The Mechanics of Falling Hailstones and Hailswaths
Kevin Vermeesch and Ernest Agee
Department of Earth and Atmospheric Sciences, Purdue University
Objectives
Cases and Model Results
•Develop a set of mechanical equations for calculating hailstone fall velocities and
characteristics for a variety of thunderstorm and atmospheric conditions
Cases Testing Fall Speeds
•Plot hailswaths for real atmospheric events and compare with model results
Hailstone Fall Velocity (m/s)
Physical Properties in the Model Equations
•Density profile of atmosphere
•Diameter and mass of spheroidal hailstones
•Drag coefficient for subcritical Reynolds number flow
Case 1 Parameters
Hailstone Fall Velocity at Ground
Parameter
35
Value
30
25
hailstone diameter (dstone)
0.75 in
hailstone density (ρstone)
900 kg m-3
drag coefficient (CD)
0.5
air density (ρair)
0.900 kg m-3
initial height AGL (z0)
7.0 km
20
15
10
5
0
1
2
3
4
5
Case Num ber
•Translational speed of supercell thunderstorm
•Updraft velocity profile of the supercell thunderstorm
Information for Cases 2-5
•Rotational velocity profile of embedded mesocyclone
•Spherical vs. non-spherical hailstone and related Re and CD
Hailstone Fall Time
Re < Rec
Fall Time (s)
450
Re > Rec
400
Case
Number
Difference from Case 1
Case 2
air density reduced by 50% (ρair = 0.450 kg m-3)
Case 3
air density exponentially increases towards
the ground [ρair = ρ0 exp(-z0 / Hρ)]
Case 4
hailstone diameter is increased to 1 inch and
updraft velocity (w) = 20.0 m s-1
Case 5
hailstone diameter is increased to 1 inch, air
density exponentially increases towards the
ground [ρair = ρ0 exp(-z0 / Hρ)] and updraft
velocity (w) = 20.0 m s-1
350
300
250
200
Hailswaths
Knight and Knight (2005), Figure 9
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.
Graphical verification of model terminal velocity with Knight and Knight (2001)
Size-sorted hailswath produced by model
1
2
3
4
5
Case Num ber
In 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
thunderstorm’s rotational velocity (vθ),
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.
References
Knight, 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.
Knight and Knight (2001), Figure 6.2
3 April 1974 tornado tracks and hailswath in Indiana