The Mechanics of Falling Hailstones and HailswathsKevin Vermeesch and Ernest AgeeDepartment of Earth and Atmospheric Sciences, Purdue UniversityObjectivesDevelop a set of mechanical equations for calculating hailstone fall velocities and characteristics for a variety of thunderstorm and atmospheric conditionsPlot hailswaths for real atmospheric events and compare with model results

Physical Properties in the Model EquationsDensity profile of atmosphereDiameter and mass of spheroidal hailstonesDrag coefficient for subcritical Reynolds number flowTranslational speed of supercell thunderstormUpdraft velocity profile of the supercell thunderstormRotational velocity profile of embedded mesocycloneSpherical vs. non-spherical hailstone and related Re and CDCases and Model ResultsCases Testing Fall Speeds3 April 1974 tornado tracks and hailswath in IndianaIn a size-sorted hailswath, the largest hail lands closest to the mesocyclone (or tornado track if present). The dimensions of the swath are a function of the thunderstorms rotational velocity (v), translational velocity (u), and mesocyclone radius. The image on the right shows a portion of the mesocyclone and underlying wall cloud, flanked by a spectacular hailshaft that produces the hailswath. Graphical verification of model terminal velocity with Knight and Knight (2001)Hailstone shapes range from being nearly spherical and smooth to very irregular, containing knobs, lobes, or spikes on their surface. The smooth stones fall under conditions of subcritical Reynolds flow, while the irregular shapes may achieve supercritical flow conditions.Re < Rec Size-sorted hailswath produced by modelReferencesKnight, C.A. and N.C. Knight, 2001: Hailstorms. Severe Convective Storms, Meteor. Monogr., No. 50, Amer. Meteor. Soc., 223-249.Knight, C.A. and N.C. Knight, 2005: Very large hailstones from Aurora, Nebraska. Bull. Amer. Meteor. Soc., 86, 1773-1781. Re > RecKnight and Knight (2005), Figure 9Knight and Knight (2001), Figure 6.2Hailswaths

Case 1 ParametersParameterValuehailstone diameter (dstone)0.75 inhailstone density (stone)900 kg m-3drag coefficient (CD)0.5air density (air)0.900 kg m-3initial height AGL (z0)7.0 km

Information for Cases 2-5Case NumberDifference from Case 1Case 2air density reduced by 50% (air = 0.450 kg m-3)Case 3air density exponentially increases towards the ground [air = 0 exp(-z0 / H)]Case 4hailstone diameter is increased to 1 inch and updraft velocity (w) = 20.0 m s-1Case 5hailstone diameter is increased to 1 inch, air density exponentially increases towards the ground [air = 0 exp(-z0 / H)] and updraft velocity (w) = 20.0 m s-1

Chart2

22.3

31.6

19.2

16.3

9.4

Case Number

Hailstone Fall Velocity (m/s)

Hailstone Fall Velocity at Ground

Sheet1

Hailstone Velocity at Ground

122.3

231.6

319.2

416.3

59.4

Hailstone Fall Time

1315.3

2224.1

3303

4433.9

5419

Sheet1

Case Number

Hailstone Fall Velocity (m/s)

Hailstone Fall Velocity at Ground

Sheet2

Case Number

Fall Time (s)

Hailstone Fall Time

Sheet3

Chart3

315.3

224.1

303

433.9

419

Case Number

Fall Time (s)

Hailstone Fall Time

Sheet1

Hailstone Velocity at Ground

122.3

231.6

319.2

416.3

59.4

Hailstone Fall Time

1315.3

2224.1

3303

4433.9

5419

Sheet1

Case Number

Hailstone Fall Velocity (m/s)

Hailstone Fall Velocity at Ground

Sheet2

Case Number

Fall Time (s)

Hailstone Fall Time

Sheet3