Growth processes and prediction of damaging ice precipitation
2026-02-27
On 20 June 2020, a supercell thunderstorm tracked across the Red Deer area of Alberta and produced hailstones up to 71 mm in diameter — golf ball size — that fell for more than 20 minutes. The resulting crop damage across Lacombe and Red Deer counties reached an estimated $82 million. Insurance adjusters were in fields within days; helicopter flights measured crop loss extent using NDVI imagery. Alberta is among the most hail-prone regions in North America: the area southeast of Calgary and along the foothills sees 7–9 significant hail days per year, driven by the combination of extreme CAPE values from Gulf moisture and strong wind shear from the Rocky Mountain jet stream. The Insurance Bureau of Canada estimates hail causes an average of $1 billion in insured losses in Alberta annually.
Hail grows inside the powerful updrafts of supercell thunderstorms. A hailstone begins as a small ice nucleus, is lofted into the cloud by the updraft, accretes supercooled water droplets and ice crystals as it rises and falls through different thermal zones, and eventually grows too heavy for the updraft to support. The maximum size it reaches is controlled by the updraft velocity (stronger updraft → more time aloft → larger hail), the liquid water content of the cloud, and the vertical temperature profile that determines the wet-growth and dry-growth regimes. The relationship between updraft velocity and hail size is not linear — it scales roughly as the cube root of the updraft kinetic energy — which means that forecasting giant hail requires accurate prediction of updraft intensity, which in turn requires accurate assessment of CAPE and shear. This model derives the terminal velocity and accretion physics, implements a hail growth trajectory model, and shows how storm parameters translate to maximum expected hail size.
Will this supercell produce baseball-sized hail?
Hail definition:
Ice particles ≥5 mm diameter (pea-sized or larger).
Damage thresholds:
Pea (6 mm): Minimal crop damage
Quarter (25 mm, 1 inch): Severe criteria, car dents
Golf ball (44 mm, 1.75 inch): Car windshields broken
Baseball (70 mm, 2.75 inch): Roof damage, vehicle totaled
Softball (114 mm, 4.5 inch): Structural damage
Record: 8 inch (203 mm), Vivian SD 2010
Annual losses (USA): ~$1-2 billion
Applications: - Agriculture (crop insurance) - Aviation safety - Severe weather warnings - Building/vehicle damage assessment - Climate studies
Balance drag and gravity:
F_d = F_g
\frac{1}{2} \rho_a C_d A v_t^2 = m g
Where: - \rho_a = air density (kg/m³) - C_d = drag coefficient (~0.6 for sphere) - A = cross-sectional area (m²) - v_t = terminal velocity (m/s) - m = mass (kg) - g = 9.81 m/s²
For sphere:
A = \pi r^2, \quad m = \frac{4}{3}\pi r^3 \rho_h
Solving:
v_t = \sqrt{\frac{8 r g \rho_h}{3 C_d \rho_a}}
Hail (ice density \rho_h = 900 kg/m³):
v_t \approx 9 \sqrt{r} (m/s, r in meters)
Example: 5 cm diameter (r = 0.025 m)
v_t = 9 \sqrt{0.025} = 9 \times 0.158 = 1.42 \times 9 \approx 14 \text{ m/s}
Updraft must exceed 14 m/s to suspend this hailstone!
Dry growth:
Temperature < -40°C or low liquid water content.
Collected droplets freeze instantly.
Opaque ice (trapped air bubbles).
Wet growth:
Warmer temperature (-10 to 0°C) or high liquid water content.
Droplets form liquid layer before freezing.
Clear ice (no air, denser).
Transition:
Depends on accretion rate vs heat removal rate.
Spongy hail:
Alternating dry/wet layers (trajectories through different cloud zones).
Mass growth:
\frac{dm}{dt} = E \cdot A \cdot W \cdot v
Where: - E = collection efficiency (~0.8) - A = cross-sectional area - W = liquid water content (g/m³) - v = relative velocity (terminal fall + updraft)
For suspended hailstone (v \approx updraft speed):
\frac{dm}{dt} = 0.8 \cdot \pi r^2 \cdot W \cdot w
Typical: W = 1-5 g/m³, w = 20-50 m/s
Example: r = 0.02 m, W = 3 g/m³, w = 30 m/s
\frac{dm}{dt} = 0.8 \times 3.14 \times 0.02^2 \times 3 \times 30 = 0.091 \text{ g/s}
Residence time in updraft: 5 minutes = 300 s
\Delta m = 0.091 \times 300 = 27.3 \text{ g}
Final mass: ~50-100 g (golf ball to baseball)
Balance condition:
Hailstone suspended when v_t = w (updraft).
From terminal velocity:
r_{max} = \frac{w^2}{81 g}
(Using v_t = 9\sqrt{r}, solved for r)
Simplified:
r_{max} = \frac{w^2}{800} (m, w in m/s)
Example: w = 40 m/s
r_{max} = \frac{1600}{800} = 2 \text{ cm} = 20 \text{ mm diameter}
Quarter-sized hail (severe threshold)
For softball (57 mm radius):
w = \sqrt{800 \times 0.057} = \sqrt{45.6} = 67.5 \text{ m/s}
Extreme updrafts required!
Radar-based estimate:
Uses vertically integrated reflectivity.
MESH = 2.54 \times W^{0.5}
Where W = integrated reflectivity (g/m²).
Severe Hail Index (SHI):
SHI = 0.1 \int_{H_0}^{H_{-20}} (Z - Z_0) \, dh
Where integration from freezing level to -20°C level.
MESH from SHI:
MESH = 2.54 \times SHI^{0.5}
MESH = 25 mm: Severe hail likely
MESH = 50 mm: Very large hail
MESH > 75 mm: Giant hail (baseball+)
Falling trajectory:
Hail falls from updraft maximum.
Lateral displacement:
\Delta x = \frac{v_t}{w} \times H \times \frac{U}{w}
Where: - H = fall distance - U = storm-relative wind
Typical: 5-20 km downwind of updraft maximum
Swath width: 1-10 km (updraft width + spread)
Problem: Predict maximum hail size and swath location.
Storm parameters: - Maximum updraft: 45 m/s - Updraft top: 12 km - Freezing level: 4 km - Storm motion: 240° at 15 m/s - Environmental wind at 6 km: 270° at 20 m/s
Radar: - Peak reflectivity: 65 dBZ - Reflectivity top: 14 km - SHI: 150
Calculate maximum hail size and impact location relative to updraft.
Step 1: Maximum suspended hail
r_{max} = \frac{w^2}{800} = \frac{45^2}{800} = \frac{2025}{800} = 2.53 \text{ cm}
Diameter: 5.06 cm = 2 inch (golf ball)
Step 2: MESH from radar
MESH = 2.54 \times 150^{0.5} = 2.54 \times 12.25 = 31.1 \text{ mm}
1.25 inch (between quarter and golf ball)
Slightly lower than maximum (radar underestimates largest stones)
Step 3: Terminal velocity for 2-inch hail
v_t = 9 \sqrt{0.0253} = 9 \times 0.159 = 14.3 \text{ m/s}
Step 4: Fall time
From updraft top (12 km) to ground:
Assume average updraft weakens: 30 m/s upper, 20 m/s lower
Ascent slowed but falls eventually.
Fall distance after release: 12 - 4 = 8 km (below freezing)
Fall time:
t = \frac{8000}{14.3} = 560 \text{ seconds} \approx 9 \text{ minutes}
Step 5: Storm-relative wind
At 6 km: 270° at 20 m/s
Storm: 240° at 15 m/s
Relative: ~15 m/s from west (simplified)
Step 6: Lateral displacement
\Delta x = 15 \times 560 = 8400 \text{ m} = 8.4 \text{ km}
Hail falls 8-9 km east of updraft maximum
Step 7: Impact assessment
Golf ball hail expected: - Car windshields broken - Roof damage possible - Crop total loss
Swath: 8-10 km east of storm core, width 2-3 km
Warning: Large hail imminent, take shelter
Below is an interactive hail growth simulator.
<label>
Max updraft (m/s):
<input type="range" id="updraft" min="15" max="70" step="5" value="40">
<span id="updraft-val">40</span>
</label>
<label>
Liquid water content (g/m³):
<input type="range" id="lwc" min="1" max="6" step="0.5" value="3">
<span id="lwc-val">3</span>
</label>
<label>
Updraft residence time (min):
<input type="range" id="residence" min="2" max="10" step="1" value="5">
<span id="res-val">5</span>
</label>
<div class="hail-info">
<p><strong>Max hail diameter:</strong> <span id="max-diameter">--</span> mm</p>
<p><strong>Hail category:</strong> <span id="category">--</span></p>
<p><strong>Terminal velocity:</strong> <span id="terminal-vel">--</span> m/s</p>
<p><strong>Damage potential:</strong> <span id="damage">--</span></p>
</div><canvas id="hail-canvas" width="700" height="400" style="border: 1px solid #ddd;"></canvas>Observations: - Hail growth accelerates (larger stones collect more droplets) - Updraft strength limits maximum size - High liquid water content accelerates growth - Longer residence time produces larger hail - Terminal velocity increases with square root of diameter - Severe threshold (25mm) reached in strong updrafts
Key insights: - Extreme updrafts (>60 m/s) required for softball hail - Growth rate nonlinear (larger stones grow faster) - Minutes in updraft sufficient for destructive hail - Terminal velocity determines suspension requirement
Hail impact on agriculture:
Pea-sized (6-13 mm): - Leaf damage, some defoliation - 10-30% yield loss
Dime-quarter (13-25 mm): - Stem breakage, severe defoliation - 30-60% loss
Golf ball+ (>44 mm): - Total crop destruction - 80-100% loss
Economic:
2017 Colorado hailstorm: $2.2 billion crop damage (corn, wheat)
Insurance:
Crop insurance covers hail explicitly (common peril).
Payouts based on damage assessment (percentage loss).
Cloud seeding:
Inject silver iodide or other nuclei.
Theory: Increase ice particles → compete for water → smaller hail
Evidence: Mixed/inconclusive
Alberta Hail Project (1956-1985): 15-20% reduction claimed
Recent studies: No significant effect detected
Operational: Some programs continue (insurance-funded)
Hail damage costs (USA):
$1-2 billion annually
Golf ball hail: - Car windshields: $300-800 replacement - Body damage: $1,000-5,000 repair - Total vehicle loss: Possible with repeated impacts
Roof damage: - Asphalt shingles: Severe damage >44 mm - Metal roofs: Denting >25 mm - Solar panels: Crack/break risk
Building codes:
Some regions require hail-resistant roofing (impact-rated).
Radar MESH underestimates largest stones.
Reasons: - Hail concentration in small area - Beam averaging - Attenuation
Example - 2010 Vivian SD:
8-inch hailstone (record), but radar MESH ~75 mm (3 inches)
4× underestimate!
Solution: Spotter reports, damage surveys verify
Wet growth hail (clear ice) has higher density.
Radar reflectivity similar to smaller dry hail.
Can be more damaging (heavier, denser) than radar suggests.
Large raindrops can produce moderate reflectivity.
Without dual-pol: Hard to distinguish
Dual-polarization: ZDR, ρHV differentiate
ZDR: - Rain: +2 to +4 dB (oblate drops) - Hail: 0 to -2 dB (tumbling, irregular)
ρHV: - Rain: >0.98 - Hail: 0.90-0.95 (mixed shapes)
Long fall distance (high cloud tops) → melting
Surface reports: Rain, but hail aloft
Melting level critical:
Low (2-3 km): Hail reaches surface
High (4+ km): Melts before impact
Wet-bulb zero height better predictor than simple 0°C level
Geographic distribution:
Hail Alley (USA): Wyoming, Colorado, Nebraska
Argentina Pampas: World’s highest hail frequency
Northern India: Large hail events
Factors: - High terrain (orographic lift) - Strong CAPE - Dry air aloft (promotes supercells)
Seasonal:
Spring-early summer peak (transitional seasons, strong shear + instability)
Trends:
Some evidence of: - Fewer hail days - But more extreme events (larger stones) - Difficult to detect trends (sparse observations)
Satellite potential:
GOES-16 Overshooting Top detection correlates with large hail
Stokes (laminar, small Re):
F_d = 6 \pi \mu r v
Newton (turbulent, large Re):
F_d = \frac{1}{2} \rho C_d A v^2
Reynolds number:
Re = \frac{\rho v D}{\mu}
Hail: Re ~ 10⁴-10⁵ → Newton drag applies
Sphere drag coefficient:
C_d \approx 0.4-0.6 (depends on surface roughness)