Distance d travelled by an object falling for time t: Time t taken for an object to fall distance d: Instantaneous velocity v i of a falling object after elapsed time t: Instantaneous velocity v i of a falling object that has travelled distance d: Average velocity v a of an object that has been falling for time t (averaged over time): Average velocity v a of a falling object that has travelled distance d (averaged over time): use g = 9.8 m/s² (metres per second squared; which might be thought of as "metres per second, per second. Assuming SI units, g is measured in metres per second squared, so d must be measured in metres, t in seconds and v in metres per second.SI units air resistance is neglected--- quite inaccurate after only 5 seconds
Particle Fallout After a very short time, ~4 seconds, particles will reach a terminal velocity in earth's atmosphere, with their gravitational attraction to the earth balanced by air resistance. Small particles have dominant air resistance (fall slowly) while large particles have dominant gravity (fall rapidly).terminal velocity
Reynolds Number Re Reynolds number is a dimensionless number (i.e. it has no units) that is a measure of the type of flow through a fluid. In the case of falling particles, this describes the way that air flows around the particle. There are three basic types: laminar where Re < 0.4, laminar intermediate where 0.4 < Re < 500, and turbulent where Re > 500. turbulent
Laminar flow; RN = 10 -2 Turbulent flow; RN = 10 6 RN = 20 RN = 40RN = 10 4 Fast-falling Large Pyroclasts Fluid dynamics applies dimensionless analysis of fall of spheres in the atmosphere, which shows that experience with large pyroclasts might not apply to smaller ones which fall much more slowly… RN =dv t / RN =dv t / Medium and small pyroclasts 10 m/s D = 1mm D = 1µm.01 cm/ s
8 Conventional Wisdom: Particle Settling particle accelerates due to gravity Drag force: (i) viscous drag (friction between the fluid and the particle surface) (ii) form drag (inertial force caused by the acceleration of fluid around the particle as it falls) Particle Reynolds number, Re p : ratio of inertial force to viscous force per unit mass Re p = V t d / v V t = particle terminal fall velocity; d = particle diameter; v = fluid kinematic viscosity Re p : > 500 turbulent 1-500 transitional <1 laminar From Sparks et al. 
Larger pyroclasts, those >2mm in diameter, fall in a turbulent flow regime (Re> 500) through the atmosphere. Small pyroclasts, <1/16 mm (62 μm or 4 Φ), fall in laminar flow regime (Re<0.4). Intermediate size particles are transitional.
10 Particle Terminal Fall Velocity For large particles (Re p > 500) – inertial forces dominate: d = particle diameter ρ p = particle density ρ f = fluid density g = acceleration due to gravity C d = dimensionless drag coefficient For small particles (Re p < 1) - viscous forces dominate: ρ p = particle density g = acceleration due to gravity d = particle diameter v = kinematic viscosity
Fall of spherical particles in earths atmosphere Schneider et al., 1999, J Geophys Res 104 4037-4050
12 Particle Terminal Fall Velocity 100 micron diameter particle has V t of ~4-7 ms -1 Mean particle size at ~330 km from MSH (Ritzville, WA) was 20 microns; V t ~0.2- 0.4 ms -1
13 Atmospheric Structure Environmental parameters determined from the radiosonde sounding taken at Spokane International Airport at 1800 UTC on 18 May 1980.