Sand - Dune Worlds: How Windblown Sand Shapes Planetary Landscapes - page 17

Table 2.1 The modified Udden-Wentworth scale of particle sizes and

windblown deposits (Udden 1898); he subsequently com-
pared aeolian sands to the sands found in rivers and on
beaches (Udden 1914). He chose to follow the lead of soil
scientists, who used sieves to separate soils into mass frac-
tions based on how the materials passed through a stacked
array of progressively smaller meshes. Charles Wentworth
(1922) codified the progressive size scale using divisions
based on powers of 2, what is now termed the 'phi scale'
(where / = log 2 d, with the grain diameter measured in
millimeters). The Modified Udden-Wentworth scale, still in
wide use throughout sedimentary geology, lists several
particle size divisions (Table 2.1 ). Many additional size
descriptors and parameters have been applied to particle size
measurements through the years (e.g., see Folk 1966; Blott
and Pye 2001), but the subdivisions within the sand size
fraction have remained constant since the introduction of the
Udden-Wentworth scale.
It is common among geologists and geographers to
document the size distribution recovered by sieving as a
histogram, and it often looks somewhat like the bell-shaped
Gaussian curve that elementary statistics is so full of. Such
a plot in linear form is a useful guide, but a more broad-
ranging and broadminded mathematical investigation can
be more instructive yet.
Note that the factor-of-two binning is a logarithmic
scale, which is what is needed to grapple with a wide range
(much as stellar magnitudes, or earthquake magnitudes, or
intensity of sound). The utility of logarithms is often for-
gotten, and in fact Bagnold himself found interest in the use
of logarithmic axes in plotting not just the size (the x-axis)
but also the the number of sand particles in each sieve (the
y-axis). When Bagnold plotted the results, he got something
like a bell-shaped curve. But it wasn't quite 'right'—at very
small and very large sizes, there were more particles than a
Gaussian or 'normal' distribution suggested there should be.
A more conventional scientist—better indoctrinated in
the tyranny of the Gaussian distribution—would have
shrugged his or her shoulders at the obvious and insignifi-
cant experimental error. But Bagnold, originally educated in
Cambridge as an engineer, was too practical a man to ignore
his own measurements.

names
Name

Size range (grain diameter) /-scale

Boulder [256 mm (/ \ -8)

Cobble 64-256 mm (/ -6to-8)

Gravel 2-64 mm (/ -1to-6)

Granule 2-4 mm (/ -1to-2)

Sand 1/16-2 mm (/ 4to-1)

Very coarse 1-2 mm (/ 0to-1)

Coarse 1/2-1 mm (/ 1to0)

Medium 1/4-1/2 mm (/ 2to1)

Fine 1/8-1/4 mm (/ 3to2)

Very fine 1/16-1/8 mm (/ 4to3)

Silt 1/256-1/16 mm (/ 8to4)

Clay \1/256 mm (/ [ 8)

perspective, as Titan's discoverer Christiaan Huygens did in
his book Cosmotheoros (1698)—it matters less what the stuff
is made of, than how it behaves.

Since 'tis certain that Earth and Jupiter have their Water and
Clouds, there is no reason why the other Planets should be
without them. I can't say that they are exactly of the same
nature with our Water; but that they should be liquid their use
requires, as their beauty does that they be clear. This Water of
ours, in Jupiter or Saturn, would be frozen up instantly by
reason of the vast distance of the Sun. Every Planet therefore
must have its own Waters of such a temper not liable to Frost.

2.1 Sand Size and Shape

A challenge with something as mundane as dirt is to for-
mally systematize something that everyone thinks they
understand. Sand is small stuff, but not really small. 1 Jon
Udden, a professor of geology at the small midwestern
college of Augustana, was the first person to publish a sta-
tistical

analysis of sand grain sizes that occur within

1 The number of grains of sand on Earth has long been a metaphor for
a quantity beyond human comprehension. Scientists—at least since
Archimedes' 'The Sand Recknoner'—however, have still attempted
estimates of the number of sand grains after making certain
assumptions. For example, one estimate for the number of sand grains
on the beaches of the world is the staggering number of *5000 billion
billion (5 9 10 21 ) grains of sand, assuming that the average sand grain
is 0.25 mm in diameter, the 'average' beach includes 50 m of sandy
beach that is 1 m deep, the sand grains are perfectly packed together,
and that the world has 1.5 million km of shoreline (Greenberg 2008,
p. 39). The vast sandy deserts present on several continents (e.g., the
Sahara), along with the sand that is now stored within sandstone
deposits exposed around the world, suggest that even this estimate of
the number of beach sand grains is likely more than a factor of ten too
small to encompass all of the sand grains on Earth.

The difficulty was that the measured frequencies along the tails
became so small that they were unplottable. Not being a stat-
istican, I concluded that this difficulty could be overcome
simply by plotting the frequencies indirectly as their loga-
rithms, thus giving every frequency, however small, an equal
prominence. The precise pattern of the complete size distribu-
tion of natural sand at once appeared. It consisted of two
converging straight lines joined by a curved summit. The curve
resembled a simple hyperbola.

A Gaussian, on logarithmic axes, falls away rapidly like
a parabola. As the numbers involved tend towards zero,
their logarithms tend towards minus infinity, and the tails of