General Comments:

This paper is the first to demonstrate the calculation of ice particle mass from measurements of ice particle maximum dimension, projected area and fall velocity, and in doing so, it represents a test of hydrodynamic flow theory.  However, as argued below, the method appears successful for graupel and quasi-spherical ice particles, but less successful for planar ice crystals (e.g., stellars or dendrites) and definitely not successful for needles and columnar ice crystals.  Similar findings were reported by Heymsfield and Westbrook (2010, JAS; henceforth HW2010), where the fall speed treatment of Mitchell (1996, JAS; henceforth M96) worked well for ice crystals having aspect ratios closer to unity (e.g., graupel, short columns, thick plates) but not well for stellars and needles having more extreme aspect ratios.  Therefore, two new approaches are offered for modifying the methodology described in this study. 

One approach is to define the Reynolds number Re in terms of a characteristic length L*, rather than maximum dimension D_max, where Re_L* = V L*/ν, where V = terminal fall speed and ν = kinematic viscosity of air = η/ρ_a where ρ_a = air density and η = dynamic viscosity of air.  This L* was found to describe the vapor mass and heat transfer from a ventilated ice crystal well and hence captures the flow effect on ice crystal growth.  L* is defined as

            L* = A/P                                                                                                                       (1)

where A = total surface area of an ice particle and P = ice particle perimeter projected to the flow.  See Pruppacher and Klett (1997), Microphysics of Clouds and Precipitation, Kluwer Academic Publishers, p. 552, for more details.

            Fortunately, Jayaweera (1971, JAS) has formulas describing L* for planar and columnar ice crystals.  For planar ice crystals, L* = (d/2) (1 + 2e) and

            Re_L* = 0.5(1 + 2e) Re_d                                                                                                  (2)

where subscript d on Reynolds number Re indicates that Re is evaluated using the diameter d of a circle having the same area as the basal face of the ice crystal.  For hexagonal plates, d = 0.910 D, where D = maximum dimension of the basal face.  Moreover, e refers to the ratio of minor to major axis.  Reasonable estimates for e can be obtained from D_max and Auer and Veal (1970, JAS).  For columnar ice crystals, L* = (π/4)d [1 + (1/(1+e))] and

            Re_L* = (π/4) [1 + (1/(1+e))] Re_d .                                                                                 (3)

Notice here that Re_L* depends on the “diameter” or thickness of a column and not its maximum dimension D_max (as was used in this ACPD paper).  This indicates V is most related to d.  By substituting Re_Dmax used in this ACPD paper with Re_L* in Eqn. 5 to calculate Best number “X”, and substituting D_max with L* in Eqn. 6 of this paper (while using this new calculation for X), the m-D_max power laws found here for columnar and planar ice crystals may be improved, having greater consistency with the body of theoretical and empirical knowledge (discussed below under Major Comments).

            The second approach is to calculate the “modified Best number” X* from Eqn. 5 of this ACPD paper (as described in HW2010) by redefining the constants used to calculate X*, where C₀ = 0.35 and δ₀ = 8.0.  However, in this case Re = Re_Dmax (as originally used in this paper) since this maintains consistency with HW2010.  Then mass m is calculated by inverting the HW2010 definition of X*:

            m = π η² X* A_r^1/2/( 8 g ρ_a )                                                                                           (4)

where m = ice particle mass, g = gravity constant and A_r = area of ice crystal normal to the flow divided by area of circle having same maximum dimension (referred to as the area ratio).

However, if the “thick columns” shape category in this study corresponds to short, thick columns, the M96 fall speed scheme should work fine for this shape category, and this second approach may not address the problem for this shape.

It is possible that neither of these alternative approaches will render improved results, but it seems worth a try.  If there is no improvement, the limitations described below will need to be mentioned in the paper.

Major Comments:

1. Lines 149-150: Please indicate which relationships in Tables 1 and 2 are based on the approach described in Sect. 3.3 vs. the approach given in Sect. 3.2.  Since this approach involves empirical relationships already having considerable uncertainty, m(D), m(A) and v(m) from this approach may have greater uncertainty than the previous approach (Sect. 3.2) due to the propagation of uncertainties in the empirical expressions used here.  This knowledge may be helpful for interpreting the results in Tables 1 and 2.

1. Lines 185-188 on stellar ice crystals: This same argument is expressed more quantitatively in Mitchell et al. (1990, Sect. 4a), where hexagonal crystal volume V is approximated by using a circular basal face so that aspect ratio k = c/a = c/r, where r = radius of this circle having the same area as the basal face.  Moreover, c and a are the semi-axes corresponding to the prism and basal faces of an ice crystal.  Thus, V = π r² k r = π k r³ for this approximation, and for constant density and constant k, the m-D relationship has a power of 3.

However, constant k is an invalid assumption for a stellar ice crystal that only forms between ~ -14 and -16 °C.  As described in Chen and Lamb (1994, JAS), k is rarely constant (definitely not at these temperatures), and depends on the ratio of condensation coefficients for the basal and prism crystal faces.  This is referred to as the inherent growth ratio (IGR), and IGR is related to m-D power laws in Sect.  7 of their paper.  I strongly recommend that the authors read this paper and then revise this commentary accordingly.  This information is also in the cloud physics text book by Lamb and Verlinde, Physics and Chemistry of Clouds (2011, Cambridge Univ. Press).  In addition, Jerry Harrington's group at Penn. State Univ. has greatly extended this work through several publications (e.g., Harrington et al., 2013, JAS, "A method for adaptive habit prediction in bulk microphysical models. Part I: Theoretical development).

Physical intuition also informs us that slope b_D is too high since b_D is a measure of the increase in mass with respect to size.   A large (i.e., steep) slope indicates a relatively large mass increase per unit size increment, but this is not true for stellar or dendrite ice crystals since their ice density decreases with increasing size.

1. Lines 190-196: These shapes are all columnar, which may be a clue to the problem here.  Reynolds Number Re is expressed by (2) in terms of D_max, but due to hydrodynamical considerations, some argue that D_max should be replaced by a "characteristic length" L* defined in Pruppacher and Klett (1997) as: L* = A/P, where A = total ice particle surface area and P = particle perimeter normal to the flow.  For needles, changes in A and P will be roughly proportional, with L* changing much less than D_max.

As shown by Jayaweera (1971, JAS), L* is strongly related to the column radius (or basal face semi-axis) and weakly related to D_max, indicating the formulation of Eqns. (2) and (5) in terms of D_max are flawed based on Re_L*.  This is indeed the case for hexagonal columns.

1. Lines 196-199: Note that b_D < 1.0 also for thick columns (C1e; group 3), with b_D = 0.81.  Compare this with Tables 1 & 3 in Mitchell et al. (1990, JAM)), where C1e b_D =6 is found to be consistent with other studies based on dimensional-density relationships.  Moreover, b_D = 2.6 is very consistent with the theoretical prediction of Chen and Lamb for C1e b_D (see their Fig. 12). This is strong evidence that the C1e b_D in this current study suffers from some limitation.  Please expose this issue for the readers.

1. Section 4.4.1 on plates: Chen and Lamb (1992) provide theoretical limits for columnar and planar ice crystals regarding b_D, where b_D for hexagonal plates lies between 2.0 and 3.0.  In this current study for plates, b_D = 1.72, indicating its value should be treated with caution; please make readers aware of this.  Since side planes grow diffusionally through a different mechanism (Furakawa, 1982, J. Meteor. Soc. Japan), it is not clear whether these limits apply to side planes.  Please mention this.

1. Figure 3 caption, last sentence: The text indicates that this should be valid for all ice particles and not just spheres; please make this clear.

1. Lines 290-291: Can it be said that Re-X represents the fall speed upper limit?

1. Summary and conclusions; 1^st bullet: Will the fact that m is derived from v produce a co-variance that contributes to the stronger correlation between v and m?  If so, please mention this wherever it is most appropriate.

1. Summary and conclusions; 2^nd bullet: Will not the power-law approximations have greater uncertainty than the relationships based on Eq. 5?  Please address this concern wherever it is most appropriate.

1. Summary and conclusions; last sentence of 3^rd bullet: But we know this is not true based on Chen and Lamb (1994, JAS) and other m-D measurements for stellar crystals.  Please remove this last sentence.

1. Line 328-330: I don't see this relationship plotted (spherical ice having a density of 0.12 g cm^-3).  If it is not shown, then please add in parentheses, "not shown".

Minor Comments:

1. Line 150: Apparent typo; Sect. 3.1 => 3.2?

1. Lines 248-9: It appears that Ma has not been defined.

1. Line 329: The second comma is not needed.

Best wishes for a successful study,

David Mitchell

I have read the manuscript “Mass of different snow crystal shapes derived from fall speed measurements” by Vázquez-Martín, Kuhn, and Eliasson and I find it an interesting manuscript.  The authors use a dataset of measurements of 2461 ice crystals including fall speed, area, and maximum dimension.  From these data they derive particle mass using standard Reynolds – Best number approaches.  The dataset is then broken down into 15 different particle habit classifications and relationships between maximum dimension and the other parameters are presented and compared to previous literature. 

While it is a nice study that merits being published, there are a few shortcomings that I feel should be addressed before full acceptance. 

The project is highly dependent on the Re – X number relationship for relating mass and terminal velocity.  Traditionally, this is used in the other direction where Vt is being calculated as opposed to using Vt to pull out mass.  I would like to suggest that the authors should consider the Heymsfield and Westbrook (2010) relationship.  In that publication, they showed that a small modification to the Re-X relationship led to much better Vt calculations.  The adjustment was a factor of the square root of the area ratio.  It should be simple to determine the area ratio for the particles in the dataset and test this.  Would the use of this modification improve the results (tighten the scatter)? 

The dataset also presents an interesting opportunity for manual comparisons, which would also assist in the selection of the traditional Re-X relationship or the Heymsfield and Westbrook modification.  The dataset is said to contain 317 needles and 103 “thick columns”.  I expect that for at least a portion of these, it should be possible to geometrically estimate the mass.  These particles will have the lowest area ratio values and thus would be impacted the most by the Heymsfield and Westbrook modification.  It would be a great addition to the manuscript to include a small closure study between Vt, mass, and area with the Re-X relationship, either the traditional relationship or the modified relationship.

The authors go to great efforts to present results by different shape categories.  As they point out though, remote sensing cannot identify the microphysical characteristics of particles, so shouldn’t we be emphasizing overall characteristics of the particle populations?  Rather than emphasizing the differences in characteristics between different particle categories, I’d suggest showing more “average values” and characterize the uncertainty that could exist.  I seriously doubt that weather modeling will get to the point where we can predict 15 different particle categories, so understanding the intimate details of each isn’t going to be too critical too soon.

Other comments:

The “Dataset” section could have an additional paragraph with a brief description / summarization of how the dataset was created.  While the reader can obviously go to the two Vázquez-Martín et al publications for more information, a one paragraph summary could potentially save the reader some time.

Fitting the dataset:  I applaud the appropriate use of data.  Reasonable binning and using the median values is something that eludes many.  Good job.

In general, it might be nice to have a table with the symbols / variables used.  You use a lot of sub scripts and super scripts and tildes etc.  While it is easy to follow what each means after reading it a few times, it took me a few times reading the equations to completely follow the variables. 

Table 1:  Some of your shape categories had as few as 37 particles.  When you separated into size bins, did you have a minimum number of particles per bin?  (Did you do six bins of 6-7 particles or 10 bins of 3-4 particles?)  You might consider noting the temperature range that particles in each type were observed if that were possible, and, if possible, the number of days when observations of each particle type were observed.  5 winters of data may not seem biased, but if all 37 graupel particles were observed on one day, then there could be a bias. 

Figure 1:  The “fit to all data” appears to have a tighter uncertainty range.  Also, the size range doesn’t go as far as one might expect.  Again, statistically, how far off would a user be if they used an average value and the particles were mostly one of the categories?

Figure 2:  I’d suggest removing this figure as it doesn’t seem important.

Figure 3:  It would be interesting to look at the ratio of the powers in the mass to D relationship versus the Area to D relationship.  Mitchell 1996 has many comparisons and Schmitt and Heymsfiield 2010 show that this ratio would be ~1.3 for fractal particles and of course, it would be 1.5 for spheres.