Summary of riming onset conditions for different crystal habits Semi-dimension: width / lateral dimension (perpendicular to c-axis)
HEAT BALANCE FOR GRAUPEL PARTICLES Consider a graupel particle growing by riming in a water saturated environment. Hence the possibility exists that the particle will also be growing by vapor deposition. Accreted droplets freeze on graupel particles and therefore release latent heat. This latent heat release effectively slows depositional growth. At some critical LWC, depositional growth will cease. At this point e v (surface)=e v (environment). At liquid water contents greater than the critical value, the particle actually falls into a state where it begins to sublime. What is W L, the critical liquid water content at which point deposition ceases? Heat balance is: specific heat of water HEAT CONDUCTION TERM Some of the latent heat released heats the surface of the particle T s = particle surface temp T o = temp of accreted water T a = ambient temperature A good approximation f v.h is a ventilation coefficient (not discussed here)
Let f v ventilation term for vapor deposition Particle x-sec area Combining above equations, Density of vapor at surface of particle The value of W L at which point deposition ceases is, Where is the temperature increment above ambient at which is assumed to be is slightly greater than (saturated with respect to water) or less
The criterion states that vapor deposition ceases. If the graupel particle is in a water saturated environment, and if supercooled drops instantly freeze, then it must mean that the surface temperature of the particle has been raised until the vapor pressure over ice at T s = vapor pressure over supercooled water at T ambient We can see that the difference will be just a few degrees So if we choose the ambient T a, we can find the ΔT, and can solve the equations for the corresponding supercooled liquid water content where vapor deposition ceases (and beyond which it reverses, to sublimation)
(Houghton 1985) See table of LWC (in Excel book): Growing cumulonimbus, 1-3 g cm -3 (Other clouds smaller) So limit may be reached only in strong cumulus clouds at warmer temperatures, and for larger particles (Small ones should grow rapidly by both mechanisms, vapor deposition and accretion) Critical liquid water contents As we will see later, the surface growth state of a graupel particle, whether it be in a depositional or sublimational state, controls the sign of electrical charge retained by the particle (in a non-inductive charging process) Interestingly: a particle may actually be in a sublimational state with respect to vapor transfer while it is growing by collecting supercooled liquid water. (We assumed here that water freezes instantly when it is collected) If the accretion rates get so large that T s rises to 0 C, the possibility of the drops NOT freezing on impact occurs!
Formation of HAIL Riming process, similar to graupel, but particles are larger Associated with convective storm systems. Provides large supercooled liquid water contents to promote hail growth Provides strong updraft velocities, that suspend and carry hail aloft, promoting growth Diameters to >13cm Weight ~1kg! Embryo required to kick-off hail growth Growth regimes: Dry growth: low to moderate liquid water contents Wet growth: Alternating dry and wet growth regimes promotes hailstone layered structure Dry growth Wet growth opaque ice clear ice graupel frozen drop
HGZ: hail growth zone EFZ: Embryo formation region FOZ: Fall-out zone
Both frozen drops and graupel particles can serve as hail embryos
(liquid) (trapped air)
DRY GROWTH Latent heat is released due to freezing of water; this heat that is liberated warms the surface of the stone. At low to moderate LWC s, this heat can be effectively dissipated to the surrounding air. Hence the stone remains below 0 C, and its surface is dry. This type of growth results in opaque ice since the rime contains quite a bit of air. The opaque layers are porous and the ice has a low bulk density. WET GROWTH At larger riming rates (higher LWC s and/or larger hail stones) latent heat release will warm the stone to 0 C, hence preventing most of the accreted liquid water from being frozen. Excess liquid water may fill in air spaces of underlying opaque ice In this case the surface of the stone becomes an ice-water mesh, promoting the term spongy ice. Even higher LWC s promote a complete liquid surface, called wet growth. Clear ice develops as this liquid layer freezes, e.g. when hailstone moves to a region of LWC and riming rate is reduced This liquid surface may be partially shed in the wake of the hailstone. The shed water produces drops that may then rapidly freeze and become new hail embryo sources.
Schumann-Ludlam Limit conditions that define the growth of an ice particle which freezes all the drops it collects and where surface temperature is 0 C. A liquid surface exists beyond SL limit. water that cannot be frozen may either be incorporated into a ice/water mix (spongy ice) or it may be shed.
Simplified models for hailstone growth rates For dry growth: the accreted water freezes instantly. Heat is released at a fast-enough rate, that although the hail particle warms, it does not exceed 0 C. The standard continuous-collection equation can be applied (same as used for graupel growth by accretion of supercooled droplets): dm h dt dry = πr h 2 V (r h )W l E c Here, r and m refer to the growing hailstone, V(r) is its fall velocity (or rather, the net rate of falling, if the strength of the updraft needs to be considered), W l is the liquid water content (the supercooled water), and E c is the collection efficiency. Since freezing is instantaneous, in dry growth, we can just apply this equation to figure out the rate of change of mass of the hailstone. For wet growth, however, this equation has to be combined with a heat balance
For wet growth: assume all excess water (that cannot be frozen) is shed Then the growth rate is determined by the rate at which collected water can be frozen (and thus retained) Have to dissipate latent heat to the environment: balance between latent heating and dissipation determines the growth rate Assumptions: Ignore heat storage in the hailstone (probably not a great assumption) Ignore collection of any other species except supercooled cloud drops Then the energy equation becomes: m h c p,h dt h dt dm = L h f dt Latent heat released by wet growth process wg + c w (T c T h ) dm h dt dry Sensible heat transferred between hailstone and collected water 0 (s.s) 4πr h L e D v (ρ v,rh 4πr h K(T h T E ) f h ρ v,e ) f v Latent heat transferred as vapor condenses or evaporates from hailstone Diffusion of heat between hailstone and its environment (conduction away from hailstone)
Further assume: The surface of the hailstone remains at 0 C (T o ) The cloud droplets are at T E Use the equation for dry growth Then the energy balance becomes: L f dm h dt wg = c w (T E T o )( πr 2 h V (r h )W l E c ) +4πr h L e D v (ρ v,rh ρ v,e ) f v +4πr h K(T o T E ) f h Solve for mass addition rate by wet growth dm h dt wg = c w (T o T E )πr 2 h V (r h )W l E c L f + 4πr h [ L e D v (ρ v,rh ρ v,e ) f v + K(T o T E ) f h ] L f
To find the critical water content, the conditions where dry growth switches over to wet growth: set dm h dt wg = dm h dt dry πr 2 h V (r h )W l E c = c w (T o T E )πr 2 h V (r h )W l E c + 4πr h [ L e D v (ρ v,rh ρ v,e ) f v + K(T o T E ) f h ] L f L f [ ] E c W l = 4 L ed v (ρ v,rh ρ v,e ) f v + K(T o T E ) f h r h V (r h )(L f c w (T o T E )) This is an effective liquid water content (since it includes the collection efficiency) As evaporation and conduction become more effective at dissipating the heat due to freezing (numerator increases), the critical liquid water content increases As the hailstone increases in size, the critical liquid water content decreases (collection of water increases with area and fallspeed)
Evaporation Conduction [ ] E c W l = 4 L D (ρ e v v,r h ρ v,e ) f v + K(T o T E ) f h r h V (r h )(L f c w (T o T E )) Note: if not all the excess water is shed, we need to modify the balance equations to account for this (see P&K) For a given T, the critical water content decreases with increasing size For a given radius, the CWC increases with decreasing T The CWC is higher for hailstones than for smooth spheres (enhanced ventilation allows heat to dissipate better) Rough Hailstones More efficient heat conduction to environment (Pruppacher & Klett 1978)
(Young 1993) Spongy ice: ice-water mixture on surface of hailstone. Most liquid water is accumulated around equator of particle. Schumann- Ludlam Limit (wet growth = dry growth)
Johnson and Rasmussen (1992) argued that once a hail particle reaches Schumann-Ludlam limit, its surface will become smoother, thereby reducing drag and increasing fallspeed. Therefore the hailstone will stay in the wet growth regime at lower LWC s compared to those required to get it into wet growth to begin with. Lower ventilation rates too----heat is dissipated less effectively. Eventually the fraction of water increases to a point where water can t be retained in ice-water matrix; SHEDDING occurs beyond this point. Shedding occurs when the overall ice fraction is <0.7.
Melting of ice particles: When ice particles fall below 0 C they begin to melt, but the process takes some time since heat transfer needs to occur (heat from ambient environment has to supply the latent heat to effect the phase change) We can apply the same simplified model as before: hailstone settling through population of cloud droplets, accreting them, but at temperature > 0 C Put in rate of dry collection here
Setting the rate of energy conduction through air equal to the rate of energy lost through evaporation, solve for the ambient temperature that yields a surface temperature = T 0 (0 C): An ice particle in cloud (RH~100%) may begin melting near 0 C, but outside the cloud, melting may not begin until ambient temperatures are a few degrees above 0 C, because at lower RH there is a driving force for evaporation, which cools the drop, and thus inhibits melting.
Points: Melting distance increases with density (more mass for given radius) Melting time increases with particle size Melting distance is shorter in water saturated environments Evaporation not allowed, thus all absorbed heat can go to phase change In subsaturated environment, some goes into latent heat of evaporation
Soviet Hail Model Start with thunderstorm model If the storm develops vigorous updrafts and high LWC during growth stage, then raindrops may form Raindrops are swept aloft, grow and ascend into supercooled regions Notice in sketch, updraft is maximum in -10 to -20 C range Many raindrops may be suspended just above the maximum: for radii > 2 mm, fall speeds are slightly > 9 m s -1 Region just above updraft maximum becomes a trap and supercooled liquid water builds up If a few of these drops freeze, they find themselves in a zone of very high LWC Hailstone formation occurs
(Prof. Cotton s notes)
Over the High Plains of the U.S. and Canada, multicell thunderstorms are found to be prolific producers of hail The mature stage of each cell provides proper updraft speeds and LWC for hailstones to grow, BUT They have to already be sizeable precipitation particles before they enter the strong updrafts (time scale issues) Weaker, transient updrafts provide sufficient time for the growth of graupel particles and aggregates of snow crystals, that then serve as hailstone embryos as the cell enters its mature stage Multicell storms thus are good environments for growing large hail Millimeter-sized ice particles settle at 8-10 m/s, compared to updrafts of 10-15 m/s at low levels and 25-35 m/s at high levels Supercell thunderstorm: A steady thunderstorm consisting of a single updraft cell that may exist for 2-6 hours Produce the largest hailstones: why aren t they swept out of anvil, given that updrafts of > 30 m/s can exist?
[Please see Cotton notes for complete discussion] Three-stage process envisioned Stage 1: embryos form in relatively narrow region on updraft edge (~10 m/s), grow to mm-sized hail embryos Can sweep around main updraft and enter the embryo curtain (trajs labeled 0 go out anvil) Stage 2: those following traj 2 experience more growth as they descend in embro-curtain region and re-enter base of main updraft Stage 3: hailstones see very high LWC during ascent in main updraft; they remain there long because their fall speeds are not so different from updraft until they get quite massive