Growth-deviation model to understand the perceived variety of falling snow

Growth-deviation model to understand the perceived variety of falling snow J. Nelson College of Science and Engineering, Ritsumeikan University, Nojihigashi 1-1-1, Kusatsu 525-8577, Japan Received: 20 December 2007 – Accepted: 9 January 2008 – Published: 4 March 2008 Correspondence to: J. Nelson (jnelson@se.ritsumei.ac.jp) Published by Copernicus Publications on behalf of the European Geosciences Union.


Introduction
Snow crystals are admired for their symmetry and variety.The source of the apparent six-fold symmetry is understood to arise from the ice crystal structure and growth mode (Frank, 1982;Nelson, 2005), but the source of the perceived variety has not been examined quantitatively.Here we use results from recent snow and cloud studies to better understand this variety.
The number of possible snow crystal shapes is estimated to be immense (Knight  and Knight, 1973;Hallett, 1984).However, these estimates are highly inconsistent 1 and they do not address important details such as the cloud conditions and growth process.Bentley (1901), addressing the cause of the variety, wrote that the various crystal features come from the various "atmospheric layers" passed through during the crystal descent.Later, Nakaya (1954), showed that the basic habit (e.g., columnar or tabular) arose from the crystal's growth temperature, and that the intricate features arose as humidity increased.But for a crystal falling through a cloud, the main cause of the perceived variety remains unknown.This cause is determined as follows.We define in Sect. 2 a growth-process-based measure of perceived shape variety, hereafter diversity, then, in Sect.3, determine the crystal-cloud influences on growth, and finally apply the measure to these influences.In Sect.4, we analyze variations to the initial crystal and trajectory parameters, then, in Sect.5, analyze deviations in updraft speed and air temperature along each trajectory.These variations and deviations are not used to predict specific crystal shapes; rather, they are used to estimate when observable changes occur to a given, yet unspecified, shape.The results in Sect.6 indicate that most of the diversity comes from the deviations, yet the magnitudes are much less than those from previous estimates.

Diversity and growth
Here we define a shape diversity S and then apply it to snow crystal growth.The key growth variable is the temperature-dependence of the prism-face growth rate.

Definition of diversity
Let S equal the logarithm of the number of possible distinguishable shapes (defined below) that can arise during growth.Hence, if each shape is nearly equally likely, then cloud conditions with a large S can produce many shapes.The logarithm is useful because the number of possible shapes can vary by many orders of magnitude.Moreover, variety is conceptually similar to entropy, which is proportional to the logarithm of the number of possible states instead of distinguishable shapes.But to proceed, we need a way to distinguish crystals.

Distinguishing features
Ideally, two crystals are perceived as distinguishable if one cannot superimpose one crystal's image over the other (viewed as in Fig. 1) without noticing a change in either image.That is, two crystals are "the same" to some resolution if the positions of the perimeter and interior lines match when viewed at that resolution.These lines mark regions, hereafter "features" (e.g., Fig. 1a-c, right side), where growth produced a sharp bend in the surface, causing a contrast in light reflection.But instead of examining images, we use the final crystal size to estimate the number of times during growth that a resolvable feature could have started.In this way, the diversity will depend, as it should, on the assumed resolution.

Focus on prism-face growth
The initial position of growth-induced features likely result from growth changes on the outermost faces.In general, these faces can be either basal or prism faces, depending on the temperature.But here we focus on the commonly viewed tabular snow crystals that form between ∼-9 and -22 • C. The prism faces grow fastest in this range, so we assume that prism growth rates determine when new features arise.This temperature restriction is useful because it simplifies the analysis, but it means that the results are valid only when the crystals remain between about -9 and -22 • C.
When the growth-rate slows, not only does the diameter increase at a slower rate, but the prism-face area changes, possibly introducing new features.For a slightly hollowed prism face (e.g., Fig. 2a-c), the pits decrease in size, producing "interior" ridges and wiggles.For a branched crystal (e.g., Fig. 2d-f), the branches widen at the tip, thus altering the perimeter.Other features can similarly arise from growth changes (Nelson, 2005).

Growth changes from temperature changes
The growth rate of the prism face r is proportional to the deviation of the vapor pressure above that of equilibrium, which is proportional to relative humidity.As temperaturedependent surface processes also affect r, both humidity and temperature can independently affect growth.But for the local, in-cloud conditions around a crystal, the humidity is often near the temperature-dependent value for liquid water in equilibrium.This is because snowfall-producing clouds often contain significant liquid water.In such mixed-phase clouds, measurements (Korolev and Isaac, 2006;Siebert et al., 2003) 2 and theory (Shaw, 2000) suggest that the humidity stays near liquid-water saturation.Hence, as a first approximation, we will assume that each crystal grows in an environment in which the humidity is at liquid-water saturation.As such, the air temperature controls r.This rate has a peak near -14.8 • C (Fig. 3), which suggests that we must carefully track the temperature to estimate when growth features can arise.

Crystal trajectories and diversity influences
If collisions and crystal-scale gradients in humidity and temperature are ignored, the above arguments mean that the initial crystal properties and temperature history determine the final crystal form.We now study these influences in detail.

Initial crystals and reference trajectories
Most snow crystals are thought to start as frozen, nearly spherical droplets.Although frozen droplets have various characteristics that may affect subsequent growth, this is poorly understood.So, we characterize the initial crystal solely by its diameter d 0 .
In contrast, we can consider details of the ensuing temperature history.Temperature changes along a crystal trajectory are due to air-temperature inhomogeneities DT ai and the altitude-dependent air temperature.The latter contribution to temperature history is the sum of that from 1) slowly varying altitudes for a crystal in an updraft of constant (path-average) speed, and 2) DT Du from altitude deviations due to updraftspeed deviations Du.Horizontal motion is ignored.
To account for these temperature deviations, we write the crystal temperature at time t as T (t)=T (z(t))+DT (t), where T (z(t)) is the temperature for a crystal lofted in an updraft with the path-averaged speed ū, in which the altitude is z(t), and DT (t) is the total temperature deviation along a trajectory (=DT ai +DT Du ).A given trajectory z(t), hereafter a "reference" trajectory, changes as d z/d t= ū−v, where v is the crystal's The duration of a trajectory depends on ū and how quickly the fallspeed v increases.
The latter depends on crystal shape, with v increasing slowest for the crystals that, surprisingly, grow the fastest (Fig. 3).These crystals fall slowest because, as tabular crystals fall broadside to the airflow, the broadest (and thinnest) crystals expose the greatest area to the airflow, thus having the greatest drag force.Hence, the crystals that grow the fastest also fall the slowest, and end up with the longest growth times.

Three influences on diversity
A reference trajectory begins when a crystal nucleates at temperature T 0 from a droplet of diameter d 0 .Here -11≥T 0 ≥-19 • C. Typically, ū exceeds the fallspeed of the micronsized frozen droplets, so the crystal initially rises and thus cools.When v reaches ū, the crystal has its maximum altitude3 and minimum T , and thereafter falls towards cloudbase at the warmer temperature T f .(We assume growth stops when the crystal reaches cloudbase, though some growth occurs below cloudbase where the humidity is between ice and liquid saturation.)The temperature deviations superimpose on the reference trajectories.Thus, we consider three influences on crystal variety: the initial diameter, the reference trajectory, and temperature deviations.
Variations in these influences are treated independently.Although not exact, this makes it easier to discuss each influence separately.In this case, the number of possible forms N≈N 0 N ref N dev , where N 0 is the number of possible forms due solely to variations of the initial crystal diameter, and N ref and N dev are the numbers for the reference trajectories and deviations.The shape diversity S≡Log 10 [N], and so where S 0 =Log 10 [N 0 ] and likewise for the other two numbers.

Influence of initial crystal and reference trajectories
To evaluate S 0 , S ref dependence on d 0 is weak).Upon exiting cloudbase, the crystal has diameter d f and total pathlength L f .Calculation details are in Appendix A.

Distinguishable classes for initial diameters and reference trajectories
For a given set of values d 0 , T 0 , ū, and T f , a variable, say ū, can be varied by up to δ ū and the final crystal forms will be indistinguishable (neglecting temperature deviations).
When |d ū|>δ ū, the resulting crystal will have a feature displaced by at least res from that on the original (|d ū|=0) crystal.To determine δd 0 , δT 0 , δ ū, and δT f , we consider changes to d f , a common feature that is relatively sensitive to the temperature history.If variable X with value x results in d f , then value x+δX results in d f ±2res.So, if d f is sensitive to X , then δX will be relatively small.Crystals within d 0 ±δd 0 , T 0 ±δT 0 , ū±δ ū, and T f ±δT f are said to be in the same "d 0 -reference" class because they all would be perceived as indistinguishable.

Number of distinguishable d 0 -reference classes
To estimate S 0 and S ref , we also need a typical range of each variable.Call these ∆d 0 , ∆T 0 ,∆ ū, and ∆T f .If the variables are independent, then the number of possible distinguishable crystals due to changes in d 0 is N 0 =∆d 0 /δd 0 , which also equals the number of distinguishable d 0 classes.That for the reference classes is • C, 0.24 m/s, 8 • C} and calculating the trajectories, the shape diversities S 0 and S ref average to 1.2 and 8.7 (see Table 1).S 0 is small because d 0 has a significant influence only at the start; for example, a crystal that begins 2-µm larger will end about 2-µm larger.This suggests that S 0 ≈Log 10 [∆d 0 /2res]=1.

Influence of temperature deviations
Let each d 0 -reference class represent a certain (yet unspecified) shape, and consider temperature deviations that may produce shape changes.As a crystal falls, the continuously varying DT will continuously alter the crystal's growth features.But there will be some average duration between relevant DT changes, that is, between DT changes that can produce distinguishable changes to the crystal features.Once a feature forms, it can grow and change.For example, if a sidebranch sprouts and no further DT deviations occur, subsequent growth is determined by the rest of the trajectory.Otherwise, subsequent relevant DT deviations will affect the shape according to when they occur.Thus, we must estimate the number of relevant deviations and when each one may occur.We first address the former.

Number of relevant deviations
We set a temperature-based criterion for the relevant deviations, then estimate how often the criterion is satisfied for a given crystal path.In time d t, a crystal falls through pathlength d L=vd t as its outermost prism faces advance by d R=rd t, where r is the prism-face growth rate (Fig. 3).(d L =d z unless ū=0.)A temperature differing by DT i produces a growth-rate change dr i =r DT i , where "i "="ai" or "Du" and r ≡dr/d T .The size of the surface perturbation thus produced is d 2 R i =dr i d t. (For a lateral deviation (e.g., Fig. 2a), the growth rate is unknown, so we use r as an approximation.)Integrating d 2 R i between depth L t at time t and L t +L gives δR i : where the T changes are small enough to remove r and v from the integrand.The number of relevant deviations in Σ i is nearly independent of L t , so we can ignore this dependence.For a surface perturbation to enlarge, eventually to exceed res, it must receive more vapor flux than adjacent regions.the vapor mean-free path, which we fix at 0.08 µm (λ varies only slightly with T and z).Thus, from Eq. ( 2), a surface perturbation can become a feature (i.e., relevant) when (3) The parameter p greatly influences S dev due to the sensitivity of v and r to T .During growth, p generally increases due to the increase of v.For crystals with T 0 =-15 • C, much of the growth occurs with p∼0.018 • C m.In contrast, crystals with T 0 near -11 and -19 • C have typical p values about 10 and 20 times larger.

Distribution functions from stratus clouds
The frequency that Eq. ( 3) is satisfied depends on p and L. To handle this dependence, we define peak distribution functions F i (p) as the number of peaks exceeding p in unit L.
To estimate F i (p), I used data from horizontal flight paths in stratus clouds with T <0 The F ai functions from two cloud datasets decayed as p −0.66 and agreed within a factor of two, even though the clouds had different temperature averages and the measurements were done differently (Fig. 5).In contrast, only one cloud dataset was available for F Du , and the values decayed as p −0.50 .The reason F i decay with exponents 2/3 and 1/2 is unclear, but, as expected, F i decrease with increasing p values.This decrease, together with the crystal size, strongly affects the diversity.

Total pathlength and crystal size
For constant p and v, the number of times n that a temperature deviation can produce a new feature is the product of the total pathlength with the distribution functions.only some fraction χ of the deviations will grow into a new feature, so Each of these n features could have been born when the crystal radius was at any one of m distinct radial positions on the crystal, where That is, there are m resolvable growth intervals.m and n are used below to calculate S dev .Although m depends on both the trajectory (through d f ) and the resolution, n depends only on the trajectory.

Estimates of L f , d f , n, and m
The reference trajectories were used to estimate L f and d f (Table 1), from which n and m were calculated (Table 2).The L f values increased as T 0 decreased, most rapidly as T 0 =-15 • C was approached from lower heights.This is because v decreases as T 0 →-15 • C (Fig. 3).Moreover, L f could greatly exceed the cloud thickness when ū was at least 0.25 m/s and T 0 was near -15 • C. In addition, near this temperature d f was largest.The reason for the peak in d f is not only the peak in r, but also the minimum in v. Similarly, the peak in n near -15 • C has two causes: the trend in L f and the peaks in r at -14.0 and -15.4 • C (Fig. 3).In Sect.6.2 below, we integrate d n=χ (F ai +F Du )d L for more precise analysis and show that a double-peak exists.

Combinatorial method for S dev
From the calculated trajectory, a crystal has n features that can arise in m resolvable growth intervals, and each feature may be born during either a growth spurt or lull, Figures

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Printer-friendly Version Interactive Discussion meaning that one of two possible features can grow from each interval4 .The resulting number of feature combinations is a standard combinatorial calculation (Feller, 1968): As m 1, n 1, and m−n 1 (Table 2), Stirling's factorial approximation can be used: When m n, N dev approaches (2πn) −1/2 (2me/n) n , showing that N dev increases rapidly when either m or n increases.In this case When res<4 µm, Eq. ( 8) is accurate to within 1-3% of the value derived from Eq. ( 6).

The main source of snow crystal variety
As Log 10 [2me/n]>1 and n 1/2 (Table 2), Eq. ( 8) suggests that S dev exceeds n, which exceeds S 0 +S ref for all T 0 .Furthermore, according to the dependence of S 0 , S ref , and S dev on res, this finding is independent of viewing resolution, provided res is not so large that m<n.Hence, of the three factors, temperature deviations caused most of the variety.This suggests that our impression of seemingly endless snow-crystal variety arises from the large number of relevant deviations, which is due to the highly fluctuating nature of T and u in clouds (See, e.g., Fig. 7).
6.2 Further analyses: the S dev "mitten" curve A plot of S dev versus T 0 reveals a curve in a mitten-like shape (Fig. 6).In particular, in the baseline case, the curve has a larger, broader peak near -15.4 • C and a smaller, narrower peak near -14.4 • C.This mitten shape comes from n, which is double-peaked because r peaks on both sides of the growth-rate peak near -14.8 • C. The lowertemperature peak is larger partly because only lower-temperature crystals experience both r peaks.In addition, crystals that start at lower temperatures have greater S dev due to their larger d f , which is partly due to their longer L f .But these effects are greatly reduced when the cloudbase is only 1 • C warmer than T 0 , making the curve more nearly symmetric about the peak in growth rate near -14.8 • C.
The results in Fig. 6 also can explain why the biggest crystals seem the most elaborate.Clearly, S dev increases with size due to the influence of m (Eq.( 8)).But there is another reason.The peak in size is due to the maximum in r (and minimum in v), whereas the S dev peak is largely due to the nearby peaks in r .This is suggested by the correlation between the double peaks in S dev and r .So, because the r peak is near the r peaks, larger crystals (larger d f ) have greater S dev and thus are more elaborate.To better understand the T 0 dependence, consider how r, r , and v affect S dev .In the baseline case, S dev changes from 41.3 at -11.0 • C to 479.9 near -15.4 • C.But when the T 0 =-11 • C case was run with r(T ) evaluated 4.

Sensisitivity and uncertainty
The analysis shows that S dev is relatively insensitive to T f but sensitive to ū.A decrease in T f decreased S dev for all crystals, particularly those that started near cloudbase.But the decrease is relatively small, and peak temperatures shifted only slightly to lower temperatures (Fig. 6, stars).The latter occurred because a raising of cloudbase decreases L f .A greater decrease in S dev occurred when ū decreased to 0.03 m/s.In this case, the peak temperatures shifted to lower values.The lower values are due to the shorter L f values, and the decrease in peak temperatures occurs because T 0 must decrease to have the same minimum temperature, where much of the growth occurs.
In contrast, S dev is sensitive to res and χ .If the resolution is coarser, then res is larger, resulting in less diversity.For example, Fig. 6 shows that the peak diversity decreases by about 170 when res increases from 1 to 4 µm.This is due to a decrease in m.Even greater sensitivity comes from χ due to the proportionality of n to χ .For example, when χ decreased from 1/2 to 1/8, the peak diversity decreased from 486 to 160.The value χ =0.5 used in the baseline case is uncertain and may depend on crystal form, feature type, and the size and rate of the deviation.Analysis suggests that χ is larger at faster growth rates, particularly when the feature is a sidebranch (Nelson, 2005), and χ may be nearly zero for small, slow-growing crystals that have not yet hollowed (e.g., Fig. 1a).So, χ =0.5 may overestimate S dev , at least at slow growth rates.

Neglected influences on diversity
By how much might the results change if we had considered the influence from humidity?If data were available on small-scale deviations of humidity from liquid-water saturation and the resulting growth-rate response, one could use the same method as that used here for T .However, unlike T , humidity has no altitude-related contribution, which is large in the temperature case: F ai >F Du for a range of v values (see Fig. 5).This difference, together with the finding that these humidity deviations are typically small, suggests that the humidity contribution should be less than that for temperature.
Other potentially major sources of variety are the initial crystal's structure and close passages/collisions of droplets.When the growth rate is low (e.g., ∼-11 and -19 • C), step-producing defects can greatly affect the growth rate of a prism face (Wood et al., 2001).Moreover, crystals nucleated under different conditions can end up with different shapes (Yamashita, 1973).But we have little knowledge of either influence.Finally, droplets may pass close to, and land on, a crystal, thus locally changing the humidity and even the temperature (when the droplet freezes).Such effects seem significant in some cases (Hallett and Knight, 1994), though quantitative treatment is not presently possible.

Errors from model idealization
The use of temperature and updraft data from horizontal paths may misrepresent the deviations experienced by rising and falling crystals.But high-resolution data along the vertical does not yet exist.Also, snow clouds can have regions with ū values much larger than the values here (Wolde and Vali, 2001); however, larger ū values would produce unrealistically large crystals, so the baseline-case ū here may be a reasonable average.Finally, data on rand v cover only idealized, constant conditions.In an actual cloud, side-to-side (leaf-like) falling motion and changing temperatures may alter r and v. So, although much remains unknown about clouds and snow, the treatment here is a realistic first step.

Comparison to previous estimates
For the above case in which res=1 µm, the maximum value of S is about 500, which, though large, is much less than previous estimates.Knight and Knight (1973) estimated the number of molecules in a typical snow crystal and then considered, but did not evaluate, the number of ways these molecules could be arranged.A lower bound of this number is derived from Eq. ( 6) by substituting for m a crystal radius divided by the water-molecule diameter and summing over all n values (0 to m).The result is 3 m , which, for a radius of 3 mm, gives an S value of 3×10 6 .In contrast, Hallett (1984) assumed a resolution 3×10 4 times larger (10 µm), yet calculated the larger number of S=5×10 6 .His treatment involved consideration of 10 4 "points" on the crystal, apparently each of which could have grown in one of 20 humidity classes and one of 50 temperature classes.Based on these assumptions, the number of possible crystals should instead be (20×50) 10000 , which gives S=3×10 4 -clearly inconsistent with the earlier estimate.Both of these earlier treatments gave S values vastly larger than those found here, mainly because they ignored limitations from the growth process.This huge difference in numbers has implications for the following question.
6.7 Comparison to observations: Should every snow crystal look unique?
An observational test of the mitten curve is impractical unless res is very large.So instead, we check to see if the numbers are reasonable by estimating the likelihood that all crystals in some collection appear unique.Consider a large, heavy snowstorm that deposits a liquid-water-equivalent snow depth of 3 cm (∼20-inches of snow) over a land area of 10 4 km 2 .In total, this equals ∼3×10 11 kg of water.Assuming each crystal is 10 −9 kg, this amount of snow contains ∼3×10 20 crystals, of which N cry ∼3×10 18 may be reasonably symmetric at cloudbase5 .
N cry could be compared to N; however, N varies with the cloud and crystal parameters (Fig. 6).So we instead consider each d 0 -reference class separately.
If all crystals are equally likely to be in each class, then the number of crystals per class is N cl ∼N cry /10 of ways that this class can have only unique crystals.This problem is equivalent to the well-known "birthday paradox", which is the (surprisingly low) probability that all people in a group have birthdays on different days of the year.Here N cl 1 and N dev (i ) 1, so we can use Feller's (1968) approximation: For all crystals to be unique, each class can have only unique crystals.Thus, pd all , the probability that all crystals are unique, should be the product of all pd(i ).This means that pd all <pd(0), where i =0 is the class with the smallest pd.From Eq. ( 9), this class has the smallest N dev ; hence, it should have a warm T 0 , a low ū, and a cold T f .For example, if T 0 =-11 • C, ū=0.02 m/s, and T f =-10.5 • C, calculation shows N dev ≈5×10 16 .Putting this N dev and N cl into Eq.( 9) gives pd(0)=0.2,suggesting that large snowstorms have enough crystals for some of the smaller crystals to appear as copies.Considering the uncertainties, this number is imprecise; however, the claim that pd all 1 strengthens if we either use a larger res or if we included all snow crystals that have ever fallen on Earth.In the latter case, N cl increases by many orders of magnitude because 10 S 0 +S ref hardly changes yet N cry increases from ∼10 18 to ∼10 33 (Knight and Knight, 1973).This greatly decreases pd(0).In contrast, if a class has crystals with T 0 ∼-15.4 • C, then pd≈1 even if we consider all such crystals that have ever fallen.(Here the exponent is ∼-10 −433 , making pd=1-10 −433 .)Finally, some crystal trajectories are probably more common than others, in which case pd could decrease; for example, some crystals may stay close together as they fall and thus experience similar conditions.The above result is consistent with our experience, but unfortunately is effectively impossible to disprove.For example, even in the first case above where N cl ∼10 9 , the number of crystal comparisons is ∼N 2 cl /2, which, if each took 1 s, would take a total of likely include some apparent copies6 .

Conclusions
To better understand snow crystal variety, we considered how snow crystals are viewed and how the crystals grow in a cloud.Specifically, we quantified the perceived crystal features using the crystal perimeter and resolvable growth intervals, and then estimated the shape diversity by analyzing the growth of a falling crystal.The resulting diversity was large, but not large enough to ensure that all crystals will appear unique at a typically viewed resolution.The main cause of the diversity was temperature deviations, arising from air-temperature inhomogeneities and deviations of updraft speed.Snow crystals can be very diverse because their growth rate is sensitive to temperature.Those that start near -15 • C are largest because the growth rate is a maximum and the fallspeed is a minimum near this temperature.Their large size gives them relatively large diversity, but the diversity peaks at slightly colder and slightly warmer temperatures (-15.4 and -14.4 • C) due to maxima in temperature sensitivity near these temperatures.Thus, despite the need for greater knowledge of snow-crystal growth, we could identify some of the important properties that contribute to their seemingly endless variety.
and b i and c i are in Table 3.The sole consideration for the above functional forms was to obtain as good a fit to the data as possible.
terminal fallspeed and T (z) decreases with z as d T (z)/d z≡T =−7×10 −3 • C/m, a typical environmental lapse rate.An actual updraft has speed ū+Du(t), but Du is used only to estimate DT Du , not for calculating trajectories.These and other model parameters are sketched in Fig. 4.
, and S dev , we track the crystal diameter d , temperature T , and pathlength L (air depth the crystal falls through) along reference trajectories.Here we use the reference trajectories to analyze S 0 and S ref .A trajectory is determined by d 0 ,T 0 , T f , ū, and v.But v depends on d and the crystal's shape, which are determined by d 0 and T (t).Thus, reference trajectories depend only on d 0 , T 0 , ū, and T f (and 3. The value is instead 1.2 because d also affects v. S ref is much larger, mainly because d f is sensitive to T 0 and ū, both of which affect T (and hence r) throughout the trajectory.In contrast, T f only affects L f , thus adding relatively little to S ref .
4• C lower (to equal that of a -15.4•C crystal), the resulting S dev was 64.0, an increase of 18.7.If instead v was evaluated 4.4• C lower, the value of S dev increased by 33.4.However, when r was evaluated 4.4 • C lower, S dev increased by 60.2, showing that r had the largest effect.(When r, r , and v were all 4.4 • C lower, S dev went up even more -by 366.1.)Hence, big crystals can appear more complex both for the obvious reason, that they have more places for features to originate, and because the growth-rate sensitivity to temperature is high near temperatures at which big crystals grow.

Fig. 5 .
Fig. 5. Temperature deviation distribution functions F ai and F Du .All functions are from measurements at 15-cm intervals except the lower F ai one, which instead had 8-mm intervals.Also shown are fits F ai =0.0287 p −0.66 and F Du =0.0262 p −0.50 v −0.5 .

Fig. 6 .Fig. 7 .
Fig. 6.Shape diversity S dev .The upper black curve, the baseline case (bold values in legend), is derived from the grey curves of m (reduced 30-fold) and n (reduced 3-fold).Solid triangles, stars, squares, and diamonds mark peak positions for other ū, T f , χ , or res values.Full curves show cases χ =1/8 and T f =T 0 +1.Vertical lines on the abscissa are peak positions and relative magnitudes for the baseline case (black lines), the value of m (long grey line), and the value of n (short grey lines).
S 0 +S ref .Using S 0 and S dev from Table 1, N cl =4×10 8 .The probability pd(i ) that all crystals in class i are unique can be estimated by counting the number Figures

Table 1 .
Trajectory results for d f , L f , S 0 , and S ref .

Table 2 .
Combinatorial parameters n and m.Based on L f from Table 1, and the average F ai +F Du for the coldest and warmest parts of the trajectory.‡ m=d f /2res, with d f from Table 1.Introduction

Table 3 .
Coefficients for fits to r and v.