Introduction
The emission of mineral dust aerosols produces important impacts on the
Earth system, for instance through interactions with radiation, clouds, the
biosphere, and atmospheric chemistry (e.g., Miller and Tegen, 1998;
Jickells et al., 2005; Cwiertny et al., 2008; Creamean et al., 2013). The
inclusion of an accurate dust cycle in climate and weather models is thus
critical. Yet, the current generation of dust modules shows substantial
disagreements with measurements (Cakmur et al., 2006; Huneeus et al., 2011;
Evan et al., 2014), and commonly uses semiempirical “dust source
functions” to help parameterize dust emission processes (e.g., Ginoux et
al., 2001; Tegen et al., 2002; Zender et al., 2003b).
Here we aim to improve the dust cycle's representation in weather and
climate models, in particular for climate regimes other than the current
climate to which most models are tuned (Cakmur et al., 2006). We do so by
presenting a physically based theory for the vertical dust flux emitted by
an eroding soil. The functional form of the resulting dust flux
parameterization is supported by a compilation of quality-controlled dust
flux measurements, and our new parameterization reproduces these
measurements with substantially less error than the existing
parameterizations we are able to test against. Moreover, our new
parameterization is relatively straightforward to implement since it uses
only variables that are readily available in large-scale models. A
critical insight from the theory is that the dust flux is substantially more
sensitive to changes in the soil state than most climate models
account for.
We derive our new dust emission parameterization in Sect. 2, after which
we compare our parameterization's predictions against a compilation of
quality-controlled vertical dust flux measurements in Sect. 3. We discuss
the implications of the new parameterization and conclude the article in Sect. 4.
Derivation of physically based dust flux parameterization
Because of their small size, dust particles in soils (<62.5 µm
diameter; Shao, 2008) experience cohesive forces that are large compared to
aerodynamic and gravitational forces. Consequently, dust aerosols are
usually not lifted directly by wind (Gillette et al., 1974; Shao et al.,
1993; Sow et al., 2009) and instead are emitted through saltation, in which larger
sand-sized particles (∼70–500 µm) move in ballistic
trajectories (Bagnold, 1941; Shao, 2008; Kok et al., 2012). Upon impact,
these saltating particles can eject dust particles from the soil, a process
known as sandblasting. Moreover, some saltating particles are actually aggregates
containing dust particles. Upon impact, these aggregates can also emit dust
aerosols (Shao et al., 1996).
We aim to obtain an analytical expression that captures the main
dependencies of the emitted flux of dust aerosols on wind speed and soil
properties. An important limitation is that, to allow its implementation into
climate models, this expression can only use parameters that are globally
available. Our approach to achieve this objective combines a theoretical
derivation with numerical simulations of dust emission. We start in the next
section by providing a basic theoretical expression for the vertical dust
flux, after which we derive the three main variables in this expression in
the three subsequent sections. We then combine all these components together
to give the full dust emission parameterization in Sect. 2.5.
Basic theoretical expression of the vertical dust flux
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).
Friction velocity and the wind stress τs on the bare soil surface
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.
The mean saltator impact speed (vimp‾)
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).
The fragmentation fraction (ffrag)
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.
Determining PEimp with the numerical saltation model COMSALT
In order to calculate ffrag with Eq. (12), we require the
probability distribution of saltator impact energies (PEimp)
for given values of u∗, β, and Ds, which we obtain
through simulations with the numerical saltation model COMSALT (Kok and
Renno, 2009). This model explicitly simulates the trajectories of saltators
due to gravitational and fluid forces, and accounts for the stochasticity of
individual particle trajectories due to turbulence and collisions with the
irregular soil surface. Moreover, COMSALT simulates the retardation of the
wind profile by the drag of saltating particles, which is the process that
ultimately limits the number of particles that can be saltating at any given
time. Finally, in contrast to many previous models, COMSALT includes a
physically based parameterization of the ejection (“splashing”) of surface
particles, based on conservation of energy and momentum (Kok and Renno,
2009). Because of this explicit inclusion of splash, as well as other
improvements over previous studies, COMSALT is the first numerical model
capable of reproducing a wide range of measurements of naturally occurring
saltation.
Since COMSALT was developed for saltation of soils made up of loose sand, it
must be adapted in order to simulate saltation over dust-emitting soils. For
soils made up of loose sand, the splashing of new saltating particles is
constrained predominantly by the momentum transferred by impacting saltators
(Kok and Renno, 2009). That is, the total momentum of splashed particles
scales with the impacting saltator momentum (Beladjine et al., 2007; Oger et
al., 2008). For dust emitting soils, this situation is likely different,
because saltating particles are more strongly bound in the soil by cohesive
forces (Shao and Lu, 2000; Kok and Renno, 2009). We therefore assume that,
for dust emitting soils, the number of particles splashed by an impacting
saltator scales with its impacting energy (Shao and Li, 1999). Furthermore,
in order for a saltating particle to eject another saltator from the soil,
the impact must be sufficiently energetic to overcome the cohesive the bonds
with other soil particles. Therefore, the larger the soil cohesive forces,
the stronger the cohesive binding energy Ecoh,s with which
sand-sized particles are bonded to other soil particles, resulting in a
smaller number of splashed saltating particles N. That is,
N∝msvimp2/ 2Ecoh,s.
Since Ecoh,s scales with βDs2 (see Eq. 9 and
Shao, 2001), Eq. (13) becomes
N=aEρpDsvimp2β,
where ρp ≈ 2650 kg m-3 is the density of the
saltating particle (Kok et al., 2012), and the dimensionless parameter
aE scales the number of splashed particles. We obtain
aE = 6.1 × 10-5 by forcing the minimum
u∗ for which saltation can occur in COMSALT with
β = β0 to equal the minimal value of u∗st
for an optimally erodible soil. We define this minimal value as u∗st0, and measurements show that
u∗st0 ≈ 0.16 m s-1 for a bed of 100 µm loose sand
particles (Bagnold, 1941; Iversen and White, 1982; Kok et al., 2012).
Other parameters of the splash process, such as the speed of splashed
particles, the coefficient of restitution, and the probability that an
impacting saltator does not rebound, are treated as described in Kok and
Renno (2009). We thus neglect any change in these parameters with changes in
soil cohesion since there is very little experimental data available to
account for any such dependences (O'Brien and McKenna Neuman, 2012). COMSALT
also computes the soil's standardized threshold friction velocity
u∗st as the minimum value of u∗ at which saltation can be sustained
for a given value of β, following the procedure outlined in Kok and
Renno (2009).
Probability distributions of the threshold impact energy per
unit area (Pψ) required for aggregate fragmentation (solid black line),
and of the saltator impact energy per unit area (PEimp) for
saltation of 100 µm particles at different values of u∗ (colored lines). Shown are results for (a) a
highly erodible soil (u∗st = 0.16 m s-1)
and (b) an erosion-resistant soil
(u∗st = 0.40 m s-1). The value of
ffrag increases with u∗ for erosion-resistant soils, but
not for highly erodible soils, as shown explicitly in (c). All plotted
energy values are normalized by ψ0, the energy per unit area of a
100 µm saltator impacting at 1 m s-1, and Pψ (ψ)
was calculated using cψ = 2 and σψ=0.2ψ‾.
COMSALT simulations of PEimp show that, although the mean
saltator impact speed (vimp‾) remains approximately
constant with u∗ (see above), the distribution of Eimp
does not (Fig. 1). Because the total drag exerted by saltators on the flow
increases with u∗, the wind profile lower in the saltation layer is
relatively insensitive to u∗ (Owen, 1964; Ungar and Haff, 1987; Duran
et al., 2011; Kok et al., 2012). Conversely, the wind speed higher up in the
saltation layer does increase with u∗ (Bagnold, 1941), which
causes the speed and abundance of energetic particles moving higher in the
saltation layer to also increase. This causes a nonlinear increase in the
high-energy tail of PEimp with u∗ (Fig. 1; also see
Duran et al., 2011 and Kok et al., 2012).
Dependence of ffrag on u* and u*st
Since we can obtain PEimp for given values of u∗, Ds,
and β (and thus u∗st) from COMSALT simulations, we can
use Eq. (12) to determine ffrag for given values of cψ
and σψ. Considering that the exact values of cψ and
σψ for any particular soil are unknown, our objective in using
Eq. (12) is to understand the functional form of the dependence of
ffrag, and thus Fd, on u∗ and
u∗st. To understand these dependencies, we consider the
distributions of Eimp and ψ for two limiting cases: a highly
erodible and an erosion-resistant soil (Fig. 1). For a highly erodible soil,
a large fraction of saltator impacts can be expected to produce fragmentation
(Rice et al., 1996 and Fig. 1a), such that
Eimp‾∼ψ‾. In this case, the value of
ffrag is thus approximately constant with u∗ (Fig. 1c).
Conversely, when the soil is erosion-resistant,
Eimp‾≪ψ‾, and only the high-energy
tail of the impact energy distribution results in dust emission through
fragmentation (Fig. 1b). Since this high-energy tail increases sharply with
u∗, ffrag also increases sharply with u∗
(Fig. 1c). Consequently, Fd scales more strongly with u∗
for erosion-resistant than for highly erodible soils. Our results thus show
that ffrag depends on both u∗ and u∗st
(Fig. 1c). Since ffrag is dimensionless, its dependency on
u∗ and u∗st should take the form of the
nondimensional ratios that capture the physical processes determining
ffrag (Buckingham, 1914). That is, ffrag should
depend only on (i) the dimensionless friction velocity
u∗ / u∗t, which sets the increase of the
high-energy tail (Fig. 1), and (ii) the dimensionless standardized threshold
velocity u∗st / u∗st0, which sets the soil's
susceptibility to wind erosion. From Fig. 1c, we infer
Simulations of the fragmentation exponent α (a)
and fragmentation constant Cfr (b) with the numerical saltation model
COMSALT (Kok and Renno, 2009) for different values of the saltating particle
size (Ds) and the threshold fragmentation energy's normal
distribution parameters (cψ and σψ). The colored dashed
lines represent the best fits of the functional forms of Eqs. (16) and (17)
to the corresponding simulation results, and the solid black lines represents
the best fit to the experimental data in Fig. 4.
ffrag=Cfru∗u∗tα.
Since this power law accounts for the dependence of ffrag on
u∗ / u∗t, the dimensionless fragmentation
constant Cfr and exponent α must depend only on the other
dimensionless number, u∗st / u∗st0
(Buckingham, 1914). Since highly erodible soils with u∗st=u∗st0 have α ≈ 0 (Fig. 1), we hypothesize that
α=Cαu∗st-u∗st0u∗st0,
where Cα is a dimensionless constant. Equation (16) is supported by
numerical simulations of ffrag for a range of plausible values of
the saltator diameter Ds and the threshold fragmentation energy's
normal distribution parameters (Fig. 2a).
The proportionality constant Cfr in Eq. (15) must decrease
sharply with u∗st (Fig. 1c), because increases in
u∗st are primarily driven by increases in soil (aggregate)
cohesion (Shao and Lu, 2000; Shao, 2008; Kok et al., 2012), for instance due
to increases in soil moisture. Such increases in aggregate cohesion reduce
the fragmentation fraction ffrag, and numerical simulations
indicate that (Fig. 2b)
Cfr=Cfr0exp-Ceu∗st-u∗st0u∗st0,
where Cfr0 ≈ 0.5 is the fragmentation fraction for
highly erodible soils (Fig. 1c), and Ce is a dimensionless
constant.
Full theoretical expression for the vertical dust flux
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).
Summary of the main characteristics of the quality-controlled data sets
used in this study. Data set names are defined in
Sect. 3.1.
Study
Event
Measurement
Range of u∗
Estimated u∗t
Fetch length
Event duration
Number of
Soil type
method
(m s-1)
(m s-1)
data points
(clay fraction in %)
GB04
16 February
Gradient method
0.26–0.43
0.24 ± 0.02
>5 km
3 h 51 min
203
Loamy sand (9.1 % clay)
GB04
20 March
Gradient method
0.33–0.62
0.31 ± 0.02
>5 km
2 h 50 min
142
Loamy sand (9.1 % clay)
ZP06
4 March
Gradient method
0.39–0.54
0.41 ± 0.03
200 m
4 h 2 min
148
Fine sandy loam (13 % clay)
ZP06
18 March
Gradient method
0.38–0.48
0.36 ± 0.03
200 m
2 h 26 min
113
Fine sandy loam (13 % clay)
FC07
Event 1
Eddy covariance
0.232–0.693
0.203 ± 0.016
>5 km
9 h 40 min
57
Sand (<1 % clay)
FC07
Event 2
Eddy covariance
0.171–0.606
0.170 ± 0.014
>5 km
11 h 50 min
54
Sand (<1 % clay)
SA09
ME1
Gradient method
0.238–0.321
0.237 ± 0.019
575 m
1 h 57 min
76
Sand (2.8 % clay)
SA09
CE4
Gradient method
0.314–0.358
0.232 ± 0.019
420 m
1 h 53 min
61
Sand (2.8% clay)
SI11
NA
Gradient method
0.164–0.246
0.161 ± 0.013
>1 km
7 h 21 min
399
Loamy sand (11 % clay)
PP11
Event 1
Gradient method
0.192–1.444
0.171 ± 0.014
>2 km
9 h 40 min
50
Sand (4 % clay)
PP11
Event 2
Gradient method
0.218–1.627
0.197 ± 0.016
>2 km
12 h 50 min
52
Sand (4 % clay)
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).
Assessment of parameterization performance using a quality-controlled
compilation of dust flux measurements
We test our proposed dust emission parameterization using a compilation of
quality-controlled literature data sets. We do so by first separately testing
the two main improvements of Eq. (18) over previous theories: the linear
increase of the dust emission coefficient a with u∗st, and
the exponential decrease of the dust emission coefficient Cd with
u∗st. This procedure also yields estimates of the
dimensionless parameters Cα, Cd0, and Ce,
subsequently allowing us to directly compare the measured dust flux against
the predictions of Eq. (18).
The following section discusses the quality-control criteria that data sets
need to meet in order to allow for an accurate comparison against our theoretical
expression. Section 3.2 then describes the various corrections applied to
bring all data sets on an equal footing, after which Sect. 3.3 describes the
procedure for determining the dust emission coefficient (Cd) and
fragmentation exponent (α) from literature data sets of dust flux
measurements. We then test the functional form of the parameterization
against the estimates of Cd and α extracted from the
literature data sets in Sect. 3.4, and test the parameterization's
predictions of the vertical dust flux against our dust flux compilation in
Sect. 3.5. Finally, we discuss the limitations of our parameterization in
Sect. 3.6.
Data set quality-control criteria
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.
The experimental field sites of the six studies in our
vertical dust flux compilation: (a) Gillies and Berkofsky (2004)
(36.48∘ N, 117.90∘ W), (b) Zobeck and
Van Pelt (2006) (32.27∘ N, 101.49∘ W), (c) Fratini
et al. (2007) (100.54∘ E, 41.88∘ N), (d), Sow et
al. (2009) (13.5∘ N, 2.6∘ E), (e) Shao et
al. (2011) (33.85∘ S, 142.74∘ E), and (f)
Park et al. (2011) (42.93∘ N, 120.70∘ E).
Correcting for differences in averaging period and measured size range
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).
Procedure for obtaining u∗t, α, and Cd
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.
Values of (a) the dust emission exponent a
(= α + 2) and (b) the dust emission coefficient Cd as a
function of the standardized threshold friction velocity u∗st,
determined from the analysis of available quality-controlled data sets. Open
symbols refer to estimates of Cd and a from the least-squares fit of the
measured dust flux to Eq. (18), whereas filled symbols refer to estimates of
a from a least-squares fit to ratios of the measured vertical dust flux and
the horizontal saltation flux (see text for details). The dashed line
indicates the best-fit forms of Eqs. (16) and (18b), and the grey shaded
area denotes one standard error from the fitted relation. Data set names are
defined in Sect. 3.1.
Test of the parameterization's functional form with dust flux measurements
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).
Test of the parameterization's predictions with dust flux measurements
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.
Root mean square error (RMSE) of the vertical dust flux
predicted by the parameterizations of Gillette and Passi, 1988 (GP88),
Marticorena and Bergametti, 1995 (MB95), and Eq. (18). RMSE values were calculated for
two separate cases and the lowest RMSE for the three different parameterizations
is underlined for each case. For each parameterization's first case, the proportionality constant was tuned
to a single value that minimized the mean RMSE for all data sets. The
resulting RMSE for this case is thus a measure of the parameterization's
ability to reproduce variations in the dust flux due to variations in both
u∗ and soil conditions (u∗st and fclay). For the second
case, the proportionality constant in each parameterization was tuned
separately for each data set. The resulting RMSE is thus a measure of a
parameterization's ability to reproduce the dust flux's dependence on
u∗ for each individual data set. Data set names are defined in Sect. 3.1.
Study
Event
GP88,
MB95,
Eq. (18),
GP88,
MB95,
Eq. (18),
case 1
case 1
case 1
case 2
case 2
case 2
GB04
16 February
0.400
0.182
0.739
0.203
0.181
0.182
GB04
20 March
0.247
0.214
0.215
0.112
0.108
0.106
ZP06
4 March
1.043
1.147
0.345
0.306
0.325
0.297
ZP06
18 March
0.390
0.566
0.137
0.088
0.111
0.085
FC07
Event 1
–
–
–
0.377
0.155
0.147
FC07
Event 2
–
–
–
0.389
0.192
0.132
SA09
ME1
0.299
0.541
0.410
0.054
0.072
0.058
SA09
CE4
0.387
0.571
0.555
0.104
0.114
0.111
SI11
NA
1.286
0.382
0.101
0.161
0.107
0.099
PP11
Event 1
–
–
–
0.609
0.347
0.295
PP11
Event 2
–
–
–
0.656
0.356
0.333
Average
0.579
0.515
0.357
0.278
0.188
0.168
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).
Comparison of measured dust fluxes with the predictions of
the parameterizations of (a) Gillette and Passi (1988), (b)
Marticorena and Bergametti (1995), and (c) this study. The
proportionality constant in each parameterization was adjusted to maximize
agreement with the compilation of measurements. To prevent cluttering of the
graph, only 15 representative measurements are shown for each data set.
Error bars denote uncertainty arising from the measurement of u∗t,
u∗, and Fd (see the Supplement). Data set names are defined in Sect. 3.1.
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).
Limitations of the dust emission theory and parameterization
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.
Discussion and conclusions
We have used a combination of theory and numerical simulations to derive a
physically based parameterization of the vertical dust flux emitted by an
eroding soil. Our new dust flux parameterization includes two main
improvements over previous schemes used in large-scale models. First, it
accounts for the predicted (Figs. 1, 2a) and observed (Fig. 4a) increasing
scaling of Fd with u∗ that occurs with increasing
threshold friction velocityt; this advance helps explain the numerous
observed scalings of Fd with u∗ (Shao, 2008; Kok et al.,
2012). Second, our parameterization accounts for a soil's increased ability
to produce dust under saltation bombardment as the soil becomes more erodible
(Figs. 1, 2b, 4b). This second improvement is especially important, as it
implies that previous parameterizations have underestimated the sensitivity
of the dust flux to the soil's dust emission threshold
(u∗st) (also see Fig. 1 in Kok et al., 2014). This
underestimation is not sensitive to the details of our parameterization
because it follows directly from the energetics of dust emission: increases
in soil cohesion both raise the dust emission threshold and cause dust
emission to require more energy, thereby reducing the dust flux for a given
saltator kinetic impact energy. Previous work by Shao and colleagues (Shao et
al., 1993, 1996; Shao, 2001) has noted that soils with stronger interparticle
forces should produce less dust per saltator impact, but this insight had not
been included in dust emission parameterizations commonly implemented in
large-scale models (e.g., Ginoux et al., 2001; Zender et al., 2003a; Cakmur
et al., 2006; Menut et al., 2013; Zhao et al., 2013).
Partially as a result of the inclusion of these two additional physical
processes, our parameterization is in better agreement with a
quality-controlled compilation of dust flux measurements than the previous
dust flux parameterizations of Gillette and Passi (1988) and Marticorena and
Bergametti (1995) (see Fig. 5). Although our parameterization thus appears to
account for more of the processes driving the dust flux than these previous
parameterizations, it is straightforward to implement as it uses only
variables that are readily available in weather and climate models (note that
the code to implement the parameterization in the Community Earth System
Model is freely available from the main author). This is made possible
because of several advances and simplifications over previous theories.
Arguably the main advance is that we use the soil's standardized threshold
friction velocity (u∗st) as a measure of soil erodibility
(i.e., the soil's ability to emit dust), allowing us to substantially
simplify the energetics of dust emission relative to previous
physically explicit schemes (Shao et al., 1996; Shao, 2001). Furthermore,
many previous parameterizations used a different threshold friction velocity
for each soil particle size bin. However, experiments, numerical modeling,
and theory all indicate that, once the saltation threshold is exceeded,
particles of a wide range of sizes are set into motion (e.g., Bagnold, 1938;
Rice et al., 1995; Namikas, 2006; Kok and Renno, 2009; Kok et al., 2012). We
therefore characterized the threshold friction velocity with a single value,
which can for instance be calculated using the models of Iversen and
White (1982), Fecan et al. (1999), or Shao and Lu (2000).
Our result that the dust flux is more sensitive to the soil's threshold
friction velocity than most current parameterizations account for emphasizes
the importance for models to accurately represent spatial and temporal
variations in soil erodibility. Our parameterization provides a convenient
way of doing so through the exponential dependence of the dust emission coefficient
Cd on the standardized dust emission
threshold u∗st. However, the parameterization of
u∗st in most models is relatively primitive (e.g., Zender
et al., 2003a). For instance, one of the main determinants of
u∗st is the moisture content of the top layer of soil
particles. Yet, the most commonly used parameterization of the effect of soil
moisture on u∗st (Fecan et al., 1999) is found to produce
unrealistic results in some models, requiring the use of a tuning constant
(Zender et al., 2003a; Mokhtari et al., 2012). Furthermore, effects of soil
aggregation and crust formation on u∗t are not included in
the most widely used global dust modules (Ginoux et al., 2001; Zender et al.,
2003a; Huneeus et al., 2011). Considering the paramount importance of
u∗st in determining dust fluxes (see Eq. 18), an effective
way to improve the fidelity of dust cycle simulations would be to develop
improved parameterizations of u∗st as a function of soil
properties, precipitation events, atmospheric relative humidity, and other
relevant parameters. Alternatively, for simulations of the current dust
cycle, u∗st could be remotely sensed (Chomette et al., 1999;
Chappell et al., 2005; Draxler et al., 2010). Doing so requires the
simultaneous determination of the threshold wind speed and the surface
roughness (Marticorena et al., 2004), such that the remotely sensed threshold
wind stress can be partitioned between the portion causing dust emission
(τs) and that absorbed by nonerodible elements (τR) (Raupach et al., 1993; Marticorena and Bergametti, 1995).
Current large-scale models commonly use semiempirical dust source functions
(e.g., Ginoux et al., 2001; Tegen et al., 2002; Zender et al., 2003b) to
help parameterize spatial variability in soil erodibility and the consequent dust emissions. The use of these source functions usually
shift emissions towards the most erodible regions. Because our
parameterization accounts for a soil's increased ability to produce dust
under saltation bombardment as the soil becomes more erodible, its
implementation in models would also result in a shift of emissions to the
most erodible regions. We therefore hypothesize that our parameterization
reduces the need for empirical source functions in dust modules. We test this
hypothesis in our companion paper (Kok et al., 2014).