Center hollows show variable behavior. Concerning their formation, and other authors e.g., have referred to the process as an instability. In the standard treatment, however, the hollow occurs when the gradient in supersaturation needed for uniform growth can no longer be compensated for by the step density e.g.,. In other words, normal growth of the entire facet becomes impossible, which is different from being unstable. In this “impossibility” case, one expects the hollow initiation and shape to (i)  be nearly identical on identical faces in a nearly uniform environment as well as (ii) being highly reproducible when other crystals grow under the same conditions. On the other hand, if hollowing is merely unstable, then a sufficiently uniform, constant condition may be expected to circumvent the hollowing indefinitely. Conversely, if hollows do form, then their initiation and shape should differ between identical faces due to minute differences in conditions. We suggest here that the inclusion of lateral-growth processes predicts qualities of unstable growth at low supersaturations, leading to hollow close-off and terracing-type features.

The standard facet-impossibility approach seems qualitatively successful in some cases of middling supersaturation e.g., and at relatively high supersaturation where the hollow tends to keep enlarging in width (e.g., hollow columns) or advance into branches (e.g., dendrites) in a generally consistent, repeatable fashion on all identical face types. But at low supersaturations, the hollow often varies in width, getting wider and then getting narrower, and may even close-off into a center pocket. Large changes in width also occur at middling to high supersaturations e.g., but are much more pronounced at the low supersaturations here. also observed the closing-off of hollows during growth at one atmosphere and supersaturations up to 33 % at -30 ∘C. At low supersaturations, otherwise identical faces can have different patterns of hollows and pockets. Thus, at least at low supersaturations, the hollow phenomenon does seem to have some qualities of an instability.

In the cases shown here, the center hollows vary considerably even though the conditions are nearly constant. For example, the hollow's size and shape can vary considerably between different faces of the same crystal in Figs. 11 and 12. Also, a given hollow's width can change suddenly, often showing periodic terrace-like features and sometimes closing off completely, leaving a center pocket. Such behavior suggests a complex process involving competing influences and a possible instability. Here we describe a simplified mechanism for such an instability between normal and protruding growth and argue that the growth behavior of adjacent faces may influence hollows, particularly at low supersaturations, possibly leading to the abovementioned observations.

First, consider the initial hollowing. The overall driver of hollowing of a surface is lateral supersaturation (σ) gradients across the surface. These gradients are influenced by growth on all crystal faces such that, for example, normal growth on the basal face produces a decrease in surface supersaturation, starting from a high value in the middle of the prisms p, as sketched in Fig. B1a, to the b–p edge, to the smallest value in the center of the basal. Normal growth only on an adjacent prism faces produces the opposite gradient on the basal (dotted lines). In general, normal growth occurs on all faces, and thus the contributions from both sets of contours in (a) are superimposed with a weight in proportion to the normal-growth rates . The normal-growth rate of a given face is proportional to the areal flux Fv to the surface, which is mainly the vapor-diffusion flux B1Fv=DNeq∇σ⋅n^, where D is the vapor-diffusion constant, n^ is the surface normal, and Neq is the equilibrium vapor density at the local surface temperature. This means that a larger growth rate of a face implies a larger normal gradient, which should positively correlate to a larger lateral surface gradient. Hence, taken together, the surface gradient in supersaturation that leads to hollowing, say on the basal face, will be weaker at low normal-growth rates of the basal and also weaker at high normal-growth rates of the adjacent prism faces. That is, the growth on one face influences the lateral gradients on the other faces.

The lateral surface gradient in supersaturation leads to hollowing via its effect on the surface steps e.g.,. Briefly, as sketched in Fig. B1b, steps originate from the crystal edge e and flow towards the face center m on the right. The sketch on the right shows the trends of vapor supersaturation along the surface and deposition coefficient function α. Near the edge, the vapor supersaturation σe is relatively high and the steps are relatively far apart, but there is a relatively high fraction that desorbs, which is described by its low deposition coefficient αe. As the steps move towards m, they slow down and become more densely packed, thus increasing the local deposition coefficient. At the edge i of the hollow, essentially all the incident molecules reach a step and the steps are clustered together to the point that they hardly move. A wall of steps builds up here, at the step-clumping region (SCR), forming the edge of the hollow. (, proposed a more detailed model of step dynamics for ice with a thick surface-disordered region, but it is not yet clear how a hollow would develop in that model.)

However, after the hollow forms, the local supersaturations may change. This change could be due to either a change in external conditions (e.g., temperature or supersaturation), a change in growth rate of a face due to a changing activity of the step source, or simply the increasing size of the crystal. For example, an increase in crystal size will generally decrease σe. Regardless of the cause, consider now the sketch in Fig. B1c in which a slight change in local conditions near the hollow edge i has occurred, causing a slight increase in the local step separation l. The normal-growth rate dz/dt at i is the step height h divided by the step-passage time τ (time between successive passings of a step at i). The latter time is the step separation divided by the step speed vs. However, for a protrusion of thickness nh (n≥1) that starts into the hollow, the protruding growth rate dy/dt=vs/2n, with the factor 1/2 due to the AST flux coming only from the top side. Comparing the two rates, B2dydt=l2nh⋅dzdt. When the hollow first forms, l may be of the order h. But an increase in l causes dy/dt to increase, further increasing l and thus further increasing dy/dt. Eventually dy/dt may greatly exceed dz/dt. Hence, in this very basic description, some change in step spacing near the hollow may become unstable, leading to a protrusion that can continue to grow, eventually sealing off the hollow into a center pocket. Initially, there will also be some direct vapor flux to the side of the hollow at the lip, but this contribution to protruding growth would vanish due to shielding from the opposite side as the pocket closes off.

This basic treatment neglects lateral supersaturation gradients and advancement of the crystal face. Briefly, these factors make the sealing-off of a hollow less likely at higher normal-growth rates because the initial protrusion will become left behind in a lower supersaturation region as the rim grow higher. As the supersaturation drops, the protrusion grows slower, amplifying the effect. Also, as the protrusion grows inward, the supersaturation should decrease (except in the case mentioned next), thus hindering or possibly preventing the instability. Thus, the above suggests a hollow instability at low normal-growth rates but not at high rates.

Concerning terracing, if the SCR develops nearer the rim and becomes elevated as in Fig. B1d, the interior region f may flatten via facet spreading. Such F growth would be aided by a reversed surface supersaturation gradient; that is, if the inside corner c becomes isolated in an effective vapor shadow (height i–c exceeding the vapor mean free path), then the steps in region f will speed up as they go towards the higher supersaturations near the center. This would produce an interior face that spreads laterally, flattening region f into terrace features such as those in Figs. 11, 12.

The growth of crystals with many terraces has been called skeletal or hopper growth, but the structure differs between that in relatively squat hollows and that in narrow columns. Referring to Fig. B2, we call the former terraced (a) and the latter banded (b). A sequence showing banding during growth at atmospheric pressure, -30 ∘C, and 8.8 % supersaturation is found in . show the banding in hollow bullet rosettes from clouds, and show numerous cases on hollow columns and sector-like forms. The bands are much denser in the latter forms. Indeed, terracing and banding are very common in natural snow and hoarfrost.

Concerning the formation of a terraced hollow, initially, the inner surface may be smoothly curved, as shown by the dashed line in Fig. B2a. Starting from such a smoothly curved surface, the closely spaced single steps may clump into large step clumps as described by . In that process, once two single steps become close enough to essentially lock together, they move more slowly, allowing steps further back to catch up. did not include supersaturation gradients, but such gradients may promote the clumping. In this way, a step-clump of two quickly becomes a clump of three, and the clumping continues. (, describes a more complex interaction for step bunching.) Once a sufficiently tall clump forms, any single step s between an inside corner c and an edge i would speed up as it approached c, flattening the terrace as described above.

The banding in a narrow column in Fig. B2b similarly starts with a smoothly curved interior surface and may form a macrostep by the same step-clumping process. But in this case, a single step s flows from the center out towards the higher supersaturations at the rim. Thus, a step clump at i does not need to be high before the flattening effect becomes large because in this case the step is speeding up due to the higher supersaturation even without an edge at i. As with the terracing case, as i grows, it sticks out into regions of higher supersaturation, meaning that the next single step may travel even faster. Thus, a later step overcomes a previous step, quickly building up a larger macrostep, which would appear as a band in the hollow. The hollowing may start with a single band, with new bands forming via the same process as the crystal grows, leading to a series of nearly equally spaced bands.

This treatment suggests that step sources and dynamics, F and P growth, the moving interface, and the shape of the supersaturation contours all likely influence hollow structure, leading to their highly variable behavior even under constant growth conditions. A similar process of banding may also apply to the “cross-rib” features on the backsides of branches on broad-branch and sector-plate crystals. The suggested mechanism in that study was instead changes in temperature or supersaturation around a crystal. These changes would cause the width of the branch to vary, and the same process may also produce some terracing and banding in hollows by temporarily changing the rim width e–i in Fig. B1d. In a changing environment, more than one mechanism may alter the hollow structure.

Hollow sizes and shapes are highly variable under low normal-growth rates (e.g., on different faces of the crystals in Figs. 11 and 12). Such variability is uncommon at high growth rates, so we outline here a few factors that may play larger roles at the low growth rates at low supersaturations.

One factor is the greater variability in the normal-growth rates. In contrast to high-supersaturation growth, the step sources at low supersaturations are thought to be crystal defects such as dislocation outcrops and stacking faults. The dislocation activities will in general be different on different faces of the same crystal and between different crystals and may also change during growth. But other factors may lead to greater variability at low supersaturation. For example, the greater relative role of lateral-growth processes at low normal-growth rates leads to the phenomena described in the previous section; that is, the interplay between the surface influence and the bulk vapor-diffusion influence may allow more complex nonlinear feedbacks on growth, leading to a greater chance of unstable behavior. This variability may be increased by the variation in dislocation activity, which would have a larger influence at low supersaturation because the surface has a larger direct influence on growth rates under these conditions. Finally, after a given duration, the smaller crystal sizes at low growth rates mean that variability in the initial droxtal size and properties would have a relatively larger influence on the later crystal form.

In contrast to the other low-supersaturation crystals shown here, the six planar pockets in Fig. 3 are remarkably similar. The reason for the pocket symmetry is likely due partly to the equal normal-growth rates of all six prism faces. This symmetry in the growth rate must arise from having equivalent step sources on all faces. Given that the crystal has an apparent stacking fault or stacking-disordered region that intersects all prism faces, the obvious step-promoting defect would be the fault. Fault-generating steps had been proposed by via a mechanism in which the fault yields a lower barrier to layer nucleation. Thus, we suggest that the six hollows opened up at the same time because the step-generation mechanism is the same stacking-fault mechanism on all six faces, producing the same normal-growth rates on all faces. (That the fault could both be a source of growth and a location of hollowing is harder to explain but possible given that the steps would start from the prism–prism edge, not the hollow location.) Concerning the hollow closing-off to form pockets, two factors occur during growth that will likely change the step separation near the pocket, thus determining whether they close off: (i) the edge supersaturation decreases due to the larger crystal areas, and (ii) the relative position of the pocket-opening changes on the prism face due to one basal face growing faster than the other (i.e., the side view in Fig. 3d shows greater advance of the right basal face than the left). As both of these factors will be equal for all six prism faces, the closing-off should occur at the same time, leading to the identical nature of the six pockets. Finally, note that the groove region, which is sublimation-rounded like the edges, did not produce a pocket during regrowth like the corners. The reason for this may be the much smaller radius of curvature in the former case.

The two-level structure occurs in planar crystals P2–P4 grown near water saturation and -15 ∘C (all crystal notation here from ). Some such crystals show a circle at the center, suggesting that the original droxtal largely remained unchanged. But it has been unclear how the original droxtal's basic shape could remain intact as vapor deposition “filled out” the crystal via normal growth of the faces. The answer appears in Fig. B3, which shows how this circle shape can remain when the top and bottom basal planes extend by protruding growth. (, also show several cases of such sprouting.) The process of formation via protruding growth has been described by Akira Yamashita (personal communication, 2014), but we include it here due to its close relation to other phenomena we describe.

The overall process is sketched in Fig. B4. In (a), the droxtal has just frozen. The latent heat release raises the droxtal's temperature to about 0 ∘C, but if the droxtal is relatively isolated, its temperature returns to the ambient value within a fraction of a second. Then, the surface quickly depletes the nearby air of vapor, driving down the surface supersaturation to a value that greatly suppresses layer nucleation on the basal faces . Thus, after the basal facets form on the top and bottom of the droxtal (a–b), they mainly spread via F growth. The growth continues as protruding growth in (c), as was the case for corner pockets. But, unlike the corner-pocket case, protruding growth does not occur on the prism faces, and thus the basal protrusions extend out from the boundaries of the initial droxtal, creating two levels. A possible reason for the lateral-growth rate being larger on the basal than the prism may be the proposed larger xs value on the basal in this temperature regime . Sketch (d) shows one level grows more than the other, which is due to asymmetry in the vapor-diffusion field around the falling crystal . As shown in Fig. B3a, and found much earlier by , branches can also occur on both levels.

Such abrupt sprouting can explain the small center circle observed in some branched snow crystals (e.g., Fig. 8c). Not all branched, two-level crystals show such a “droxtal center”. The two levels in these other cases likely arise instead via the hollowing-type (“lacunary”) process observed by see also. wrote that “at least half” of the 4200 crystals he had photographed in 40 years had such a center circle and argued that it was the frozen droplet (droxtal) upon which the crystal formed. What factors may cause the droxtal sprouting in some cases but not others? Akira Yamashita (personal communication, 2014) observed sprouting on larger-than-average droxtals. Large initial droxtals may favor sprouting because their larger areas would depress the surface vapor density more than that of a much smaller droxtal, essentially shutting off the normal growth on the basal face. This effect of size may also lead to a greater vapor-density gradient where the protrusion starts, particularly at higher ambient supersaturations that would tend to produce higher supersaturation at the face edge. In addition, the AST flux should be higher when the basal face has larger xs values. found xs to peak in -9 to -15 ∘C, a temperature near which such two-level crystals sprout. Also, high supersaturations likely promote such sprouting: in the recent vertical wind-tunnel experiments of , nearly all the images of planar snow crystals clearly show the center droxtal as described here. In those experiments, the crystal nucleated and grew in a droplet cloud of various liquid-water contents and thus grew near liquid-water saturation. The mean droplet diameter (before freezing) was 8 µm, but inspection of the images indicates that the droxtals that sprouted the two-level crystals had a slightly larger diameter (∼9–13 µm). Thus, both relatively high initial supersaturations and relatively large droxtals may favor two-level initiation via protruding (P) growth.

The snow-crystal image in Fig. 8c appears to show corner pockets on both levels of a two-level crystal. In addition, the smaller inset shows a sequence of small circles along the centerline of a branch, a pattern reproduced on the other branches as well, ruling out the possibility that these are rime. Similar series of circles appear in crystals 11, 12, and 22 in . In most cases, the circles appear before the crystal branches sprout. These circles may be corner pockets by the following mechanism. During growth, the backside of the plates and branches of two-level crystals show rounded features even without sublimation e.g., Fig. B3d,. A short slowdown in growth may allow facet spreading and protruding growth from the rounded backside region via steps similar to those shown in Fig. 4b–e but on one side only. The protruding growth from the side is more difficult to picture in this case due to the positioning on a ridge. Nevertheless, the basic process may be like that sketched in Fig. B4e–f. If correct, the existence of each pocket marks the time when growth temporarily slowed down.

Capped columns (CP1a, CP1b) and multiple-capped columns (CP1c) form when a columnar crystal quickly moves into a high-supersaturation region with temperature in a tabular regime. For example, a column growing near -8 ∘C can be quickly lifted in a vigorous updraft to the thin planar regime starting below about -10 ∘C. The resulting form is similar to the droxtals with two levels (e.g., Fig. B3), but with a column between two basal extensions such as a thin plate or dendrite. The left crystal in Fig. B5 shows two thin end caps and another thin, but shorter, central plane in between. On the right is a case where the central plane is longer and all three plates are thicker than those in the other crystal. The planes in this case may have passed back into the columnar temperature regime. These two crystals are multiple-capped columns.

A previous model of capped-column formation involves an extreme form of hollowing in which the step-clumping region (SCR) on the prism faces forms near the step origin at the basal–prism edge . However, that model cannot readily explain the central thin basal extensions that occur in the multiple-capped column (type CP1c). Instead, the similarity to the two-level case in Fig. B4 suggests an AST contribution to cap formation. If so, what is the source of the originating basal plane on the interior of the column for the CP1c case?

The basal protrusions on rime in Fig. B3a, e (thin arrows) suggest one possible source of an interior basal extension. As sketched in Fig. B6a (top), a rime droxtal could develop a basal face aligned along that of the column (assuming the droplet freezes with the same orientation). The face would grow laterally and then protrude via AST as shown in (b). Once the basal extension starts, it can grow both outward and around the column. If the two end caps have a head-start on growth, they would deplete the nearby vapor, making the rime droxtal nearest the center the more likely one to have sufficient vapor to develop a significant basal extension. (Otherwise, two basal extensions may form relatively closely together, competing for vapor until one grows significantly larger, stunting the other.) The image labeled CP1c in Fig. 1 of shows other rime droplets along the column, suggesting this mechanism. Without the rime, it may be unlikely that a high density of new layers could nucleate in the middle of the column, produce an SCR, and sprout a new plate.

In addition, the examples of CP1c in Fig. 5 show no rime on the column, but a shift in the column that provides an interior basal plane upon which to sprout the extension. Figure B6, bottom, shows how this case may proceed. Although we know neither how common such crystals are nor how the interior planes arise, we show here one such case in Fig. 9. Such an interior basal plane could also arise from a small bundle-type column such as a bundle of needles (C1b) or bundle of sheaths (C2b), which are discussed in Appendix B9. Other cases include a column with a small double twin e.g., Sect. 4.10 of, crystals such as those in Fig. 12 (possibly twins), columns with prism hollows, or a column that underwent previous transitions in the tabular regime.

Similar to the capped columns, the protruding growth process may also influence the initiation of tabular extensions on “florid” crystals or side planes . The base crystals (before sprouting) are squatter than the columns for the capped columns and sometimes polycrystalline (see type P8b “complex multiple plates”; ). Also similar to the capped columns are the bullets with plates (CP2c), which probably form by the same processes. The left crystal in Fig. B5 shows smaller planes on the end caps in different directions. Such new crystal orientations likely formed on rime droplets that froze in these other orientations and then extended like those in Fig. B6b (top).

To estimate the contribution of the AST flux to lateral growth of several growth forms, consider a simple treatment based on the BCF model of crystal surfaces. Assume here that the region over the face edge (i.e., the lateral-growth front) is rough, and thus this region can be treated as BCF for a step edge, that is, as having an equilibrium concentration of mobile surface molecules. Assume further that, as suggested by step-motion experiments , molecular migration over the edge encounters no barrier. In this case, straightforward use of BCF gives a flux of molecules fL (per edge length) that equals B3fL=xsv4Neqσ, where xs is the surface migration distance, v is the mean molecular speed in the vapor, Neq is the equilibrium vapor density, and σe is the vapor supersaturation near the edge. This result suggests that we can view the adjoining face region within xs of the edge as a collection region of molecules impinging from the vapor. Assuming that this face edge has n adjacent facets from which to draw the flux and the edge has thickness t over which the AST flux is distributed, the effective flux (per area) FAST is B4FAST=nxstv4Neqσe, an amount we compare to the direct vapor flux Fv e.g.,, B5FV=v4Neqσe, where the deposition coefficient is assumed to be unity (effectively rough surface), consistent with the assumption of a step edge in the derivation of Eq. (B3). Thus, as a crude estimate, the ratio of AST flux to standard vapor flux is just n⋅ xs/t, with n∼1 or 2 depending on whether the thin edge is bound by one facet, as in a dendrite branch, or two facets, as in a thin disk. In general, n can vary between 1 and 2 when two faces meet on one edge, such as the two prism faces along the edge of a sheath, and may exceed 2 in the case of a thin whisker. In the next two subsections, it will be convenient to view the total flux as equivalent to having an effective supersaturation of σeff≡σe⋅(1+n⋅xs/t).

Concerning xs/t, early estimates of xs from step-motion experiments on the basal face gave a range of values, depending on temperature, of about 1–6 µm . A more recent measurement gives a value of about 5–10 µm at -8.6 ∘C . For the thickness t, a recent study shows that the tip region of a dendritic snow crystal has a tapered tip . The measurement does not give a precise value of t exactly at the edge, but an average value within ∼10 µm of the edge gives about 0.3 µm. Thus, the estimates of xs/t here suggest that the AST flux could be up to ∼3–60 times the direct vapor flux in certain cases. The resulting increase in growth rate from this flux would be less than this factor due to the vapor-diffusion process. Further analysis (to be included in part II) suggests that a partial barrier to migration over the edge may exist, and the resulting reflection of some ad-molecules at the edge would reduce this AST flux. Nevertheless, such a flux could significantly increase the rate of growth (maximum dimension) of thin planar growth forms such as the dendrite (P3b) and fern (P3c) crystals, as well as sheath (C2) and needle (C1) forms. For a thin disk, a more complete calculation is at the end of Appendix A.

Fast-growing crystals have leading growth fronts that can appear rounded, and some thin tabular crystals can be disk-shaped or scalloped. For example, the sheath extensions in Fig. B7 include some with tips that appear rounded and some that appear flat-faced. Similarly, observed sheath needles with rounded tips. shows rounded sheath-needles sprouting from prism corners that later flatten upon becoming larger. An increase in supersaturation would then cause smaller, round tips to sprout on the larger, flat tips. Round tips also seem to appear on the faster-growing dendrites near -15.0 ∘C (P3b, c). even found such rounding on dendritic crystals grown with their basal face against glass. (However, for the fast-growing dendrite cases, the small scale of the tips makes it hard to discern small facets with limited image resolution, and a slight amount of rounding may quickly occur in brief undersaturated conditions, so the phenomenon is not well-established yet.) Away from the tip, rounding of the side vertices has been attributed to SCR forming due to a decaying gradient in surface supersaturation . In slower-growing tabular crystals, observed disk crystals faceting as they became larger and thicker, as did . In our previous experiments , we also saw small disk crystals develop facets as they grew. Also, grew rounded tips of “serrated” dendrites at about -2.0 ∘C and noted that they were thinner and grew faster than the thicker, facetted tips around -3.0 ∘C. Thus, rounding can occur in several situations. But, as argued in , the supersaturations are too low for the phenomenon to arise from kinetic roughening, so we ask if AST may contribute to such rounding.

Consider first a thin tabular crystal with two basal faces, focusing on the region near a prism–prism edge as sketched in Fig. B8a. As shown in the top sketch, the collection area for AST flux at the tip c has an angle of just 120∘ compared to the 180∘ further down (about ∼xs or more) at e, making the AST flux ∼ two-thirds the value at the tip. If the total molecular flux is dominated by the AST flux, then this effect would move the point of highest total flux away from the tip. Equivalently, we can view position e as having higher effective supersaturation than c. Thus, as shown in the bottom sketch, the point of new-layer nucleation has moved down from the tip to e. Moreover, if the edge region is effectively rough, then the regions of higher effective supersaturation will advance faster than regions with lower values, changing the vertex or tip shape until a steady-state shape emerges. Such a shape would be rounded, as shown in the sketch. Thus, if the crystal edge is nearly rough, then AST flux can cause the corner or tip to round as steps traveling from their source at e to further down the tip towards c can readily clump. In this case, n=2 due to AST flux from the top and bottom basal faces. Thus, the edge plan view may be similar to that shown at the top in Fig. B8b.

A dendrite branch has a different tip shape, with the edge prism faces bound by just one basal face, the other being non-crystallographic. Thus, the edge plan view may be more like that at the bottom in Fig. B8b. In this case, a second factor may cause the vertices to round. As sketched in (c), the leading prism faces are expected to be thicker at the vertices; that is, t2>t1,3 and t4>t3. (Although the thickness variation right at the perimeter was not detected by recent interferometry studies , the formation of a main ridge and side ridge, which are clearly visible in nearly all tips, appears to require such an increase in thickness.) Their formation mechanism, as described in , relies upon the influence of the direct vapor flux. But if their growth is also significantly influenced by AST flux and they are thicker at the vertices, then Eq. (B4) shows that their growth rates will be slower. In this case, they may advance more slowly at the vertices, leading to the rounding shown in (c). In this latter case, a small region of prism facet remains (dark patch in c), but may be indiscernible in photomicrographs.

These rounding mechanisms depend upon the prism faces being nearly rough. One possible factor is illustrated by the bottom sketch in Fig. B8b. Here, layers nucleate at one basal–prism edge and can reorient parallel to the edge with a separation l of roughly the crystal thickness t, which may make the edge effectively rough. For a face to essentially collect all surface ad-molecules, the step spacing l need only be smaller than xs. But if the spacing becomes significantly smaller than xs, the normal-growth rate becomes proportional to the local effective supersaturation (as all the flux is incorporated into the crystal), meaning that a decrease in this supersaturation would cause the surface to respond by slowing, producing rounding. Such a case is akin to the kinetic roughening that is driven by high supersaturations , but in this case the supersaturations are relatively low.

Although AST may be a key factor in some of these cases of crystal rounding, rounding in general on vapor-grown ice is more complex than our simple treatment here. In different situations, crystal rounding on vapor-grown ice may involve a combination of the above AST mechanism, thermal roughening, a thin solute layer, and perhaps yet undiscovered factors.

Other aspects of tip shape may also influence the normal-growth rates of needle crystals. Needle crystals (C1a) are long, thin, columnar crystals with “tops shaped like a knife edge” that form near -5 ∘C. In their initial growth, they appear as narrow prism planes that sprout from the edges of the basal plane, similar to the sprouting of basal planes during the formation of two-level crystals examined above (Sect. B3). Examples shown in Fig. B7, as well as in , suggest that they initiate via P growth. observed both sheath needles and, less often, a newly reported type he called sharp needles. The sheath needles would grow at a rate of about 0.3–1.0 µm s-1, whereas the sharp needles grew about twice as fast (about 1.5–2.5 µm s-1). The sharp needles also had a smaller diameter and appeared more nearly round in cross section. Why did they have different tips and why the distinct growth rates?

The different tip shapes may be the reason for the bimodal growth rate. If each side (prism plane) of the needle tips are of the order of 10 µm or less, the AST flux can be significant, perhaps even dominant. The net effect of this flux should depend on the ratio of the collection area (on the adjoining prism faces) to the growth-front area. Consider the two tip shapes in Fig. B9. Figure B9a, showing a 120∘ interior angle, is like the sheaths in Fig. B7, whereas Fig. B9c shows a 60∘ angle like that of “iii” in the right scroll of Fig.  B10b in which one prism face is missing. If the arrows represent the AST flux, then Fig. B9c has almost twice the AST flux to the growth-front area at the tip (dashed circle) as Fig. B9a; that is, n is nearer 1 in Fig. B9a but nearer 2 in Fig. B9c. Another factor is the influence of the backside of the tip on the surrounding vapor density. The backside, being non-crystallographic, is effectively rough and thus efficiently draws in vapor. Neither the front-side prism faces (p1, p2 or p1, p5) nor the tip are collecting much mass from the vapor. But for a given length of needle, the open sheath Fig. B9a has a greater backside area than the proposed sharp needle Fig. B9c. This backside may dominate the mass uptake, just as it appears to do on dendritic crystals (). With the smaller area for the sharp-needle case comes a smaller mass uptake and with a smaller mass uptake, a higher surface supersaturation and higher normal-growth rate. In this way, the sharp needle should grow significantly faster than the sheath needle, as was observed. An approximate calculation (to appear in part II) finds that these effects cause an increase in rate by about 50 %. A possibly larger effect, though harder to accurately model, may come from the direction of the supersaturation gradients between the two needle cases. This gradient is likely larger in the sharp-needle case due to the smaller interior angle, which would have the effect of sharpening the needle, which in turn could greatly increase its growth rate.

Consistent with our hypothesis that the two case have different interior angles, found that when the temperature of the sharp needle was raised to slow down growth, it thickened enough to discern that it had a triangular shape (Fig. 4 of ). This 60∘ interior angle of the proposed sharp needle is a feature of trigonal crystals. In Appendix B11 below, we argue that this angle is stable in the columnar regime, which includes the needle case.

A perplexing growth feature is the scroll (C3c). With a scroll feature, thin prism-face “sheets” or side-planes tend to curl inward while maintaining a prism orientation, somewhat resembling a paper scroll as shown in Fig. B10. (For formation sequences, see Figs. 3 and 5 in .) In the atmosphere, the original Nakaya diagram e.g., shows scrolls forming at and above water saturation near -7 ∘C, which is consistent with later diagrams by other authors. Later, found scrolls to be very common features of certain polycrystals (Gohei twins, CP7a). But perhaps one of the earliest descriptions of a scroll is in , where he finds large examples in crevasse hoar. However, with the aid of a standard macro lens on a common digital camera, one can observe them frequently in hoarfrost. A description of their formation is briefly mentioned in , but their proposed mechanism differs from that presented next.

The scroll may start as a protrusion, like a sheath needle, except on a larger-area basal face as shown in Fig. B11a. When this protrusion is thin, it can grow rapidly because the AST flux is deposited onto a small-area region at the growth front. (Growth may be more rapid normal to the page, but we focus on the side growth.) This rate is marked by the large arrow pointing across prism p2 in (a). But normal growth is also occurring on the backside (or inside), causing it to gradually thicken. (In this description, we assume that the backside of the scroll is mostly non-crystallographic.) This normal-growth rate is marked by the smaller arrow pointing down, towards the basal interior. On the leading front of the protrusion, direct vapor flux also contributes to growth, but when the feature is thin, this flux is overcome by the larger AST flux from prism p2 (Eq. B4). Eventually, this protrusion thickens enough to reduce the protruding growth rate, at which point prism facet p3 can form and spread there, essentially ceasing the lateral growth of p2 as shown in the sketch. Now the process starts on the edge of these new prism faces, causing new P type growth at 60∘ to the old P type growth (right side), and the process repeats, later curling around another 60∘ into a scroll-like shape. Each new “wing” of the scroll may have a smaller area due to it moving towards lower vapor density, allowing the edge front faceting to start sooner. Such a process would produce a curling feature. In brief then, the mechanism involves normal growth, F growth, and P growth.

A scroll may also initiate from a cup-type crystal form or full sheath (C2a) after part of its rim forms a break, which may in turn be due to an axial asymmetry in the nearby vapor-density field. As sketched in Fig. B11b, the sides of the break (at the bottom) could then curl around via the same process as that sketched for (a). Supersaturations high enough to produce cup crystals are rare in the atmosphere but are very common near hoarfrost. As hoar cup crystals tend to be closely clustered, and thus have large local variations in vapor density via crystal competition, this initiation process may be likely for hoar scrolls. Large examples of this type are in . This mechanism may have produced the crystal in Fig. B10a.

Another perplexing growth form is the bundle of sheaths (C2b) and, similarly, the bundle of needles (C1b). These habits form in a narrow temperature regime near -5 ∘C where growth is almost exclusively in the c axis direction e.g.,. However, unlike the needles, these crystal forms have widened significantly perpendicular to the c axis. Such widening is hard to reconcile with our knowledge of growth driven by layer nucleation, which may effectively shut off all normal growth of the prism faces. Two mechanisms may overcome this nucleation barrier. One is riming. A rimed drop on a prism plane may then sprout a new sheath protrusion along the c axis. The second mechanism is sketched in Fig. B11c. In this mechanism, the protruding sheath widens as in the scroll form in (a), but overshoots the base crystal, thus advancing the crystal width perpendicular to the c axis. The protrusion will thicken and may then develop a new prism face by the mechanism suggested in (a) for the scroll. In this way, a new sheath or needle can develop to the side of the original, producing a “bundle”. Such overshooting of the prism planes in this case is analogous to the overshooting of the basal planes in the two-level crystal (Sect. B3).

The backsides of branches on tabular crystals appear to be largely non-crystallographic during growth under constant conditions. But they are not gently curved; rather, various ridges and ribs are common, which show up as dark interior lines in images. However, when part of a relatively fast-growing crystal branch slows down, due to either a change in conditions or to its gradual drift towards the crystal interior as the outer parts grow out, the ridges and cross ribs may form planar protrusions. For example, such protrusions appear relatively common on the slower-growing planar crystals in at temperatures of -12.5 and -16.3 ∘C. refers to them as “aftergrowth plates”, though they form while the crystal is growing. Examples are marked with arrows in Fig. B13a. Also, fairly common are long pockets that we call ridge pockets. Ridge pockets include the main ridge pockets aligned towards the vertex and coming in a pair as well as the side-ridge pockets aligned towards each side and generally having numerous pairs. Several examples appear in Fig. B13b. This image shows main ridge pockets at A (enlarged inset upper left) and side-ridge pockets at B.

The dendrite branch in Fig. B13c shows numerous pocket features. (This crystal has undergone significant sublimation just before imaging, and thus most of the pockets appear to have reopened. But it provides clear examples of pockets seen on other crystals.) Here, A and B mark main ridge pockets. C marks an example in which pockets can form where a main ridge meets a thickened cross rib. At the bottom is a pocket and at the top is a position that may produce a pocket by such an intersection. D appears to be an elongated edge pocket (Sect. 4.6). E and G appear to be side-ridge pockets, and F is a center ring pocket (Akira Yamashita, personal communication, 2018). H may be a center pocket or an edge pocket like D. The main ridge pockets at B fade towards the main branch, indicating that the base of the pocket has a downward slope towards the tip. In images such as these, which show only one view, it is difficult to be sure that the features are pockets as opposed to channels or small hollows. Nevertheless, they appear to be common and whether or not they are pockets or channels, their effect on light scattering in clouds may be similar.

The ridge pockets may form as sketched in Fig. B14. The branch backside often has a main ridge from the tip, shown in cross section at (a). As pointed out by , the ridge produces a vapor-shadowing effect leading to the two parallel channels on both sides in the next sketch. These channels are clearly seen in images of most branched snow crystals and were noticed much earlier by . With a local growth slowdown, which may arise simply due to the branch tips growing further out, thus depleting the crystal inner regions of vapor, facet spreading may start to dominate, transitioning to protruding growth in the next two sketches. This may continue and eventually close off the channels making the two main ridge pockets at the bottom in (a). The mechanism for side-ridge pockets is sketched in (b). The process is like that of the main ridge, except side ridges can form closely together, creating a central channel gap. Protrusion growth at the crest starts bridging the gap and completes the pocket at the bottom. These sketches do not show the 3-D structure of the branch, in which the top surface slopes downward towards the thinner outermost prism faces at the tip. Thus, the protrusions likely also grow from the interior region towards the tip (as in the corner-pocket case), eventually covering the entire backside.

Trigonal crystals have only three clearly observable prism faces, a striking feature that begs for an explanation. Another crystal type with three-fold symmetry is the scalene hexagonal, which is shown together with the trigonal type in Fig. B15. finds both types rare in precipitation, but reports on a very cold cloud in which roughly 50 % of the crystals had three-fold symmetry. In the laboratory, found that numerous trigonal and scalene forms would result from seeding with an adiabatic-expansion method that creates submicron ice nuclei, but only hexagonal crystals would result from nucleating cloud droplets of much larger size. This finding may explain the difference in Bentley's and Heymsfield's findings because the latter observations were of crystals that likely formed on submicron droxtals. In Akira Yamashita's submicron seeding experiments, the crystals grew at temperatures down to about -26 ∘C. The trigonal forms appeared stable when columnar; for example, about 10 %–20 % of all crystals were trigonal in the columnar regime above -10 ∘C. But the trigonal tabular forms appeared to transition to the scalene hexagonal at small sizes, with the latter types occurring in over 40 % of the crystals at all temperatures except around -12 to -18 ∘C. In all cases the trigonal were more common than rhombohedral and pentagonal forms.

A recent review proposes three possible explanations for trigonal formation . In one, they suggest that stacking disorder can lead to the growth of trigonal forms, but they do not give a specific formation mechanism. Stacking faults in cubic crystals can produce trigonal forms e.g.,, but the mechanism involves growth on alternating reentrant corners that have not yet been shown to occur in stacking-disordered ice. Also, it is not clear how such a mechanism would explain Akira Yamashita's observations above. Our observations here (discussed in Sect. B2) suggest that regions of stacking disorder may instead lead to near-symmetric hexagonal forms (e.g., Fig. 3). The second explanation involves having equivalent dislocation step sources on just three alternating prism faces e.g.,. Such a mechanism cannot explain the preponderance of trigonal over rhombohedral and pentagonal forms. The third explanation involves aerodynamic factors that would influence habit more for tabular and larger crystals. Such an explanation also appears inconsistent with Akira Yamashita's findings above, specifically the greater stability of the trigonal form on columnar habits and the transition from trigonal to scalene hexagonal as the tabular forms grew larger. Instead of these proposed explanations, we suggest a closer look at the growth mechanism, focusing on two factors: a mechanism for their initial formation in submicron droxtals, first proposed by Akira Yamashita, and a mechanism for their stability as they grow larger.

The possible mechanism for the formation of an initial trigonal habit from a submicron droxtal is sketched in Fig. B16a–d. When the submicron droplet freezes, one prismatic plane must start forming first. Assume, as in sketch (b) that it is p1 on the left side. If the crystal is smaller than xs, facet spreading of p1 may be dominated by AST yet increase in rate as the face expands. This rate would increase in proportion to the increase in area because all the vapor impinging on the area can migrate to the edge. Thus, facet spreading would greatly favor the first face that develops. Moreover, this growth may overshoot and effectively bury the neighboring faces p2 and p6. If the next face that develops is either p3 or p5, then face p4 would be similarly buried as shown in (c). As p3 and p5 expand, the crystal fills out as a trigonal form with only p1, p3, and p5 faces as shown in (d). In this way, if after step (b), the three remaining faces were equally likely to form next, then the likelihood of a trigonal would be twice that of a crystal with four prism faces. Experimentally, the trigonals formed much more frequently than those of the rhombohedral and pentagonal (except at -4.2 ∘C), and thus the formation of the second prism face may depend on the formation of the first. For example, the region marked u in sketch (b) may immediately develop into p3 (ditto for p5 on the other side), leading to trigonal in all cases. Or, a small crystal with four prisms may be unstable compared to one with three. Regardless, the basic mechanism would lead to trigonal crystals in most cases.

The trigonal form may then be maintained via an effect of AST and the vapor supersaturation contours on layer nucleation. The supersaturation contours around the crystal, viewed in the plane of Fig. B16e, should have the same symmetry as the crystal. In the sketch, the contour is a circle, but in general needs to only have reflection symmetry about the dashed lines from the center to points RP2 (reflection-symmetry point, face 2) and RP3 as shown. (Far from the crystal, the contour will be a circle, but closer to the surface, the lines will bend closer to the surface as suggested in (f)). The consequence is the asymmetry in the contour about the vertex between p2 and p3, as shown. Assuming that the growth is via layer nucleation, with layer nucleation points near to, but not exactly at, the vertex, then such points on either side of the vertex will experience different vapor supersaturations. In particular, point NS2 on face p2 will have greater supersaturation and thus nucleate new layers at a faster rate than at the corresponding point on face p3. This factor will increase the normal rate of growth of p2 over that of p3.

This factor, though small, can be amplified by AST. Consider that the rate of layer nucleation depends on the density of surface-mobile molecules, not the adjacent vapor density directly. Thus, as indicated in (f), the slightly faster production of new layers at NS2 will draw a net AST flux from face p3, thus reducing the surface ad-molecule concentration there. As a result, the layer nucleation point NS3 must move further away from the vertex, as shown in the sketch, further reducing the layer nucleation rate on face p3. In this way, the normal-growth rate of p2 can significantly exceed that of p3 even with a relatively small vapor-density asymmetry, driving its area lower. The difference in growth rates between the faces may lead to the smaller face becoming relatively smaller or larger, depending on the ratio of the rates. (With a little trigonometry, you can readily work out the condition for a relative decrease.) But if the face area shrinks, the effect here may increase, causing further shrinkage; conversely, if the face area grows, the effect may weaken, leading first to the scalene hexagonal and then to fully hexagonal. This AST effect on the relative layer nucleation rates between adjoining faces was previously proposed by to explain the abruptness of primary-habit change with temperature. As he suggested, it should apply in general to the basal–prism edge as well, influencing the primary habits (aspect ratios) of snow crystals in general.

About the higher stability of columnar trigonal forms, consider the magnitude of the effect. The magnitude should depend on the size of the mean migration distance xs on the prism faces compared to the crystal size. When the value of xs is a significant fraction of the large-face diameter (e.g., 0.1 or more), the effect is likely to be stronger as NS3 is pushed further from the vertex. In contrast, when xs is much smaller, then the shift of NS3 will be insignificant. The values of xs for the prism face are unknown, but argued that they should be relatively large in the columnar regime (compared to the tabular regime). Such a trend in xs, if verified by experiment, could the explain the higher stability of trigonal columns as well as the transition to scalene hexagonal for the tabular case. The instability of the tabular case here is also consistent with the argument that an imposed gradient in supersaturation has little effect on the direction of tabular branches ; that is, prism faces adjacent to a supersaturation maximum should grow at the same normal-growth rate (unlike the case in Fig.  B16f) because in the tabular regime, xs for the prism face would instead be relatively small.

In addition, the AST flux from the basal to the prism should be smaller towards the narrow prism in a scalene hexagonal crystal than to the wider prism (following Sect. B7 above). This effect would further destabilize the tabular trigonal and scalene hexagonal forms, particularly for the thinner tabular crystals, but have less effect on the columnar crystals. Concerning the role of the vapor mean free path, this mechanism for the stability would have a vanishing role when the vapor mean free path exceeded the crystal size. But in an atmosphere of air, this condition would require crystal diameters below a few tenths of a micron. Finally, a greater sensitivity of the layer nucleation rate to supersaturation would increase the influence of the AST flux, strengthening the mechanism. In general, the supersaturation in a cloud is higher when the first crystals nucleate and is also higher at the crystal surface when the crystal is small, but as each crystal grows and more crystals nucleate, the supersaturation drops. This effect also predicts a transition from trigonal to scalene hexagonal, and eventually, to hexagonal. However, if the crystal develops branches while still scalene hexagonal, the nearly three-fold symmetry should remain as the branches grow independently of one another. This may explain the large, branched crystals with nearly 3-fold symmetry in Bentley's collection .

Finally, consider what would result if p4 developed right after p1 (instead of p3 or p5) in Fig. B16c. In this case, one can argue that the resulting crystal would have two large-area prisms p1 and p4, with the latter smaller than the former, and just two other equally sized faces p3 and p5. That is, the shape in cross section would be an isosceles trapezoid. Such a shape falling with p4 side down could generate the suncave Parry arc in a thin cloud. Upon growing larger, the p2 and p6 faces would likely develop, and then regardless of whether the falling orientation had p4 side down or up, a suncave Parry arc would result e.g.,. A sampling of crystals from a Parry arc display found no evidence of the trapezoid form , so such a form may transition to the six-sided form while still small.