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Triangular Snowflakes: Growing Structures with Three-fold Symmetry using a Hexagonal Ice Crystal Lattice

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Abstract and Figures

Snow crystals growing from water vapor occasionally exhibit morphologies with three-fold (trigonal) symmetry, even though the ice crystal lattice has a molecular structure with six-fold symmetry. In extreme cases, thin platelike snow crystals can grow into faceted forms that resemble simple equilateral triangles. Although far less common than hexagonal forms, trigonal snow crystals have long been observed both in nature and in laboratory studies, and their origin has been an enduring scientific puzzle. In this paper I describe how platelike trigonal structures can be grown on the ends of slender ice needles in air with high reliability at -14 C. I further suggest a physical model that describes how such structures can self-assemble and develop, facilitated by an edge-sharpening instability that turns on at a specific combination of temperature and water-vapor supersaturation. The results generally support a comprehensive model of structure-dependent attachment kinetics in ice growth that has been found to explain many of the overarching behaviors seen in the Nakaya diagram of snow crystal morphologies.
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1
Triangular Snowflakes:
Growing Structures with Three-fold Symmetry
using a Hexagonal Ice Crystal Lattice
Kenneth G. Libbrecht
Department of Physics, California Institute of Technology
Pasadena, California 91125, kgl@caltech.edu
Figure 1: This paper considers the formation and development of snow crystals with large-scale trigonal symmetry,
such as the two laboratory
-grown examples shown above. Ignoring minor surface markings, both
these crystals are
essentially equilateral
-triangular ice plates growing on the ends of c-axis ice columns, and each structure consists of
a single crystal of
hexagonal ice Ih bounded by basal and prism facets.
Abstract. Snow crystals growing from water vapor occasionally exhibit morphologies with
three
-fold (trigonal) symmetry, even though the ice crystal lattice has a
molecular structure
with six
-fold symmetry
. In extreme cases, thin platelike snow crystals can grow into faceted
forms that resemble simple equilateral triangles. Although far less common than hexagonal
forms, trigonal snow crystals have long been observed both in nature and in laboratory
studies,
and their origin has been an enduring scientific puzzle
. In this paper I describe how
platelike
trigonal structures
can be grown on the ends of slender ice needles in air with high
reliability at
-14 C. I further suggest a physical model that describes ho
w such structures can
self
-assemble and develop, facilitated by an edge-sharpening instability that turns on at
a
specific combination of temperature and
water-vapor
supersaturation. The results generally
support
a comprehensive model of structure-
dependent attachment kinetics in ice growth
that
has been found to
explain many of the overarching behaviors seen in the Nakaya diagram
of snow crystal morphologies.
2
Observations of
Trigonal Snow Crystals
The earliest mention of trigonal snow crystals
I have found is by William Scoresby [1820Sco],
who included one example in his sketches of
arctic snow crystals shown in Figure 2. Wilson
Bentley presented many trigonal crystals in his
photographic album [1931Ben], devoting a
small morphological section to platelike forms
with three-fold symmetry. Nakaya presented
some photographic examples as well
[1954Nak], although many of these were split
stars that had broken into three-branched stars
[2021Lib]. While the latter do exhibit three-
fold symmetry, their formation involves a
breakage step that is not present in the trigonal
snow crystals discussed below. (For this
reason, I exclude broken split stars from the
discussion of true trigonal forms below.)
Murray et al. [2015Mur] have summarized
many subsequent observations of trigonal
snow crystal forms in the Earth’s atmosphere
[2002Hal, 2013Shu, 2021Mag].
Trigonal snow crystals are not especially
difficult to find in natural snowfalls in
temperate climates if one examines numerous
small specimens, but of course they are far less
common than hexagonal forms. As illustrated
in Figure 3, simple plates with surface markings
are the most common trigonal crystals, with
the most basic design exhibiting alternating
long and short prism facets. In my experience,
trigonal forms in nature tend to cluster in time,
meaning that some snowfalls bring enough to
provide for multiple observations, while other
snowfalls provide essentially none. This basic
observation suggests that some atmospheric
conditions are especially conducive to trigonal
formation, a statement than can be made for
most snow crystal morphologies.
Figure 3: Photographs of several natural trigonal snow
crystals taken by the author
. The
small platelike forms
(top)
range in size from 0.6 mm to 1.4 mm, while t
he
branched crystal (bottom) measures
about 2.5
mm from
tip to tip.
Figure 2: Snowflake sketches by William Scoresby
[1820Sco] included one
specimen with three-
fold
symmetry (next to last in the diagram), which may be
the first recorded
observation
of a trigonal snow
crystal.
3
Nomenclature for Simple
Platelike Crystals
The focus of this paper is on simple platelike
trigonal forms (without branching), and for
these crystals I have found it useful to define
a “hexagonality” parameter
1=16
and a “triangularity” parameter
1= 1 − 34
for each individual specimen, where the six
prism facets are ordered by length from 1
(shortest) to 6 (longest). A perfect hexagon
then has (1,
1) = (1,0), while a perfect
equilateral triangle has (1,
1) = (0,1). For a
typical sample of simple snow-crystal plates,
the vast majority are clustered near (1,
1)
(1,0) when plotted in the (1,
1) plane,
indicating basic hexagonal forms. As a rule of
thumb, any crystal with 1> 0.85 looks
essentially hexagonal by eye.
If one removes high-1 crystals from the
sample, the remaining crystals tend to be
clustered on a line with 1+
1= 1, as
illustrated in Figure 4. I call this the “trigonal
line,” as platelike crystals near this line
generally exhibit three long and three short
prism facets that alternate around the
perimeter. Figure 5 shows some examples of
trigonal platelike crystals (the top six photos)
and irregular crystals (the lower six photos).
After excluding hexagonal crystals from the
sample, trigonal crystals are the most common,
while irregular crystals are rare.
By placing crystals on the (1,
1) plane, it
becomes clear that the most common simple,
platelike forms span a continuum along the
trigonal line, from perfect hexagons at one end
to perfect equilateral triangles at the other end.
With this continuum in mind, I often speak of
“hexagons” with (1,
1)(1,0),
“triangular” crystals with (1,
1)(0,1),
and I use “trigonal” crystals to mean anywhere
near the trigonal line with 10.85.
A basic statistical analysis [2009Lib4]
reveals that trigonal crystals are indeed much
more common than one would expect if the
different facets grew at random perpendicular
velocities. These observations suggest that
there is some deterministic physical
mechanism that guides the formation of
trigonal crystals, and this paper is mainly
focused on understanding this mechanism.
on the (1,
1)
crystal grown at -10 C in a free-
2009Lib4].
1) crystals were excluded from
These data illustrate that most
1
, where the morphology exhibits a clear three-
Figure 5). This clustering suggests
deterministic physical mechanism
promoting the growth of trigonal forms [2009Lib4]
morphologies
may be influenced
terogeneous nucleation processes from
-
ions that may arise during growth.
4
Trigonal Snow Crystals in
the Laboratory
In perhaps the first laboratory study focusing
on trigonal snow crystals, Yamashita
[1973Yam] surveyed earlier observations and
examined the probability of finding trigonal
crystals at different temperatures in a free-fall
cloud chamber, where crystals grew in air with
a background supersaturation roughly equal to
that of supercooled water droplets. These data
revealed sharp peaks in the formation of
trigonal crystals near -7 C and -25 C, with up
to 60 percent trigonal forms at both peaks.
These fractions are higher than observed by
subsequent researchers, but this may depend
on the 1 cutoff used to separate trigonal
crystals from hexagonal forms. Irregular forms
were less common than trigonal crystals in this
study, supporting earlier photographic
observations of atmospheric snow crystals.
Yamashita also pointed out that many larger
snow crystals likely went through an early
trigonal phase before growing out to nearly
hexagonal morphologies, as indicated by
trigonal surface markings near the crystal
centers (see also Figure 3).
Takahashi et al. [1991Tak] produced
another substantial data set of snow crystals
growing in air with a supersaturation near that
of liquid water, reporting that trigonal crystals
were especially prevalent near -4 C and -8 C,
but with lower probabilities than [1973Yam].
Libbrecht et al. [2008Lib1] again found
relatively low numbers of trigonal crystals over
a broad temperature range, reporting that
simple trigonal plates were especially prevalent
with low supersaturations at -2 C and -10 C
[2009Lib4].
The various observations, both in nature
and in the lab, convincingly show that trigonal
snow crystals can be found over a broad range
of growth conditions. The statistical analysis in
[2009Lib4] confirms that trigonal forms are
more common than one would expect if the six
prism facets all grew at different but randomly
determined rates. These data all describe
growth in air at a pressure near one
atmosphere, and, to my knowledge, there have
been no reports of trigonal forms appearing in
low-pressure growth data. Unfortunately, the
relative paucity of laboratory data does not
provide an especially clear picture of how
trigonal growth varies with temperature and
perhaps nucleation, and there are little data
investigating growth as a function of
supersaturation.
What these earlier measurements lack are
unambiguous clues as to the underlying
physical processes that promote trigonal
growth over the usual hexagonal
morphologies. Most snow crystal phenomena
are largely deterministic, so a careful
experiment should, in principle, provide a
dependable and reproducible sample of
trigonal forms. Indeed, the laboratory data do
Figure
5:
Photographic examples of small platelike
snow crystals grown at
-10 C in a free-
fall convection
chamber [2009Lib4].
Basic hexagonal crystals
were
the norm in this data set, and trigonal forms (top six
examples) were
the most common low-1 crystals
, as
see
n in Figure 4
. Irregular forms (bottom six
examples) were relatively rare
.
5
indicate that trigonal forms appear more
readily under certain conditions, although the
published observations are not consistent with
one another to a satisfactory degree, making it
difficult to ascertain just what optimal
conditions most reliably result in the creation
of trigonal forms.
This situation changed somewhat when I
began an experimental investigation into the
growth of platelike snow crystals on ice needles
at -14 C [2020Lib1]. During a series of
observations at this temperature, I discovered
that trigonal crystals grow quite readily with
nearly 100 percent probability over a
remarkably narrow range in background
supersaturation (denoted , which is
generally much higher and better determined
than  at the crystal surface). I had not
looked with sufficient resolution in
supersaturation to notice this localized
phenomenon earlier, but it prompted the
detailed investigation of trigonal growth on ice
needles presented here.
Trigonal Snow Crystal
Growth on Ice Needles
To begin with some background on the
peculiar physics of ice crystal growth from
water vapor, I recently developed a
Comprehensive Attachment Kinetics (CAK)
model that can explain the abrupt changes in
snow crystal growth as a function of
temperature often summarized in the Nakaya
diagram [2021Lib, 2019Lib1]. A key feature of
the CAK model is the concept of Structure-
Dependent Attachment Kinetics (SDAK), in
which the attachment kinetics on faceted
surfaces may depend on the mesoscopic
structure of the crystal, especially the width of
a faceted surface [2003Lib1, 2021Lib]. A
microscopic physical mechanism that can
produce the SDAK phenomenon is described
in detail in [2021Lib, 2019Lib1]. The CAK
model has been well supported by subsequent
targeted experimental investigations
[2019Lib2, 2020Lib, 2020Lib1, 2020Lib2],
suggesting that many of the overarching tenets
of the model provide a reasonably accurate
physical picture of snow crystal growth
dynamics over a broad range of environmental
conditions.
To further test the CAK model, I have
been looking with particular care at snow
crystal growth on slender ice needles at -14 C,
because the SDAK effect on prism facets is
especially strong at this temperature, leading to
an Edge-Sharpening Instability (ESI) that
promotes the formation of thin plates
[2021Lib, 2020Lib1]. In a nutshell, the prism
attachment coefficient  depends
especially strongly on facet width at -14 C,
Figure 6: To obtain this photograph, a set of c-axis
“electric” ice needles
was first created
on the end of
a frost
-
covered wire (the black structure at the
bottom of the frame).
The
slender needles were then
transported to an attached chamber where hexagonal
platelike snow crystals
grew on the needle tips.
Here
the plate diameters are
roughly 200 µm.
The wire can
be rotated about the vertical axis to bring each
plate
into focus for cl
oser observation.
6
because the effective nucleation barrier drops
precipitously as the facet width decreases to
molecular scales at that temperature. The CAK
model provides a physical foundation for
understanding the dynamics of snow crystal
growth as well as a narrative and nomenclature
for better discussing both the formation of thin
plates and trigonal platelike crystals on ice
needles.
The apparatus used to make these
observations is a dual-diffusion-chamber
system described in [2014Lib1, 2021Lib]. The
first diffusion chamber was used to produce
“electric” c-axis ice needles by applying a high
voltage to a frost covered wire, with the wire
tip placed at -6 C in highly supersaturated air.
Once the ice needles grew to 1-2 mm in length,
the voltage was removed and the needles were
transported to an adjoining diffusion chamber
to observe their growth under adjustable and
carefully controlled growth conditions, always
in air at one atmosphere. Figure 6 shows a
typical set of ice needles, here with thin
hexagonal plates growing on the ends of the
needles. Figure 7 shows a closer view of a
single plate from a different set of needles.
Using slender ice needles as seed crystals
has a distinct experimental advantage
compared to normal seed crystals, in that there
is only a single basal surface initially present at
the needle tip, accompanied by a single set of
basal/prism corners [2021Lib]. This provides a
simpler platform for investigating the
formation of thin platelike crystals, as the
diffusion-limited growth dynamics on a typical
small seed crystal is significantly complicated
by the presence of two basal surfaces with
competing sets of basal/prism corners.
Figure 8 shows the results of a single data
run in which the supersaturation was slowly
lowered as a function of time over several
hours while new needle crystals were
periodically grown and inserted into the second
diffusion chamber for observation. The
resulting platelike crystals clustered mainly near
the trigonal line, so the value of 1 is roughly
equal to 11− 
1 for each data point. Note
that this is a complete data set for this run,
where every crystal with a reasonably well-
formed morphology was recorded. (A few
asymmetrical, clearly abnormal structures were
not counted.) Additional data runs showed that
the general trends seen in Figure 8 were quite
reproducible.
At high in Figure 8, thin hexagonal
plates grew on the ends of all the ice needles,
and Figures 6 and 7 show examples of this
happening. As dropped below about 8
percent, however, trigonal plates appeared
instead, including some nearly perfect
triangular plates with
11. Remarkably,
Figure 7: This image shows closer view of a hexagonal
platelike snow crystal growing on the end of an ice
needle.
The plate diameter is about 400 µm.
The top
basal surface
is essentially flat and featureless,
presumably with a slightly concave structure (not
visible
in the photo) required for diffusion-
limited
faceted growth
.
The hexagonal tips show the initial
development of “ri
dgestructures [2021Lib]
, which
form on the underside of the
plate.
(Figure 6 shows
narrower ridges that grow at somewhat higher
supersaturations.) The faint “billowing” features near
the plate center are macrostep
s
that form where the
lower
basal surface meets the needle tip
, as there is
no
basal nucleation barrier at tha
t location. These
macrosteps propagate outward as the crystal grows,
which is easily seen in timelapse videos.
The c-
axis
needle is initially quite slender, but then thickens with
time
with an overall hexagonal columnar structure.
7
essentially no hexagonal plates (depending on
where one places the
1 cutoff) grew along
with the trigonal plates. When was reduced
below 6 percent, platelike crystals no longer
grew from the ice needles, and the initially
slender ice needles grew slowly into blocky
forms or into simple columns. Figure 9 shows
several examples of platelike crystal
morphologies with different values of
1.
The behavior seen in Figure 8 provides an
important clue for understanding the
formation of trigonal plates at -14 C. The
“SDAK dip” on the prism facet is most
pronounced at -14 C [2020Lib1], meaning that
the Edge-Sharpening Instability (ESI)
responsible for plates emerging on the end of
ice needles is especially sensitive to growth
conditions. I observed trigonal plates forming
at other temperatures near -14 C, but the drive
to form trigonal plates appeared to be strongest
at precisely the same temperature that plate
formation is also strongly driven. This is almost
certainly not a coincidence, and it means that
the ESI responsible for plate growth near -14 C
likely plays an important role in the
development of trigonal forms. This
correspondence does not explain every aspect
of trigonal growth, specifically at other
temperatures, but it provides an important clue
for ascertaining the physical mechanisms
underlying trigonal growth from ice needles at
this specific temperature.
Triangular Plates
Figure 10 shows the development of a nearly
perfect triangular snow crystal on an ice needle.
In this example, the triangular structure
Figure
8: A data run in which the supersaturation
was
slowly lowered over several hours while
crystal growth
on
needle crystals was observed at -14 C.
To avoid
selection bias,
all growing crystals were recorded
during
this run
(except for a few malformed outliers)
. The data
clearly show a transition from hexagonal plates to
trigonal plates
as decreases, followed by
additional
transition
s to blocky forms and then to simple columns.
Figure 9: Examples of platelike crystals growing on ice
needles at different measured values of
1. Note that
the crystals are not viewed face
-
on, as evidenced by
the orientation of the c
-axis ice needles.
8
emerged quite quickly when the crystal was too
small to image clearly, and the equilateral-
triangle morphology was maintained as the
crystal grew larger. This series of images
provides another important clue for
understanding triangular growth, because all
the triangular crystals I observed seemed to
develop very quickly, when the plate was quite
small. Once a trigonal crystal was observed
with
1< 1, it never developed into a
triangular crystal with
11. This was true
not only for full triangular crystals, but also for
individual triangular corners (that is, a sharp
corner consisting of a single, exceedingly small
prism facet flanked by two larger prism facets).
Figure 11 shows an example where two
sharp triangular tips appeared when the crystal
was small and maintained their sharp-tip
morphology as the crystal grew. The third tip
did not establish itself fully when the crystal
was small, and it retained its truncated-tip
morphology during further growth. Both
Figures 10 and 11 show that sharp triangular
tips (where a prism facet with near-zero width
is flanked by two large prism facets) can grow
stably under the right conditions once this
morphology is established. With plate-on-
needle growth at -14 C, however, the sharp-tip
morphology seems to develop only when the
plate first emerges from the columnar needle,
and not at later times.
Figure 10: This series of photos illustrates the
development of a nearly perfect triangular plate on
the end of an ice needle. The first image shows a view
perpendicular to the c
-
axis needle, showing the plate
edge
-on. The other images show the same crystal
from
a different angle
to better view
the plate morphology
as it grows
. In all my observations, I never witnessed
a
trigonal
plate with
1< 1
transforming into a
triangular plate with
11. Rather, t
he three sharp
points on triangular crystal
s always emerged early
,
when the overall plate size was 20 μm or less.
Conversely, triangular plates frequently transformed
into trigonal plates with
1< 1, which occurred
both
when
was
above or below the optimal value that
maintained
triangular growth.
Figure 11: In this example, a plate with two sharp
triangular corners develops over time.
The
longest prism
facet in the bottom
image is 320 μm
in length, and the
top images are shown at the same scale. The two main
triangular tips developed early and maintained their
sharp morphology as the crystal grew
, over
an elapsed
time of 33 minutes in this series
. The bottom facet
,
however,
did not develop into a third sharp
triangular
tip.
9
Diffusion-Limited Growth
and Anisotropic Attachment
Kinetics
The growth of a sharp-tipped triangular snow
crystal necessitates some degree of anisotropic
attachment kinetics to maintain, specifically
where  on the tip facet must be higher
than  on the much larger neighboring
facets. Diffusion-limited growth in the absence
of this kind of anisotropic attachment kinetics
will not yield a stably growing tip structure. To
see this, consider the region near a sharp tip
illustrated in Figure 12.
Considering the large prism facets first,
these maintain an essentially flat structure as
they grow, requiring a constant perpendicular
growth velocity at all points on the facet.
Particle diffusion produces a larger  at the
tip compared to the large-facet centers,
however, so faceted growth requires that
 be larger at the facet centers. The
 needed for stable constant-velocity
growth is provided by molecular steps on the
large-facet surface, and the step density
automatically adjusts itself to maintain a
perpendicular growth velocity that is equal at
all points on the facet surface. The large facets
are thus always somewhat concave in shape,
although these surfaces appear flat in
photographs. This self-regulating process is
well known as the primary mechanism for
maintaining a faceted morphology when the
growth is limited by diffusion.
The situation becomes more interesting
near the sharp tip, however, where all three
prism facets exhibit their top molecular
terraces (see Figure 12). Nucleation on these
top terraces, free from molecular steps, is
generally what limits the growth of the entire
faceted structure. In the case of a sharp tip
structure, the top terraces on all three facets
will be positioned quite close to the tip.
Because the length scale over which 
changes appreciably will likely be much larger
than the spacing between the three top
terraces, one can assume that  is
essentially constant near the growing tip.
From simple geometrical considerations
with a 60-degree tip, the narrow prism facet at
the apex in Figure 12 must grow at exactly
twice the perpendicular growth velocity of the
adjacent large facets. Moreover, because 
is nearly constant at the tip, this means that
, on the top terrace of the apex facet
must be about double that on the top terraces
of nearby large facets. In fact, the stable tip
growth suggests , > 2, ,
as this allows the width of the top terrace to
automatically adjusts itself until the proper 2:1
growth velocity is achieved via the Gibbs-
Thomson effect. Explaining a factor-of-two
difference in the attachment coefficients on
these surfaces is a nontrivial challenge,
however, because the molecular structures of
all three of these top prism terraces are
identical. This is the crux of the trigonal-
growth problem explaining how ostensibly
identical prism surfaces can have markedly
different attachment kinetics.
Figure 12: A sketch of a growing triangular crystal near
one of the sharp tips.
Far from the tip,
terrace steps
provide
a higher attachment coefficient,
thus
balancing
the lower supersaturation
to yield normal
faceted growth.
Near the tip, however, the top
prism
terraces
will be step-free, so
some other form of
anisotropic attachment kinetics on the different prism
facets is necessary to maintain the sharp
-tip structure.
10
I believe that Structure-Dependent
Attachment Kinetics (SDAK) can again
provide a ready explanation for the anisotropic
attachment kinetics that is required to maintain
a growing sharp-tip structure. In the SDAK
model I have proposed [2003Lib1, 2019Lib1,
2020Lib1, 2021Lib],  depends on the
width of a growing prism facet at -14 C,
because the broad-facet nucleation barrier is
greatly diminished by surface-diffusion effects.
This SDAK model is essentially a two-
dimensional (2D) effect, operating to promote
the growth of thin platelike crystals with sharp
prism edges.
For triangular snow crystals at -14 C, I
envision a three-dimensional (3D) version of
the same SDAK effect, this time operating on
prism facets that are small in both lateral
dimensions (as opposed to a plate edge, where
the prism facet is only small in one dimension).
Given that the 2D SDAK effect yields large
increases in  for prism edges at -14 C
[2020Lib1], it would not be surprising to see an
increase of a factor of two or more in 
on the sharp triangular tip, where the top prism
terrace becomes a small island. Developing a
comprehensive model of this new 3D SDAK
effect is a nontrivial task but hypothesizing its
existence to explain the growth of triangular
plates at -14 C seems like a reasonable next step
toward a fuller understanding of the complex
phenomenon of snow crystal growth.
Continuing this reasoning, I hypothesize
that the 3D SDAK effect then yields a Tip
Sharpening Instability (TSI) that corresponds
to the Edge Sharpening Instability (ESI)
resulting from the 2D SDAK effect. As with
my previous efforts to understand the CAK
model including the SDAK effect, creating an
unambiguous molecular model of all the
underlying physical processes will likely remain
an unsolved problem for many decades. There
is hope that molecular-dynamics simulations
will shed some light on these issues, and we
have seen recent progress on this front
[2021Lib]. In the meantime, however, I have
found it quite beneficial to develop the semi-
empirical CAK model using qualitative
physical insights coupled with targeted
experimental investigations. At the very least,
this avenue of model development suggests
additional targeted experiments that can shed
light on the underlying model assumptions. I
proceed with the SDAK model hypothesis,
therefore, as it provides a framework for
guiding our discussion of trigonal snow
crystals.
Morphological Development
of Triangular Plates
While I have never witnessed a non-sharp tip
on an established platelike trigonal crystal
develop into a sharp triangular tip, I have
frequently observed sharp triangular tips
evolve into trigonal shapes where the tips are
not sharp. Lowering the supersaturation will
easily produce this evolution, and Figure 13
shows one example of this happening.
Figure 13: Lowering the supersaturation sufficiently
will
disable the Tip-Sharpening Instability
so sharp
triangular tips develop into
larger prism facets.
The
image on the left shows a near
-
perfect triangular
plate
with prism facet lengths of 165 μm.
The surface
markings on the underside of the plate
appear to
have
a negligible effect on its overall triangular structure.
The image on the right shows the same crystal (at the
same scale) after an additional 35 minutes of growth
after lowering the
supersaturation. The
resulting
slower growth
deactivates the TSI mechanism,
so
then
all six prism facets grow at roughly equal rates
,
causing
1 to decrease with time.
11
As with the ESI, the TSI will only operate
above some threshold supersaturation, below
which  on the sharp-tip facets will be
less than twice that of  on the larger
facets. When this happens, the plate
morphology will slowly evolve to lower
1
values. If  becomes equal on all six
facets, then the shape will slowly evolve toward
that of a simple hexagon, even if it started out
with a trigonal shape.
This overall behavior reflects the general
maxim in diffusion-limited snow-crystal
growth that anisotropic growth morphologies
on large scales require anisotropic attachment
kinetics [2021Lib]. As discussed with Figure
12, diffusion alone, specifically the Berg effect,
will not promote the growth of triangular
plates, but will instead result in evolution
toward a basic hexagonal shape. The
observation demonstrated in Figure 13,
therefore, supports the idea of a new form of
SDAK at the triangular tips, resulting in a Tip-
Sharpening Instability that will drive the
formation of triangular plates.
Increasing the supersaturation will also
cause triangular plates with
11, to evolve
into trigonal plates with
1< 1, and this
phenomenon is illustrated in Figure 14. My
interpretation of these images is that a TSI was
initially present, yielding the two sharp
triangular tips seen in the first image. As
described above, the TSI was necessary to
maintain the sharp-tip structure during small
changes in with time. Increasing
substantially then caused an abrupt jump in the
growth rates of the broad facets, while having
a smaller effect on the tip growth. This
changed the growth ratio sufficiently that the
tip was no longer advancing at twice the rate of
the broad facets. The narrow tip then
broadened so the TSI was inoperable and
1
soon dropped below unity.
A better analysis would require full 3D
numerical modeling of this complex structure,
which is not yet possible with existing
techniques. Nevertheless, the example does
illustrate that increasing the supersaturation
Figure 14: This series shows how triangular tips can
evolve to
larger
facets when the supersaturation is
increased. The top image show
s
a partial triangular
plate (
with two sharp tips) growing at 7%
.
Turning the supersaturation up to
12%
yielded the second and third images after additional
growth,
showing ridges dev
eloping at the sharp tips
and thin
splitting as the sharp tips evolved into larger
prism facets.
12
can disrupt the steady-state growth of a sharp-
tipped triangular crystal.
From these and other observations not
presented here, it appears that triangular plates
forming on c-axis needles at -14 C can be
understood as arising from a 3D form of
Structure-Dependent Attachment Kinetics
(SDAK), as this could provide a sufficiently
high degree of anisotropy in the attachment
kinetics. Specifically, the growth of triangular
crystals with
11 requires that  on
the sharp tips be at least twice that of 
on the large facets. The required SDAK effect
then leads to a Tip Sharpening Instability that
establishes and maintains the sharp triangular
tips. This physical effect is a somewhat natural
extension of the Comprehensive Attachment
Kinetics model [2021Lib, 2019Lib1], thus
further supporting that model and its
explanation of the Nakaya diagram.
Morphological Development
of Trigonal Plates
Trigonal plates span a broader spectrum of
possible morphologies, covering the whole
trigonal line in the (1,
1) plane, so any
analysis will not be as clean as with the extreme
case of sharp tips on triangular crystals.
Moreover, many trigonal morphologies
depend on the entire growth history of the
crystal, which further complicates the
discussion. Nevertheless, Figure 15 shows one
illustrative example of a basic trigonal plate and
how its growth changed when the
supersaturation was increased.
In the first image in Figure 15, the trigonal
form grows in an essentially stable fashion with
the short facets having a 60% higher
perpendicular growth velocity than the long
facets. One can perform a qualitative diffusion
analysis like that in Figure 12, except this time
focusing on a small region around a single
corner between a short and long facet. Once
again, because the top facet terraces are both
quite close to the corner, one expects that the
surface supersaturation will be nearly identical
on each. The difference in growth velocities,
therefore, indicates that  on the short
facet must be about 60% higher than that on
the long facet. And the TSI no longer applies
in this case because both top prism terraces are
long and narrow.
One possible explanation for the
difference in  is that the short facet is
slightly thinner than the long facet, so we can
then invoke the normal 2D SDAK effect. The
two sets of facets do have a slightly different
appearance in Figure 15, but this is not
sufficient to say much about the top terrace
widths. Nevertheless, the measurements in
[2020Lib1] show that  at -14 C increases
roughly 100-fold as thin edges of platelike
crystals develop, so it would not be surprising
to see a 60% increase in  between the
short and long facets in Figure 15.
Raising the supersaturation (second photo
in Figure 15) increases the growth rates of both
the long and short facets, but the long facets
started out slower and exhibit a larger net
velocity increase compared to the short facets.
Figure 15: The image on the left shows a trigonal plate
with a relatively high
1,
and measurements reveal
that the short facets have a perpendicular growth
velocity
about 60% faster than the long facets
. The
growth ratio
can also be ascertained from
the ridge
structure, as each ridge traces
the position
of a corner
between facets
as a function of time. After this
first
picture was taken, the supersaturation was increased
and the photo on the right was taken after additional
growth.
Now the long and s
hort facets have similar
overall appearances, and the
ridges indicate that they
have nearly equal growth rates.
13
As with triangular crystal in Figure 14, it
appears that the higher reduced the 
difference between facets, equalizing the
growth rates and lowering
1 in the process.
Soon, as seen in Figure 15, the long and short
facets develop similar appearances and grow at
similar rates. Extending this growth behavior
forward, the overall shape would evolve to
become more hexagonal with time.
My main conclusion from this example is
that a substantial series of quite careful
measurements would be required to draw any
quantitative conclusions along these lines.
Moreover, the ridge structures present on these
platelike crystals may substantially affect the
growth dynamics and edge thicknesses.
Understanding this better would thus require a
full 3D numerical analysis of the growing
crystals, which is not technically possible at
present.
A secondary conclusion, however, is that
the trigonal growth behavior seen in this
example is at least consistent with the basic
tenets of the proposed SDAK mechanism. The
crystal morphology may be too complex for
detailed analysis, but the overarching behaviors
seen in this and other trigonal plates-on-
needles generally supports the CAK model
interpretation.
A Physical Model
Explaining the Growth of
Trigonal Snow Crystals
on Ice Needles at -14 C
A fundamental mystery surrounding trigonal
snow crystals since their first observations
relates to how they form and develop.
Focusing on platelike crystals, the basic
hexagonal morphology is certainly expected
because the six prism facets are all essentially
identical and would be expected to exhibit
identical growth velocities. And, indeed, simple
hexagonal forms are commonly found
whenever platelike crystals grow.
But why would a trigonal plate form,
especially a sharp-tipped triangular plate? What
physical processes differentiate the growth
rates of the short and long prism facets, and
why do they alternate around the crystal to
yield a trigonal shape? With these new
observations of trigonal plates growing on ice
needles, I have created a developmental model
that can explain the many of the overarching
features of how these enigmatic crystals
originate and develop.
There have been some previous attempts
to explain the formation of trigonal snow
crystals, but I feel that they fall short of
achieving a reasonable understanding of the
underlying physical processes. For example,
Murray et al. [2015Mur] recently suggested that
trigonal ice crystals in the atmosphere may not
be made from normal crystalline ice Ih, but
rather from a stacking-disordered ice lattice
with trigonal symmetry. This seems quite
unlikely, in my opinion, especially when cubic
ice Ic the only other form of ice clearly
identified under atmospheric conditionshas
never been observed in single-crystal form (to
my knowledge). Moreover, small snow crystals
are often essentially dislocation-free single
crystals, and even twinning is relatively rare. It
is difficult to imagine the occurrence of near-
perfect stacking disorder in macroscopic snow
crystals. X-ray crystallographic measurements
could prove the existence of a triangular lattice
structure, and potential trigonal samples for
such a study are not hard to fine. Until such
evidence appears, however, I consider the
stacking-disorder hypothesis somewhat
improbable.
Libbrecht and Arnold suggested that
aerodynamic stability considerations could
result in trigonal forms [2009Lib4], but I now
believe that this is not likely a viable
explanation either. In a subsequent
investigation of the aerodynamics of snow
crystal growth more broadly, I found that
diffusion effects on snow crystal growth (also
known as the ventilation effect [1982Kel]) are
probably too insignificant around small snow
crystal plates to affect their early growth
dynamics [2009Lib3]. Moreover, the reasoning
in Figure 12 suggests that anisotropic
14
attachment kinetics are needed for trigonal
growth, which is separate from aerodynamics
considerations. Plus, of course, aerodynamic
effects are completely absent in the plate-on-
needle observations described above, which
take place in static air.
In thinking about this problem more
carefully, I found that it was not difficult to
conjure up physically plausible models that
could facilitate the formation of sharp
triangular tips. Given that these tips will grow
stably via the TSI, assuming the 3D SDAK
effect described above, all that is required is
some sequence of events that yields three such
tips in an alternating pattern to make a
triangular crystal. A major problem with most
such scenarios, however, is that non-triangular
forms should also appear with sharp triangular
tips, especially the diamond-shaped form
illustrated in Figure 16. Both diamonds and
triangles possess stable triangular tips, and both
these forms would grow stably with the TSI.
However, observations clearly reveal that
triangles form much more readily than
diamonds. Explaining this disparity proved to
be an important consideration in the formation
problem. If sharp triangular tips emerged
somewhat randomly, one would expect to see
diamonds appearing with some regularity,
whereas few to none are observed in
experiments or in nature. This observation
suggests that there must be some deterministic
path that guides the formation of triangular
plates with much higher probability than
diamond-shaped plates.
Origin and Development
of thin Plates on Ice Needles
Before describing a model for the formation of
trigonal plates on ice needles, it is beneficial to
consider first how hexagonal plates arise,
which is illustrated in Figure 17. The first
sketch (a) in this figure depicts the initial ice
needle, created using high electric fields in our
dual-diffusion-chamber apparatus [2014Lib1]
and then transported (with the electric fields
removed) to the second chamber for
Figure
17: This series of sketches illustrates how
a thin
hexagonal plate can emerge from the tip of a
hexagonal needle crystal. As described in the text, the
SDAK mechanism
and the resulting ESI naturally cause
a plate segment to form at some random location, and
the plate segment will quickly
“wrap around”
the
column to
form a full hexagonal plate. The slight
initial
deviation from a perfect hexagonal
shape will soon
become insignificant
as the six facets
continue
growing
outward at nearly identical rates.
Figure 16: Triangular snow crystals grow readily from
hexagonal needle precursors at
-
14 C (left), but
diamond
-
shaped crystals clearly do not (right).
Explaining this disparity is an important and nontrivial
requirement for any physical model describing the
formation of triangular plates on ice needles.
15
observation. The radius of this initial column is
no more than a few microns at its tip.
As the needle grows, its radius increases
and the basal/prism edges on the needle tip
will develop thin plates via the ESI, being
driven by the SDAK phenomenon, which is
part of the CAK model of snow crystal growth
[2021Lib, 2019Lib1]. I assume that the CAK
model is correct in this discussion, and note
that the growth of such plates, once
established, is described in some detail in
[2020Lib1].
Referring to Figure 17 (b), we assume that
the ESI does not produce a thin plate on all
prism edges simultaneously, but instead a
single plate segment likely first appears at
random on one of the basal/prism edges. Once
the plate segment forms,  increases
dramatically on the thin prism facet, as this is
the nature of the underlying SDAK effect. This
increases the plate edge growth relative to the
broad needle facets, so the plate segment grows
out rapidly from the needle, as illustrated in
Figure 17 (b).
As the plate segment extends out from the
column, we see that three plate edges soon
develop – the central edge flanked by two side
edges. Because the side edges are also quite
thin,  will soon become high on these
also, via the SDAK effect, so the side edges will
begin to grow outward, causing the plate
segment to “wrap around” the column, as
shown in Figure 17 (c). Put another way, the
narrow side edges have a much-reduced
nucleation barrier compared to the broad
needle facets, and this enhanced nucleation
yields a train of molecular steps that propagate
onto the needle facets.
This process continues with the other
prism surfaces, so the plate quickly wraps
around the column completely, yielding a full
platelike crystal on the end of the columnar
needle. Once the six segments start growing
outward at roughly equal rates, the plate
evolves to become essentially hexagonal in
shape, so it is no longer obvious that one
segment had a brief head-start relative to the
others. The result is a near-perfect hexagonal
plate growing out from the ice needle, which is
commonly observed.
Figure 18: This series of sketches illustrates how a thin
triangular plate can emerge from the tip of a
hexagonal needle crystal. The main difference from
Figure 17, as described in the text, is that the wrap
-
around
process proceeds more slowly when
the
supersaturation is so low that the ESI is just barely
operational.
In this case,
plate segments can grow out
to become triangular tips before
the neighboring
plate segments develop. If the timing is
right
, this
process
can yield full triangular plates, as are
observed. If the timing is somewhat off, “failed”
triangular plates with only one or two triangular tips
can result, and these are also commonly seen in
observations. It would be highly unlikely for a
diamond
-shape
d crystal to result from this process,
however, thus explaining why
diamond-like
shapes
are
not seen in plate-on-needle data.
16
Next consider the formation of a plate
when the supersaturation is somewhat lower,
so the ESI is weaker and just barely able to
produce the transition from a basal/prism edge
to a thin plate. The formation of the first plate
segment eventually takes place, and this is
shown in Figure 18 as the transition from (a) to
(b). The faster growing plate will tend to lower
the supersaturation around the crystal,
however, so the first “wrap-around” event may
be quite slow. While this is happening, the first
plate segment may grow out so far that it
becomes a triangular tip, activating the TSI
because of its small apex width, as illustrated in
Figure 18 (c). Note that, because the plate
segment is attached to the needle, the plate
thickness will be smallest on the outermost
facet, which facilitates the tip formation. Once
the tip forms, the TSI then ensures that it will
grow stably outward thereafter.
After some additional growth, the nascent
plate will continue to wrap around the
perimeter of the needle, proceeding from (c) to
(d) in Figure 18. Once again, however, it will
take time to make these transitions with the
weakened ESI, and the final step to the sixth
facet may occur after the 4th and 5th plate
segments have grown out to form triangular
tips, as indicated in Figure 18 (e). If this chain
of events proceeds as indicated, the result will
be a full triangular snow crystal plate growing
out from the ice needle. From there, as
illustrated in Figure 11, the tip morphology can
remain stable for long periods, yielding a large
triangular plate with sharp tips.
Of course, the process depicted in Figure
18 is likely somewhat fragile, leaving numerous
avenues for creating a “failed” triangular crystal
with fewer than three sharp tips. Indeed,
Figures 11 and 14 show plates with only two
sharp tips, which would be a likely failure mode
as the plate wraps around the needle. In some
circumstances, I have found that these
incomplete triangles are more likely than full
triangular crystals, which is not surprising with
this model.
Importantly, it would be somewhat
difficult to create a diamond-shaped plate using
the progression shown in Figure 18. A
diamond would require a substantial imbalance
in the wrap-around steps, or perhaps the
simultaneous emergence of plate segments on
opposite sides of the initial needle. These
scenarios seem somewhat unlikely in this
model, thus explaining why diamond-shaped
plates are much less likely than triangular
plates. Overall, therefore, this model does a
reasonably good job explaining, at least
qualitatively, why triangular plates and related
“failed” triangular plates would be relatively
common while diamond-shaped plates would
be unlikely.
This model also provides a ready
explanation for the sharp peak in the
production of triangular crystals as a function
of supersaturation at -14 C. A high
supersaturation would yield a strong ESI and a
fast wrap-around process, thus yielding
hexagonal plates with high probability, as is
observed above below =8% in Figure 8.
In contrast, a low supersaturation would mean
that the ESI is too weak to produce any
platelike growth, and this is observed at
supersaturations below =6%. Only when
the ESI is just beginning to turn on will the
conditions be conducive to reliably producing
trigonal and full triangular plates, which
happens over a narrow range in
supersaturations.
A particularly pleasing feature of this
trigonal origin scenario is how well it fits with
the CAK model. If one embraces the CAK
model from the outset and then asks how then
plates will emerge from ice needle precursors,
the growth behaviors in Figures 17 and 18
emerge quite naturally. The formation of
trigonal plates is indeed a deterministic process
that becomes likely when the ESI is just turning
on, so over a narrow range in supersaturation,
as is observed. Thus, although the CAK model
was not developed with trigonal growth in
mind, it seems to do a satisfactory job
explaining many observed behaviors observed
in the formation of trigonal plates on ice
needles at -14 C.
17
Progress Toward
Understanding the
Formation of Trigonal
Snow Crystals
Since they were first documented over 200
years ago, trigonal snow crystals have remained
something of an enduring scientific puzzle.
Basic crystallography tells us that the six prism
facets on a trigonal crystal have identical lattice
structures, yet the alternating long and short
facets grow at different rates in trigonal
crystals. Equilateral-triangular forms are
especially enigmatic in this regard, being the
most extreme examples of trigonal snow
crystals.
In this paper, I describe a focused
investigation of trigonal snow crystals that
makes substantial progress toward a better
understanding of their origin and growth
dynamics. Primary results include:
I developed a novel “recipe” for growing
trigonal and triangular snow crystals on the
ends of thin ice needles. Trigonal plates form
most readily near -14 C in a narrow range of
supersaturations, as shown in Figure 8. The
trigonal yield peaks at nearly 100 percent,
and sharp-tipped triangular plates can be
made quite reliably. This discovery enabled a
detailed experimental investigation of the
growth behaviors of these crystals under
controlled environmental conditions.
The formation of plate-on-needle trigonal
crystals at -14 C in air occurs precisely when
the supersaturation is just high enough to
drive the formation of thin plates. Blocky
crystals appear at lower supersaturations,
while thin hexagonal plates appear at higher
supersaturations, as observed in Figure 8.
Triangular plates always appear quickly, as
the plate first forms on the needle tip. I have
never observed a triangular plate with
1
1 evolving from an established trigonal plate
with
1< 1. The opposite evolution, from
triangular to trigonal, it commonly observed
by either raising or lowering the applied
supersaturation.
Basic modeling considerations reveal that
diffusion-limited growth alone cannot
explain the growth of sharp triangular tips,
as illustrated in Figure 12. The three prism
facets (defined by their top molecular
terraces) exist in close proximity near the tip,
so the surface supersaturation must be nearly
identical on all three. Yet the tip facet grows
at twice the rate as is broad-facet neighbors,
thus requiring , > 2, .
This result supports a general maxim in
snow-crystal growth that large-scale
morphological anisotropy requires a
corresponding anisotropy in the attachment
kinetics [2021Lib].
The Comprehensive Attachment Kinetics
(CAK) model can provide a physically
reasonable explanation for the stable growth
of triangular tips. While the 2D SDAK effect
describes how  on a thin plate edge
can be higher than  on a broad prism
facet [2019Lib1, 2021Lib], a straightforward
3D extension of the SDAK effect could
yield the required higher , on sharp
triangular tips. The 2D Edge-Sharpening
Instability (ESI) then turns to a 3D Tip-
Sharpening Instability (TSI) on the narrow
prism facet, stabilizing the sharp-tipped
structure during growth. Creating a full,
quantitative 3D model of this phenomenon
is a challenging task, but the overarching
physical processes in the CAK model
provide a plausible physical explanation for
the observed stable tip growth.
I further describe a scenario where triangular
crystals can emerge from the tip of a needle
crystal via a deterministic process, as
illustrated in Figure 18. While the chain of
events in this model is speculative, it
provides at least one plausible mechanism by
which triangular and trigonal forms can
emerge in a predictable fashion. Moreover,
the model further suggests that diamond-
shaped crystals would be rare in comparison
to triangular plates, as is observed.
18
While these results provide a reasonably
self-consistent and physically plausible
explanation for the origin and growth of
triangular and trigonal crystals on ice needles,
many questions remain. The investigations
presented here all focused on trigonal plates
growing on ice needles at -14 C, and the ideas
presented in this paper are somewhat limited to
this experimental circumstance. Trigonal
crystals have also been created in controlled
laboratory conditions at other temperatures,
however, as summarized at the beginning of
this paper, and these likely require one or more
different explanations. One particularly
pertinent example, continuing with our focus
on platelike trigonal crystals, is the relatively
common growth of trigonal plates at -2 C
[2008Lib1].
Snow crystal growth at -2 C is quite
different compared to -14 C, and the SDAK
mechanism proposed in the present paper does
not apply at the higher temperature. The
“SDAK dip” on the prism facet is localized
near -14 C [2020Lib1], and its effects are
completely absent at -2 C. However, additional
growth data at -2 C suggests that there is some
new physics at play when surface melting
becomes more pronounced near the melting
point [2020Lib2, 2019Lib2]. To explain the
growth data, I have hypothesized the existence
of a second SDAK effect on prism facets at
high temperatures, a phenomenon that is,
unfortunately, only poorly understood at
present. I further hypothesize here that this
second SDAK mechanism may play a role in
producing platelike trigonal crystals at -2 C.
Much additional study is needed to better
understand even the basic behaviors of snow
crystal growth at such high temperatures,
however, let alone trigonal growth. The data do
suggest, however, that there is much left to be
learned by additional studies of snow crystal
growth at temperatures near the melting point,
which is a relatively poorly explored region of
phase space.
In summary, I have demonstrated an
experimental procedure for readily creating
platelike trigonal snow crystals on the ends of
ice needles at 14 C. The technique can be
used to produce nearly perfect triangular
specimens, which are particularly interesting as
an extreme special case. I also describe a
qualitative physical model that can explain the
stable growth of triangular tips, various aspects
of trigonal morphological development, and a
deterministic chain-of-events that will plausibly
yield triangular crystals, specifically without
producing diamond-shaped crystals that are
not observed. The phenomenon of trigonal
snow crystals, therefore, is perhaps somewhat
less puzzling than it was before, at least under
these restricted experimental conditions.
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Often overlooked in studies of ice growth is how the crystal facets increase in area, that is, grow laterally. This paper reports on observations and applications of such lateral facet growth for vapor-grown ice in air. Using a new crystal-growth chamber, we observed air pockets forming at crystal corners when a sublimated crystal is regrown. This observation indicates that the lateral spreading of a face can, under some conditions, extend as a thin overhang over the adjoining region. We argue that this extension is driven by a flux of surface-mobile molecules across the face to the lateral-growth front. Following the pioneering work on this topic by Akira Yamashita, we call this flux “adjoining surface transport” (AST) and the extension overgrowth “protruding growth”. Further experiments revealed other types of pockets that are difficult to explain without invoking AST and protruding growth. We develop a simple model for lateral facet growth on a tabular crystal in air, finding that AST is required to explain observations of facet spreading. Applying the AST concept to observed ice and snow crystals, we argue that AST promotes facet spreading, causes protruding growth, and alters layer nucleation rates. In particular, depending on the conditions, combinations of lateral- and normal-growth processes can help explain presently inexplicable secondary features and habits such as air pockets, small circular centers in dendrites, hollow structure, multiple-capped columns, scrolls, sheath clusters, and trigonals. For dendrites and sheaths, AST may increase their maximum dimensions and round their tips. Although these applications presently lack quantitative detail, the overall body of evidence here demonstrates that any complete model of ice growth from the vapor should include such lateral-growth processes.
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I examine a variety of snow crystal growth measurements taken at a temperature of -5 C, as a function of supersaturation, background gas pressure, and crystal morphology. Both plate-like and columnar prismatic forms are observed under different conditions at this temperature, along with a diverse collection of complex dendritic structures. The observations can all be reasonably understood using a single comprehensive physical model for the basal and prism attachment kinetics, together with particle diffusion of water vapor through the surrounding medium and other well-understood physical processes. A critical model feature is structure-dependent attachment kinetics (SDAK), for which the molecular attachment kinetics on a faceted surface depend strongly on the nearby mesoscopic structure of the crystal.
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Trigonal ice crystals, ice crystals with the basal faces of a regular triangle, are formed by seeding I of using adiabatic cooling as well as by seeding II of using a very cold body. When ice crystals are made by the former seeding and are grown in free fall for about 200 seconds in a dense supercooled cloud, the production rate of trigonal ice crystals depends considerably on temperature of the cloud. At about -7°C it runs up to about 20%. Scalene hexagonal ice crystals, which are obtained in abundance at the same time as trigonal ice crystals, are inferred to be grown from trigonal ice crystals. Rhombic ice crystals, scalene pentagonal ice crystals and trigonal dendrites are also observed occasionally. These results mean that the growth directions <1120> and <1010> are the first and the second favourable directions respectively in the basal plane for the growth of ice crystals. In free fall growth, therefore, ice crystals having trigonal constructions seem to be obtained because growth toward<1010> initiate at the time of nucleation.
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The characteristics of snow crystal growth by vapor diffusion at water saturation and in free fall were quantitatively investigated in a vertical supercooled cloud tunnel for periods up to 30 min at temperatures from -3 to -23°C. The results obtained are as follows: 1) the basic growth habits were plates (>-4.0°C), columns (-4.0--8.1°C), plates (-8.1--22.4°C) and columns (<-22.4°C), respectively. At about -5.5, -12, -14.5, -16.5 and -18°C, crystal shapes were enhanced with time; 2) for an isometric crystal, the slope of a log-log plot between the crystal mass and the growth time showed the Maxwellian value of 1.5. The mass growth rate of a shape-enhanced crystal was larger than that of the isometric crystal, indicating more effective vapor transfer on the former; 3) in the case of shape-enhanced planar crystals grown at around -12, -14.5 and -16.5°C, ventilation effects became recognizable, whereas the effect was not evident for needle crystals grown at about -5.5°C. This suggests that the characteristic length of the flow field even around a needle crystal is along the a-axis. The ventilation effect became significant when the Reynolds number exceeded about 2 (sector) and 5 (dendrite); 4) linear relationships between the drag coefficient and the Reynolds number were found in log-log plots.
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We propose a new type of crystal growth phenomenon—structure-dependent attachment kinetics—to explain the growth of thin ice plates from the vapor phase at temperatures near −15°C. In particular, we propose that the condensation coefficient for growth of ice prism facets increases dramatically when the width of the facet approaches atomic dimensions. This model reconciles several conflicting measurements of ice crystal growth and makes additional predictions for future growth experiments. Other faceted crystalline materials may exhibit similar morphological instabilities that promote the diffusion-limited growth of thin plate-like or needle-like crystal structures.
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Recent laboratory experiments and in situ observations have produced results in broad agreement with respect to ice crystal habits in the atmosphere. These studies reveal that the ice crystal habit at -20°C is platelike, extending to -40°C, and not columnar as indicated in many habit diagrams found in atmospheric science journals and texts. These diagrams were typically derived decades ago from laboratory studies, some with inherent habit bias, or from combinations of laboratory and in situ observations at the ground, observations that often did not account for habit modification by precipitation from overlying clouds of varying temperatures. Habit predictions from these diagrams often disagreed with in situ observations at temperatures below -20°C. More recent laboratory and in situ studies have achieved a consensus on atmospheric ice crystal habits that differs from the traditional habit diagrams. These newer results can now be combined to give a comprehensive description of ice crystal habits for the atmosphere as a function of temperature and ice supersaturation for temperatures from 0° to -70°C, a description dominated by irregular and imperfect crystals. Cloud particle imager (CPI) habit observations made during the Second Alliance Icing Research Study (AIRS II) and elsewhere corroborate this comprehensive habit description, and a new habit diagram is derived from these results.
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Cloud and atmospheric properties strongly influence the mass and energy budgets of the Greenland Ice Sheet (GIS). To address critical gaps in the understanding of these systems, a new suite of cloud- and atmosphere-observing instruments has been installed on the central GIS as part of the Integrated Characterization of Energy, Clouds, Atmospheric State, and Precipitation at Summit (ICECAPS) project. During the first 20 months in operation, this complementary suite of active and passive ground-based sensors and radiosondes has provided new and unique perspectives on important cloud–atmosphere properties. High atop the GIS, the atmosphere is extremely dry and cold with strong near-surface static stability predominating throughout the year, particularly in winter. This low-level thermodynamic structure, coupled with frequent moisture inversions, conveys the importance of advection for local cloud and precipitation formation. Cloud liquid water is observed in all months of the year, even the particularly cold and dry winter, while annual cycle observations indicate that the largest atmospheric moisture amounts, cloud water contents, and snowfall occur in summer and under southwesterly flow. Many of the basic structural properties of clouds observed at Summit, Greenland, particularly for low-level stratiform clouds, are similar to their counterparts in other Arctic regions. The ICECAPS observations and accompanying analyses will be used to improve the understanding of key cloud–atmosphere processes and the manner in which they interact with the GIS. Furthermore, they will facilitate model evaluation and development in this data-sparse but environmentally unique region.
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The effect of air velocity on the growth behavior of ice crystals growing from water vapor was investigated at temperatures between 0 and -35 C and at supersaturation levels ranging from 2 to 40 percent, using a laboratory chamber in which it was possible to make these variations. It was found that crystal growth was most sensitive to changes in the air velocity at temperatures near -4 C and -15 C where, near water saturation, the introduction of only a 5 cm/s air velocity induced skeletal transitions (columns to needles near -4 C and plates to dendrites near -15 C). The experiments provide conditions which simulate growth of ice crystals in the atmosphere, where crystal growth takes place at or somewhat below water saturation.
  • Benjamin Morrison
  • Faber
Morrison, and Benjamin Faber, Measurements of snow crystal growth dynamics in a free-fall convection chamber, arXiv:0811.2994, 2008.