1. Understanding the origins of different crystal “habits” and sizes Even for crystals that grow by diffusion processes, we see from observations that a wide range of different crystal shapes are possible What controls the shape? What controls the rate of growth? Only the driving force (vapor supersaturation)? What happens in a liquid water cloud that cools below 0 ˚C? Can liquid and ice coexist? Important points made in the following slides: Typically, supersaturations over ice are very high (we will look at reasons for this) The “crystal lattice” of ice is hexagonal in its symmetry under most atmospheric conditions Varying conditions of temperature and vapor pressure can lead to growth of crystalline forms in which the simple hexagonal pattern is present in widely different habits (a thin hexagonal plate or a long thin hexagonal column) trigonal symmetry can sometimes be observed, suggesting an influence of a cubic symmetry Growth rates depend on supersaturations and on the crystal structure itself (i.e., they are not diffusion-limited, as modeled for drop condensational growth)

2. LARGE SUPERSATURATIONS OVER ICE ARE TYPICAL The large ice supersaturations mean that mass transfer of vapor is much faster to the ice phase than the liquid phase We have already noted that, in water saturated clouds, ice crystals have their largest growth rates near -12 ˚C (offset a bit colder due to heat budget)  The most complex crystal geometries also occur near -12 ˚C Vapor pressure difference is plotted  gradient that is related to growth rate Also related to supersaturation with respect to ice, if we know water vapor pressure If we assume the environment is water saturated, then ice supersaturation is approximately 10% at -10 ˚C 20% at -20 ˚C 50% at -40 ˚C supersaturation

3. Pruppacher & Klett (1978) prismlike (c-axis growth dominates) columns, needles platelike (a-axis growth dominates) As S increases, sector plates, stellar and dendritic crystals form Here, prismlike said to dominate, but Bailey & Hallett (2009) suggest that these commonly-accepted forms below -20 are in error: should be platelike down to -40 C Note Bailey & Hallett (2009) suggest that these commonly-accepted forms below -20 are in error: should be platelike down to -40 C

4. General overview The vapor pressure curves explain why growth of the ice phase is faster than diffusional growth of liquid drops, and they also explain why growth rates are fastest near -12 ˚C However, a molecular-kinetic approach is required to explain the different growth habits, and the transition from one habit to another At a microscopic level, the ice surface consists of flat terraces of different heights, terminating at ledges and separated by steps Molecules that impinge on the surface and are weakly attached (adsorption) are bound more strongly at the ledges than on the terraces, so unattached molecules move (surface diffusion), fixing preferentially at the ledges, causing growth by lateral motion of surface steps (step propagation)

5. Formation of a new step Either by nucleation (two-dimensional), or by growth at crystal defects Nucleation: –Critical cluster is envisioned as a disk, one molecule deep –as local supersaturation with respect to ice increases, probability of nucleation increases –On / off mechanism Defects: –The dislocation creates a spiral in the lattice planes, creating a step on the surface –If supersaturations are low, growth is slow and steps are widely spaced –If supersaturations are high, fast growth and closely spaced steps –No on / off, variable rates instead

6. Linear growth rate The liner growth rate is the rate of advancement of a step If steps have height h and pass a point on the surface with frequency f step, then the linear growth rate R (µm s -1 ) is R = f step h h is the dimension of a single molecule packed in the crystal; compute from molar volume of ice, The step frequency depends on –Mechanism of step origin –Local supersaturation immediately over the facet –If supersaturation is high, a new layer can begin even before the prior layer is complete Lamb and Verlinde explain crystal growth theory in detail in Section 8.3 Figure below shows that growth rate (pure vapor, no air  i.e., no gradients!) increases proportionally to supersaturation; s 1 depends on surface (including mean migration distance)

7. Local (microscopic scale) picture Describes the growth of each facet of a crystal; always varies with interfacial supersaturation as shown here – however, we don’t know the local supersaturation field at each facet (derivation assumes uniform, no gradient) Also – need the transitional supersaturation on each facet as a function of T (depends on the surface properties) Correct the theory by adding in the gradient in the air / vapor binary system  slows down the growth process (K = 0 is back to dislocation growth – no limitations due to gas-phase diffusion, only surface; larger K increases important of gas-phase diffusion limitations) Equations also show that as deposition coefficient increases, vapor is removed more effectively from the immediate vicinity (affects habit development)

8. Ice crystal habit formation Note the definitions of the c and a axes The “aspect ratio” is c/a The “primary habits” are one of the two “basic shapes” shown “Secondary habits” are growth features driven primarily by supersaturation We will look first at what controls the primary habit …. Plate-like (a-axis growth, prism-face growth) Column (or prism)-like (c-axis growth, basal face growth) typo

9. Controls on primary habits Experiments were done at low vapor pressures (gradient effects minimized) Temperature is the main control switching between dominant rates  Reason for this still unclear! Note T range

10. Controls on primary habits (continued) The growth ratio can’t be explained simply by ratioing the deposition coefficients on the prior plot, because The total rate also depends on local supersaturations, which are different over the basal and prism faces:

11. Putting it together: The shape of the crystal dictates the distribution of vapor, and the vapor is redistributed preferentially toward the fastest-growing faces (because they deplete it more effectively) In far field, crystal is like a point sink, which is how we treated it in capacitance model In near field, gradients over two faces are quite different  here the prism face grows faster Gradients at sharp corners can be large Secondary habit: At high S i, shape instability occurs: the shape change modifies vapor field and ends up reinforcing the initial shape change  bumps see higher S i, preferentially grow

12. Updated diagram suggested by Bailey & Hallett, JAS, 2009

14. INHERENT GROWTH RATES Given the linear growth rates for the based and prism faces, the mass growth rate of a crystal can be found from a where = ice crystal density = area of both basal faces = area formed by 6 prism faces = linear growth rate for basal face = linear growth rate for prism faces Geometry: H a= radius of inscribed sphere H= length of prism face Therefore;

15. Since the linear growth rates are independent of time at fixed temperature, we have So a and H are expressed as linear growth rates multiplied by growth time t. Hence Growth rate of crystal referred to as ‘inherent’ growth rate since growth is governed by kinetic processes on surface of crystal and not by diffusion of water vapor towards crystal and conduction of heat away from crystal. Recall that these linear growth rates are observed growth rates at very low water vapor pressures, rendering diffusivity and conduction processes unimportant in controlling crystal growth. Measured inherent growth rates Peaks near -6°C and -12°C Min ≈ -8°C

16. ‘Capacitance’ of ice crystals Spherical crystal c = a a= radius Columnar crystal -Assume prolate spheroid model a= semi major axis b= semi minor axis Column a Needle For b<<a Plate (hexagonal) With HABIT + Capacitance  compute GROWTH RATE A = (a 2 -b 2 ) 1/2 Please also refer to new values of C presented by Westbrook et al. (Lecture 22)

18. Depositional growth very rapid- Dendrite of “radius” 1000 µm forms in 10 3 seconds @ T=-15°C, water saturation. Approximately a drizzle drop upon melting.

19. Pruppacher & Klett(1978) Small crystals: similar growth rates for various geometries Large crystals: growth rates sensitive to particle geometry For example, a dendrite of given mass has more surface area than a column with the same mass, hence larger growth rates. flux lines- vapor flux

20. Chen’s “adaptive” growth model Use classic capacitance model, BUT compute capacitance as crystal evolves Use spheroid geometry for mass rate of change calculations (spheroid volume is V) Can derive Solve as coupled set of equations to model evolution of total ice mass and shape φ is aspect ratio

21. Pruppacher & Klett(1978) OBSERVED ICE PARTICLE DIMENSIONS Hex. plates: 100 µm to 1 mm, thickness 10-40 µm needles

22. Pruppacher & Klett (1978) PLATE SECTOR PLATE DENDRITE STELLAR We see from this figure that crystals with more pronounced dendritic features exhibit smaller fall speeds relative to equal-size (same radius) disks / plates. Since a dendrite or stellar of radius r has less mass than a plate of the same radius, mg is smaller and terminal fall speed is lower. Drag on dendrite also reduces the fall speed. The loss of sensitivity to diameter at large diameters for dendrites is because the increasing drag balances the change in gravitational force. FALL SPEEDS

23. Fallspeed of hexagonal plate decreases with diameter. Larger area (same mass) implies larger drag force. Columns behave in similar fashion. Fallspeed maximums for plates ~1 m s -1 Fallspeed maximums for columns ~50 cm s -1 Drag force decreases with height, associated with lower air density PLATESCOLUMNS

24. Melting Melting transforms snow and graupel into rain By cooling the air, melting affect thermodynamic stability of lower atmosphere  dynamics of convective systems Seen as “bright band” in radar because dielectric properties change ice  water Increases density and fall speeds of particles How fast does a particle melt? –Limited by heat transfer to the liquid-ice interface –Need conservation of energy equation again

25. More on habits Equations for ice crystal growth theory

26. Change of habit occurs at -4, -10, and -22°C. We see that ice crystal shapes are rather complicated, which leads us to suspect that heat and vapor exchange alone can’t adequately explain crystal growth/ habits. Must examine surface forces at surface of ice crystal and their ability to control incorporation of vapor molecules into the ice lattice. T basic habit; vapor excess degree of skeletal growth Surface processes

27. Pruppacher & Klett(1978) INCREASING SKELETAL FEATURES WATER SATURATION ICE SATURATION SOLID COLUMN HOLLOW COLUMN- INCOMPLETE BASAL FACES AT HIGHER VAPOR DENSITIES NEEDLES REPRESENT SHEATHS WHOSE PRISM FACES ARE INCOMPLETE.

28. Updated diagram suggested by Bailey & Hallett, JAS, 2009

29. Equiaxed means the crystal has similar dimensions along the c- and a-axis. Wallace and Hobbs, Second Edition

30. -THEORY FOR ICE CRYSTAL HABITS- Laboratory studies indicate that ice crystal growth preferentially occurs at STEPS on the crystal’s face. This figure illustrates the situation h H O H XsXs h= step height (200 – 1000 Angstroms) X s = collection distance, typically order of few microns X s has been found to be strongly dependent upon T. If X s large, molecules have chance to desorb from crystal face prior to being incorporated into ice lattice. Velocity of ice step measured experimentally indicate that the step velocities - are inversely proportional to step height - increase with increasing vapor supply - exhibit a characteristic dependence on temperature H 2 O molecule

31. If then Step velocity inversely proportional to step height. The following figure illustrates the variation of x s (collection distance) with temperature for the basal face. These measurements were made by Mason and colleagues in the early 60’s. f n = net flux of H 2 O molecules to crystal’s surface h = step height; u= step velocity U has been found to be strongly dependent on temperature U B has been measured in lab experiments. ----------step velocity on basal face

32. PRISM FACE GROWTH Basal face collection distance, x SB is highly sensitive to T. When large, its expected that H 2 O molecules will effectively diffuse to prism faces, hence formation of steps on basal face, and growth of existing steps on basal face, are reduced. Molecules diffuse to prism faces Formation of steps on basal face reduced Basal face growth reduced Molecules diffuse to basal faces. Basal face growth preferred. Small c-axis growth a-axis growth This figure illustrates the variation of x s (collection distance) with temperature for the basal face. These measurements were made by Mason and colleagues in the early 60’s.

33. steps sweep across crystal face Growth step velocity Formation of new steps x s = collection distance of a step (T sensitive) X s also interpreted as mean “migration” distance of a H 2 O molecule before bonding occurs.

34. Hobbs and Scott (1965, J. Atmos. Sci.) offered a theory for ice crystal habits, step growth, etc. They assumed that planar steps originate near crystal edges Postulated that when a step is near the edge of a face, (within distance x s of edge), probability of another step forming on that face is reduced. This follows from the fact that the concentration of adsorbed H 2 O molecules is low within a distance x s of the step. (molecules are being absorbed into the lattice) Assuming “fast steps” move away from the edge quickly, when More steps will be nucleated on prism faces plate-like ice crystals Opposite occurs for basal growth, leading to needle- like crystals. Step velocities linear growth rates (step velocities exhibit same behavior with T as linear growth rates, supporting the role of step velocity in determining the habit of the growing ice crystal)

35. Once surface forces have determined the basic habit of the crystal, further growth at low to moderate vapor excess will not alter the original habit (provided temperature remains constant). Water vapor arriving at the crystal edges and corners can be quickly distributed over the crystal surface by diffusion. At larger vapor densities, surface diffusion cannot keep pace with the supply of vapor molecules to the crystal. Growth will then preferentially occur at the corner and edges of the crystal, which results in formation of spikes and branches (which themselves grow in the basic habit at the temperature). Flux lines branches Analogous to flux (field) lines on a conductor.