Mechanics of Sand Transport - Dune Worlds: How Windblown Sand Shapes Planetary Landscapes - page 42

Fig. 4.8 A schematic of the threshold speed and how it depends on
diameter (shown on unlabelled, but logarithmic, axes). Weight varies
as the cube of diameter, whereas lift and drag vary roughly as the
square, thus the threshold windspeed that balances them is propor-
tional to size. At very small sizes, a cohesive force proportional to
diameter (i.e., a cohesive stress inversely proportional to diameter) will
dominate and the threshold will be inversely proportional. Where the
cohesion and weight are about the same, the overall resistance to
motion is a minimum, and thus the threshold speed is the lowest: the
saltation 'optimum'. For different gravity, different cohesion param-
eters, different fluid density, etc., the numbers will all be different, but
the overall shape of the curve will be the same

Fig. 4.7 Measurements of the friction velocity (i.e., the slope of
speed vs. logarithm of height) are linearly correlated with winds at
9.5 m height. These measurements are from Death Valley (from
Greeley et al. (1991b)), and show u**0.05U 9.5 . Image NASA

(like a wing) can be written as Eq. 4.1 , i.e., D/S =
0.5C d q f U 2 with all the parameters as before, except here the
drag coefficient is not that of a spherical or similar body, but
is the friction coefficient of a surface. That (as before)
depends on roughness and Reynolds number, but broadly
speaking is around 0.001 for 'smooth' surfaces and maybe
0.01 for rough ones. Now, we can rearrange Eq. 4.1 to
calculate the wind stress on the ground as s = D/S, and
rearrange Eq. 4.3 to say s =u 2 /q f , and we find then simply
that u*/U*(0.5C d ) 0.5 (some definitions of C d may result in
the 0.5 not appearing, but the square root means this doesn't
matter terribly). In other words, u * /U*20, give or take a
little. While this rule of thumb lacks rigor, it is a handy way
to grapple with the numbers at least approximately. And,
indeed,

A variety of mechanisms can play a role, with different
behavior. If we imagine the cohesion is due to some sort of
weak strength like a glue between grains, then it would have
a constant force per unit area. In that case, the threshold
speed as a function of size would be flat—constant—cor-
responding to the speed at which the aerodynamic stress
equals the failure stress of the glue. On the other hand, and
damp sand may behave a bit like this, imagine the cohesion
force is a linear function of diameter. Surface tension of
liquids can behave this way. In this case, the cohesion force
grows linearly with diameter, whereas the aerodynamic
stress (ignoring the Reynolds number effect) grows with the
square of diameter. Thus the windspeed at which these two
will balance, and thus the threshold windspeed for motion,
will vary as the reciprocal of size. This is shown schemat-
ically in Fig. 4.8 .
Bagnold conducted extensive wind tunnel measure-
ments, both in the field and in the laboratory, to understand
the flow conditions that led to motion. Greeley, Iversen and
White in the 1970s conducted many further experiments,
using particles made of different materials and using air of
different density. These latter experiments tend to report
their results with the drag coefficient formulation, and the
cohesion

field measurements show just this sort of

relationship.
For large particles, it is the weight that dominates, and so
the threshold stress (i.e., friction velocity, or the corre-
sponding windspeed) relates closely to the terminal veloc-
ities shown in Sect. 4.1 . Bigger particles have larger
weights and thus higher threshold speeds, and in principle
arbitrarily small particles would have arbitrarily small

threshold speeds.
However, at small sizes, adhesion (or cohesion) forces
take over. This tendency is evident in the kitchen: icing
sugar and granular sugar are the same stuff, just with dif-
ferent particle size. Dig some out with a spoon, and granular
sugar (with a grain size usually 0.5 mm or more) will
slump, making a conical pit with an angle of about 30, set
by the friction between particles. But dig a hole in icing
sugar, and it can have vertical walls—the cohesion between
the smaller grains is enough to prevent the walls failing.
This cohesion is also what means small particles with slow
terminal velocities do not easily get lifted of a surface.

formulation, buried together in a 'threshold

parameter' A.
They use a rather ugly empirical formulation for this—it
fits the data, although it is not clear how applicable it may
be to ices or organics on Titan. Since both the drag and