Vertical variation of particle speed and flux density in aeolian saltation: Measurement and modeling

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1 JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 113,, doi: /2007jf000774, 2008 Vertical variation of particle speed and flux density in aeolian saltation: Measurement and modeling Keld R. Rasmussen 1 and Michael Sørensen 2 Received 13 February 2007; revised 26 July 2007; accepted 10 October 2007; published 14 March [1] Particle dynamics in aeolian saltation has been studied in a boundary layer wind tunnel above beds composed of quartz grains having diameters of either 242 mm or 320 mm. The cross section of the tunnel is 600 mm 900 mm, and its thick boundary layer allows precise estimation of the fluid friction speed. Saltation is modeled using a numerical saltation model, and predicted grain speeds agree fairly well with experimental results obtained from laser-doppler anemometry. The use of laser-doppler anemometry to study aeolian saltation is thoroughly discussed and some pitfalls are identified. At 80 mm height the ratio between air speed and grain speed is about 1.1 and from there it increases toward the bed so that at 5 mm it is about 2.0. All grain speed profiles converge toward a common value of about 1 m/s at 2 3 mm height. Moreover, the estimated launch velocity distributions depend only very weakly on the friction speed in contrast to what has often been assumed in the literature. Flux density profiles measured with a laser-doppler appear to be similar to most other density profiles measured with vertical array compartment traps; that is, two exponential segments will fit data between heights from a few millimeters to mm. The experimental flux density profiles are found to agree well with model predictions. Generally, validation rates are low from 30 to 50% except at the highest level of 80 mm, where they approach 80%. When flux density profiles based on the validated data are used to estimate the total mass transport rate results are in fair agreement with measured transport rates except for conditions near threshold where as much as 50% difference is observed. Citation: Rasmussen, K. R., and M. Sørensen (2008), Vertical variation of particle speed and flux density in aeolian saltation: Measurement and modeling, J. Geophys. Res., 113,, doi: /2007jf Introduction 1 Department of Earth Sciences, University of Aarhus, Aarhus, Denmark. 2 Department of Mathematical Sciences, University of Copenhagen, Copenhagen, Denmark. Copyright 2008 by the American Geophysical Union /08/2007JF [2] When the wind blowing over an immobile bed of cohesionless grains becomes sufficiently strong, grains are set in motion. Grains in the size range of mm are hopping or dancing over the surface in saltation, which is the primary mode of movement [Bagnold, 1941]. Saltation is an important link by which momentum is transmitted from the air to the bed through grain impact. [3] Several approaches have been taken to study the saltation process. In a large number of studies grain fluxes and their vertical variation and wind profiles within the saltation layer have been measured in wind tunnels or in the field [e.g., Horikawa and Shen, 1960; Williams, 1964; Jensen et al., 1984; Rasmussen and Mikkelsen, 1991; White and Mounla, 1991; Rasmussen et al., 1996; McKenna Neuman and Maljaars, 1997; Iversen and Rasmussen, 1999; Namikas, 2003; Liu and Dong, 2004]. Several mathematical models have been proposed to explain such empirical findings, either numerical models [e.g., Sørensen, 1985; Anderson and Hallet, 1986; Jensen and Sørensen, 1986; Anderson and Haff, 1988, 1991; Werner, 1990; McEwan and Willetts, 1991, 1993; Shao and Li, 1999; Spies and McEwan, 2000; Spies et al., 2000] or analytical models that result in explicit formulae [e.g., Owen, 1964, 1980; Sørensen, 1991, 2004; Sauermann et al., 2001]. [4] To verify the assumptions made in such mathematical models, direct observations are needed of fluid shear stress and particle behavior in the saltation layer, including the splash process when a grain impacts on the bed, rebounds and ejects other grains. However, the momentum transfer and the splash process take place in a very shallow layer at the air-bed interface with large grain concentration and velocity gradients where it is very difficult to make reliable measurements. Therefore existing experimental data are not made close to the surface or are made under somewhat artificial conditions. The splash process has been studied by computer simulations [e.g., Werner and Haff, 1988; Anderson and Haff, 1991] and experimentally in wind tunnel experiments with a controlled very small transport rate [e.g., Willetts and Rice, 1985, 1989; Rice et al., 1996] or by propelling single metal particles [Mitha et al., 1986] or sand grains [Werner, 1990] into a static bed of similar particles. Particle trajectories have been studied using highspeed photography [e.g., White and Schulz, 1977; Nalpanis, 1of12

2 1985] or stroboscope [Mitha et al., 1986] that enable data on the variation of grain speed with height, but unfortunately only for small particle concentrations. In recent years, laser based methods (e.g., Phase Doppler Analyzers, PDA and laser Doppler) have become available for measuring directly particle speed in the saltation layer [Dong et al., 2002; Rasmussen and Sørensen, 2005]. Measurement of particle speed using optic Doppler sensors can potentially be used at higher particle concentrations nearer to the bed, and despite the fact that misinterpretation of such data is possible [Rasmussen and Sørensen, 2005], they may potentially give precise values for the variation of grain speed with height [Rasmussen, 2002; Dong et al., 2002]. [5] The objective of the present study is to gain further insight in the saltation process by combining an experimental study and theoretical modeling. Data on air and particle speed and flux density are obtained at a number of heights above the bed using laser-doppler technology, and the numerical model of Jensen and Sørensen [1986] is further developed and shown to predict the observations well. The model results are also compared to results from Rasmussen and Sørensen [2005] obtained using an approximate analytical model. [6] Firstly a description of the experimental setup is given followed by a brief explanation of the mathematical model. Then data on the vertical variation of horizontal grain speed and grain flux density above beds of quartz grains having either diameter Dp = 254 mm or Dp = 320 mm are presented. For a range of friction speeds observed profiles of wind velocity, grain velocity, and grain flux density are compared to each other, to findings by other researchers, and to predictions made using the mathematical model. 2. Instruments and Methods [7] The general setup of the wind tunnel at Aarhus University is depicted in Figure 1. The working section of the tunnel is 15 m long and has a rectangular cross section of width W = 0.60 m and height H = 0.90 m. The tunnel is constructed from 22 mm plywood, with large glass panes inserted in the front and at three positions in the rear side of the working section. All front glass panes can be opened a thus provide easy access to the tunnel interior. A small bell mouth followed by turbulence spires and a 3-m-long replaceable array of roughness blocks provides a turbulent boundary layer in fair equilibrium with the boundary layer induced by ongoing saltation in the main part of the working section. Thus it is possible to avoid the overshoot of saltation otherwise typical for wind tunnel studies [Shao and Raupach, 1992]. Except for the larger cross section, the tunnel is almost similar to that described by Rasmmussen and Iversen [1993]. Grains are being fed into the tunnel 1 m before the end of the roughness array and caught in a 4-mwide expansion (sand collector) before the axial fan at the end of the tunnel. The speed of the airflow can be varied continuously between zero and approximately 20 m/s. [8] In the experiment presented here, the bed was covered by a 25 mm thick layer of uniform sand grains. During the first part of the study grains having diameter D p = 320 mm were used while profiles of air and grain speeds together with grain flux densities were measured at three friction speeds. Then the bed was replaced by one composed of 242-mm grains, and a new set of speeds and flux densities were obtained for four friction speeds. While recording a set of speeds and grain fluxes the average mass transport was determined as the total mass of grains collected in the sand collector divided by the duration of the experiment and the effective tunnel width (550 mm). Thus the procedure is similar to the one used in earlier experiment on effective mass transport [Iversen and Rasmussen, 1999]. [9] Air speed above the 320 mm bed was measured at 10, 20, 40, and 80 mm with a pitot-static tube connected to an electronic micro-manometer (Barocell with a 1.5 mm Hg high-resolution differential pressure cell). Above the 242 mm bed air speeds were recorded at 5, 10, 15, 20, 30, 40, 50, 60, 70, and 80 mm. However, in both cases measurements were made only after an equilibrium bed texture, i.e., a bed with a uniform pattern of wind ripples, had developed. The velocity profile was determined as the average of one set of data recorded before and one set recorded after a set of grain speeds and flux densities were measured. From data on the vertical profile of the horizontal wind speed (u) the bed shear stress can be calculated using the logarithmic wind law u ¼ u k ln y y 0 where u * is friction speed, y 0 is aerodynamic roughness height, and k is von Karman s constant. [10] Grain speed was measured with a one dimensional Dantec Flowlite integrated nm laser-optics system. All measurements were taken with the laser head placed outside the glass panes in the tunnel front wall at about 3 m upwind of the sand collector. Laser Doppler anemometry (LDA) configured for backscatter measurement is a nonintrusive optical measurement technique for measuring the instantaneous velocity components of a particle. In aeolian studies the aim is to measure the instantaneous velocity of a saltating (ballistic) particle which normally has velocity different from that of the surrounding fluid. The measurements are performed at the intersection of two coherent laser beams, where there is an interference fringe pattern of alternating light and dark planes. Particles traversing this pattern scatter the light, which appears to flash, as the particles pass through the bright planes of the interference pattern. The back-scattered light is collected by the front lens of the probe head and focused on the end of a fiber optical cable. This cable carries the collected light to a photomultiplier which then converts the light intensity fluctuations to electrical signals which are in turn converted to velocity information in the Flow Velocity Analyzer (FVA) signal processor. The frequency of the flashing light (Doppler frequency) is proportional to the particle velocity at the measurement point, and by placing the probe head so that the light and dark planes of the interference fringe pattern are vertical and perpendicular to the tunnel axis we are able to record the u-component of particle velocity. The LDA-probe is configured with a beam separation of 38 mm and focal length of 400 mm which, according to the technical manual, results in a measuring window having height 0.2 mm and width 4.22 mm. Validation of the signal was chosen at 2 db thus achieving at the same time a reasonable validation level and an acceptable signal to noise ð1þ 2of12

3 Figure 1. The horizontal wind tunnel with main sections indicated: 1, entry with screen (S) and bell mouth (B); 2, boundary layer modification with turbulence spires (T), roughness array (R), and sand feed (SF); 3, working section with laser Doppler (LD) and Pitot-static tube (P); 4, expansion box (sand collector) with screens (S); and 5, fan section. ratio. We determine the average horizontal particle velocity of the total grain population passing the measuring window since the 1-D laser system is unable to detect whether a grain is impinging or ejected. Grain rates can be calculated as the number of validated grains per time unit. [11] During the first part of the study, where the bed was covered by 320-mm grains, the duration of a run typically lasted from 15 to 60 seconds. Runs were short for measurements made close to the bed. Here the grain flux is high so that many measurements can be obtained quickly. Moreover, the local height of the sensor relative to the bed may vary when ripples move, which necessitates short runs. Longer runs were made farther away from the bed where grain concentrations are small and the influence of ripples is negligible. Mostly 2000 to 3000 grain speed values were measured below 20 mm height; several hundreds were recorded at 20 mm and 40 mm, while only grain speeds were recorded at higher elevations. Because analysis of the 320 mm data revealed rather large scatter, in particular in the recorded grain rates, multiple runs at the same friction speed were made during the second part of the experiment where the bed was covered by 242 mm grains. Thus more than 500 samples were mostly recorded at the uppermost level of 80 mm while from 500 to more than 10,000 measurements were recorded at lower levels. The processed data contains information about particle arrival and transit time, but only for those data where the particle speed was validated by the Dantec software. 3. Modeling Grain Speed and Flux in Aeolian Saltation [12] The model of the trajectories of saltating grains used in this paper is a standard model. It is a slight modification of the model used by Jensen and Sørensen [1986]. As in most other recent models, lift forces on the grains and effects of turbulent fluctuations are ignored when the grain has left the bed. The magnitude of the influence of lift forces is relatively small [see McEwan and Willetts, 1993], and turbulent eddies are mainly of relevance for grains smaller than 100 mm. All sand grains are assumed to be identical. In particular, they have the same shape, so that the drag on a grain is a function only of the relative speed between the grain and the air. The drag on a grain at the relative speed w is denoted by D(w). The grain mass is denoted by m. Thus the equations of motion are x ¼ Hw ð ÞðUðyÞ _x Þ y þ g þ Hw ð Þ_y ¼ 0 where x and y denote the horizontal and the vertical position of the grain, respectively, H(w) =D(w)/(mw) and U(y) isthe mean wind speed at height y. The reciprocal of the quantity H(w) has the dimension of time and can be interpreted as the response time of a grain to changes in the wind speed. The wind profile is taken to be logarithmic with friction speed and roughness height as measured during the experiments. Following Bagnold [1936] and many others we calculate the drag D(w) as the drag on an aerodynamically equivalent sphere, i.e., a sphere with the same terminal velocity of fall in air as the grains. Bagnold found the diameter of this sphere to be 0.75 times the diameter of the sand grains. We use the Schiller and Naumann [1933] formula cr ð Þ ¼ 24 R 1 þ 0:15R0:687 for the drag coefficient of a sphere, where R = 0.75d w/n is the Reynolds number with d denoting the grain diameter and n the kinematic viscosity of air. This formula has excellent accuracy for Reynolds numbers up to about 700, which is sufficient for aeolian saltation. It follows that " Hw ð Þ ¼ 13:5m d 2 1 þ 0:123 dw # 0:687 s n where s is the density of the grains, and m denotes the viscosity of air. The equations of motion were solved numerically by keeping the wind speed and H constant in small height intervals of 1 mm. This corresponds to using the explicit trajectory model from Sørensen [1991] in each of these small height intervals and is a very quick way of solving the equations. [13] It is assumed that (x(0), y(0)) = (0,0) and that the launch velocity (_x(0), _y(0)) is random. The following rather 3of12

4 simple assumptions are made about the probability distribution of the launch velocity vector. The vertical component _y(0) is taken to be gamma distributed. This distribution has the probability density function pw ð Þ ¼ wa 1 e w=b GðaÞb a ; w > 0 and its mean value is ab. The parameter a is called the shape parameter because it determines the general shape of p(v). The probability distribution of the vertical lift-off speed was also modeled by a gamma distribution by Anderson and Hallet [1986] and Namikas [2003]. In work by Rasmussen and Sørensen [2005], _y(0) was taken to be exponential distributed, which is a particular case of the gamma distribution with a = 1. The generalization to the gamma distribution was made because it gives a much better fit to the observed vertical variation of the grain flux. The launch angle (relative to a vector in the flow direction) is assumed to be normal distributed and independent of _y(0). The angle is however restricted to be positive and smaller than 150. [14] For a given choice of the gamma distribution and the normal distribution that model the launch velocity of the grains, we can at any given height calculate the distribution of the horizontal component of the grain velocity and the grain flux. For a given launch velocity, we can obviously calculate the horizontal component of the grain velocity at a given height. By making a fine discretization of the launch velocity distribution and calculating a grain trajectory for each velocity vector in the discretization, we obtain a very accurate approximation to the probability distribution of the horizontal component of the grain velocity at the given height. A grain with a particular launch velocity makes two contributions to this distribution: one on the way up and one on the way down, but only if the top of the grain trajectory is above the given height. The mass of grains that per time unit move through a plane surface with unit width, perpendicular to the mean wind direction and going from height y 1 to height y 2 is fdðy 1 ; y 2 Þ where f is the mass flux of grains from the sand bed into the air, and D(y 1, y 2 ) is the horizontal displacement of a grain while its altitude is between the heights y 1 and y 2 on its way up and down. This quantity can easily be calculated using the numerical model. The bar indicates mean value with respect to the distribution of the launch velocity. In particular, the total transport rate is given by f, where is the mean jump length of a grain. For details about these calculations see Jensen and Sørensen [1982] and Sørensen [1985]. 4. Results and Discussion 4.1. Wind Speed [15] Before any grain data were collected the boundary layer in the tunnel was investigated by recording profiles of wind speed between 10 mm and 200 mm height. The profiles revealed a slight wake region around 140 mm, while between 20 mm and 80 mm they strictly followed the law of the wall [White, 1991]. Although not always visible, below 20 mm wind profiles are expected to be influenced by shear stress partitioning between grains and fluid so that during moderate or high mass transport, lowlevel air speeds deviate from the law of the wall [Owen, 1964; Sørensen, 1985, 2004; Anderson and Haff, 1991; McEwan and Willetts, 1993]. Therefore the wind profiles that were collected during the first part of the investigation were only sampled between 20 mm and 80 mm above the bed. However, during the second part of the investigation data were sampled both from lower levels and with smaller vertical increment. This not only increased precision of the estimated friction speeds, but also allowed a more detailed comparison of the measured/predicted air and grain speeds. [16] Air speed profiles obtained for the four friction speeds at which grain dynamics were studied above the 242-mm grain bed are presented in Figure 2. Firstly, it is noticed that between 20 mm and 80 mm the data follow a log-linear profile with little scatter. Secondly, the aerodynamic roughness length (y 0 ) increases steadily from about 10 4 m at u * = 0.27 m/s (i.e., just above the saltation threshold) to about 10 3 matu * = 0.69 m/s. The observed systematic increase in y 0 with increasing friction speed corresponds well to predictions by Owen [1964] as well as observations by, for example, Raupach [1991], Rasmussen et al. [1996], and McKenna Neuman and Maljaars [1997]. Thirdly, it is noticed that for the three higher friction speeds, the observed air speeds in the region 5 10 mm above the bed are systematically higher than values extrapolated to the same heights from the log-linear wind profile. This probably indicates shear stress partitioning Grain Speed Laser-Doppler Measurements [17] In most industrial and environmental applications the laser-doppler instrument is used to measure the velocity of particles which are considerably smaller than saltating quartz grains. Caution must therefore be exerted in the interpretation of the recorded data. Thus early in the experiment it was observed that the horizontal grain velocities recorded at any level contained a small fraction of negative velocities, even when the velocity of the airflow was large. At 40 mm height, for instance, several grains had velocities smaller than 6 m/s when the average air speed was just over 6 m/s. Although negative speeds may occur occasionally [Bagnold, 1941] it seems unphysical that such negative values will be found at relatively large distance above a soft sand bed. However, for the large sand particles it is likely that a grain will only partially penetrate the control volume of the laser beams or that particle spin or facets on grain surfaces may produce artifacts in the recorded signals. Therefore the recorded data are likely to contain some unrealistic large positive or negative speeds. This interpretation of the negative velocities as mainly erroneous is further supported by the fact that negative values are only recorded for those grains that spend the shortest time in the control volume, i.e., particles which have the lowest transit times [Rasmussen and Sørensen, 2005]. Obviously such artifacts will influence the tails of the distribution of grain speed, but since they are rare, they 4of12

5 Figure 2. Profiles of air speed recorded at four different friction speeds above a bed composed of 242-mm quartz grains. are unlikely to bias the calculated average grain speeds much. [18] Another cause for concern is the possible existence of low-frequency temporal variation of transport characteristics caused, for instance, by changes in bed texture or secondary flow. For the speeds used in this study an air parcel will typically travel through the tunnel within 3 5 s. Since the duration of the runs varies from 30 s to 400 s, which is much more than the travel time for an air parcel, we do not expect low-frequency variations in our data. However, as a check we have plotted (Figure 3) the average grain velocity and flux density at 40 mm height above the 242 mm as function of the number of validated grains found in each run (as a measure of the duration of the run). All runs were made under identical conditions (same setting of fan speed) except that a change in the setup caused a small change in the average speed between run 1 8 and run The scatter of the average grain speed is about 0.5 m/s for runs with less than 1000 grains whereas it is only about half of that value when the number of grains is somewhat larger than 1000 grains. The scatter in grain flux density seems to be around grains s 1 m 2 for the short runs with few grains, but it also decreases to about half or less that value when the number of grains considerably exceeds 1000 grains. Thus averages tend to stabilize as the sampling time increases, indicating temporal stability of the wind tunnel Variation of Grain Speed in the Saltation Layer [19] Grain speed is rarely measured in neither field nor laboratory experiments. However, in the present investigation we have measured profiles of the average horizontal grain speed (V g ) for grains above a 242 mm for four friction speeds (Figure 4a), and for a 320-mm bed for 3 friction speeds (Figure 4b). The data represent conditions in different parts of the saltation layer. Thus data recorded in the interval 5 10 mm above the bed represents the intense part of the saltation layer; those in the interval mm are a bit above its most vigorous part, while those above 40 mm are in the upper part of the saltation layer where the grain concentration is low. The data for the lowest friction speed u * = 0.27 m/s represent conditions not far above the saltation threshold, while the data for the higher values represent conditions at vigorous (u * = 0.39 and 0.56 m/s) and intense transport (u * = 0.69 m/s). It is interesting to see Figure 3. Grain speed and flux density at 40 mm above a bed composed of 242-mm quartz grains as function of number of validated samples. A change in setup between run 1 8 and run 9 30 decreased the average grain velocity. 5of12

6 Figure 4. Mean horizontal grain speed profiles above a bed of (a) 242-mm quartz grains for four different friction speeds and (b) 320-mm quartz grains for three different friction speeds. that all profiles converge toward a particle speed of about 1.3 m/s at 2 3 mm height above the bed. Like wind profiles [Bagnold, 1941], grain speed profiles seem to have a focus point. The profiles increase smoothly with height except for the 0.56 m/s profile which is based on fewer measurements and is therefore also expected to show more scatter. The ratio between measured air and grain speed is about 1.2 for all friction speeds at the highest elevation of 80 mm (Figure 5). Below 80 mm the ratio gradually increases to about 2 or slightly more at 5 mm height. Close to the bed there is considerable scatter. This is to be expected because both wind speed and grain speeds are less well determined here. Moreover, because of the passing ripples, the height is also less well determined here so that the wind and grain speed measurements may have been made at slightly different heights. For the two particle sizes used here, there is no apparent influence from grain size in the ratio between air and grain speed. [20] The probability distribution of the grain speed was calculated for each of the heights where measurements were made and for the different friction speeds and grain diameters using the saltation model. To do so the parameters, i.e., f (the mass flux of grains from the bed into the air) and the parameters of the gamma distribution and the normal distribution that model the probability distribution of the launch velocity, were chosen so that the best possible fit was obtained to the observed grain fluxes and to the observed probability distributions of the horizontal velocity component of the grains. Specifically, the distance between measured and theoretical values of the mean and standard deviation of the horizontal grain velocity distribution and the distance between observed and model grain fluxes were minimized. It turned out that the best fit was in almost all cases obtained for a value of the shape parameter of the gamma distribution close to three. Therefore the shape parameter was in all cases taken to equal three. This differs from the results of Anderson and Hallet [1986] and Namikas [2003] who, using only flux data, obtained the best fit with the shape parameter equal to one (the exponential distribution). With the shape parameter fixed, the general picture was that the magnitude of the mean value of the vertical component of the launch velocity was mainly determined by the vertical variation of the grain flux, the mean launch angle was then determined essentially by the mean values of the horizontal grain speeds, and the variance of the launch angle by the variances of the horizontal grain speeds. The observed variances of the horizontal grain speeds appear to have a considerable random variation, and it is not unlikely that the actual variances vary much less with height. It was therefore not attempted to fit in detail the vertical variation of the observed variances. The quantity f was chosen so that the sum of the observed fluxes was equal to the sum of the Figure 5. Ratio between air speed and mean grain speed as a function of height for two different grain sizes and different friction speeds. 6of12

7 Table 1. Model Parameters Friction Speed (m/s) Number of Grains Mean Vertical Launch Speed (m/s) Mean Launch Angle Variance of Launch Angle D p = 242 mm D p = 320 mm model fluxes. The parameters which produced the best fit to the observed grain speeds and grain flux densities are given in Table 1. [21] The estimated values of the mean vertical launch velocity are consistently close to 0.38 m/s in good accordance with the values found by Rasmussen and Sørensen [2005] and with those found by White and Schulz [1977] and Nalpanis [1985] using high-speed film. Sørensen [1985] and Namikas [2003] found mean vertical lift-off speeds that were somewhat larger. The mean launch angles are smaller than those found in other studies. White and Schulz [1977] found a mean angle of 50 degrees, Nalpanis [1985] reported mean angles in the range degrees, and Sørensen [1985] in the range degrees. Willetts and Rice [1985] found the range degrees for ricochet angles and for ejected grains. Our smaller values might be due to the fact that we have observations very close to the bed (5 mm above the bed). It should be noted that the estimated mean values of the vertical component of the launch velocity and of the launch angle do not vary much with the friction speed. This was also concluded by Jensen and Sørensen [1986] in an analysis of wind tunnel observations by Williams [1964] and is in accordance with findings by Namikas [2003] where field observations were analyzed. The result may explain the focus point of grain velocity profiles mentioned above, and it gives support to the arguments of Sørensen [2004] that the empirical parameters in the transport rate formula derived in that paper do not depend on the friction speed, an assumption that is essential for the formula to make sense. The finding is, however, contrary to the assumption that the mean launch speed scales with the friction speed. This assumption, which goes back to Bagnold [1941], has often been made in the literature. [22] With the parameters given above the calculated mean horizontal grain speed for the two particle sizes are given in Table 2, together with ratios between predicted and measured speeds. Overall there is reasonable agreement between the values. However, below 10 mm height the model systematically predicts higher grain speed than the measured speeds, while at the highest levels the model predictions are systematically lower than measured values. Possibly this is due to deficiencies in the model, but it may also result from measurements error in the grain laden layer close to the bed. [23] Initially, grain speeds were also predicted assuming that the vertical lift-off speed of the grains is exponentially distributed (gamma distribution with shape parameter one). For this distribution the tendency for overestimation at low levels and overestimation at high levels is the same, but the numerical differences between measured and estimated values were found to be larger, i.e., about 20 70% underestimation at the lowest levels and 10% overestimation at the highest levels. [24] Values of the observed standard deviations (s) of the horizontal grain speed distribution and the ratio between predicted and observed standard deviations (s p /s) are given in Table 3 for the two grains sizes. There is much scatter in the ratios in the table, which is to be expected since estimated standard deviations are known to have a larger random variation than estimated means and are much more sensitive to erroneous extreme measurements (outliers), which we know are present. There is a clear tendency for the standard deviations to increase with the friction speed. Perhaps there is a tendency that the standard deviations are largest at the middle heights (around 4 cm). The observed variation of the standard deviation with height is not well predicted by the model because the model values vary in a much more smooth way. However, it should be noted that for the majority of the observations the discrepancy between the predicted and measured variance is less than 15%. [25] The observed probability distribution of the horizontal grain speed is presented for two heights and two friction Table 2. Model Predictions of the Mean Horizontal Grain Speed (V gp ) and Ratios Between Predicted and Measured Mean Particle Speeds (V gp /V g ) Particle Diameter D p = 242 mm u * Particle Diameter D p = 320 mm u * Height (m) 0.27 (m/s) 0.39 (m/s) 0.56 (m/s) 0.69 (m/s) 0.27 (m/s) 0.47 (m/s) 0.74 (m/s) of12

8 Table 3. Observed Standard Errors (s) and the Ratio Between Predicted and Observed Standard Errors (s p /s) for Different Friction Speeds and Two Particle Sizes Particle Diameter-D p = 242 mm u *, s 2, s 2 p /s 2 D p = 320 mm u *, s 2, s 2 p /s 2 Height (mm) 0.27 (m/s) 0.39 (m/s) 0.56 (m/s) 0.69 (m/s) 0.27 (m/s) 0.47 (m/s) 0.74 (m/s) , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 0.93 speeds for the 242-mm grains in Figure 6 together with model calculations of the same distribution. The examples plotted in Figure 6 were chosen to illustrate what happens at a high and a low wind speed and at a height with intense saltation and a height with relatively low grain concentration. Generally there is fair agreement between the observed and predicted distributions. For the low friction speed, the majority of grains have speeds between 1 and 3 m/s. While grains having speeds between 1.0 and 1.0 m/s are also quite common at the low height, these grains are rarely found at 20 mm height where the second most common group are grains with speeds between 3 and 5 m/s. Grains in the range 1 3 m/s are also the most common group at 5 mm height for the high friction speed, but at 40 mm height these grains are rare and the velocity spectrum is dominated by grains having speeds in the wider range from 3 and 7 m/s. Qualitatively the observed variation agrees with the perception that during the splash only a limited number of grains will receive a large vertical momentum through the conversion of forward momentum of the impinging grain, while several grains will receive a relatively small momentum and make low, short jumps. The few grains that jump as high as 40 or 80 mm are all accelerated during ascent by the increasing wind speed further away from the bed. As the friction speed increases, saltating grains will receive an increasing amount of forward momentum which will in- Figure 6. Predicted (grey) and measured (black) distributions of horizontal grain speed of 242-mm grains for two friction speeds and three different heights. 8of12

9 crease not only the chance that more grains will be set in motion by the collision of an impinging grain, but also the chance that the ejected grains will jump higher. The result is that the distribution near the bed is not changed much while the distribution far from the bed changes much more. [26] Grain velocity data measured close to the bed have been reported by Dong et al. [2002]. These authors observed many grains with negative velocities and interpreted this to be caused by midair collisions. However, besides the methodological difficulties already mentioned at the beginning of the Grain Speed section, we believe that observation by LDA quite close to the bed poses additional technical difficulties. It is possible that a bias may be caused by interference from multiple grains in the measuring volume and by large relative changes of bed elevation. During data collection it was noted that the validation rates, and hence the results, are sensitive to the signal-to-noise criterion which is set on the instrument before each run. This sensitivity is weak far from the bed, but increases dramatically close to the bed. In our data the number of grains with negative velocity increases near the bed, but many of these have relatively poor statistics as determined by the LDA software as already reported by Rasmussen and Sørensen [2005]. With the criterion that we apply ( 2 db), a small proportion of the measured grain speeds are negative even at 5 mm or higher. Our saltation model allows grains to start at an angle of 150 (i.e., 60 against the wind direction). Even so, the model predicts that negative grain speeds are extremely unlikely as close to the bed as 5 mm. As discussed above we therefore think that most of the measured negative grain speeds in our data are erroneous and caused by grain spin, grain facets, effects of the relatively large particle diameter, and multiple grains in the measurement window. The possible bias caused by such artifacts in our data has been further investigated by measuring the grain velocity distribution at 3 mm height above the bed composed of 242-mm grains. For a friction speed of 0.27 m/s about 2% of the grains were found to have a negative speed which will only insignificantly influence the average speed presented here. But because we are uncertain and unable to experimentally verify whether the increase in particles with negative velocities near the bed is an artifact or is caused by midair collisions as suggested by Dong et al. [2002], we generally omit presenting any data measured below 5 mm height. [27] Previous experimental studies as well as numerical modeling indicate that immediately above the bed the speed of an impinging grain is typically in the range of 2 8 m/s [e.g., McEwan and Willetts, 1991; Anderson and Haff, 1991]. Comparison of the data in Figure 4 shows that the speeds we have measured at 5 mm height are within the range of expected speeds for an impinging grain. Our data include ejected grains too, so it is not surprising that our mean speeds are in the lower end of the interval. Experimental data on the variation of grain speed close to the bed are few, but in a recent study, Dong et al. [2002] presents data on the distribution of impact speed for different grain sizes and free stream velocities. For the same particle sizes their study generally records impact speeds which are much lower than the bulk speeds found here, typically an order of magnitude. Apart from the serious problems connected with measurements very close to the bed, the difference might partly be due to the fact that close to the bed the proportion of (slow) ejected grains is much larger than 5 mm from the bed. We take the consistent behavior of our measurements and model predictions combined with the fact that our experimental data are in fair agreement with observed particle trajectories [White and Schulz, 1977; Nalpanis, 1985] and numerical predictions [Anderson and Haff, 1991] as an indication that the grain speed data recorded with the laser-doppler instrument are not severely influenced by particle spin, influence from large particle diameter, grain facets and other measurement errors which may significant influence results if not handled properly Grain Flux Density [28] Vertical profiles of mass transport are commonly recorded in aeolian research for estimation of the total sand transport by wind although this is difficult given the nonlinearity of the profile and the vigorous nature of transport close to the bed [Butterfield, 1999, 1991; Rice et al., 1996]. Vertical profiles are also commonly used for calibration of numerical saltation models. Profiles of the raw rate of grains for which the velocity estimation was validated by the LDA software are presented in Figure 7a for the 242-mm grains and in Figure 7b for the 320-mm grains. For transport rate purposes the grain flux density is found as the rate divided with the nominal area of the measuring window, and those values are typically in the range from about grains s 1 m 2 at 80 mm height to 10 8 grains s 1 m 2 or more at 5 mm height. Also model predictions of the grain flux are plotted in Figure 7 and are in good agreement with the measurements. [29] In the LDA results there are large discrepancies between the number of grains apparently encountered by the laser (attempted samples) and the number of grains for which the estimation of particle speed was successful (validated samples). The ratio between validated and attempted samples typically change from about 35% at 5 mm and about 50% at 20 mm height to about 85% at 80 mm height. To investigate to what extent this implies that we underestimate grain fluxes, the LDA results were compared to the directly observed total transport rate. To do so, we have fitted two exponential segments to the recorded flux density profiles and using this curve calculated the total mass transport for the different friction speeds and plotted the data versus measured mass transport rate (Figure 8). Generally measured and profile based mass transport results for the 242-mm particles agree well, although the flux profile slightly underestimates the total transport rate at the high friction speed. For the 320-mm particles, the two transport rate determinations are reasonably close at the low levels near the bed where many grains have been recorded, but the values differ considerably at the higher levels where much fewer observations have been recorded. Nevertheless there is no general tendency that the profile based transport rates underestimate the directly measured transport rates, and there is no doubt that the validation level chosen in the present experiment is reasonable and the number of validated samples reflects fairly well the actual flux whereas the number of attempted samples overestimates the flux. [30] Generally flux density profiles sampled with vertically segmented sand traps up to mm above the 9of12

10 while [Rasmussen et al., 1985] for another beach recorded the same fraction below 150 mm. Similarly in wind tunnel studies about 95% of the flux was measured below respectively 100 mm [McKenna Neuman and Nickling, 1994], and 150 mm [Horikawa and Shen, 1960]. Contrary to this, in a wind tunnel study with particles of a diameter similar to those used here, Dong et al. [2002] found than many grains jumped considerably higher that mentioned above. Depending on free stream velocity they found that sampling must be made as high as mm in order to record about 95% of the total flux. However, a fully developed saltation layer may not have formed in their experiment since they used no sand feed and saltation was measured downwind of a sand bed which was only 4 m long. Figure 7. Profiles of horizontal grain rates measured and modeled above beds of (a) 242-mm and (b) 320-mm quartz grains. Individual symbols refer to different friction speeds. 5. Conclusions [32] The dynamics of saltating particles has been investigated by measurements in a wind tunnel and by simulations using a numerical saltation model. The height of the boundary layer is more than 150 mm and wind profiles follow the log-linear law up to at least 80 mm. Turbulence fluctuations and random variation of the launch velocity of the grains impede the recording of precise vertical profiles of grain speed and grain flux density and requires that about 2000 samples must be taken before a reasonably small uncertainty is achieved. The recorded profiles of average horizontal grain speed are almost log linear, but not quite. Thus the ratio between wind and grain speed is of the order of 2 at some millimeters above the bed, but decreases to approximately above 40 mm height. For the two bed particle diameters D p = 242 mm and 320 mm investigated in the present experiment there is no particular difference in these ratios. Grain speed profiles recorded at different friction speeds seem to converge toward a focal point at about 2 4 mm above the bed somewhat similar to the convergence of wind profiles in a saltation cloud [Bagnold, 1941]. Horizontal grain speeds were predicted bed, show profiles with two regions where the flux density decays exponentially [Butterfield, 1999; Rasmussen and Mikkelsen, 1998]. In the profiles sampled in the present investigation a single exponential curve (or power function) cannot be fitted well through the data points. Two segments fit the data well, but the array of measurement points is too sparse to give information about the detailed shape of the profiles near the bed. [31] Extrapolation based on flux density values measured above 30 mm shows that for the 320-mm grains the horizontal flux above 250 mm height is very small, while for the 242-mm grains the flux most likely is insignificant above mm height. Vertical profiles of flux density have been observed in several experiments, primarily in order to assess total transport rate, and the profiles recorded in the present investigation are comparable to profiles recorded in both field and laboratory investigations. Thus for 250-mm saltating grains on a beach, Namikas [2003] recorded about 95% of the total flux below 200 mm height Figure 8. Mass transport calculated from the flux density profile and plotted versus measured mass transport for the 242-mm and 320-mm beds. 10 of 12

11 quite well using the saltation model with estimated parameters. Thus measured and calculated mean values differ by less than 10% except at 5 mm height where in some cases as much as about 50% deviation is observed. However, near the bed both our experimental data as well as our numerical predictions differ significantly from data published recently by Dong et al. [2002] who find much lower values for the horizontal grain speed. Overall measured and modeled probability distributions of grain speed are fairly similar, except for a tendency of the model to predict more grains at the dominant grain speed and fewer at the higher speeds than actually observed. At low friction speeds the majority of grains have horizontal speeds in the interval 1 3 m/s in most of the saltation cloud. However, at moderate to high friction speeds the upper part of the cloud is characterized by faster grains in the wider range 3 7 m/s. It is noteworthy that the estimated launch velocity distributions depend only weakly on the friction speed. This finding is in contrast to what has often been assumed in the literature, but in support of field observations by Namikas [2003]. Grain rates increases with increasing friction speed and decrease with height. They are well predicted by the numerical model. The observed decrease of flux density with height can be well approximated by two regions where the flux density decays exponentially. When flux density profiles based on the validated data are used to estimate the total mass transport rate, results are in fair agreement with measured transport rates and certainly do not underestimate these. References Anderson, R. S., and P. K. Haff (1988), Simulation of eolian saltation, Science, 241, Anderson, R. S., and P. K. Haff (1991), Wind modification and bed response during saltation of sand in air, Acta Mech., 1, suppl., Anderson, R. S., and B. 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