F. Comola1 , J. F. Kok1 , M. Chamecki1 , and R. L. Martin1
1Department of Atmospheric and Oceanic Sciences, University of California, Los Angeles, Los Angeles, CA, USA
Abstract Wind-blown sand is the main driver of dune development and dust emission from soils andis thus of fundamental importance for geomorphology, ecology, climate, and air quality. Even thoughsand transport is driven by nonstationary turbulent winds, and is thus inherently intermittent, currentparameterizations in atmospheric models assume stationary wind and continuous transport. We draw onextensive field measurements to show that neglecting saltation intermittency causes biases in the timingand intensity of predicted fluxes. We present a simple parameterization that accounts for saltationintermittency and produces substantially improved agreement against measurements. We investigate theimplications of accounting for transport intermittency in atmospheric models by analyzing 35 years ofhourly wind speed data from climate simulations. We show that accounting for intermittency leads tosignificantly different predictions of sand mass fluxes throughout the year, with potential implications fortiming and intensity of dust emission.
Plain Language Summary The wind-driven transport of sand, known as saltation, drivesatmospheric emission of mineral dust and thus plays a critical role in climate and air quality. Existing sandtransport parameterizations in atmospheric models assume that saltation is a continuous process drivenby steady winds. In reality, however, sand transport is highly intermittent and is driven by fluctuatingturbulent winds. Here, we analyze a comprehensive data set of sand transport to show that neglectingintermittency may lead to significant errors in the estimated time and intensity of sand transport and dustemission. We thus propose a novel formulation of intermittent sand transport that provides much betteragreement with measured data and could improve the quantification of dust emission in atmosphericmodels.
1. IntroductionWind-driven sand transport, known as aeolian saltation (Bagnold, 1941; Shao, 2008), affects the Earth sys-tem in numerous ways, including by causing agricultural wind erosion (Zobeck & Van Pelt, 2006) andshaping arid and coastal landscapes (Nickling & McKenna Neuman, 2009). Moreover, saltation is the prin-cipal mechanism for the emission of dust minerals from arid surfaces (Shao et al., 1993), which is offundamental relevance for air quality, ecology, and climate (Chin et al., 2007; Kok et al., 2017; Okin et al.,2004). Saltation is driven by near-surface turbulent winds, characterized by strong spatial and temporal fluc-tuations in speed and direction. Consequently, saltation is highly intermittent, with pronounced variabilityon time scales of seconds to hours (Dupont et al., 2013). In contrast, existing sand transport parameteriza-tions describe saltation as a process that is uniform in time and space and driven by a constant downwardflux of flow momentum. This clear disconnect between theory and reality is likely one of the main causesof the poor performance of parameterizations of sand transport and dust emission (Barchyn et al., 2014;Todd et al., 2008).
Turbulent winds initiate saltation when the downward flux of flow momentum, represented by the windfriction velocity u*, exceeds the so-called fluid threshold u*,ft, which accounts for the resistance to particlelifting due to gravity and interparticle forces (Shao & Lu, 2000). Airborne grains accelerate under the effectof wind drag and hop along the surface, rebounding and splashing other grains upon impact with the bed.Once initiated, saltation mainly sustains itself through particle rebound (Pähtz & Durán, 2018) and granularsplash (Comola & Lehning, 2017; Kok & Renno, 2009), which allow transport to occur at friction veloci-ties below the fluid threshold. The minimum friction velocity at which saltation can be sustained throughrebound and granular splash is known as the impact threshold u*,it (Bagnold, 1941). It follows that, for fric-tion velocities larger than the fluid threshold, the turbulent flow provides sufficient momentum flux to keep
RESEARCH LETTER10.1029/2019GL085739
Key Points:• Accounting for transport
intermittency in parameterizations ofaeolian saltation improves mass fluxpredictions
• We propose a novel parameterizationof intermittent saltation forimplementation in atmosphericmodels
• Saltation intermittency affectstiming and intensity of sandtransport at annual scale
Supporting Information:• Supporting Information S1
Citation:Comola, F., Kok, J. F., Chamecki, M.,& Martin, R. L. (2019). Theintermittency of wind-driven sandtransport. Geophysical ResearchLetters, 46, 13,430–13,440. https://doi.org/10.1029/2019GL085739
Received 8 OCT 2019Accepted 29 OCT 2019Accepted article online 14 NOV 2019Published online 16 NOV 2019
saltation active. Conversely, when the friction velocity lies between the impact and the fluid threshold, salta-tion is active only if transport was initiated (u* > u*,ft) more recently than it was terminated (u* < u*,it)(Martin & Kok, 2018), a process known as hysteresis.
Saltation parameterizations implemented in atmospheric models commonly assume that sand mass fluxesQ are proportional to the excess of friction velocity with respect to either u*,ft or u*,it. A sand mass fluxparameterization widely used in atmospheric models (Marticorena & Bergametti, 1995; Zender et al., 2003)is that proposed by Kawamura (1951) and White (1979), which posits that Q scales with the cube of thefriction velocity as
Q = Aft𝜌a
gu3∗
(1 −
u2∗,ft
u2∗
)(1 +
u∗,ft
u∗
). (1)
In equation (1), Aft is a dimensionless proportionality coefficient, g is the gravitational constant, and 𝜌a isthe air density. The main drawback of equation (1) is that it predicts zero flux when the friction velocitylies below the fluid threshold and thus neglects the contribution of saltation in the hysteresis regime. Infact, compelling physical considerations and comprehensive field-based analyses (e.g., Durán et al., 2011;Ho et al., 2011; Martin & Kok, 2017; Sørensen, 2004; Ungar & Haff, 1987) indicate that the turbulent flowsustains saltation when the friction velocity exceeds the impact threshold, with Q scaling with the square ofthe friction velocity as
Q = Aitu∗,it
g𝜌a
(u2∗ − u2
∗,it
). (2)
Although equation (2) represents a conceptual improvement over equation (1), it assumes that saltation iscontinuous above the impact threshold and may thus overestimate intermittent mass fluxes.
Erroneous estimations of intermittent sand transport may significantly affect dust emission simulations inatmospheric models, which commonly assume linearity between mean vertical dust fluxes and mean hori-zontal sand fluxes (Marticorena & Bergametti, 1995). Even though modeling approaches to incorporate theeffect of wind speed variability on dust emission have been proposed (Cakmur et al., 2004; Engelstaedter& Washington, 2007), accounting for transport intermittency in sand mass flux parameterizations remainsa challenge. The major reason for the neglect of transport intermittency by existing parameterizations isthat the time scales at which saltation turns on and off in response to fluctuating turbulent winds, typi-cally of few seconds (Stout & Zobeck, 1997), are much shorter than the 10–30 min time intervals requiredto account for the entire turbulence spectrum in friction velocity calculations (Namikas et al., 2003; VanBoxel et al., 2004). Here, we move beyond the paradigm of continuous saltation and propose a viable wayto account for saltation intermittency in a mass flux parameterization. We test the proposed parameteriza-tion against a comprehensive field-based data set of sand saltation in three different locations in Californiaand Brazil. Finally, we investigate the implications of accounting for saltation intermittency in predictionsof sand transport at annual scale.
2. Sand Mass Flux Parameterization That Accounts for IntermittencyWe propose a modification to equation (2) to account for saltation intermittency. Let us assume that withinthe time interval over which the friction velocity is calculated (typically 10–30 min), saltation is active for afraction of time 𝜂q ∈ [0, 1] and inactive for a fraction of time 1 − 𝜂q. When saltation is inactive, the instan-taneous mass flux is equal to zero. Conversely, when saltation is active, the instantaneous mass flux is wellpredicted by equation (2). It follows that the mass flux Q averaged over the entire time interval reads
Q = A𝜂
u∗,it
g𝜌a
(u2∗ − u2
∗,it
)𝜂q. (3)
The variable 𝜂q, which we refer to as the intermittency factor, allows us to represent both continuous salta-tion (𝜂q = 1) and intermittent saltation (𝜂q < 1). Equation (3), in fact, coincides with equation (2) for𝜂q = 1.
We test the performance of equations (1)–(3) using a comprehensive saltation data set (Martin et al., 2019),which provided experimental support for recent investigations on sand transport (Martin & Kok, 2017;Martin & Kok, 2018; Martin et al., 2018; Martin & Kok, 2019). The data were collected during field
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Figure 1. Predicted versus measured mass fluxes at the Oceano, Jericoacoara, and Rancho Guadalupe field sites. (a)Mass fluxes are predicted with parameterizations for continuous saltation based on either the fluid threshold(equation (1); red and orange symbols) or the impact threshold (equation (2); blue and cyan symbols). (b, c) Massfluxes are predicted with the new parameterization that accounts for intermittent saltation (equation (3)), with 𝜂qeither measured directly from our field data (panel b) or estimated via equation (6) (panel c). Binning is performed on10 min averaged data, markers indicate geometric means, vertical bars indicate standard errors, and the dashed linesindicate equality between measurements and predictions. In panel (a), the axes are linear from 0 to 10−2 andlogarithmic from 10−2 to 102, to allow the representation of zero-valued mass fluxes. The shaded gray area ofcontinuous saltation includes all data points with measured 𝜂q = 1.
campaigns at three different locations, namely Oceano, California, Jericoacoara, Brazil, and RanchoGuadalupe, California (see Figure S1 in the supporting information for their geographic positions). TheJericoacoara site is located on a gently undulating sand sheet 750 m from the Atlantic Coast and 250 mdownwind of a patch of gravel and vegetation under dominantly easterly winds. The Rancho Guadalupesite is located on a flat sand patch 950 m from the Pacific Coast, whereas the Oceano site lies 650 m fromthe Pacific Coast on a gently sloped surface in landward direction. The Rancho Guadalupe and Oceanosites are separated by about 10 km and are shaped by dominantly westerly winds. The median surface graindiameters are 0.398, 0.526, and 0.533 mm at Oceano, Jericoacoara, and Rancho Guadalupe, respectively; dif-ferences in grain size characteristics and their effects on transport occurrence are further described in Martinet al. (2018) and Martin and Kok (2019). The total duration of the measurements during active saltation isapproximately 60 hr in Oceano and 10 hr each in Jericoacoara and Rancho Guadalupe.
In order to evaluate equations (1)–(3), we need to estimate the values of u*, 𝜂q, u*,it, and u*,ft. We calculatedthe wind friction velocity using the eddy-covariance technique, that is, u2
∗ = −u′w′, where u′ and w′ are thehorizontal and vertical turbulent fluctuations around the mean wind speed over time intervals of 10 min(Namikas et al., 2003). The wind speed time series were obtained from high-frequency (25–50 Hz) sonicanemometers at elevation zw ≈ 0.64, 0.48, and 0.45 m at Oceano, Jericoacoara, and Rancho Guadalupe,respectively. Further, we calculated the intermittency factor 𝜂q from the high-frequency (25 Hz) time seriesof vertically integrated mass flux Q(t), calculated by Martin and Kok (2018) using arrays of high-frequencyoptical particle counters and low-frequency sand traps. Specifically, 𝜂q =
∑Ni=1 𝛿q(ti)∕N, where N is the total
number of flux values within the averaging time interval and 𝛿q(ti) is either 0 or 1 depending on whetherQ(ti) = 0 or Q(ti) > 0. Although Martin and Kok (2018) indicated that single optical particle counters maybe affected by potential underdetection of particles (false negatives), we argue that the problem is mitigatedin our analysis because 𝜂q is based on the combined detection of a large number of particle counters (3–9,depending on the field site). Finally, we estimated the impact and fluid thresholds u*,it and u*,ft with the dualthreshold methodology proposed by Martin and Kok (2018). This method, described in section S1 of the sup-porting information, yields u*,it = 0.26, 0.32, and 0.33 m/s and u*,ft = 0.36, 0.42, and 0.41 m/s for Oceano,Jericoacoara, and Rancho Guadalupe, respectively. We show in the supporting information (Figure S2)that the values of u*,it and u*,ft obtained from our analysis are consistent with those estimated by Martin andKok (2018) at the same field sites and with those calculated from literature formulas (Bagnold, 1941; Greeley& Iversen, 1985; Shao & Lu, 2000).
We evaluate equations (1)–(3) using the estimated values of u*, 𝜂q, u*,it, u*,ft, and the proportionality coef-ficients Aft, Ait, and A𝜂 fitted to the data at each location with the least-squares-error method. The results
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suggest that equations (1) and (2) are accurate for continuous saltation (gray area in Figure 1a). How-ever, equation (1) is not capable of reproducing the measured mass fluxes below the fluid threshold andequation (2) clearly overestimates the saltation mass fluxes in periods of intermittent saltation. In contrast,equation (3) provides accurate mass flux estimations across several order of magnitudes (Figure 1b), sug-gesting that the intermittency factor accounts for the saltation physics that previous parameterizations failto represent. The results also indicate that all parameterizations leave a significant amount of statistical vari-ability unexplained, especially for small values of the mass flux (vertical bars in Figure 1; see the supportinginformation Figure S3 for the scatter plots of the raw data).
The results thus indicate that equation (3) improves estimations of sand mass flux for conditions of inter-mittent saltation. To implement equation (3) in atmospheric models, we propose in the next section amethodology to estimate the intermittency factor 𝜂q based exclusively on turbulence characteristics.
3. Prediction of Saltation Intermittency From Turbulence CharacteristicsTo estimate the intermittency factor 𝜂q at the time scales of interest, we draw on the cumulative probabilitydensity function of the wind speed to infer the fractions of time that saltation is continuous, inactive, or inthe hysteresis regime (i.e., u*,it < u* < u*,ft). For this purpose, we first identify the turbulent scales in the windspeed signal that are relevant to the saltation dynamics. Second, we define a representative statistical modelfor the wind speed based on essential information, such as the mean wind speed and wind speed variance.Finally, we adopt this statistical model to assess the fraction of time of active saltation and compare thepredictions of this model against our field measurements. In what follows, we assume negligible buoyancyeffects in the near-surface atmosphere, as the wind shear required to trigger saltation generally dominatesthe production of turbulence kinetic energy (Martin & Kok, 2018).
In atmospheric models, the wind speed time series is available at elevation zw equal to the height of the firstgrid node (∼10 m), which is likely to be much larger than the saltation layer height zs ≈ 0.1 m (Martin &Kok, 2017). To illustrate how the wind speed at height zw can be predictive of the wind speed that drivessaltation near the surface, we rely on a well-known heuristic description of turbulent flows (Richardson,1922). Specifically, we can think of atmospheric flows as given by the superposition of turbulence structureswith a wide range of length and time scales. Classical turbulence scaling laws suggest that the character-istic time scale 𝜏 l of a turbulent eddy scales with its size l as 𝜏 l ∼ (l2∕𝜖)1/3, where 𝜖 is the dissipation rateof turbulence kinetic energy (Kolmogorov, 1941). It follows that the smaller structures, which generate thehigh-frequency components of the wind speed signal, present weak correlation in time and space, and theiroccurrence is not felt at large distances or after long time intervals. Conversely, the larger structures, whichcontain most of the turbulence kinetic energy and are responsible for the low-frequency components of thewind speed signal, are long lasting in time and far reaching in space. The low-frequency wind speed, drivenby large and very-large scale motions that impinge from above, is known to be height invariant (Marusicet al., 2010), as also confirmed by our wind speed measurements at different heights at Oceano (see support-ing information Figure S4). It follows that the low-frequency turbulent fluctuations generated by large andvery large turbulence structures at zw are highly correlated with the low-frequency fluctuations at the salta-tion layer height. In contrast, the high-frequency fluctuations at zw are likely uncorrelated to those at thesaltation layer height. We thus argue that only the low pass-filtered wind speed uw(t) at height zw modulatesthe mass flux time series at time scales that are relevant for calculation of the intermittency factor 𝜂q.
In order to apply a low pass filter to the wind speed time series, we need to identify a relevant cutoff fre-quency 𝛺. Let us assume that, of all the turbulent eddies at height zw, only the large-scale and very largescale motions with characteristic length l > zw are capable of impinging on the saltation layer. To estimatethe characteristic time scale 𝜏l ∼ (z2
w∕𝜖)1∕3 of such eddies we rely on the classic parameterization of the dis-
sipation rate in the neutral atmospheric boundary layer 𝜖 ≈ u3∗∕kzw (Stull, 2012), where k ≈ 0.387 is the
experimental value of the von Kármán constant for high Reynolds number flows (Andreas et al., 2006). Itfollows that, for typical values zw ≈ 10 m and u* ≈ 0.3 m/s, the characteristic eddy time scale is 𝜏 l ≈ 25 s.We thus take the cutoff frequency of the low pass filter 𝛺 = 1∕25 = 0.04 Hz. Our sensitivity analyses, shownin the supporting information (Figure S5), indicate that the results are not sensitive to the exact value of thelow pass filter frequency. The low pass-filtered wind speed signal at the saltation layer height can be writtenas us(t) = ⟨us⟩ + u′
s(t), that is, as the sum of the mean speed ⟨us⟩ and the turbulent fluctuations around the
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mean u′s(t) due to large-scale atmospheric motions. Since the low-frequency wind speed is height-invariant
(Marusic et al., 2010), it follows that the large-scale turbulent fluctuations at height zs are equal to those atheight zw, that is, u′
s(t) = u′w(t). Furthermore, the mean wind speed over fully rough surfaces follows the
well-known logarithmic profile described by the law of the wall
⟨u⟩ = u∗
klog
(zz0
), (4)
where z0 is the value of aerodynamic roughness at each field site, which was estimated by Martin and Kok(2017) (z0 = 0.993×10−4, 0.707×10−4, and 1.420×10−4 m, at Oceano, Jericoacoara, and Rancho Guadalupe,respectively).
The application of equation (4) to aeolian transport is subject to major restrictions. These include unidi-rectional winds in a neutrally stable atmosphere (Salesky et al., 2012), sufficiently long averaging intervalsto capture the full spectrum of turbulent motions in the atmospheric boundary layer (Van Boxel et al.,2004), and the accounting for apparent variations of the aerodynamic roughness with saltation intensityin the parameter z0 (Sherman, 1992). Martin and Kok (2018) addressed these major assumptions underly-ing equation (4) to draw support for use of the law of the wall during intermittent saltation. Furthermore,the largest uncertainty in the value of z0 occurs in periods of intense saltation, for which the proposedintermittency correction is least important. Equation (4) implies that, for a constant friction velocity,
⟨us⟩ = log(
zs∕z0)
log(
zw∕z0) ⟨uw⟩. (5)
After obtaining the time series of the low-frequency component of the wind speed at the saltation layer, wenow seek to derive the saltation intermittency factor 𝜂q from the cumulative distribution function of the lowpass-filtered wind velocity P
(us). Let uit and uft be the wind speeds at height zs associated with the impact
and fluid thresholds through equation (4). Further, P(
us < uit)= Pit is the probability that the low-pass
filtered wind speed does not exceed uit. Similarly, Pft is the probability that the low pass-filtered wind speeddoes not exceed u*,ft. Within averaging time intervals 𝛥t = 10 min, saltation is continuous for the fractionof time (1 − Pft) that the wind speed exceeds the fluid threshold. Conversely, saltation is in the hysteresisregime for the fraction of time
(Pft − Pit
)that the wind speed is intermediate between the fluid and impact
thresholds. We thus estimate the intermittency factor as
𝜂q = 1 − Pft + 𝛼(
Pft − Pit), (6)
where 𝛼 ∈ [0, 1] is the probability of saltation occurrence when the wind speed is in the hysteresis regime.We expect 𝛼 to approach 1 when the hysteresis is mostly initiated by downward crossings of the fluid thresh-old, that is, when the wind speed goes from larger values to values smaller than uft. Conversely, we expect 𝛼to approach 0 when the hysteresis is mostly initiated by upward crossings of the impact threshold (Martin& Kok, 2018), that is, when the wind speed goes from smaller values to values larger than uit. Accordingly,we estimate 𝛼 as the fluid threshold crossing rate over the total crossing rate, that is, 𝛼 ≈ Cft∕(Cft + Cit). Itis reasonable to assume that the low pass-filtered wind speed at height zs at short time scales and neutralatmospheric conditions is well represented by a Gaussian process of mean ⟨us⟩, variance 𝜎2
u, and autocorre-lation function 𝜌u(t) (Chu et al., 1996). We can thus estimate the crossing rate of a generic wind speed valueu as (Barnett & Kedem, 1991; Kratz, 2006)
Cu = 1𝜋Δt
cos−1 (𝜌u (Δt))
exp
(−(
u − ⟨us⟩)2
2𝜎2u
), (7)
where 𝛥t is the wind speed time step and 𝜌u (Δt) is the normalized autocorrelation at lag-time equal to thetemporal resolution of the wind speed signal (25–50 Hz in our case; (Martin et al., 2018)). Using equation (7),we can express 𝛼 as
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Figure 2. Comparisons between predicted and measured variables in the parameterization of saltation intermittency.Shown are the predicted versus measured crossing rates of the impact and fluid threshold (panel a), the fraction ofactive saltation in the hysteresis regime (panel b), and the intermittency factors (panel c) at Oceano, Jericoacoara, andRancho Guadalupe. Binning is performed on 10 min-averaged data. Markers indicate arithmetic means in panels (a)and (b) and geometric means in panel (c). Vertical bars indicate standard errors and the dashed lines indicate equalitybetween measurements and predictions.
𝛼 ≈
[exp
(u2
ft − u2it − 2⟨us⟩ (uft − uit
)2𝜎2
u
)+ 1
]−1
. (8)
We verified that equation (7) accurately reproduces the crossing rates (Figure 2a) and that equation (8) wellapproximates the fraction of active saltation measured at all three field sites (Figure 2b).
In order to use equation (6) to predict saltation intermittency factors, we further need to determine Pft andPit. For this purpose, we rely once more on the assumption of Gaussianity for the probability distribution ofthe low pass-filtered wind speed at height zs (Chu et al., 1996), that is, us ∼ (⟨us⟩, 𝜎2
u), such that
Pft =12
[1 + erf
(uft − ⟨us⟩√
2𝜎u
)], (9)
Pit =12
[1 + erf
(uit − ⟨us⟩√
2𝜎u
)]. (10)
The implementation of equations (6)–(10) in atmospheric models may require a model for the lowpass-filtered wind speed standard deviation 𝜎u. We propose to estimate 𝜎u using available literature formu-las (e.g., Banerjee et al., 2015; Panofsky et al., 1977; Townsend, 1980) and scaling laws of classic turbulencetheory. Specifically, we adopt the formula by Panofsky et al. (1977) to predict the standard deviation 𝜎u ofthe total (unfiltered) wind speed time series and rely on turbulence theory to derive a downscaling factorf(𝛺) such that 𝜎u = 𝑓 (Ω)𝜎u. Our proposed parameterization for 𝜎u then reads
𝜎u = 𝑓 (Ω)u∗
(12 + 0.5
zi
−L
)1∕3, (11)
where zi is the boundary layer height. Further, the Monin-Obukhov length L expresses the height in theatmosphere where shear-generated and buoyancy-generated turbulence are equal, and is thus a proxy for thestrength of buoyancy-generated turbulent kinetic energy (see the supporting information for its expression).Equation (11) coincides with the parameterization by Panofsky et al. (1977) for f(𝛺) = 1, that is, when thefull variance in the wind speed is accounted for. In order to derive a theoretical value of the downscalingfactor f(𝛺), we express the variance of the low pass-filtered wind speed as the integral of the wind energyspectrum Suu(𝜔) up to frequency 𝛺, that is, using a spectral cutoff filter (Pope, 2001),
𝜎2u ∝ ∫
Ω
𝜔i
Suu(𝜔)d𝜔. (12)
In equation (12), 𝜔i ≈ 1∕Ti is the frequency of the large turbulent structures, for which the characteristictime scale Ti is known as the integral time scale (Dosio et al., 2005). Let us call 𝜔𝜂 ≫ 𝜔i the frequency of the
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smallest turbulence structures. The range of frequencies 𝜔i ≪ 𝜔 ≪ 𝜔𝜂 , commonly referred to as the inertialsubrange, presents the well known scaling of the turbulent spectrum Suu(𝜔) ∼ 𝜔−5/3 (Kolmogorov, 1941).For the purpose of this analysis, we assume that this scaling holds in the entire frequency range between 𝜔iand 𝜔𝜂 . It follows that 𝜎2
u ∝ 𝜔−2∕3i − Ω−2∕3, which, for 𝛺 = 𝜔𝜂 , gives the variance 𝜎2
u of the unfiltered windspeed. Assuming that the turbulence spectrum presents a very large scale separation, that is, 𝜔i∕𝜔𝜂 ≈ 0, wereadily obtain a simple expression for the factor f(𝛺)
𝑓 (Ω) =√
1 −(𝜔i
Ω
)2∕3. (13)
We show in the supporting information (Figure S6) that equation (13) allows us to correctly downscale thestandard deviation of the unfiltered wind speed at Oceano (typically by a factor f(𝛺) = 0.6 − 0.8), leading toreliable predictions of the low pass-filtered wind speed standard deviation 𝜎u with equation (11).
We used equation (6) to predict the intermittency factors 𝜂q at our field sites. We find that the proposedmethod to estimate the intermittency factor 𝜂q well approximates the values measured in the field sites acrossseveral orders of magnitude (Figure 2c). We note that, because the proposed method assumes normallydistributed wind speed values, predicted 𝜂q values (unlike field-observed 𝜂q) only asymptotically approach0 and 1 for inactive and continuous saltation, respectively. The large uncertainty in the estimation of 𝜂qin highly intermittent conditions (low sand fluxes) is partially due to experimental noise and occurrenceof false negatives (Martin & Kok, 2018), which becomes increasingly important for smaller values of theintermittency factor. We also show in the supporting information (Figure S5) that the estimation providedby equation (6) is robust with respect to the choice of the saltation height zs, the sonic height zw, the cutofffrequency 𝛺, and the averaging time step 𝛥t used to compute the friction velocity.
We further assess the accuracy of the predicted intermittency factor 𝜂q for use in the proposed mass fluxparameterization. For this purpose, we compare again the mass fluxes measured in the field to those pre-dicted with equation (3), but this time using the intermittency factors estimated with equation (6). The newmass flux parameterization seems to provide accurate predictions of the intermittent mass fluxes (Figure 1c),with an overall bias reduction compared to existing parameterizations (Figure 1a).
4. Relevance of Sand Transport Intermittency at Annual ScaleOur analyses thus far indicate that parameterizations for continuous saltation may produce large errorswhen sand fluxes are intermittent, and that an explicit accounting for intermittency in the saltation fluxprediction can help to reduce this error. The field data set we used to test this intermittency treatment,however, covers a limited number of days and is thus not representative of sand transport frequency andintensity throughout the year. To evaluate the implications of our findings on predictions of sand fluxes onlonger time scales, we analyzed 35 years of wind speed data at Oceano, Jericoacoara, and Rancho Guadalupe,extracted from the Meteoblue History+ archived numerical weather model predictions (Meteoblue, 2019).These numerical simulations provided us with hourly wind speed at 10 m elevation and spatial resolutionfiner than 30 km. To better represent the atmospheric conditions at each field site, we applied a scalingfactor to the simulated wind speed to match the average wind speed measured during the measurementcampaigns. We then calculated the friction velocity from the wind speed data through the logarithmic law ofthe wall (equation (4)) and the time series of intermittency factors with equation (6). This procedure allowedus to predict 35 years of mass flux time series at the three field locations with equations (1)–(3). In seeking arepresentative reproduction of long-term mass flux time series in atmospheric models, we assumed constantgeomorphic conditions, that is, constant surface particle size distributions, thresholds, topographic settings,and surface roughness.
We drew on this long sand mass flux data set to investigate the differences in the annual frequency and inten-sity of saltation predicted with equations (1)–(3). The probability of the intermittency factors P
(𝜂q < ��q
),
that is, the monthly fraction of 𝜂q smaller than a specific level ��q (Figures 3a–3c), indicates that, for most ofthe year, saltation occurs in highly intermittent conditions. Moreover, the temporal variability of this prob-ability is correlated with that of the monthly mean wind speed ⟨uw⟩ at 10 m (black dots in Figures 3a–3c).Specifically, saltation is highly intermittent in months of low wind speed, with 50% of the calculatedintermittency factors smaller than 0.25 at Oceano and Rancho Guadalupe and 80% smaller than 0.25 atJericoacoara.
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Figure 3. Effect of saltation intermittency on the timing and magnitude of predicted sand fluxes. (a–c) Histogramsrepresent the probability P
and 0.75 (scale on the left axis). Black dots represent monthly mean wind speed ⟨uw⟩ at 10 m elevation (scale on theright axis). (d–f) Histograms (scale on the left axis) represent monthly mass fluxes estimated with theparameterizations based on continuous saltation (equations (1) and (2)) and the proposed parameterization thataccounts for saltation intermittency (equation (3)). Black dots (scale on the right axis) represent monthly mean windspeed ⟨uw⟩ at 10 m elevation. The results are obtained from analyses of 35 years of wind speed data at Oceano,Jericoacoara, and Rancho Guadalupe.
We further investigate the differences in the monthly mass fluxes estimated with parameterizations forcontinuous saltation (equations (1) and (2)) and our new parameterization that accounts for saltationintermittency (equation 3). The results highlight significant differences, with the impact threshold-basedparameterization predicting the largest mass flux values, the fluid threshold-based parameterization pre-dicting the smallest values, and the intermittency factor-based parameterization predicting intermediatevalues (Figures 3d–3f). Moreover, periods of low wind speed (black dots in Figure 3) are characterized bythe largest differences among mass flux predictions, suggesting that intermittency can significantly affectpredictions of sand transport outside of the windy season.
We also find substantial differences in the probability distributions of mass fluxes calculated with the con-tinuous and intermittent saltation parameterizations (Figure 4). Specifically, we find that the assumptionof continuous saltation narrows down the range of estimated mass fluxes toward large mass flux values(blue and orange histograms) and neglects the large range of “background” sand transport in intermittentconditions (green histogram).
5. Discussion and ConclusionsWe have shown that existing sand mass flux parameterizations produce biased results when saltation isintermittent. Specifically, parameterizations based on the fluid threshold u*,ft, such as equation (1), predictnull fluxes when the wind friction velocity lies between the impact and the fluid thresholds and thus under-estimate frequency and intensity of sand transport in the hysteresis regime. Conversely, parameterizationsbased on the impact threshold u*,it, for example, equation (2), assume that sand transport is continuous evenin the hysteresis regime and thus overestimate the resulting mass fluxes.
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Figure 4. Annual probability distribution of sand mass fluxes, estimated with the parameterizations based oncontinuous saltation (equations (1) and (2)) and the proposed parameterization that accounts for saltationintermittency (equation (3)). The distributions do not account for zero mass flux values, which occur when u* < u*,ft inequation (1) and u* < u*,it in equations (2) and (3). The results are obtained from analyses of 35 years of wind speeddata at (a) Oceano, California, (b) Jericoacoara, Brazil, and (c) Rancho Guadalupe, California.
We have proposed a simple mass flux parameterization that accounts for intermittency and produces muchbetter agreement with measured mass fluxes across several orders of magnitude (Figure 1). The proposedformulation is based on the excess friction velocity with respect to the impact threshold and includes anintermittency factor 𝜂q ∈ [0, 1] to quantify the degree of saltation activity. We have shown that 𝜂q can beaccurately estimated based on variables and parameters that are commonly available in atmospheric models(Figure 2). Specifically, the inputs required are the mean wind speed ⟨us⟩ at a saltation-relevant height, thefluid and impact thresholds u*,ft, u*,it, the aerodynamic roughness length z0, the boundary layer height zi,and the Monin-Obukhov length L.
To assess the implications of the inability of existing sand mass flux parameterizations to account for salta-tion intermittency, we analyzed 35 years of wind speed data colocated with our field sites. Our resultsindicate that, throughout the year, saltation is very often intermittent (Figures 3a–3c). Consequently, param-eterizations based on the fluid threshold and impact threshold respectively underestimate and overestimatethe monthly sand fluxes (Figures 3d–3f). Even though there is little difference between the three parame-terizations for saltation mass flux in periods of mostly continuous transport, the predicted monthly massfluxes can be off by nearly an order of magnitude during periods of low wind speed and mostly intermittenttransport. Moreover, over the 35 years of observation, the probability distributions of mass fluxes predictedby continuous saltation parameterizations are narrower and clustered around larger values (Figure 4).
Our results demonstrate the value of implementing intermittent saltation parameterizations for sandy soilswith a variety of surface particle diameters. The application of the proposed parameterization to nonsandysoils has not been tested and may thus be subject to limitations. For nonsandy soils that are cohesive becauseof soil moisture, a large soil fraction of clay particles, or the presence of soil crusts, the separation betweenfluid and impact thresholds might be larger (Comola et al., 2019), potentially increasing the occurrence ofintermittent sand transport.
Because sand transport is the major driver of dust emission from arid soils, atmospheric models imple-menting continuous saltation parameterizations may not be capable of predicting the “background” dustemission that occurs during seasons with lower wind speeds. The errors that result from adopting param-eterizations for continuous saltation in atmospheric models may thus cause miscalculations in a numberof processes that depend on accurate predictions of airborne dust. These include the absorption and scat-tering of solar radiation by suspended dust, the supply of dust-borne nutrients to a variety of ecosystems,and the atmospheric levels of PM10 and PM2.5 in urban areas outside of the windy season. More accuratepredictions of intermittent sand transport could also improve applications of landscape evolution modelsto dune fields on other planetary bodies with thin atmospheres, such as Mars, where the effect of transporthysteresis is even more pronounced (Kok, 2010). The proposed parameterization that accounts for saltationintermittency may thus provide deeper insights into geomorphological processes on Earth and other plan-ets, as well as more accurate quantifications of the global dust cycle and its implications for climate, ecology,and air quality.
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AcknowledgmentsWe thank the two anonymousreviewers for their constructivecomments. The authors acknowledgethe support of the Swiss NationalScience Foundation (GrantP2ELP2_178219), the United StatesNational Science Foundation (GrantsAGS-1358621 and AGS-1358593), theCold Regions Research andEngineering Laboratory (Contract No.W913E520C0001), and the U.S. ArmyResearch Laboratory (GrantW911NF-15-1-0417). The views andconclusions contained in thisdocument are those of the authorsand should not be interpreted asrepresenting the official policies, eitherexpressed or implied, of the ArmyResearch Laboratory or the U.S.Government. The data presentedin this paper are available at thissite (https://doi.org/10.17632/kbh94r44zd.2).
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