [Re-X] predicts well these measurements. The straight line for the shape group (15) of VM21 is at somewhat lower fall speeds below approximately 10 µg. All data but two particles in shape group (15) have m below that mass. For those two particles heavier than 10 µg the fit line VM21 350 over-predicts mass (see Fig. A3 in the Appendix). While VM21 represents the power-law fit to our measurements of droplets and spherical and almost spherical ice particles, the two curved lines of G49 and [Re-X] represent only liquid droplets, and, thus, an upper limit in fall speed.

Summary and conclusions
This manuscript presents new mass and fall speed parameterizations derived from D-ICI measurements of natural snow, ice 355 crystals and other hydrometeors, covering sizes from 0.1 mm to 3.2 mm. Using the dataset and fall speed vs maximum dimension and vs cross-sectional area relationships from Vázquez-Martín et al. (2021), where fall speeds cover ranges from 0.1 m s −1 to 1.6 m s −1 , in this study, we have added particle masses to our dataset of measured maximum dimension, cross-sectional area, and fall speed of individual particles. The calculated values of individual particle masses range from close to 0.2 µg to 450 µg Mitchell (1996) presented fall speed relationships derived from power laws of cross-sectional area and mass vs maximum 360 dimension using a relationship between Re and X. We calculate particle mass data from our measurements of maximum dimension, cross-sectional area, and fall speed using the same Re-X relationship. With this new extended dataset, mass vs maximum dimension relationships, mass vs cross-sectional area, and fall speed vs mass, given by Eq. 8-Eq. 10, have been derived and studied for different particle shapes. We present the conclusions that our results led to below.
• As seen in Figs. A1-A3 in Appendix A, and discussed in Section 4.1, the data's large spread is apparent. However, when 365 fitting m vs D max , m vs A, and v vs m relationships to binned data, there are high correlation coefficients for most shape groups, with values between 0.9 and 1. The only exceptions are shape groups (1) Needles, (2) Crossed needles, (3) Thick columns, (6) Stellar, and (10) Spatial plates for the m vs D max relationship with R 2 D 0.7, as well as for the m vs A relationship shape groups (2) and (6) with R 2 A 0.8 and (6) with R 2 A 0.5. While for all other shape groups R 2 D and R 2 A are similar, for these groups with lower R 2 , R 2 D is lower than R 2 A for all but shape group (10), for which R 2 A 370 is lower. For v vs m, there is a good correlation for all 15 shape groups (see Table 1). The fact that m is derived from v contributes to a stronger correlation between both quantities.
• For the three shape groups related to columnar or elongated shapes, i.e. shape groups (1)-(3), width rather than length or D max is more closely related to a suitable characteristic length to determine Re (see Sect. 4.2.1 and Appendix C).
Consequently, mass and relationships with it are not reliable. For these shape groups,b D is close to or smaller than 1.

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Additionally, contrary to expectationsb D is larger than b and the ratio of exponentsb D tob A is too low for these groups.
For most other shape groups it is similar to b, as theoretically expected. Shape groups (9) and (10) (the latter with low   Gunn and Kinzer (1949) [G], Locatelli and Hobbs (1974) [Lo], Heymsfield and Kajikawa (1987) [H], Kajikawa (1989) [K], Mitchell (1996) [M], and this study [VM] are shown. In c), the line by [G] corresponds to the mass of spheres given by m = π 6 · ρw · D 3 max that was shown also in Fig. 1a). In d), for comparison, a line for speeds determined from Eq. 2 using Re from Eq. 3 and Eq. 4 for spherical particles with density ρw = 1 g cm −3 is added as a red dashed line. This line is referred to as  Table 2). number of particles and low correlations in relationships) show similar limitations when comparing with b. Therefore, as long as a more suitable size parameter is not available in our dataset for these shapes, mass derived from Re for these shape groups should only be used with great caution.

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• On a selection of 75 simple columns from shape group (3), we have done a closure study (see Appendix C) to confirm the Re-X relationship, which is central in our method (see Sect. 3.1) and used by many other studies. For this, the widths and lengths of the columns have been determined in addition to D max . From these geometric dimensions, the masses of the columns have been estimated directly. Then, from each column mass, the Best number X has been determined using Eq. 3. Thus, Re and X have been determined independently and consequently compared to the X-Re relationship given 385 by Eq. 5. This closure showed the superiority of the characteristic length L * (Jayaweera, 1971) over D max , confirming that D max is not suitable to calculate Re and X using Eq. 2 and Eq. 3, respectively, for columns. The closure study also showed that using the modified Best number X * (Heymsfield and Westbrook, 2010) instead of the Best number X improved the agreement. The best closure for our subset of simple columns was achieved when using both characteristic length L * and modified Best number X * together.

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• When deriving the m vs D max , m vs A, and v vs m relationships analytically from A vs D max , v vs D max , and v vs A given from a suitable dataset (see Section 3.3), the results are equivalent to fitting to the same dataset after adding m for individual particles derived from v (see Sect. 3.1). On the one hand, fitting m vs D max , m vs A, and v vs m relationships to data has the advantage that the X-Re relationship from Eq. 5 can be used rather than power-law approximations required for the analytical derivation of the same relationships (see B in Appendix). On the other hand, if a suitable 395 dataset is not available but power-law relationships for A vs D max , v vs D max , and v vs A are, the analytically derived mass relationships Eq. 14-Eq. 16 can be used.
• The parametersb D andb A , i.e. the slopes of the m vs D max , m vs A power laws, respectively, are highest for the shape groups (6) Stellar, (11) Spatial stellar, (12) Graupel, and (15) Spherical. For groups (12) and (15)  • We compared our m vs D max and v vs m relationships with other mass relationships given by previous studies. The 405 shape groups compared in this study are (5) Plates, (6) Stellar, (12) Graupel, and (15) Spherical. Our results agree reasonably well with the references used.
• For graupel and spheres, (Section 4.4.3), Locatelli and Hobbs (1974) Lo74 reported m vs D relationships for lump graupel (R4b) with different particle densities (high, medium, and low). Our relationship for graupel is between Lo74's low and medium density relationships, and it is well approximated by the mass of spherical particles with a density of 410 0.12 g cm −3 (not shown in Fig. 3c).
• Looking at v vs m, the two lines for spherical particles of G49 and [Re-X], corresponding to a line for speeds determined from Eq. 2 using Re from Eq. 3 and Eq. 4 for spherical particles with density ρ w = 1 g cm −3 , are very close to each other. We report somewhat lower speeds for the shape group (15) Spherical VM21. This difference may be due to shape group (15) in VM21 consisting of any spherical or almost spherical particle, including ice, whereas the two lines of G49 415 and [Re-X] are exclusively for liquid droplets.
These resulting parameterizations may improve our understanding of precipitation in cold climates and improve the microphysical parameterizations in the climate and forecast models. Through these relationships, we can determine particle masses based on fall speed and particle sizes.
Data availability. The presented data will be available at the Swedish National Data Service (DOI will be available). Table 2. The m vs D and v vs m relationships of previous studies given by Locatelli and Hobbs (1974) L74, Heymsfield and Kajikawa (1987) H87, Kajikawa (1989) K89, Mitchell (1996) M96, andErfani andMitchell (2017) E17 are shown for some shapes that were selected for the comparison and correspond to (5) Plates, (6) Stellar, (12) Graupel, and (15) Spherical. The power laws for M96 have been determined by using equation [15] in Mitchell (1996). The relationships in this study (see Table 1) have been found by fitting Eq. 8-Eq. 10 to our data as described in Sect. 3.2. Those selected for comparison are also shown here as VM21. The snow particles type, the total number of particles N , ranges of particle sizes D, mass m, fall speeds v, the m vs D, v vs m relationships, the correlation coefficient (R 2 ), and the reference of the studies are displayed. In these references, the particle sizes are defined somewhat differently. In VM21, as well as H87 and M96, D is defined as Dmax. Magono and Lee (1966)'s symbols are sometimes added for shape clarification. These m vs D, v vs m relationships are shown in Fig. 3. The power laws from the literature have been converted in order to have the same units, so that mass m is in µg, particle size D in mm, and fall speed v in m s −1 .
Shape group (5)   Appendix B: Mass derivation using power laws The particle mass relationships are derived analytically from a relationship between the Reynolds and Best numbers, in addition to A vs D max , v vs D max , and v vs A power laws given by Eq. 11a-Eq. 13a. Section 3 has briefly presented this approach of deriving the particle mass analytically. The m vs D max , m vs A, and v vs m relationships given by this approach are equivalent 435 to fitting to individual data. Indeed we get identical results in theã D ,b D ,ã A ,b D , a m , b m parameters if using X * vs Re as power law where γ and δ are the parameters in the power law. We determine these parameters by fitting the power law to Eq. 5 over ranges of Re corresponding to each shape group. For this, we first calculate Re for all particles in a shape group and determine X * 440 using Eq. 5 for this set of Re values. Then, we do a linear fit to the logarithm of X * vs logarithm of Re. Consequently, for each shape group, we get one set of γ and δ.
We express Re as a power law in D max using Eq. 2 and replacing v with the power law given by Eq. 12a Now we can determine the particle mass m using Eq. 7 and express it as a function of particle size D max , area A, or fall 445 speed v. Consequently, the mass relationship as a function of particle size D max given by Eq. 8 can be derived as follows (using Eq. B1, Eq. B2, Eq. 11a, and the area ratio The mass relationship as a function of A given by Eq. 9 can be derived as follows (using Eq. B1,Eq. 11b,and Eq. 13a,and expressing Re as a power law in A) The mass relationship as a function of v given by Eq. 10 can be derived as follows (using Eq. B1, Eq. 12b, and Eq. 13b, and expressing Re as a power law in v) From Eq. B5, we can determine v(m) as follows (B6) Appendix C: Closure study-Reynolds and Best numbers for simple thick columns Selecting a simple shape with area ratio noticeably below 1, we can test if the modified Best number approach by Heymsfield and Westbrook (2010) yields better results than using Best numbers and the approach by Böhm (1989) (see Sect. 3.1 for details about these approaches). If Reynolds and Best numbers (Re and X, respectively) can be determined independently, i.e., without using any X-Re relationship, then they can be compared with the X-Re relationship by Böhm (1989) given by Eq. 5. Thus, it represents a closure study that can confirm the X-Re relationship, which this and other studies rely on. The following explains the method and results when applied to simple columns, a small subset of our data.
For a simple-geometry shape we can calculate the particle mass from the geometrical dimensions and, thus, determine both X and Re independently. Then X vs Re, or alternatively X * vs Re can be compared to the empirical relationship given by 465 Eq. 4 or Eq. 5. Needles or columns would be suitable shapes as they have low area ratios and a simple geometry. Looking at particles in the shape group (1) Needles reveals that it contains many bundles of needles and only few pristine needles. Shape group (3) Thick columns, on the other hand, contains many simple columns. Therefore, we have selected 75 columns from shape group (3) for this comparison study. Figure C1 shows examples of the selected columns.
Most columns fall horizontally so that width and length can be easily determined from the top-view images. We estimate 470 that the length may be underestimated on the order of up 15% due to deviations from alignment of the column axis in the image plane. On the other hand, the geometrically determined mass, m geom , may be overestimated for part of the columns that show signs of cavities or hollowing of faces (see Fig. C1).
For columns, D max , which is similar to the column's length, is not a suitable representative size parameter to determine Re, as we have discussed in Sect. 4.2.1 and Vázquez-Martín et al. (2021). A characteristic length L * = A t /P , where A t is the total 475 surface area and P the perimeter of the particle projected to the flow (see Eq. 13-81 in Pruppacher and Klett, 2010), can be used instead. For columns, L * can be determined from width and length (Jayaweera, 1971) and is more closely related to the width.
Now, Re can be determined from measured fall speed and L * . The Best number, according to Eq. 3, can be determined from measured cross-sectional area A and D max . Note that here, D max represents the same size parameter best suited to calculate Re as in Eq. 2. Thus, not only Re should be determined from L * (instead of D max ), but also X. Then, X can be determined 480 from measured A in addition to calculated m geom and L * .
Consequently, X vs Re can be plotted and compared to the X-Re relationship (Eq. 5). Figure C2 shows X vs Re determined either using D max or L * . The points related to D max (blue triangles) do not match well the empirical relationship X-Re (with δ 0 = 5.83 and C 0 = 0.6) by Böhm (1989) based on a theoretical treatment by Abraham (1970). This confirms that, as argued above, D max is not suitable to determine Re or X for this shape. The points X vs Re determined using L * , on the other hand, 485 are much closer to the empirical relationship. Thus, this closure experiment comparing independently determined Re and X to the X-Re relationship demonstrates the superiority of characteristic length L * over D max as a particle size parameter when dealing with particle mass m.
The points X vs Re can be transformed into X * vs Re where X * = X · A 1/2 r is the modified Best number suggested by Heymsfield and Westbrook (2010). The resulting points (using L * ) are also shown in Fig. C2 (green 'x') and provide an even 490 better closure to the empirical X-Re relationship. Heymsfield and Westbrook (2010) used D max and not characteristic length L * (they focused on shapes with open geometries for which characteristic length is difficult to determine). The closure for our columns using X * and D max (cyan crosses in Fig. C2) is not as good as using X * and L * , but still better than using the unmodified Best number X and D max . Thus, for columns we can conclude that the modified Best number represents an improvement over the Best number. In addition, the superiority of characteristic length L * over D max for columns is given also 495 when working with the modified Best number X * .
In addition to the empirical relationship X-Re by Böhm (1989), also the relationship by Heymsfield and Westbrook (2010) (δ 0 = 8.0 and C 0 = 0.35) for their the modified Best number approach, used in our study, is shown in Fig. C2. The two lines are relatively close to each other. Thus, the above conclusions of superiority of characteristic length L * over D max and improvement when using modified Best number rather than Best number remain valid regardless of which relationship is used 500 as comparison.  Figure A1. Mass vs particle size (m vs Dmax) relationships given by Eq. 8-Eq. 10 for all the shape groups are shown. Individual data (coloured symbols) and binned data (blue symbols with error bars) are displayed. Median values in the respective bins represent the binned data. The total length of the error bars represents the spread in mass data, which is given by the difference between the 16 th and 84 th percentiles. The relationships fitted to binned data are shown. The 68% prediction band and the 68% confidence region for the fits are also shown. The same data are shown in Table 1.  Figure A2. Same as Figure A1, but mass vs cross-sectional area (m vs A) relationships given by Eq. 8-Eq. 10 are shown here.   Using D max and X * Using L * Using L * and X * Figure C2. X vs Re and X * vs Re for simple thick columns selected from shape group (3) Thick columns. X and Re are determined either using Dmax or L * . The points X vs Re using L * are much closer to the empirical relationship (Eq. 5) than the points using Dmax. Using the modified Best number X * instead of the Best number X leads to a better agreement with Eq. 5. For comparison, the empirical relationship given by Eq. 5 is shown with parameters from Böhm (1989) and Heymsfield and Westbrook (2010), respectively. They are, however, very similar.