The starting point of our theory is the insight that a saltator impact will produce dust emission only if a threshold impact energy is exceeded (Rice et al., 1999), with the nature and value of this threshold depending on the soil type and state. For instance, for a soil with only a small fraction of suspendable particles, much of the dust is present as coatings on larger sand particles (Bullard et al., 2004), such that the relevant threshold is likely the energy required to rupture these coatings (Crouvi et al., 2012). Conversely, for a soil containing a large fraction of suspendable dust particles, the threshold for fragmentation of brittle dust aggregates could be most important (Kok, 2011b). Since the theoretical size distribution predicted by brittle fragmentation theory is in good agreement with dust size distribution measurements (Albani et al., 2014; Mahowald et al., 2014; Rosenberg et al., 2014), and its implementation into large-scale models improves agreement with other measurements of the dust cycle (Johnson et al., 2012; Nabat et al., 2012; Li et al., 2013; Evan et al., 2014), the threshold for fragmentation of soil dust aggregates might be the most relevant threshold for dust emission under many conditions. For simplicity, we thus assume that the energy required for dust aggregate fragmentation is globally the most relevant dust emission threshold, but we note that the functional form of the dust flux parameterization derived below is likely relatively insensitive to the chosen threshold process (see further discussion in Sect. 3.6). Following the discussion above, the vertical dust flux Fd (kg m-2 s-1) generated by a soil during saltation can be written as Fd=fbarensffragmfragε, where fbare is the fraction of the surface that consists of bare soil; ns is the number of saltator impacts on the soil surface per unit area and time; ffrag is the average fraction of saltator impacts resulting in fragmentation of either the impacted soil dust aggregate, or the saltator if that is an aggregate itself; mfrag is the mean mass of emitted dust produced per fragmenting impact; and ε is the mass fraction of emitted dust that does not reattach to the surface and is transported out of the near-surface layer where it can be measured (Gordon and McKenna Neuman, 2009). Since ε likely depends predominantly on the flow immediately above the surface, which remains relatively constant with wind speed (Ungar and Haff, 1987; Shao, 2008; Kok et al., 2012), we expect ε to be approximately constant for different wind conditions for a given soil. Finally, we obtain ns from the balance of horizontal momentum in the saltation layer (Shao et al., 1996; Kok et al., 2012): ns=Cnsτs-τstmsvimp‾, where τs denotes the wind stress exerted on the bare soil, and τst denotes the threshold value of τs above which saltation occurs. Furthermore, ms and vimp‾ are the mean saltator mass and impact speed, and the constant Cns ≈ 2 (Kok et al., 2012). Substituting Eq. (2) into Eq. (1) yields Fd=fbarefclayγεCnsτs-τstvimp‾ffrag, where we assumed that mfrag / ms=γfclay. That is, we assumed that mfrag / ms scales with the volume fraction of the soil that contributes to the creation of dust aerosols (Sweeney and Mason, 2013). The size limit of dust relevant for climate is usually taken as ∼10 µm (Mahowald et al., 2006, 2010), but since the mass fraction of soil particles ≤10 µm is not available on a global scale, we instead use the soil clay fraction (fclay; ≤2 µm diameter), which is globally available (FAO, 2012). The dimensionless coefficient γ likely depends on the relative sizes of soil dust aggregates and saltators. Because many saltators are aggregates (Shao, 2008), we expect only modest variations in γ between soils and take it as a constant.

Since we thus expect variations of γ and ε with wind and soil conditions to be less important (see above), we seek to understand the dependence of τs, vimp‾, and ffrag on wind and soil conditions in order to complete our theoretical expression for Fd. In the next three sections, we derive these dependencies through a combination of insights from previous studies, new theoretical work, and simulations with the numerical saltation model COMSALT (Kok and Renno, 2009).

The dust flux emitted by an eroding soil depends on both the soil's properties and on the wind shear stress τ exerted on the surface (Marticorena and Bergametti, 1995; Shao et al., 1996; Alfaro and Gomes, 2001; Shao, 2001; Klose and Shao, 2012; Kok et al., 2012). This shear stress is characterized by the friction velocity, which is defined as (e.g., Bagnold, 1941; Shao, 2008; Kok et al., 2012) u∗′=τρa, where ρa is the air density. Dust emission often occurs in the presence of nonerodible elements such as rocks and vegetation. Thus, τ can be partitioned between the stress τR exerted on nonerodible roughness elements and the stress τs exerted on the bare erodible soil; only τs produces dust emission (Raupach et al., 1993; Shao et al., 1996). In analogy with Eq. (4), we define the soil friction velocity corresponding to τs as

u∗=τsfbareρa, where fbare is the fraction of the surface that consists of bare, erodible soil (note that fbare corresponds to the quantity S′ / S in the terminology of Raupach, 1992). The soil friction velocity u∗ can be derived from u∗′ using knowledge of the soil's roughness elements – fbare, the aerodynamic roughness length, and/or the spatial distribution and size of roughness elements – through the use of a drag partitioning model (e.g., Raupach et al., 1993; Marticorena and Bergametti, 1995; Okin, 2008) that yields the stress exerted on the bare erodible soil.

Equation (5) thus accounts for the effect of wind momentum absorption by nonerodible roughness elements on aeolian transport through the wind stress on the bare soil, as captured by the soil friction velocity u∗. However, with the exception of Okin (2008), most previous studies have accounted for the effects of roughness elements by using the ratio of τs / τ to scale the value of the threshold friction velocity u∗t′ at which transport is initiated (Raupach et al., 1993; Marticorena and Bergametti, 1995). Although phenomenologically correct, the result of this approach is that, in the presence of nonerodible roughness elements, the quantity ρa u′∗2 overestimates the wind shear stress exerted on the bare soil. For instance, Marticorena and Bergametti (1995) equate the wind stress driving saltation to τsand = ρau′∗2-u′∗t2 (in their Eq. 24), rather than τsand = τs-τst=ρau∗2-u∗t2 (Owen, 1964), where the soil threshold friction velocity u∗t is defined in more detail in the next paragraph. Therefore, using u∗′ and u∗t′ to parameterize saltation properties likely results in an overestimation of aeolian transport in the presence of nonerodible roughness elements (Webb et al., 2014), which our approach avoids.

In analogy to the threshold friction velocity u∗t′, the soil threshold friction velocity u∗t is the minimum value of u∗ for which the bare soil experiences erosion. u∗t depends on both the properties of the fluid and on the gravitational and interparticle cohesion forces that oppose the fluid lifting of sand particles that initiates saltation (Shao and Lu, 2000; Kok et al., 2012). In principle, u∗t can be estimated from dust or sand flux measurements, as long as a correction is made for the presence of nonerodible elements, as discussed above and in the Supplement. However, the theoretical interpretation of this threshold is complicated by several factors. For instance, the threshold friction velocities at which saltation is initiated (the fluid or static threshold u∗ft) and terminated (the impact or dynamic threshold u∗it) are not equal. For most conditions, the impact threshold is thought to be smaller than the fluid threshold, of the order of ∼85 % (Bagnold, 1941; Kok, 2010). Moreover, spatial and temporal variations in soil conditions (Wiggs et al., 2004; Barchyn and Hugenholtz, 2011), as well as large variations in instantaneous wind speed for a given friction velocity (Rasmussen and Sorensen, 1999), make it such that there is generally not a clear value of u∗ above which saltation does occur and below which it does not (Wiggs et al., 2004). Despite these problems, we neglect here for simplicity the temporal and spatial variability of u∗t and also assume that u∗t = u∗ft=u∗it, as previous dust emission parameterizations have also done (e.g., Gillette and Passi, 1988; Shao et al., 1996; Marticorena and Bergametti, 1995).

In addition to u∗t, we define the standardized threshold friction velocity (u∗st) as the value of u∗t at standard atmospheric density at sea level (ρa0 = 1.225 kg m-3). Consequently, u∗st is not only independent of the presence of roughness elements, but is also invariant to variations in ρa, and is thus equal for similar soils at different elevations. Therefore, u∗st is a measure of the soil's susceptibility to wind erosion that depends on the state of the bare soil only. Since u∗t∝ρa (e.g., Bagnold, 1941), u∗st≡u∗tρaρa0. We hypothesize that u∗st is a proxy for many of the soil properties known to affect dust emission, including soil cohesion, size distribution, and mineralogy (Fecan et al., 1999; Alfaro and Gomes, 2001; Shao, 2001). That is, although we do not understand in detail the effect of each of these soil properties on the dust flux (Shao, 2008), changes in soil properties that decrease the dust flux tend to also increase u∗st. Consequently, it is possible that u∗st can be used to partially account for the poorly understood effect of these soil properties on the dust flux.

After saltation has been initiated by the aerodynamic lifting of surface particles, new particles are brought into saltation primarily through the ejection, or splashing, of surface particles by impacting saltators (Ungar and Haff, 1987; Duran et al., 2011; Kok et al., 2012). (Note that this is only correct for soils with a sufficient supply of loose sand particles. The present theory is not valid for soils that instead are supply-limited, which we discuss in further detail in Sect. 3.6.) Saltation is thus in steady state when exactly one particle is ejected from the soil bed for each particle impacting it. Since the number of splashed particles increases with the impacting saltator's speed (Kok et al., 2012), this condition for steady state is met at a particular value of vimp‾. Consequently, theory and measurements indicate that, while the shape of the probability distribution of vimp changes with u* (Fig. 1), vimp‾ is independent of u∗ for steady-state saltation (Ungar and Haff, 1987; Duran et al., 2011; Kok, 2011a; Kok et al., 2012) (Supplement Fig. S1). Although vimp‾ is independent of u∗, it does depend on soil properties. In particular, the soil's saltation threshold sets the wind speed in the near-surface layer (Bagnold, 1941), which in turn determines the particle speed (Duran et al., 2011; Kok et al., 2012). Then, to first order, vimp‾=Cvu∗st, where Cv ≈ 5 since vimp‾ ≈ 1 m s-1 for loose sand with u∗st ≈ 0.20 m s-1 (Supplement Fig. S1).

An impacting saltator can fragment a dust aggregate in the soil if its impact energy exceeds a certain threshold (Kun and Herrmann, 1999; Kok, 2011b). The threshold impact energy per unit area ψ (J m-2) required to fragment a soil dust aggregate scales with the sum of the energetic cohesive bonds Ecoh between the constituent particles that make up the aggregate (Kun and Herrmann, 1999). That is, ψ∝∑EcohDs2, where Ds is the saltator size, and the sum is over all interparticle bonds in the aggregate. Measurements and theory suggest that (Shao, 2001) Ecoh∝βDc2, where Dc is the typical size of a constituent particle of the dust aggregate. The parameter β (J m-2) scales the interparticle force, which is the sum of a complex collection of individual forces, including van der Waals, water adsorption, and electrostatic forces (Shao and Lu, 2000). Consequently, β depends on the state of the soil, including soil moisture content, mineralogy, and size distribution. Since the number of bonds in the aggregate scales with Dag3Dc3, where Dag is the aggregate size, Eq. (8) becomes ψ∝βDag3Ds2Dc. For highly erodible, dry soils, β=β0 ≈ 1.5 × 10-4 J m-2 (Shao and Lu, 2000; Kok and Renno, 2006). Experiments suggest that most typical saltator impacts (i.e., Ds = 100 µm and vimp = 1 m s-1) eject dust for such highly erodible, dry soils (Rice et al., 1996), yielding ψ0 ≈ 0.1 J m-2. Thus, ψ̃=cψβ̃, where ψ̃=ψ / ψ0 and β̃=β / β0. The dimensionless parameter cψ is of order unity and depends on the soil size distribution since it scales with Dag3Ds2Dc. In particular, because saltators are often aggregates (Shao, 2008), with both Dag and Ds having typical sizes of the order of 100 µm (Shao, 2001), the leading order scaling is likely cψ ∼ Dag / Dc. Here we take cψ as a constant, both because there are insufficient vertical dust flux data sets available that report a detailed soil size distribution, and because global soil data sets are not nearly detailed enough to represent spatial and temporal variability in the soil size distribution.

Since the soil's standardized threshold friction velocity (u∗st) depends on the strength of interparticle forces (Shao and Lu, 2000), ψ must increase monotonically with u∗st (Shao et al., 1996). This is intuitive: soils that are more erosion resistant, for example with strongly bound soil aggregates due to surface crusts or high moisture content, require a larger impact energy to fragment (Rice et al., 1996, 1999). For such soils, wind tunnel experiments show that only a small fraction of saltator impacts produce dust emission (Rice et al., 1996).

We calculate the fragmentation fraction ffrag from the overlap between the probability distributions of ψ and the saltator impact energy per unit area Eimp. Since ψ is the sum of a large number of individual cohesive bonds, its probability distribution Pψ(ψ) is normally distributed per the central limit theorem (Kallenberg, 1997), with a mean ψ‾ and standard deviation σψ. The total fraction of saltator impacts that produces dust emission through fragmentation then equals ffrag=∫0∞∫0EimpPEimpEimpPψψdψdEimp=∫0∞PEimpEimp12+12erfEimp-ψ‾2σψdEimp, where erf is the error function, which results from the integration of the normally distributed ψ.

We complete our theoretical expression by substituting Eqs. (2), (5), and (15)–(17) into Eq. (3), yielding Fd=Cdfbarefclayρau∗2-u∗t2u∗stu∗u∗tCαu∗st-u∗st0u∗st0,u∗>u∗t, where Cd=Cd0exp⁡-Ceu∗st-u∗st0u∗st0, with Cd0=γεCnsCfr0Cv. Equation (18) thus predicts that the dust flux (Fd) scales with the soil friction velocity (u∗) to the power a≡α+2. We determine the dimensionless coefficients Cα, Ce, and Cd0 through comparison against a quality-controlled compilation of vertical dust flux data sets in Sect. 3. The dimensionless dust emission coefficient Cd is independent of the soil friction velocity u∗, and is thus a measure of a soil's ability to produce dust under a given wind stress. This susceptibility to dust emission is termed the soil erodibility in the dust modeling literature (e.g., Zender et al., 2003b), which is not to be confused with the identical term in the soil erosion literature referring more generally to the susceptibility of soil particles to detachment by erosive agents (e.g., Webb and Strong, 2011).

The increase in the dust emission coefficient Cd with decreasing u∗st accounts for a soil's increased ability to produce dust under saltation bombardment as the soil becomes more erodible (i.e., its threshold friction velocity decreases). This is an important result, as this process is not included in the previous dust flux parameterizations of Gillette and Passi (1988) and Marticorena and Bergametti (1995) that dominate dust modules in current climate models (e.g., Ginoux et al., 2001; Zender et al., 2003a; Huneeus et al., 2011). In particular, this result implies that the dust flux is more sensitive to the soil's threshold friction velocity than climate models currently account for. We further discuss this result and its implications in Sect. 4 and in the companion paper (Kok et al., 2014).

Note that the dust flux parameterization of Eq. (18) is considerably simpler than previous physically based dust emission models (Shao et al., 1996; Shao, 2001). This was achieved in large part by using u∗st as a measure of soil erodibility, which allowed us to substantially simplify the energetics of dust emission. Furthermore, since our parameterization's main variables (u∗,u∗t, and fclay) are available in weather and climate models, its implementation is relatively straightforward, in contrast to these more complex models (Darmenova et al., 2009).

We strive to obtain a compilation of high-quality vertical dust flux measurements that we can use to test our new parameterization. We thus apply several quality-control criteria that data sets need to meet in order to be included in our compilation; these criteria are designed to ensure that the measured dust flux is governed by a soil in an approximately constant state. This is critical, because any changes in the soil state affects u∗t, which is one of the main parameters in our parameterization. Since changes in the threshold friction velocity can occur on timescales as short as an hour (Wiggs et al., 2004; Barchyn and Hugenholtz, 2011), we only use data sets for which all data were taken within a limited time period of up to ∼12 h. This requirement excludes many of the data sets on which previous dust flux schemes were based, in particular data sets by Gillette (1979), Nickling and colleagues (Nickling, 1978, 1983; Nickling and Gillies, 1993; Nickling et al., 1999), and Gomes et al. (2003). In addition, we require that a data set contains sufficient measurements to reliably determine the threshold friction velocity for the measurements. Furthermore, we only use data sets of natural dust emission taken in the field, because the characteristics of saltation and dust emission simulated in (portable) wind tunnels have been shown to, in some cases, be substantially different from the characteristics of natural saltation (Sherman and Farrell, 2008; Kok, 2011a). Finally, the measurements should be made for relatively homogeneous terrain, such that the soil state is spatially approximately constant. This last constraint is only required for predicting the dust emission coefficient Cd. Therefore, data sets that meet all criteria except that of homogeneous terrain (i.e., the data sets of Fratini et al., 2007 and Park et al., 2011) are not used for comparison against the theoretical equations for Cd and Fd, but are still used for assessing the fragmentation exponent α.

Our literature search for vertical dust flux measurements that met the above quality-control criteria resulted in the identification of six studies: Gillies and Berkofsky, 2004 (hereinafter referred to as GB04), Zobeck and Van Pelt, 2006 (ZP06), Fratini et al., 2007 (FC07), Sow et al., 2009 (SA09), Shao et al., 2011 (SI11), and Park et al., 2011 (PP11). Images of the experimental sites of these six studies are shown in Fig. 3, and the main properties of each data set are summarized in Table 1. We used the original data for each of these six studies, and extracted 11 individual data sets from them. We describe the general procedures for correcting for differences between data sets and for extracting estimates of u∗t, α, and Cd in the next two sections. A detailed description of the analysis of each individual data set is provided in the Supplement.

A critical property of dust flux data sets is the time period over which measurements are averaged. In particular, since the vertical dust flux is nonlinear in the friction velocity, the averaging period needs to be consistent among the data sets (Sow et al., 2009; Martin et al., 2013). In setting the averaging period, an important consideration is that the friction velocity, being a turbulence parameter, is only meaningful when obtained over averaging periods long enough to sample a sufficient range of the turbulent eddies contributing to the downward flux of horizontal fluid momentum (Kaimal and Finnigan, 1994; Namikas et al., 2003; van Boxel et al., 2004). Moreover, the averaging period needs to be short enough such that the meteorological forcing of the boundary layer, which partially sets the downward momentum transfer, remains approximately constant. A compromise between these constraints is an averaging period of 30 min (Goulden et al., 1996; Aubinet et al., 2001; van Boxel et al., 2004; Fratini et al., 2007), which conveniently is also of the order of the typical time step in global models. We thus reanalyzed each data set using a 30 min averaging period. In order to get maximum use out of each data set, the data were averaged over 30 min with a running average (e.g., a 60 min continuous data set with 1 min resolution yielded 31 data points).

In addition to using the same averaging period for each data set, we also need to correct for differences in the measured dust size range between the data sets. We therefore corrected each data set to represent the mass flux of dust aerosols with a geometric diameter Dd between 0 and 10 µm, which is a size range commonly represented in atmospheric circulation models (Mahowald et al., 2006). Several of the dust flux data sets (e.g., GB04, ZP06) reported size ranges not in terms of the geometric diameter Dd, which is defined as the diameter of a sphere having the same volume as the irregularly shaped dust aerosol, but in terms of the aerodynamic diameter, Dae, which is defined as the diameter of a spherical particle with density ρ0 = 1000 kg m-3 with the same aerodynamic resistance as the dust aerosol (Hinds, 1999). Therefore, depending on the data set, two separate corrections need to be made: one to correct from aerodynamic diameter to geometric diameter, and one to correct the measured geometric size range to 0–10 µm.

The geometric and aerodynamic diameters are related by Hinds (1999) and Reid et al. (2003) as Dd=χρ0ρpDae, where ρp ≈ 2.5 ± 0.2 × 103 kg m-3 is the typical density of a dust aerosol particle (Kaaden et al., 2009), and χ is the dynamic shape factor, which is defined as the ratio of the drag force experienced by the irregular particle to the drag force experienced by a spherical particle with diameter Dd (Hinds, 1999). Measurements of the dynamic shape factor for mineral dust particles with a geometric diameter of ∼10 µm find χ ≈ 1.4 ± 0.1 (Cartwright, 1962; Davies, 1979; Kaaden et al., 2009). Inserting this into Eq. (19) then yields that Dd ≈ (0.75 ± 0.04) Dae, where the standard error was obtained using error propagation (Bevington and Robinson, 2003).

After converting each data set's measured aerodynamic particle size range to a geometric size range as necessary, we corrected the measured dust flux by assuming that the size distribution at emission is well-described by the theoretical dust size distribution expression of Kok (2011b), which is in good agreement with measurements Mahowald et al. (2014). For instance, Eq. (6) in Kok (2011b) predicts that 71 ± 5 % of emitted dust in the geometric 0–10 µm size range lies in the aerodynamic 0–10 µm size range (which is equivalent to the geometric 0–7.5 ± 0.4 µm size range). We thus apply a correction factor of (0.71 ± 0.05)-1 = 1.42 ± 0.10 in order to correct a measured aerodynamic PM10 flux (e.g., GB04, ZP06) to a geometric ≤ 10 µm flux. Note that the uncertainty in the correction factor is propagated into the uncertainty on the value of Cd extracted from each data set (see the Supplement).

In addition to correcting for differences between data sets in the averaging time and the measured size range, we also corrected for differences in the fetch length when possible (see the Supplement).

After putting all data on an equal footing using the above procedures, we extracted the parameters u∗t, α, and Cd from the dust flux data sets. Because u∗t is required to determine the other parameters, we first determined the soil's threshold friction velocity for each data set.

Since many field experiments did not report the threshold friction velocity, and because of differences in the definition of threshold between data sets that did report a threshold friction velocity, we estimated u∗t in a similar manner for each data set as described in detail in Sect. B in the Supplement. In brief, we estimated u∗t using least-squares fitting of a second-order Taylor series of Eq. (22) below to saltation flux measurements within a limited range around the threshold (Barchyn and Hugenholtz, 2011). If the data set did not contain sand flux measurements, we instead used a least-squares fit of a second-order Taylor series of Eq. (18) to measurements of the dust flux.

After determining u∗t in this manner, we used the following procedure to extract Cd and α from each data set's dust flux measurements. Following Eq. (5), we start by calculating the dimensionless dust flux for each measurement of Fd at given values of u∗ and u∗t (obtained as described below) as F̃d=Fdfbarefclayρau∗2-u∗t2/u∗st. Through substitution of Eq. (18) we now obtain an analytical expression for F̃d as a function of Cd and α: F̃d=Cdu∗u∗tα. We then use least-squares fitting of Eq. (21) to the values of F̃d calculated from dust flux measurements to determine the dust emission coefficient Cd and the fragmentation exponent α, as well as their uncertainties, for each data set. The least-squares fitting procedure and the calculation of uncertainties is described in more detail in the Supplement.

In addition, we obtain an independent estimate of the fragmentation exponent α, and thus the dust emission exponent a=α + 2, by using measurements of the sandblasting efficiency, which is defined as the ratio of the vertical dust flux to the horizontal saltation flux (Gillette, 1979). The sandblasting efficiency is thus defined for the data sets that reported measurements of both the dust flux and the (impact) flux of saltators at a certain height (i.e., ZP06, SA09, and SI11). This latter variable was usually measured with the Sensit piezoelectric instrument (Stockton and Gillette, 1990), which has been shown to provide a good measure of the horizontal saltation flux (Gillette et al., 1997; van Donk et al., 2003).

We extract α from measurements of the sandblasting efficiency as follows. We start with the saltation mass flux, which is given by (Bagnold, 1941; Kok et al., 2012) Q=ρau∗2-u∗t2LΔv, where L is the typical saltation hop length, and Δv is the average difference between the saltators' impact and lift-off speeds. The ratio L / Δv is thought to scale with the friction velocity, LΔv∝u∗r, where the exponent r ranges from 0 (Ungar and Haff, 1987; Duran et al., 2011; Ho et al., 2011; Kok et al., 2012) to 1 (Owen, 1964; Shao et al., 1993), such that we take r = 0.5 ± 0.5. We now obtain an analytical expression for the sandblasting efficiency by combining equations (Eqs. 18, 22, 23): FdQ=Csu∗α-r, where the dimensional constant Cs contains all parameters that do not depend on u∗. We then obtain α and its uncertainty by fitting measurements of the sandblasting efficiency to the power law in u∗ of Eq. (24); this procedure is described in more detail in the Supplement. Note that an important advantage of the calculation of α from the sandblasting efficiency is that, unlike the calculation of α from the dimensionless dust flux described above, the result does not depend on the determination of the threshold friction velocity u∗t. Therefore, errors that arise due to the procedure for assessing u∗t do not affect the estimate of α derived from the sandblasting efficiency.

All 11 data sets from the six studies that met the quality-control criteria discussed in Sect. 3.1 were used to determine the fragmentation exponent α through nonlinear least-squares fitting of Eq. (21) to the vertical dust flux (see Supplement Fig. S5). Furthermore, five data sets featured simultaneous dust flux and saltation flux measurements, which we used to determine α by fitting Eq. (24) to the ratio of the vertical dust and horizontal saltation (impact) fluxes (see Supplement Fig. S6), and seven data sets were taken over spatially homogeneous terrain and thus were used to determine the dust emission coefficient Cd (see Supplement Fig. S5).

The resulting analysis of the compilation of quality-controlled dust flux data sets shows an approximately linear increase in the dust emission exponent α with u∗st (Fig. 4a), as predicted by Eq. (16). We obtain the dimensionless constant Cα using least-squares fitting of Eq. (16), yielding Cα = 2.7 ± 1.0. Moreover, the literature-extracted data sets show an approximately exponential decrease of the dust emission coefficient Cd with u∗st, as also predicted from our theory (Eq. 18) and numerical simulations (Fig. 4b). We obtain Ce = 2.0 ± 0.3 and Cd0 = (4.4 ± 0.5) × 10-5 from least squares fitting of Eq. (18b).

After testing the parameterization's functional form and determining the values of its dimensionless coefficients, we can compare the predictions of Eq. (18) against our quality-controlled compilation of dust flux measurements. To avoid testing the model with the same data used to obtain its dimensionless coefficients (see previous section), we use the cross-correlation method (e.g., Wilks, 2011; p. 252–253). That is, we use the following method for each data set: first, we obtain the dimensionless coefficients using the procedure in the previous section, but without using that particular data set or any other data sets from the same study. We then use the obtained dimensionless coefficients, which are thus specific for each of the six studies in our compilation, to predict the dust flux for each of the 11 data sets in our compilation. The resulting comparison between model and measurements is reported in Fig. 5c and Table 2.

For reference, we also compare against the predictions of the previous dust flux parameterizations GP88 (Gillette and Passi, 1988) and MB95 (Marticorena and Bergametti, 1995). Note that we unfortunately cannot compare our measurements compilation against the physically explicit dust flux parameterizations of Shao and colleagues (Shao et al., 1993, 1996; Shao, 2001), because these parameterizations use detailed soil properties that are unavailable for most data sets.

The MB95 dust flux parameterization is given by Fd=CMBηfbareρagu∗′31+u∗t′u∗′1-u∗t′2u∗′2,u∗′>u∗t′, where the dimensionless parameter CMB is a proportionality constant, and the sandblasting efficiency η (units of m-1) depends on the clay fraction following η=1013.4fclay-6. Note that Eq. (25) simplifies Eq. (34) in Marticorena and Bergametti (1995) by using a single value of u∗t′ for the soil rather than different thresholds for different soil particle size bins. This is a common simplification necessary for the implementation of MB95 into most large-scale models (e.g., Zender et al., 2003a). Moreover, measurements, numerical models, and theory indicate that this simplification is actually more realistic (Bagnold, 1938; Rice et al., 1995; Namikas, 2006; Kok et al., 2012). Also, note that u∗t′ in MB95 is calculated through a drag partition parameterization (Eq. 20 in MB95), which we use for consistency for the comparison of MB95 against the measurement compilation (see the Supplement).

The GP88 parameterization is given by Fd=CGPfbareu∗41-u∗t/u∗,u∗>u∗t, where CGP (kg m-6 s3) is a proportionality constant. Note that GP88 is thus formulated in terms of the soil friction velocity u∗ since it converts wind speed measurements taken over an airport with approximate roughness length of 1 cm to the u∗ over a bare eroding field with roughness length of 20 µm (p. 14 234 in GP88).

Our new parameterization reproduces the compilation of dust flux measurements with substantially less error than the parameterizations of GP88 and MB95 (Figs. 5a–c, S3 in the Supplement, Table 2). Equation (18) also produces better agreement when each parameterization's proportionality constant is tuned to each individual data set (Table 2, Fig. S2).

We derived the dust emission parameterization of Eq. (18) for dust emission occurring primarily through the fragmentation of either soil dust aggregates or saltating aggregates by the energetic impacts of saltators. Nonetheless, the main assumption used in deriving Eq. (18) is the existence of a normally distributed threshold controlling dust emission. Consequently, Eq. (18) theoretically applies to any dust emission processes controlled by an approximately normally distributed threshold. This point is underscored by the insensitivity of the functional form of Eqs. (16) and (17) to the threshold's normal distribution parameters and the saltator size (Fig. 2). Examples of dust emission processes other than aggregate fragmentation that are controlled by a normally distributed threshold could include dust emission from crusted soils (Rice et al., 1996) and from sand particles with clay coatings (Bullard et al., 2004). Since we do not know what the relative contribution of different dust emission processes is to each of the dust flux data sets used to calibrate the dimensionless coefficients in Eq. (18), it is likely that the obtained values of these coefficients represents some weighted average of the relative contribution of each dust emission process. As discussed in Sect. 2.2, we consider it most likely that the fragmentation process contributes the largest fraction of the dust flux for each data set. Thus, although our parameterization theoretically applies to dust emission from soils dominated by processes other than fragmentation, the dimensionless coefficients in Eq. (18) could be quite different for such soils. We are not aware of any experimental data sets that meet our quality-control criteria that could be used to estimate the dimensionless coefficients for soils for which dust emission is dominated by any specific process other than fragmentation.

Furthermore, as mentioned in Sect. 2.2.1, our theory applies only to soils for which the saltation flux is limited by the availability of wind momentum, and are thus transport limited (e.g., Nickling and McKenna Neuman, 2009). The present theory is thus not valid for soils for which the horizontal saltation flux at a given point in time is limited by the availability of sand-sized sediment. Such supply-limited soils are inherently inefficient sources of dust aerosols (Rice et al., 1996), and are thus probably less important in the global dust budget. Note that dust emission from some prominent sources can be limited by the sediments supplied to these sources, for instance through the deposition of fluvially eroded sediment (Bullard et al., 2011; Ginoux et al., 2012). However, when substantial emission occurs from such regions, the soil is generally not supply limited at that point in time (Bullard et al., 2011), such that Eq. (18) could be used to parameterize the dust flux.

Our parameterization attempts to include only the most important processes affecting the dust flux. Thus, Eq. (18) does not explicitly account for many other processes that might affect dust emission, including changes in the parameters γ and ε with u∗ and u∗t, and the dependence of cψ and σψ on the soil size distribution, mineralogy, and other soil properties. Future studies should consider these effects, especially if more extensive global (or regional) soil data sets become available, or if more dust flux data sets that sufficiently characterize these soil properties become available. However, as mentioned above, many of these processes partially affect the dust emission flux Fd by increasing or decreasing u∗st, such that some of their effect might be captured in the calibration of the dimensionless coefficients of Eq. (18) to our compilation of vertical dust flux data sets.

Another limitation of our theory is that it does not account for dust emission due to saltator impacts that do not produce fragmentation but that nonetheless produce dust by “damaging” the dust aggregate (Kun and Herrmann, 1999). It also does not account for the lowering of an aggregate's fragmentation threshold through the rupturing of cohesive bonds by impacting saltators. These effects might dominate for very erosion-resistant soils, such as crusted soils. A further limitation of our theory is that it simplifies the energetics of dust emission by considering u∗st the prime determinant of soil erodibility (Shao and Lu, 2000). Although the threshold for saltation (u∗st) and the threshold energy required to fragment dust aggregates (ψ) are likely strongly coupled for many soils (Shao et al., 1993; Rice et al., 1996, 1999), increases in ψ might not produce corresponding increases in u∗st for some soils. An example of such a soil is a sandy soil for which dust emissions occur primarily from the removal of dust coatings on sand grains (Bullard et al., 2004); thus, emission from such soils might be poorly captured by the present theory.