Department of Earth System Science zender@uci.edu
University of California Voice: (949) 824-2987
Irvine, CA  92697-3100 Fax: (949) 824-3256

Figure 1: Mineral dust lifecycle. Illustration by Robynn Zender.

We gratefully acknowledge The Annotated Grateful Dead Lyrics site by David Dodd.
Lyric to “Box of Rain” (p. 169), © Ice Nine Publishing Company. Used with permission.

Contents

List of Figures

List of Tables

Chapter 1
Introduction

This monograph describes mathematical, physical, chemical, and computational considerations pertinent to understanding and simulating the distribution and effects of natural aerosols in Earth’s atmosphere. Much of the content applies generically to any aerosol, but the majority of the aerosol-specific sections focus on mineral dust. There are also minor sections on sea salt mobilization and gaseous uptake on sulfate particles.

1.1 Acknowledgements

This monograph benefits from my discussions with many scientists. Their names appear in citations whenever possible. However, many of the their ideas, presented at meetings or in private conversations, are recapitulated here without acknowledgement. These people include Drs. Stephane Alfaro (Université Paris), Richard Arimoto (New Mexico State University), Vicki Grassian (University of Iowa), Zev Levin (Tel Aviv University), Natalie Mahowald (National Center for Atmospheric Research), Bill Nickling (Guelph University), Greg Okin (University of Virgina), Kevin Perry (University of Utah), Yaping Shao (University of New South Wales), and Richard Washington (Oxford University).

1.2 Literature Review

Heintzenberg (1989) reviewed the state of knowledge of tropospheric aerosol composition gleaned mainly from boundary layer observations. Gomes et al. (1990) report the observed size distributions and elemental compositions of mineral aerosols measured by a cascade impactor in the Sahara. Alfaro et al. (1997) describe results of wind-tunnel experiments to deduce the dependence of the emitted dust size distribution on the saltation intensity and u*. Prigent et al. (1999) and Lacaze et al. (1999) show how multi-angle and microwave satellite sensors can adequately retrieve land surface properties such as LAI and roughness length, key to determining dust mobilization. Sokolik et al. (1998) show the important role of infrared absorption by mineral dust. Sokolik and Toon (1999) analyze the effects of mineral composition on dust optical properties. Chiapello et al. (1999) compared the in situ observations of mineral dust with TOMS satellite retrievals. Lohmann et al. (1999) counted mineral dust particles smaller than 2 μm as cloud droplet condensation nuclei. Reader et al. (1999a) and Reader et al. (1999b) analyzed changes between mineral dust climatologies during the Last Glacial Maximum and the present. Gillette (1999) describes the factors contributing to the recurrence of dust emission “hot spots” as seen from TOMS. Hamonou et al. (1999) characterize the vertical structure of Saharan dust exported to the Mediterranean basin. Claquin et al. (1999) combined the FAO soil map of the world with surface mineralogy of specific samples to create predictive relationships linking soil type to surface mineralogy on a global scale. Li et al. (1999) showed is it possible to identify specific soil types from as few as six narrow-band infrared measurements. Batt and Peabody II (1999) measured threshold friction velocities for beds of pebbles 5–50 mm in diameter. Ichoku et al. (1999) describe an intensive field campaign in which radiative, microphysical, and chemical properties of various aerosols in the Negev desert were measured and inter-correlated. King et al. (1999) present an overview of the potential of current and future space-borne platforms to measure tropospheric aerosols including dust. Rillig et al. (1999) discovered that the proportion of soil aggregates larger than 250 μm increases linearly with CO2 concentration in certain grasslands due to biological effects. Wang et al. (2000) describe a Kosa (yellow dust) deflation model and evaluations its fidelity over East Asia. Alfaro and Gomes (2001) describe how to estimate the size distribution of the emitted dust by accounting for the size-dependent binding energy of the saltating particles. Arimoto (2001) present a broad overview of the climate factors influencing the abundance of atmospheric dust, as well as the radiative properties controlling the climate impact of dust. Myhre and Stordal (2001) performed sensitivity tests of the global radiative forcing of anthropogenic mineral dust. Grini et al. (2002b) discuss the stability, accuracy, and behavior of sandblasting fluxes determined by the Alfaro and Gomes (2001) model. Lunt and Valdes (2002) develop the Hadley Centre dust model and evaluate it against the standard suite of observations available on Earth. Léon and Legrand (2003) combined visible and infrared satellite measurements to identify dust sources and track dust plumes near the north Indian Ocean. VanCuren (2003) directly measure chemical composition of aerosol, including mineral dust, from Asia which dominates the mass concentration of remote, high altitude sites as far east as the western United States. van Donk et al. (2003) examine anthropogenic erosion on military bases in the Mojave Desert. Kurosaki and Mikami (2003) discovered that increased frequency of strong winds explains much of the observed increased in Dust Storm Frequency (DSF) in East Asia from 2000–2002 relative to the previous decade. Cakmur et al. (2004) show how sub-gridscale gustiness, driven largely dry convection, explains dust emissions in regions where mean winds are otherwise too weak to generate observed emissions. Kurosaki and Mikami (2004) derive an empirical Snow Cover Factor (SCF) that accounts for the influence of snow on the threshold wind velocity for dust mobilization. Grini and Zender (2004) apply show that accounting for saltation, sandblasting, and wind-speed PDFs improves the simulated size distribution of long range transported dust in a global model. Menut et al. (2005) further discuss the problems with determining sandblasting fluxes highlighted by Grini et al. (2002b), and present a new numerically stable scheme for their evaluation. Brooks et al. (2005) describe the interaction between climate and society in the Sahara. Arimoto et al. (2006) summarize dust measurements and modeling during ACE Asia. Many studies examine the possible role of dust as a vector for disease organisms affecting humans (Zender and Talamantes, 2006) and downwind ecosystems such as coral reefs (Shinn et al., 2000; Prospero et al., 2005). Yang et al. (2006) quantify the sensitivity of global dust mobilization, loading, and deposition to assumed size distribution. S. et al. (2006) characterize the effects of iron oxides on dust optical properties.

Many researchers have investigated the Martian dust cycle. In fact, until the 1990s, probably more dust research was performed by researchers more concerned with Mars than Earth. Recent global dust simulations on Mars are described in Pankine and Ingersoll (2002), Newman et al. (2002), and Basu et al. (2004).

Chapter 2
Boundary Layer Physics

2.1 Definitions

2.1.1 Wind Stress

Surfaces dissipate the momentum of the wind blowing over them. This dissipation is the result of tangential shear stress between the wind and the surface elements. The rate of change of atmospheric momentum Ma defines a stress force and the magnitude of this stress force τ = (⋅)^1∕2 expresses the total momentum the surface extracts from the wind per unit surface area per unit time. Hence the surface wind stress is also called the surface momentum flux. Some fraction of this wind stress τ does work on the surface in the form of moving the surface elements, e.g., moving leaves, or causing waves. Over bare or nearly bare ground much of the wind stress must go into aeolian abrasion (over stony surfaces) or soil movement since there is little else to absorb the force. The remainder of the wind stress may be converted to frictional heating of the surface, or small scale atmospheric turbulence.

We define the horizontal wind stress τ by appealing to basic principles of fluid dynamics. A fluid of density ρ moving at speed U exerts a pressure p (force per unit area) of ρU² on a stationary object transverse to the flow. The wind stress τ tangential to the surface takes a similar form,

The total stress to the surface is the vector sum of individual components representing stress dissipated by the plant canopy, stress dissipated by airborne (saltating and suspended) particles, and, finally, wind stress dissipated by the solid surface itself. This stress partition or drag partition has important implications for dust studies.

Using (2.5) to express the wind stress τ (2.1) solely in terms of u* we obtain

2.1.2 Friction Velocity

We now consider the wind speed profile U(z) between the free atmosphere and the surface. The planetary surface is the interface between the fluid atmosphere and the “solid” surface (soil, ocean, etc). A solid land surface requires a no slip boundary condition, i.e., the wind speed is zero exactly at the surface. To a good approximation, the ocean may also be treated with the no slip boundary condition since the atmospheric wind speed Ua is usually much larger than the surface current in the ocean uo, i.e., Ua ≫ uo. Let us assume that we know the measured or predicted wind speed Ua at a height z above the surface.

Knowing the wind with speed U at height z exerts a stress τ on the surface,

Substituting (2.1) into (2.4) we see that

The friction velocity u* is the fundamental quantity determining the flux of dust into the atmosphere. Nevertheless, it is difficult to attach a simple physical interpretation to the friction velocity. However we now demonstrate two important physical properties of u*. First, the mean horizontal wind speed at the top of the laminar layer is u*. Thus immediately after uplift, a particle is embedded in a horizontal wind of speed u*. In §3.3 we use this property of u* to explain the observed cubic dependence of the horizontal mass flux of saltating particles on the wind speed.

Secondly, u* is proportional to the mean velocity gradient near the surface.

2.1.3 Conversions

There are many other useful relations which can be established between U, u*, rm, Cm, and τ. These relations are simple, but tedious, to derive. Table 2.1 lists many of the relations between frequently occurring boundary layer parameters.

---

Table 2.1: Conversion Factors Between Quantities In The Boundary Layera U u* rm Cm τ m s-1 m s-1 s m-1 N m-2 U - u*Cm-1∕2 ln             u* UCm1∕2 -             rm ln - (CmU)-1             Cm (rmU)-1 -             τ - ρu*2 -C mρU2 -

---

2.1.4 Neutral Stability

The thermodynamic properties of the boundary layer determine the vertical gradient of fluxes within the boundary layer. In neutral conditions the wind speed varies logarithmically with height according to

Finally, it is sometimes useful to invert (2.7) in order to obtain z0,m in terms of Cn m

2.2 Surface Fluxes

The surface fluxes for momentum, heat, and vapor transfer are coupled by micrometeorological exchanges between the surface and the atmosphere in the surface (constant flux) layer. Determination of these fluxes from observation is possible via eddy flux correlation techniques. From a modeling perspective, the fluxes may determined by solving coupled non-linear differential equations in the surface layer. This technique is employed in Large Eddy Simulation (LES) models. LES solution resolve, as exactly as practical, the complex, turbulent eddies which determine the thermodynamic behavior of the boundary layer. However, large scale atmospheric models cannot afford to solve the continuous equations of motions throughout the boundary layer. Instead, a class of bi-level solutions for boundary layer fluxes has been developed based on Monin-Obukhov similarity theory.

Monin-Obukhov similarity theory is usually applied in terms of resistance r and conductance C ≡ r^-1 which describe the transfer of scalar quantities between two levels within the boundary layer. For simplicity, one of these levels is taken as zatm the midpoint height of the lowest atmospheric layer in the large scale atmospheric model. A host model provides the potential temperature θatm, pressure patm, specific humidity qatm, and meridional and zonal winds vatm and uatm. The subscript atm indicates the quantities are defined at the height zatm. The momentum fluxes τx and τy [kg m-1 s-2], sensible heat flux H [W m-2], and moisture flux E [kg m-2 s-1] are defined by the vertical gradient of the appropriate thermodynamic quantity between z = zatm and z = zs, where zs is the “surface height” (defined below). The fluxes are expressed as

Solutions to (2.9)–(2.12) must balance the surface energy budget. In other words, power absorbed by the surface must be dissipated by surface heating/cooling, and energy divergence to the atmosphere or soil in the form of latent, sensible, and radiative heating or cooling.

2.2.1 Bulk Aerodynamic Approximation

The turbulent surface fluxes, also called Reynolds fluxes, are the fluxes of heat, moisture, and momentum between the surface and the atmosphere. These fluxes arise as the atmosphere and the surface attempt to reach equilibrium states with the prevailing conditions. Because they are usually unresolved, the turbulent fluxes are usually determined by the application of Monin-Obukhov theory to the prevailing mean conditions. One simple and popular method, the bulk aerodynamic approximation, is of particular utility to large scale atmospheric models. We shall describe the essential physics for determining the turbulent surface fluxes, and related parameters, using the bulk aerodynamic approximation and more complex approximations.

There are three turbulent fluxes of interest: the momentum flux (also called the surface stress or wind stress) τ [kg m-1 s-2], and the sensible and latent heat fluxes H and L, respectively, both measured in W m-2. These fluxes are defined in terms of the eddy fluxes of the appropriate fields. Any scalar field x(t) may be decomposed into time-mean and fluctuating components, and x′, respectively

	=	(2.14a)
x′(t)	= 0	(2.14b)

Eddy fluxes arise from the fluctuating components of state variables. Consider the vertical fluxes of the scalar quantity x (2.13). For concreteness, imagine that x represents horizontal wind speed U, temperature T , or specific humidity q. Using (2.13) we see that the instantaneous vertical flux of x is

The eddy fluxes are multiplied by a pre-factor to obtain the conventional units

The bulk aerodynamic approximation for turbulent fluxes defines the eddy fluxes in terms of the time-mean state variables. The eddy correlations are assumed to be proportional to the product of the horizontal wind speed U and change of the appropriate state variable (U, θ, or q) between the surface and the height of interest.

Combining (2.17) with (2.18) we obtain

(2.19)

Each of the variables in (2.19) is height dependent. However, it is very common to evaluate the exchange coefficients at a particular height known as the reference height zr. The reference height is usually taken to be 10 m. Shifting the exchange coefficients between zr an arbitrary height z is useful for putting measurements from a variety of heights into a common framework for analysis.

Table 2.2 describes the defining relations of many of the related quantities which prove useful in boundary layer meteorology.

---

Table 2.2: Drag Coefficientsa b Name Symbol Defining Relationship Drag coefficient Cm τ = CmρU2       Drag coefficient at reference height Cm,r Cm,rUr2 = τ∕ρ = -w′U′       Neutral drag coefficient at zr = 10 m Cn m Cn m = k2-2       Wind speed at z = zr Ur U -       Neutral wind speed at z = zr Un r Un r = U = u*

---

Garratt (1977) reviews drag coefficient terminology, relationships, measurements, and constraints.

2.2.2 Roughness Length

We assume aerosol has the same flux properties in the boundary layer as momentum. The roughness length for momentum transfer z0,m is a property of the surface characteristics only, i.e., z0,m is independent of wind speed when the following conditions are met:

1. All roughness elements are inelastic. Inelastic elements, such as rocks, do not deform under wind stress whereas elastic elements, such as some vegetation does. For flexible vegetation, z0,m increases with wind speed U up to a critical speed Uc, and decreases thereafter. Ordinary lawn grass displays this behavior as it flutters in gentle breezes and bows before stiff winds.
2. Saltation is not occuring. Saltation roughens the boundary layer once it commences (Section 3.6).

Rougher surfaces tend to absorb more wind stress into non-erodible elements. Thus saltation decreases as z0,m increases, and visa versa. The frequency of saltation events follows the same pattern, since smooth surface initiate saltation more readily.

Raupach (1994) derived simple analytic relations for the roughness length z0,m and the zero-plane displacement D of vegetated surfaces as functions of vegetation height h and area index Λ. Microwave radar data may be inverted to obtain high resolution roughness length (and soil moisture) data of bare ground globally (e.g., Prigent et al., 1999; Lacaze et al., 1999; Zribi and Dechambre, 2003).

There are two roughness lengths pertinent to wind erosion over bare ground. The first is the aerodynamic roughness length of the bare ground including the non-erodible elements such as pebbles, rocks, and vegetation. This is what is traditionally known as the roughness length for momentum transfer, z0,m. The second roughness length is the so-called “smooth” roughness length, zs 0,m (Marticorena and Bergametti, 1995). zs 0,m is the roughness length of a bed of potentially erodible particles without any non-erodible elements. The roughness length most easily measured in laboratory wind tunnel experiments is zs 0,m. Wind tunnel experiments over uniform beds comprised of known particle sizes show that

It is useful to distinguish between the susceptibility of soil to erosion, called erodibility, from the power of the wind (or some other force) to cause erosion, called erosivity. Erodibility depends on the microphysical, chemical, and mechanical properties of the the soil, vegetation, and topography (D’Odorico et al., 2001). Erosivity depends on the wind speed, intermittency, shear, and turbulence.

Discounting erodible particles which are sheltered by non-erodible elements, the roughness length felt by the atmosphere over erodible particles is zs 0,m. Moreover, our theoretical understanding of threshold wind velocities is based on zs 0,m, while most large scale atmospheric models are concerned with total momentum flux, and thus tend to compute z0,m. Thus a theory is necessary to connect the zs 0,m to z0,m. This is the theory of drag partition. The increase fd in threshold friction velocity for saltation u*t due to drag partition is (Marticorena and Bergametti, 1995)

Strong saltation can modify z0,m because the saltators provide a sink for momentum distinct from the surface. Consider a saltator ejected from the surface with an initial speed proportional to u*. It is common assumption that, after launching, a saltating particle experiences no vertical acceleration except gravity. Such trajectories are called ballistic. In a ballistic trajectory, the vertical velocity decreases linearly with time, and the initial upwards velocity equals the final downwards velocity. It is easy to show the height reached by a ballistic saltator is proportional to u*²∕g. Strong saltation causes an effective thickening of the roughness length also in proportion to u*²∕g. For strongly saltating surfaces with u * ≫ u*t, Chamberlain (1983) suggests

Table 2.3 shows typical roughness lengths of non-vegetated surface types.

The roughness length of fluids depends on the surface wind shear. The shear stress deforms the fluid and generate ripples or waves at higher wind speeds. These waves, in turn, modify the surface roughness length. Chamberlain (1983) pointed out that, in certain circumstances, wind drag entrains very similar amounts of surface mass into the atmosphere over many different surface types. He assembled a variety of observational data which showed that saltation of sand and snow was analogous to entrainment of sea-spray over the ocean. This agreement could be explained by assuming z0,m was proportional to u*²∕g over each of the surfaces. Over oceans, we adopt the dependence of z0,m on U proposed by Large and Pond (1982)

Note that Cn m (2.28) is the not the drag coefficient but is the neutral drag coefficient. Stability-based corrections must be applied to Cn m in order to obtain Cm. Large and Pond (1981) summarize the procedure used to convert Cn m to a (non-neutral) drag coefficient, shifted to any height z:

Exchange coefficients such as the drag coefficient Cm (2.29) are positive-definite by definition, e.g., (2.1). Some care must be taken to ensure numerical procedures do not erroneously predict negative-valued exchange coefficients. For example, (2.29) predicts Cm(z) < 0 when . . . fxm

2.2.3 Stability Corrections

As mentioned above, the vertical profile of momentum in the boundary layer is, to a first approximation, logarithmic with height (2.6). However, the stability properties of the atmosphere introduce a correction

The stability parameter ζ is the ratio of the height z to the Monin-Obukhov length L [m] (defined below).

In the surface layer, the stability parameter equals the flux Richardson number, ζ = Rf. The flux Richardson number is the ratio of the production (or loss) of turbulent kinetic energy (TKE) by buoyancy to the production of turbulent kinetic energy by shear stresses. When Rf = 1, turbulence is consumed by buoyancy as fast as it is produced by shear stress. This is why the similarity functions defined below have discontinuous first derivatives at ζ = 1. This behavior is well-documented in nature.

It is convenient to define separate stability parameters for the processes of momentum, vapor, and heat transfer in the boundary layer:

The correction factor ψm is defined in terms of the similarity function φ via

2.2.4 Flux-Gradient Relationships

Monin-Obukhov similarity theory is a more physically accurate description of surface turbulent fluxes than the bulk aerodynamic formulation (2.19). We may re-write in terms of the mean gradients of the scalar properties and their respective transfer resistances.

The Monin-Obukhov similarity theory relates surface turbulent flux to the mean gradients of the scalar quantities in the surface (constant flux) layer.

τ	= ρ	(2.35a)
H	= -cpρ	(2.35b)
L	= -lρ	(2.35c)

2.2.5 Similarity Functions

Many studies have constructed empirical similarity functions from boundary layer experiments. We adopt forms for the similarity functions summarized by (Brutsaert, 1982, p. 71) and Zeng et al. (1998):

Zeng et al. (1998) find that the similarity functions for very stable (ζ > 1) and very unstable (ζm < -1.574, ζh < -0.465) conditions approach different limits than (2.36). For completeness, we present the Zeng et al. (1998) recommendations for the entire range of ζ for momentum

2.2.6 Monin-Obukhov Length

The Monin-Obukhov length L [m] characterizes the stability of a fluid. There are many equivalent definitions of L, such as

The procedure for estimating the surface turbulent fluxes often begins with an initial estimate of the Monin-Obukhov length L. The following method is based on Zeng et al. (1998) and is used in CLM (Dai et al., 2003). We assume use five quantities to estimate L: First, the wind speed at reference height Ur. Second and third, the virtual potential temperatures at reference height θvr and at surface θvs [K] (or, equivalently, their difference Δθv ≡ θvr - θvs). Fourth, the reference height zr [m]. Fifth and finally, the roughness length for momentum z0,m [m]. These five quantities may be measured or, in a model, their values from the previous timestep may be used.

Given these quantities, we first estimate a flux wind speed Uf [m s-1] which depends on stability. We define the term “flux wind speed” operationally as the wind speed used to compute surface fluxes. In stable conditions, the flux wind speed is simply the vector magnitude of the mean zonal and meridional components, u and v.

After the flux wind speed is estimated (2.40), we obtain the bulk gradient Richardson number Ri

We use Ri to form an initial guess for ζm (2.32)

Finally, we estimate L from ζm (2.42) and z0,m using (2.32)

2.3 Wind Speed

2.3.1 Reference Level Wind Speed

The single most important parameter determining surface erosion is the wind speed. In practice we predict or measure the wind speed U at a certain height above the ground and wish to relate U to an observed or predicted dust flux. It is convenient to define a reference height zr so that U and Q may be intercompared at disparate locations. By convention zr is set to 10 m, which is a standard height both for field observations and for numerical model output. As discussed below, it is actually the wind friction speed which is directly linked to dust mobilization.

Chapter 3
Dust Source Processes

This section describes the wind driven processes which determine the flux of long-lived mineral dust into the atmosphere. The atmospheric dust burden is the end result of a chain of processes which begins with saltation. Saltation is the wind-initiated movement of large soil particles in the downstream direction. Particles large enough to be entrained into motion directly by wind are called sand. Sand ranges in texture (size) from very fine to coarse, 1∕16 < D < 2 mm (62 < D < 2000 μm) respectively. Smaller particles of crustal material, those susceptible to long term suspension and transport in the atmosphere, are collectively referred to as dust. When the drag on the particles is large enough to overcome the inertial and cohesive forces attaching the particle to the soil bed, the particle suddenly lifts from the bed in a nearly vertical trajectory. Blown from behind and unable to resist gravitational settling for very long, the particle arcs back to the surface along a shallower trajectory. The energy dissipated by the impact of the saltating particles on the surface, may, in turn, break adhesive bonds and liberate much smaller dust particles known as dust from the surface. Although the distinction is somewhat arbitrary, dust particles are smaller than the population of saltators which liberate them.

The size of dust particles is generally taken to be D < 50 μm. Since the time a particle requires to settle out of the atmosphere under the influence of gravitation alone is proportional to the square of the particle diameter (see §5 below), dust particles have significantly longer atmospheric residence times than sand.

3.1 Literature Review

Two books are devoted to physical mechanisms of wind erosion, Bagnold (1941) and Shao (2000). Raupach and Lu (2004) provide an excellent review of all aspects of dust mobilization and dry deposition.

3.2 Threshold Wind Friction Speed

The conditions necessary to initiate or maintain the movement of particles depend on many properties, including wind speed, particle size, surface roughness, stability, sheltering effects, surface moisture, and interparticle forces. Laboratory experiments have been performed over a wide range of ambient pressures and particle densities to isolate and define these dependencies (Iversen and White, 1982; Shao et al., 1996). We now characterize the aerodynamic forces on particles which initiate and maintain particle motion which in turn causes saltation, dust mobilization and wind erosion.

3.2.1 Bagnold’s Original Theory

Bagnold (1941) first derived the size dependence of the threshold wind velocity by considering the balance of forces on a particle. His lucid derivation, though lacking in completeness, nevertheless provides the starting point for more complete theory of mobilization. Consider a particle of density ρp resting atop a bed of similar particles in a fluid of density ρ traveling above and around the particles with speed u*. Define the packing angle α as the angle from the vertical subtended by the downstream point of contact P of the particle with the bed it lies atop. If the particles are all spherical and regularly packed, then α is the angle from the center of the upper sphere to the center of one of the lower spheres.

The forces considered by Bagnold (1941) are drag and gravity. Bagnold assumes particle movement initiates when the aerodynamic drag exerted by the wind on the particle overcomes the component of the particle’s weight directed opposite to the wind stream. A more complete physical statement of the balance of forces leading to saltating is that particle motion commences when the sum of the forces acting on the particle result in a net moment of zero at the downstream point of contact P . We shall use this more general principle to include the cohesive and lift forces later; for now, we retrace Bagnold’s original derivation.

The mass of the particle is πD³ρ p∕6 and its net weight relative to the surrounding fluid is πgD³(ρ p -ρ)∕6. The contact point P is the pivot, or axis of support, about which gravity attempts to pull the particle downwards and backwards. The gravitational torque acts through the center of mass of the particle to create a moment of force. The gravitational moment MW is the product of the gravitational force and the distance between the axis through which gravity acts (i.e., the center of mass) and the point P . This distance is (D sin α)∕2, and thus

The horizontal drag force MD exerted on the particle by the fluid is proportional to the exposed cross sectional area of the particle normal to the fluid flow, MD ∝ D². The drag per unit particle surface area is proportional to ρu*². Assuming the net drag force is directed through the center of mass of the particle, then the distance from the axis of drag force to P is (D cos α)∕2, and thus the drag moment is

At the threshold velocity u*t, the gravitational moment equals the drag moment on the particle so that any small perturbation in wind speed may initiate particle motion, i.e., saltation. Equating MW (3.1) and MD (3.2)

(3.4)

The preceding discussion applies to the initiation of particle motion in a bed of grains initially at rest. Thus u*t defined by (3.3) is called the fluid threshold friction velocity or aerodynamic threshold friction velocity. Once particle motion begins, the threshold velocity actually decreases since some of the momentum needed to initiate further particle motion is supplied by particles already in motion. The friction velocity in an environment of particles already in motion is known as the impact threshold friction velocity u*ti. For large sand grains, Bagnold (1941) found that u*ti ~ 0.8u*t The effect of particle motion on the surface wind speed and its feedbacks to further particle motion will be discussed further in §3.3 below.

Subsequent developers of this theory have continued to use the parameter A as it appears in (3.3) and in (3.4), i.e., as the proportionality factor between u*t and the factor containing the square root of the particle diameter.

(3.5)

A is called the dimensionless threshold friction speed, or simply the threshold parameter (Iversen and White, 1982). Theoretical approximations and empirical parameterizations of A will be discussed in §3.2.3 below.

Bagnold (1941) developed (3.3) for particles large enough to appear as isolated elements to the fluid flow. In particular, experiments confirm the predictions of (3.3) only for particle sizes larger than about 200 μm. A ~ 0.1 for large particles in air and A ~ 0.2 for particles in water (Bagnold, 1941, p. 88). In this size range u*t increases linearly with D^1∕2. For smaller particles, however, it is found that cohesive and lift forces may not be neglected, and that these forces combine to produce an optimum size for particle saltation, i.e., a minimum in u*t(D) not suggested by (3.3).

3.2.2 Reynolds Number

In order to develop a more complete theory of saltation and dust emissions, let us first examine the physics of airflow around a single particle. The balance of these forces determine when particle motion is initiated by airflow. Consider an isolated particle of size D at rest in a Newtonian fluid of density ρ moving with velocity . For simplicity, we consider the component of motion in the x direction. As shown in §18.14, the equation of continuity (conservation of mass) for an incompressible fluid (18.80) requires

(3.6)

and momentum conservation in the x direction (18.90) requires (18.92)

(3.7)

We can gain insight into the relevant physical processes by non-dimensionalizing the equations. We do this by changing to a non-dimensional coordinate space which is scaled by the relevant physical dimensions of the fluid flow. For fluid flow around an isolated particle at rest, the relevant velocity is the upstream velocity v∞ far from the particle. The relevant length scale is D.

(3.8)

The dimensionless time and pressure variables are

(3.9)

Thus, by definition, none of the transformed coordinates has any physical dimensions. Substituting the non-dimensional coordinate definitions into (3.6) we obtain

Multiplying each side by D∕v∞² we find

(3.12)

Note that all physical scales in the problem appear in a group which multiplies the diffusion term. This single dimensionless factor must determine the behavior of the entire system. The Reynolds number Re is defined as the inverse of this factor

(3.13)

The Reynolds number expresses the ratio of inertial to viscous forces. The nature of the solutions to (3.12) is strongly sensitive to whether Re < 1 or Re > 1. When Re ≪ 1, viscous forces dominate and the LHS of (3.12) is small relative to the RHS (because Re^-1 ≫ 1) and may be neglected. The steady state behavior of (3.12) then approaches

(3.14)

The solution to (3.14) for a sphere at rest in a fluid is one of the central results of fluid mechanics.

The Reynolds number of an arbitrary flow is defined as

(3.15)

so that Re varies with position. The gravitational settling speed vg of particles smaller than 10 μm is less than 10^-2 m s-1. This implies Re < 10^-2 for D < 10 μm (3.15). Thus inertial effects may be neglected for most aerosols falling at terminal velocity. However, vg increases as the square of the particle diameter so that Re > 1 for D > 50 μm. When Re ≫ 1, then inertia plays a large in the motion. In this case, the flows described by (3.12) become highly turbulent.

The relative velocity between strong winds and an undisturbed sand grain at the surface, however, easily exceeds vg. The relevant wind speed for entraining particles from the surface is not the atmospheric wind speed, but the friction velocity u*. Section 2 defines and describes the friction velocity as the characteristic wind velocity dissipated by shear stress and small scale turbulent interactions between the surface and the atmosphere. In complete analogy with (3.15) we define the friction Reynolds number as

(3.16)

Windy conditions in the free atmosphere, e.g., wind speeds of 5–10 m s-1, typically correspond to u * between 10–50 cm s-1 over barren surfaces. In such conditions, Re > 1 for particles larger than about 20 μm, and inertial forces should not be neglected.

The friction Reynolds number at the threshold friction velocity is called the threshold friction Reynolds number

(3.17)

The literature of particle mobilization quite often uses B instead of Re*t. The symbols are equivalent, and the following sections shall use both notations.

3.2.3 Iversen’s Theory

Our presentation follows the detailed summary of Greeley and Iversen (1985), which remains the state-of-the-art in theoretical modeling of particle mobilization in planetary atmospheres. Most of the empirical elements were presented in a unified theory in Iversen and White (1982). We begin by reconsidering a loose particle at rest atop a bed of similar particles. We shall account for the effects of five moments acting on this particle. The forces creating these moments are drag FD, gravity FW, lift FL, interparticle cohesion FC, and rotational inertia. Drag and gravity were explicitly considered by Bagnold (1941) and were discussed in §3.2.1.

The lift force is due to the extremely large velocity gradient above the particle. For Re smaller than about 5, the particle resides in a thin (1–10 mm) quasi-laminar sublayer where the velocity profile is

(3.18)

Thus the velocity shear is constant in the quasi-laminar sublayer

(3.19)

In typical dust-prone conditions on Earth, u* = 0.25 m s-1 and ν = 1.3 × 10^-5 m2 s-1, so that the velocity shear (3.19) is nearly 5 × 10³ m s-1 m-1. The strong velocity shear is an indication that the pressure distribution along the top of the particle generates a lift force akin to an airfoil. The lift force acts through the center of mass of the particle in the vertical direction, i.e., opposite to the gravitational force.

Once the particle is lifted out of the quasi-laminar sublayer the lift force is expected to decrease significantly. Some particles receive their initial vertical momentum from the impact of being struck by other saltating particles. However, the vertical momentum of particles not released due to ballistic impact is thought to be due to the lift force. Experiments (Chepil, 1958; Einstein and El-Samni, 1949, MFC) show that lift and drag forces on hemispheres are of the same order of magnitude. Experiments show qualitatively similar behavior in the lift force occurs in fully turbulent boundary layers (Re* > 5).

Originally, it was thought that the observed optimal size for initiation of particle saltation was due to aerodynamic effects. The observations suggested the existence of forces which act in opposition to the drag on small particles (3.2). Iversen and White (1982) demonstrated that interparticle cohesive forces are important for small particles. Interparticle cohesive forces include a number of processes including moisture, suction, electric charge, and chemical reactions. Many of these forces may be conceived as acting along the line connecting the centers of adjacent particles. The sum of all interparticle cohesive forces is denoted FC.

The angle of repose of a soil is the maximum inclination at which the soil does not undergo spontaneous slippage. Thus the angle of repose is a qualitative estimate of the importance of interparticle cohesion. The greater the tilt, the more important interparticle cohesion is in preventing slippage. The angle between the leeward face of a sand dune and the horizontal is an excellent proxy for the angle of repose. For ordinary dune sand, the angle is 34∘ but, for very small particles, the angle can approach the vertical (Iversen and White, 1982). Bagnold (1941) conjectured that the packing angle α (§3.2.1) is, on average, close to the angle of repose.

The forces described above act on the particle along axes at varying distances from the downstream point of contact P . The distance between the axis of force and P is called the moment arm. From the figure we see that the moment arm for drag is a, for gravity and lift is b, and for interparticle cohesion is c. Thus the moments associated with each force are

It is possible to measure the relative strength of lift, drag, and rotational inertia by non-dimensionalizing the forces in (3.20) and (3.22). Following Greeley and Iversen (1985), the small particle shear flow force coefficients KD, KL, KI, for drag, lift, and rotational inertia, respectively, are defined as

All three moment arms in (3.20) are proportional to D, and so may be defined as a = D, b = D, and c = D. The non-dimensional moment arms , , and may be obtained for a specified packing geometry. Coleman (1967, MFC) showed that, for closely packed spheres of identical size,

Empirical constraints on KD, KL, and KI were inferred by Iversen and White (1982) using data from the Mars Surface Wind Tunnel, MARSWIT. Wind tunnels allow direct measurement of T , p, D, ρp, U(z), and, most importantly, Ut(z). From these, one uses theory to infer z0,m, ρ, ν, u*, u*t, and Re*t. f(Re*t). Iversen and White (1982) performed wind tunnel observations with MARSWIT over wide ranges of ρ, U, and D. The data were expressed in terms of A, the non-dimensional threshold friction speed from (3.3).

Substituting (3.20), (3.23) and (3.24) into (3.21) we obtain

Substituting (3.24) into (3.27) we obtain

Iversen and White (1982) defined the cohesionless threshold coefficient A₁ by setting the cohesion force FC = 0 in (3.28)

Greeley et al. (1980, MFC) and Iversen and White (1982) used experiments with MARSWIT to infer KD, KL, and KI (3.23). They removed the effects of cohesion from the data and performed a least squares fit to find

If the measured values in (3.30) are used in (3.29), we find

(3.31)

Equation (3.31) predicts A₁ > 0.66. As previously mentioned, many observations suggest A ≈ 0.2 for large particles (e.g., Bagnold, 1941, p. 88) where cohesive force are presumably small. Thus the assumptions leading to (3.31) have rendered the theory a qualitative, rather than an exact, description of saltation initiation.

In order to reconcile theory with observations, Iversen, Greeley, and colleagues set forth to isolate each functional dependence in (3.28) which could be independently measured. They assumed A could be expressed as the product of three factors: the cohesionless threshold A₁ (3.29), a function f(Re*t) (3.17), and a function g(D) which accounts for all interparticle cohesive forces (Iversen et al., 1976a,b, MFC)

(3.32)

The absence of interparticle cohesion corresponds to g = 1.

Based on the theoretical influence of FC on A (3.28), g is assumed to have a square-root relationship with A and to be a function of particle size to an unknown power n

(3.33)

where n and I are to be determined empirically. Clearly g(D) is a generalized version of the final term in (3.28): the factor 6 FC∕(π ) is combined into I, and n allows cohesion to depend on a non-integer power of D. The latter assumption is intuitively appealing because cohesive forces depend on surface properties (e.g., van der Waal’s forces, capillarity) as well as volume properties (e.g., electrostatic charge).

In order to determine I and n, Iversen and White (1982) grouped together many observations of A²(D) for a fixed value of Re *t. Each such curve showed A was strongly dependent on particle size for D < 80 μm, but confirmed that A approaches an asymptotic value of about 0.02 for D > 150 μm. Iversen and White (1982) found that these data were best fit by I = 6 × 10^-7 kg m^1∕2 sec^-2 and n = 2.5

(3.34)

where all quantities are expressed in MKS units. Taken together with (3.28), the observed of best fit value n = 2.5 implies the FC ∝. The quality of the fit in (3.34) adds confidence to the form of parameterization chosen for A (3.32).

Once g(D) was known from (3.34), Iversen and White (1982) were able to infer the product A₁f(Re*t) from A (3.32). The data were found to obey different functional relations depending on the value of Re*t,

Using (3.32), (3.34), and (3.35) to compute u*t(D,B) (3.4),

Rather than solving (3.36) iteratively, Marticorena and Bergametti (1995) parameterized Re*t(D) using ρ and ν typical of dust source regions

The numerical solution of (3.36) shows that u*t(D) has a fairly shallow minima at u*t0 which defines the optimal particle diameter for saltation, D0. For a steady friction velocity u* > u*t0 over dry, bare ground, we expect saltation to initiate with particles of size D0. For typical arid regions of Earth, D0 ~ 75 μm (Iversen and White, 1982; Pye, 1987; Shao et al., 1996; Marticorena et al., 1997). The minima is not symmetric in D or u*, however, due to the rapid increase of cohesive forces with decreasing size. Broadly speaking, particles in the range 40 < D < 200 μm are susceptible to saltation.

Predictions of (3.36) agree remarkably well with wind tunnel observations. Iversen and White (1982) tested (3.36) over a wide range of Re*t for particles as small as D = 12 μm. Agreement was within 5% for particles larger than D = 40 μm. Uncertainties in both the model and the measurements become significant for D < 40 μm, which is outside the saltation range. Thus, these uncertainties need not be worrisome. As previously mentioned (3.36) agrees with Bagnold’s formulation for particles larger than 200 μm.

For particles smaller than D0, u*t increases very quickly. In fact, for particles D < μm, (3.36) predicts u*t > 1 m s-1, i.e., the threshold speed exceeds values plausible for terrestrial conditions.

Shao and Lu (2000) derive an alternative theory for u*t that fits the wind tunnel data of Iversen and White (1982) but which also has the virtue of resulting in simpler expressions than (3.36).

To summarize, Bagnold’s formulation (3.4) predicts that u*t decreases with D so that very small particles should be most efficiently mobilized. Noting, however, that all observations show that u*t(D) decreases with D until a critical particle size D = D0, Iversen and colleagues developed a semi-empirical formulation for u*t (3.36) which accounts for lift and cohesive forces. Their formulation agrees well with wind tunnel measurements for 12 < D < 1000 μm.

3.3 Horizontal Dust Flux

Until now we have concentrated on the determination of the threshold friction velocity u*t required to initiate motion of particles of diameter D initially at rest. A useful theory of dust mobilization also requires specification of the horizontal mass flux of all particle sizes and at all heights, since this quantity can be measured in a wind tunnel.

(3.38)

where n is the number distribution of particles, M their mass, and v their horizontal velocity. The units of q are kg m-2 s-1. In addition to u *t(D), a useful theory of dust mobilization requires specification of the vertically integrated horizontal mass flux due to saltation

(3.39)

where zs is the height of the saltation layer,

The units of Q are kg m-1 s-1 rather than kg m-2 s-1 since a vertical integration of the streamwise mass flux has been performed. The integration over D ensures Q includes all sizes of saltating particles, while the integration over z ensures Q includes particles (including suspended dust) at all levels. Thus Q is the mass crossing orthoganally through an infinitely tall column per unit width of the column. Q is often called the streamwise mass flux since it measures the movement of crustal material in the direction of the prevailing winds.

The total streamwise movement of surface material Q is the result of three intertwined processes, saltation, suspension, and surface creep. Saltation includes the movement of all airborne particles which are too large to become suspended. Suspension, on the other hand, includes the movement of all particles The prevailing aerodynamic conditions play a role in determining what particles are susceptible to long term suspension and transport, so there is no single size below which particles are always suspended and above which never suspended. However, as a rule of thumb we call particles larger than 60 μm in diameter sand, other particles are collectively referred to as dust.

Surface creep accounts for the movement of particles which are pushed along the surface. The creep may be caused by the wind directly or by nudges from saltator impacts. The important distinction is that saltating particles remove momentum from the near surface wind which decreases the friction velocity felt by the surface. Creeping particles, on the other hand, do not directly remove any momentum from the near surface wind, and thus do not lessen the surface friction velocity.

With the above discussion in mind, we may also write Q (3.39) as the sum of the individual streamwise fluxes due to saltation Qs, surface creep Qc, and suspended dust Qd thusly

A primary goal of mineral dust studies is the prediction of the lifecycle of dust particles which begins with Qd. As shown above, wind tunnel studies (e.g., Iversen and White, 1982) show that the threshold velocity u*t (3.36) increases rapidly (due to cohesive forces) for particles smaller than about 60 μm, and exceeds terrestrial conditions for D < 10 μm. Thus most long-lived dust particles are thought to be initially lofted by impacts from more massive saltators rather than directly lofted by the wind. Hence particles in suspension are said to be secondary particles. For this reason Qd depends intimately on Qs. Thus we concentrate on Qs before turning our attention to the link between Qs and Qd in §3.4.

Many expressions have been proposed which express Q in terms of u* and u*t. Greeley and Iversen (1985) summarize these expressions in their Table 3.5. These expressions are based on theories or observations of particle saltation. The theories must predict the concentration and motion of particles set into motion by u*, as well as particles released by the impact of saltating particles. Complicating these theories is the interaction between particles of different sizes.

Theoretical and empirical evidence strongly suggests the horizontal flux of saltating particles Q varies with the cube of the wind friction velocity during saltation u*,s Owen (1964); Shao et al. (1993)

Since the wind speed is related to the wind friction speed by U = Cm^1∕2u *, we may rewrite (3.41) as

Leys and Raupach (1991) used a portable wind erosion tunnel to measure u*t(D) and total streamwise mass flux Q in field conditions in Australia. The observed Q was well described by (3.39). They also predicted u*t with four competing methods, including Equation (3.36). Unfortunately, no method, including (3.36), adequately predicted the observed u*t.

We now show the development of many theories for Q, including those of Bagnold, Kawamura, Owen, and the Australian school. Unfortunately, authors in this field have often inadvertently incorporated typographical errors into their papers (and models) (Namikas and Sherman, 1997). Baas (2005) covers this subject with more up-to-date examples, and intercompares some of these theories. Papers with known typographical errors include Blumberg (1993, equation 3.2), Lancaster (1995, equation 2.12), White (1979) (equation 22), Greeley et al. (1996, table 2, equation 4), Pye and Tsoar (1990, equation 4.48), Pankine and Ingersoll (2002), and Zender et al. (2003a) (equation 10). Researchers are urged to verify the correctness of their saltation formulations by comparison to the original formulations.

3.3.1 Original Theory

Bagnold (1941) developed the original theory relating Q to wind speed based on the energetics of an idealized, steady state, linear, saltation zone. A few qualitative observations motivate the strategy of Bagnold’s theory of saltation. The first, alluded to in §3.2.1, is that once saltation has initiated by winds exceeding the fluid threshold (u* > u*t), the downstream surface wind speed drops because some fraction of the drag is now exerted on suspended particles rather than directly on the surface. Downstream saltation continues as long as the surface wind speed exceeds the impact threshold, i.e., u* > u*ti.

Once saltation initiates, moreover, the ambient wind speed U may reach any strength but the surface wind speed is relatively constant. In other words the velocity gradient near the surface can grow without bound, but the surface wind speed at about 3 mm height is insensitive to this gradient. These qualitative observations strongly suggest that drag in excess of that necessary to maintain saltation is dissipated in the atmosphere by the saltating grains. Note that this observation is the basis of the second hypothesis of Owen (1964) (cf. §3.3.2).

These observations underpin Bagnold’s simple energetic explanation of the cubic dependence of Q on u*. Consider the following idealized scenario: A bed of particles of identical size D and mass Ms saltates in a steady state wind with friction velocity u*. The wind (rather than impacts by other particles) is responsible for lifting each particle into the atmosphere with an initial horizontal velocity u0. The wind transports each particle a mean horizontal distance L before the particle fall to the surface. During this transport, the wind accelerates the particle to a final horizontal velocity u1 which is completely dissipated in the impact.

We now impose energy conservation constraints on this system. The initial and final momenta of the particle are Msu0 and Msu1, respectively. The difference between these momenta, Ms(u1 - u0), is extracted from the atmosphere over the distance L. Therefore the rate of loss of atmospheric momentum per unit area due to a total horizontal mass flux of saltating particles Qs must be

(3.43)

Newton’s second law (18.81) tells us that the rate of change of momentum is equivalent to the applied force. In this case, the applied force is the surface wind stress due to particle the drag of the particles on the wind τp.

To proceed further than (3.48), is necessary to discover or formulate a relationship between w0 and u*. Bagnold argued that, on average, w0 = C₁u* where C₁ is called the impact coefficient. In a perfectly elastic reflection, or ricochet, C₁ = 1. An elastic collision may also eject multiple saltators as products, in which case C₁ < 1 for each product. However, it is quite possible to have C₁ > 1 when large particles eject smaller particles. Bagnold reasoned that particles are ejected with a velocity proportional to the incident velocity of the impacting particle which, he argued, ought to be u* on average. This reasoning is somewhat difficult to defend, but the assumption turns out to be correct. A better justification for this assumption is that fxm. This leads to our first derivation of the well known phenomena that the horizontal mass flux due to saltation is proportional to the cube of the friction velocity

Based on careful analysis, Bagnold empirically modified (3.49) to include a factor of (D)^1∕2. With this correction, and subsuming C₁ into a new empirical constant cs

Expressing (3.50) in terms of U rather than u*, Bagnold found

3.3.2 Owen’s Theory

Owen (1964) developed a physical theory describing the saltation of uniform particles in air. His theory continues to serve as the best mathematical definition and description of saltation, as well as being a valuable exposition of mathematical physics in its own right.

Owen’s two hypotheses are:

3.3.3 Kawamura/White Formulation

Kawamura’s work on sand transport in the late 1940s and early 1950s is not widely known. During this time Kawamura developed a rather complete theory for the streamwise saltation flux (1951) which appeared in translated form in 1964. White based his formulation for the streamwise saltation flux (White et al., 1976; White, 1979) squarely on Kawamura’s work. Thus, White appears to be the first to recognize the efficacy of Kawamura’s work for modern dust models. Unfortunately (but understandably), many references inadvertently attribute Kawamura’s formulation of streamwise mass transport to White (1979). In recognition of White’s association, we sometimes refer to this as the Kawamura/White formulation.

Rather uniquely, White and colleagues performed numerical simulations of the saltation jumps of individual particles based on first principles, i.e., the equations of motion. These simulations, in aggregate, later serve to validate simplifying assumptions he makes in his bulk theory. The equations of motion for a particle saltation take the form

The lift and drag forces on the particle are expressed in terms of the lift and drag coefficients CL and CD as

The drag force on a particle is usually expressed in terms of the density of the medium, the projected area of the particle, and the square of the particle’s velocity.

As presented in §2.1.1, the total stress to the surface τ is the sum of a particle drag and an aerodynamic drag. The particle drag on the surface, τp, is caused by the horizontal deceleration of the impactors by the surface. The aerodynamic drag on the surface, τa, is the drag directly due to gas flow over the surface (2.2). We know τ (2.3) and can obtain τa using Owen’s second hypothesis (§3.3.2)

Equation (3.61) defines τp as a residual between the total surface stress and the aerodynamic stress. We may also define τp directly from kinematic considerations as the rate of deposition of streamwise momentum to the surface by the saltating particles. If Fs is the downward mass flux of saltating particles per unit area per unit time, then the total streamwise momentum deposited to the bed by the particles is

Let us denote the mean initial vertical velocity of saltating particles as w0. Then the mean initial vertical momentum of saltating particles is w0Fs. Clearly the initial vertical momentum of saltating particles varies with the intensity of saltation. One possibility is that w0Fs obeys a functional relationship with the surface saltation stress τp. For example, one might hypothesize that τp is converted into vertical momentum with some non-unity efficiency. More specifically, we shall assume that w0Fs is linearly proportional to τp. Then (3.61) and (3.62) imply

The relationship in (3.63) assumes that the kinetic energy of bombardment converts to vertical momentum of the product saltators with an imperfect, but constant, efficiency. In other words the bombardment process is inelastic. The disposition of the unaccounted-for energy is not specified. The theory of Shao et al. (1993), presented below, extends this treatment of energy conversion in developing a theory for the vertical flux of small dust particles.

The crucial advance of Kawamura’s theory is made possible by the assumption of a relation between Fs and the friction speeds. MARSWIT data and the saltation model of White (1979) (described above) both support the following empirical relationship

The total streamwise saltation flux is simply the product of Fs and L. Using (3.64) and (3.66),

Namikas and Sherman (1997) and Baas (2005) document that Equation (19) in White (1979), which corresponds to Equation (3.67) has a typographical error which propagated into some dust emission models apparently including Pankine and Ingersoll (2002). If adopted, this error would cause models to overpredict dust emissions. In practice, however, the erroneous equation performs no worse than other (correct) mobilization implementations. By coincidence, Zender et al. (2003a) also contains a typo in its version of Equation (3.67)¹ .

White (1979) used MARSWIT to replicate a variety of Earth and Martian saltation conditions in order to determine cs. With small glass beads (D = 0.208 mm) as saltators, they found cs = 2.61 under a wide range of conditions. This differed by only 6% from Kawamura’s original estimate of cs = 2.78 (Namikas and Sherman, 1997; Baas, 2005). From this we may infer that (3.67) contains all the relevant physics of saltation for both the Earth and Mars simulations. Of course, such complicating factors as moisture, heterogeneous soil sizes, and vegetation were not considered in the tests. Nevertheless, the consistency of the experiments of White (1979) with (3.67) are very encouraging as a point from which to begin to include the effects of more complicating factors.

3.3.4 Australian School

Researchers in Australia published theories for saltation and sandblasting beginning in the 1990s. This research involved many groups, though Michael Raupach and Yaping Shao appear to be most consistently behind it. I refer to their approach as the Australian School, since it is nicely synthesized and originally evaluated with Australian models. Shao et al. (1993) present a theory for the streamwise saltation flux which differs slightly from the Kawamura/White formulation (§3.3.3). Lu and Shao (1999) and Shao (2001) summarize the full development of this theory. Raupach and Lu (2004) includes these theories in the context of more general treatments of mobilization and dry deposition.

The Australians’ approach considers uniform particles of mass Ms in steady state saltation. In steady state, the mean rates of saltator bombardment and ejection per unit surface area are equal, and denoted by N↑ s which has units of m-2 s-1. With these assumptions the vertically integrated streamwise saltation flux defined in (3.39) may be rewritten as

As in the Kawamura/White formulation (§3.3.3), we assume τ and τa are given by (3.60). Combining Bagnold’s expression for τp (3.44) with (3.68) we obtain

The parameter cs (3.72) appears in most theories of Qs (e.g., Greeley and Iversen, 1985, p. 100). As presented in §3.3.1, Bagnold (1941) estimated cs ~ 0.8. Equation (47) of Owen (1964) estimates cs empirically to be

Note that N↑ s (3.70) can be expressed in terms of Qs using (3.68) and the ballistic approximation for L

3.3.5 My Theory

If a saltating particle undergoes constant streamwise acceleration while airborne then its mean horizontal velocity is = (u0 + u1)∕2. Constant acceleration is a plausible assumption for some particles (fxm: which???). The total streamwise distance traversed is then

3.4 Vertical Dust Flux

Until now we have concentrated on predicting the wind-initiated motion of saltating particles but we have put aside consideration of the mass flux of the smaller, suspended particles known as dust. Bombardment by saltating particles, or sandblasting, is thought to be the ultimate source of most fine dust emissions. The theory of sandblasting has been extensively developed by Alfaro, Gomes and coworkers (Gomes et al., 1990; Alfaro and Gomes, 1995; Alfaro et al., 1997, 1998; Alfaro and Gomes, 2001). Applications of these theories in regional scale models is infrequent (Shao et al., 1996; Shao and Leslie, 1997; Gong et al., 2003). Saltation-sandblasting is beginning to appear in global scale models (Grini and Zender, 2004; Grini et al., 2005). The small size and long atmospheric residence time of dust particles causes them to exert significant influences over climate.

Saltation is summarized by the streamwise saltation flux Qs. Likewise, the streamwise mass flux of dust is denoted Qd. The vertical mass flux of dust particles through a horizontal plane is

(3.77)

When z > zs, then (3.77) includes only dust mass so that F = Fd. The units of F are kg m-2 s-1. To distinguish dust particles from saltating particles, we shall use Md and Ms, respectively. Dust, by definition, is suspended in the atmosphere and does not immediately settle back to the surface. Thus F is somewhat insensitive to z once z > zs. Many large scale atmospheric models assume turbulence uniformly mixes all dust emissions into the lowest atmospheric layer, so that the net dust source term is taken to be F(z = zs). Note that the mean streamwise distance from ejection to impact L, which proved useful in defining Qs (3.68), does not appear in (3.77).

Prediction of Fd is the crux of mineral dust aerosol models. To date, all theories are based on establishing a relation between Q (3.41) and Fd (3.77). There is good observational evidence to support this link (Shao et al., 1993; Gillette et al., 1997, e.g.,), but the lack of theoretical support is somewhat discomfiting.

The most successful theories are based on the energetics of the impact-ejection mechanism. Each ballistic impact a saltating particle with the surface results in a transfer and conversion of momentum and kinetic energy from the saltating particle to the surface. We have seen that some horizontal momentum is transferred into soil creep. The vertical momentum may be reflected into the next bounce of the saltator, or it may initiate the ejection of another saltator. When the product of an impact is the ejection of dust, however, it is likely that some energy has been used to break the cohesive bonds binding the dust particle to the surface. Thus some fraction of the energy from particle bombardment is responsible for the injection of dust into the atmosphere. The theory of Shao et al. (1993) explains many, but not all, of the observed features of the Fd-Qs relationship.

3.4.1 Australian School

Shao et al. (1993) developed a theoretical framework for the rupture of the interparticle bonds between dust particles and the surface. They allow for a mean interparticle binding energy of ψ. ψ is the depth of the energy potential well which must be surmounted in order to free the dust grain from the surface. Thus ψ accounts for the forces of cohesion FC and gravity FW discussed in §3.2.3. ψ is also related to the modulus of surface rupture discussed by Gillette and Passi (1988).

They consider a scenario where a saltator impacts the surface and ruptures the bonds of, on average, Π dust grains which are then ejected from the surface along with zero or more saltators. We call this the impact-rupture-ejection scenario. Let the kinetic energy of the impacting saltator be E1 and the total kinetic energy of the product saltators (which include the original when reflection occurs) be E2. Thus the energy available for freeing dust bonds is ΔE = E1 - E2. Let us define cε as the mean fraction of ΔE which is channeled to rupturing dust bonds. Energy conservation requires cε < 1 since, in addition to other sinks, some of E1 is converted into the kinetic energy of the ejected dust particles. Then the energy balance of each saltator impact is

By definition, dust is suspended once it is emitted, so there is no downward flux of dust particles in the saltation layer. Therefore the net vertical dust flux is simply the dust emission flux, which is the product of the areal rate of saltator impacts N↑ s and the dust production efficiency per impact Π. With these assumptions the areal rate of emission of dust mass from the surface defined in (3.77) may be rewritten as

Let us now examine the ΔE term. If u1 and w1 are the mean streamwise and vertical components of a saltator’s velocity at impact, then its kinetic energy is E1 = Ms(u1² + w 1²). Shao et al. (1993) assume that u1 ≫ w1 so that E1 ~Msu1². Then, they assume each bombardment ejects only one saltator. We call this conservative bombardment since, on average, it is equivalent to an inelastic saltator reflection at each impact-rupture-ejection event. In conservative bombardment every sand grain is the sole product of one bombardment, i.e., E2 = E0. Clearly this assumption is only plausible for steady state saltation. In these conditions

If the conditions for conservative bombardment are met, then we may substitute (3.83) for ΔE so that

Inserting (3.72) back into (3.84), Shao et al. (1993) defined a new parameter α₁

The factors defining α (3.85) and α₁ (3.86) are similar and bear further examination. Consider first the dimensionless speed factor γ = (u1 + u0)∕(2u*). Equations (19) and (20) of Owen (1964) are formulae for u0 and u1 in terms of u* and D. Shao et al. (1996) evaluated these expressions numerically and found that γ ≈ 2.5 for most conditions. Hence the proportionality parameters from (3.85) and (3.86) are, respectively,

Shao et al. (1996) introduced a semi-empirical method for obtaining cε and ψ. First, they assumed that ψ is the product of a length scale determined by the dust particle size and the mean drag force due to the friction wind

We may invert (3.90) to solve for β

Shao and Li (1999) further developed these theories and applied them in a Large Eddy Simulation.

3.4.2 Marticorena’s Theory

Marticorena and Bergametti (1995) have synthesized many theories and observations into a comprehensive global mineral dust emission model. Many components of their model are described in the sections above. We focus now on the components which are native to their model.

Marticorena and Bergametti (1995) begin by adapting the streamwise saltation results of White (1979) for Qs (3.67) to account for more realistic soil conditions. First, they accounted for the fraction of surface that actually consists of erodible soils. This fraction, E, includes only bare ground or sparsely vegetated surfaces susceptible to saltation. For example, E excludes bare stone surfaces, swamps, and lakes. However, E does not exclude surfaces which require only large friction speeds to initiate saltation, e.g., wetted soils. Clearly Qs is linearly proportional to E. Defining R as the ratio of threshold friction speed to friction speed, (3.67) may be rewritten as

In order to apply (3.93) to a continuous size distribution n(D) in the source soil, some simplifying assumptions are required. First, Marticorena and Bergametti (1995) assume that the value cs = 2.61, empirically derived from monodisperse saltation experiments in MARSWIT (White, 1979), is size-independent and applies equally to heterogeneous, polydisperse size distributions. Second, they assume that the mass flux Qs(D) arises only from motions of particle sizes between D and D + ΔD. In other words they assume that interactions between particles of different sizes do not contribute significant errors to (3.93), which is based on monodisperse assumptions. This approximation breaks down in the limit of dust production by ballistic impacts, since the impact of one large particle is assumed to eject many small dust particles. Shao et al. (1996) contains a more thorough discussion of the validity of these approximations. Nevertheless, with these two assumptions,

Marticorena and Bergametti (1995) assume that p(D) is best represented by the fractional cross-sectional area distribution of the soil. This appears to be a reasonable assumption because, in the absence of information to the contrary, the fractional surface area covered by grains of a given size should vary linearly with the cross-sectional area of the grains. Moreover, the exposure of saltation grains to wind also is proportional to the cross-sectional area of the grains. Another plausible assumption is made by Shao et al. (1996), who, as described below, set p(D) to the fractional mass distribution of the soil.

The discretization of (3.94) proceeds as follows. Continuous soil size distributions are most often approximated as multi-modal log-normal distributions, i.e., The three parameters required to define each mode. The parameters usually available from soil sieving techniques are the mass median diameter v, the geometric standard deviation σg, and the mass fraction M. Thus it will be convenient to express p(D) in terms of M(D). To do so, we first put down the geometric relations between the cross-sectional area, mass, and volume of spherical particles

(3.95)

We now take the limit of (3.95) to define the differential changes of area and mass in terms of eachother

(3.96)

We may normalize the increments of area and of mass by dividing (3.96) by A and M, respectively

(3.97)

By definition, ∫ dA = 1 and ∫ dM = 1. Thus A and M are properly normalized PDFs and either may take the place of p(D) in (3.94).

Using (3.97), we may discretize (3.94)

To convert this to a vertical dust flux Fd, Marticorena and Bergametti (1995) assume the relation

Gillette (1979, 1981) describes a dataset comprised of measurements of Fd and Qs at numerous (anywhere from two to ten) friction velocities for each of nine distinct soils. These data are reproduced in Table 3.1.

All soils except soil 9 (a clay) are very dry, with volumetric water contents θ < 0.013 (θ is defined in §3.7.1). These data show that more finely textured soils (i.e., with higher clay content) produce more dust per unit Qs than coarser soils (i.e., with higher sand content).

Marticorena and Bergametti (1995) showed that, in this dataset, the percentage of clay particles (D < 2 μm) in the source soil explains more than 90% of the covariation of Fd with Qs. Their best linear fit is

In their regional dust model, Marticorena et al. (1997) used a blended average approach rather than using (3.100) directly. First they defined four distinct characteristic populations of coarse particles found in arid and semiarid regions based on Chatenet et al. (1996). The mineralogical and geometric features of these populations are described in Table 3.2.

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Table 3.2: Characteristic Arid Soil Populationsa Type Code Minerological Size GSD Clay Efficiency Features n σg Mclay α μm % cm-1 Aluminosilicated silt ASS Clay minerals dominant 125 1.6 9.7 1.0 × 10-5               Fine sand FS Quartz dominant 210 1.8 3.6 1.0 × 10-6               Coarse sand CS Quartz only 690 1.6 0 1.0 × 10-7               Salts Sa Salt and clay minerals 520 1.6 3.2 3.3 × 10-6

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The value of α assigned to each population was arrived at by blending the observed α from the datasest of Gillette (1979, 1981). The first soil type, Aluminosilicated Silt, refers to many compounds such as Al2O3, SiO2, . . . The second soil type, Fine sand, includes . . . The third soil type, Coarse sand, includes . . . The fourth soil type, Salts, includes . . . fxm

Alfaro and co-workers find that soil texture and composition affect soil crusting and aggregation in wind-tunnel experiments on soils of widely varying composition and texture. In particular, CaCO3 affects aggregation. Interestingly, they found no detectable influence of texture and composition on sandblasting efficiency per se. Thus, composition and texture may influence the threshold velocity of saltation, but have no direct affect on sandblasting. This surprising result de-couples composition from sandblasting and helps to simplify models.

Shao represents the size distribution of saltators in air, ns m(D) as a weighted combination of the size distribution of the source soil in a “natural” (undisturbed) state, nn m(D), and in a “fully disturbed” state, nf m(D).

Aeolian soils are those formed with significant contributions from upwind dust sources. These soils are often called loess, although the specific definition of loess is context-sensitive (fxm: add note loess commission in INQUA). Soils accumulate and form downwind of dust sources when bioclimatic constraints such as vegetation and incisions trap near-surface clay and silt particles. Muhs (1983) shows that regional winds called Santa Anas deposit silty loam soils on the Channel Islands in California. Sweeney et al. (2005) and Stroeve et al. (2005) describe the evidence for loess-production in the the Eureka Flat flat region of the Palouse in the northwest U.S.A.

The composition of many of the most common minerals comprising dust is given in Table 3.3.

Quartz (SiO2) is most commonly associated with sand. Opal (SiO2(H2O)n) content is often measured in oceanographic cores (e.g., Adkins).

For lack of other information, many studies assume mineral dust comprises the same elemental distribution as is found, on average, in Earth’s mantle. Table 3.3 list the elemental composition of Earth’s crust according according to Lide and Frederikse (1995).

Marticorena et al. (1997) found that the Western Sahara desert could be adequately described by eight different soil types. Each of the eight possible soil types is a predefined blend of the four soil populations described in Table 3.2. Table 3.5 defines the populations composing each of the eight soil types.

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Table 3.5: Western Sahara Soil Typesa Soil Type Code Population 1 Population 2 Population 3 Flux Ratio n σg M n σg M n σg M α μm % μm % μm % cm-1 Silty fine sand SFS 210 1.8 62.5 125 1.6 37.5 - - - 4.5 × 10-6                         Medium sand MS 690 1.6 80 210 1.8 20 - - - 5.5 × 10-7                         Coarse sand CS 690 1.6 100 - - - - - - 1.0 × 10-7                         Coarse medium sand CMS 690 1.6 90 210 1.8 10 - - - 3.3 × 10-7                         Fine sand FS 210 1.8 100 - - - - - - 1.0 × 10-6                         Silty medium sand SMS 125 1.6 37.5 210 1.8 31.25 690 1.6 31.25 4.2 × 10-6                         Moderately salty sand SEM 125 1.6 50 520 1.5 50 - - - 4.1 × 10-6                         Highly salty sand SEF 520 1.5 75 125 1.6 25 - - - 3.1 × 10-6

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