Particle Fall through the Particle Fall through the atmosphereatmosphere

Lecture #5Lecture #5Ashfall Class 2009Ashfall Class 2009

Distance d travelled by an object falling for time t:

Time t taken for an object to fall distance d:

Instantaneous velocity vi of a falling object after elapsed time t:

Instantaneous velocity vi of a falling object that has travelled distance d:

Average velocity va of an object that has been falling for time t (averaged over time): Average velocity va 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.

air resistance is neglected--- quite inaccurate after only 5 seconds 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).

Reynolds NumberRe

• 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, • intermediate where 0.4 < Re < 500, and • turbulent where Re > 500.

Laminar flow; RN = 10-2

Turbulent flow; RN = 106

RN = 20 RN = 40 RN = 104

Fast-falling Fast-falling Large Large PyroclastsPyroclasts

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 =dvRN =dvtt//

Medium and Medium and small pyroclastssmall pyroclasts

10 m/s

D = 1mmD =

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, Rep:

ratio of inertial force to viscous force per unit mass

Rep = Vtd / v

Vt = particle terminal fall velocity; d = particle diameter; v = fluid kinematicviscosity

Rep :

> 500 turbulent

1-500 transitional

<1 laminarFrom Sparks et al. [1997]

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 (Rep > 500) – inertial forces dominate:

fd

fp

C

gdtV

)(

3

4

d = particle diameter

ρp = particle density

ρf = fluid density

g = acceleration due to gravity

Cd = dimensionless drag coefficient

For small particles (Rep < 1)- viscous forces dominate:

18

2gdV pt

ρp = particle density

g = acceleration due to gravity

d = particle diameter

v = kinematic viscosity

Fall of spherical particles in earth’s atmosphereFall of spherical particles in earth’s atmosphere

Schneider et al., 1999, J Geophys Res 104 4037-4050

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Particle Terminal Fall Velocity

100 micron

diameter particle has Vt of ~4-7 ms-1

Mean particle size at ~330 km

from MSH (Ritzville, WA)

was 20 microns; Vt ~0.2-0.4 ms-1

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Atmospheric Structure

Environmental parameters determined from the radiosonde sounding taken at Spokane International Airport at 1800 UTC on 18 May 1980.

Bonadonna et al., 1998

Bonadonna et al., 1998

Bonadonna et al., 1998

Bonadonna et al., 1998

Bonadonna et al., 1998

Bonadonna et al., 1998

Bonadonna et al., 1998

Bonadonna et al., 1998

  Figure 3. Digital elevation map produced from stereo-pair in Figure 2.

Figure 2 Typical stereo-pair taken at 8o tilt angle.

Owen P Mills, MS thesis, Michigan Tech, 2007

Ash is NOT Ash is NOT spherical!spherical!

Riley et al., 2003

Augustine ash P Izbekov

Riley et al., 2003

Rose W I, C M Riley and S Dartevelle, 2003, J Geology, 111:115-124.

Riley et al., 2003

Riley et al., 2003

Riley et al., 2003

Riley et al., 2003

Rose W I, C M Riley and S Dartevelle, 2003, J Geology, 111:115-124.

Rose W I, C M Riley and S Dartevelle, 2003, J Geology, 111:115-124.

Riley et al., 2003