HETEROGENEOUS SALTATION: THEORY, OBSERVATION AND COMPARISON

Transcription

1 Boundary-Layer Meteorology (25) 115: Ó Springer 25 DOI 1.17/s HETEROGENEOUS SALTATION: THEORY, OBSERVATION AND COMPARISON YAPING SHAO* and MASAO MIKAMI 1 Department of Physics & Materials Science, City University of Hong Kong, HK SAR, China 1 Meteorological Research Institute, Japan Meteorological Administration, Tsukuba, Japan (Received in final form 5 November 24) Abstract. In the theory of saltation, under development since the 194s, it is often assumed that saltation is homogeneous, i.e., saltating particles are uniform in size and follow identical trajectories. This assumption is a limitation to the development of saltation theory. In this paper, as in some other studies made since the 198s, we are concerned with the saltation of multi-sized particles in turbulent flows, a process that we refer to as heterogeneous saltation. The theory deals with several questions, including the variation of particle size distribution with height, the entrainment rates of particles in different size ranges, and the associated profiles of saltation flux, particle momentum flux and particle concentration. It is hypothesised that saltation is dynamically similar and universal similarity functions can be established. The similarity function for saltation flux is presented. Field observations are carried out at the southern fringe of the Takla Makan Desert in April 22. Measurements of streamwise saltation flux of 32 particle size groups are made using a sand particle counter, together with measurements of wind speed and other atmospheric variables. These data are used to validate the saltation theory, by examining whether the observed saltation flux and particle size distribution can be reproduced. It is shown that the theory is promising in predicting these quantities. Key words: Aeolian process, Saltation, Sand drift, Similarity theory, Wind erosion. 1. Introduction Saltation, the process of wind-propelled hop motion of sand-sized particles over an aeolian surface, has been a focus of wind erosion research since Bagnold (1941). Three key issues can be identified in the subject: (1) streamwise saltation mass transport; (2) threshold condition for saltation and (3) particle flow interactions. An assumption is made in most of the earlier studies, explicitly or implicitly, that saltation is homogeneous, i.e., the saltating particles are uniform in size and follow identical trajectories (e.g., Bagnold, 1941; Owen, 1964). This assumption has been for a considerable period of time the basis of the theoretical framework for saltation from which much of our understanding on the subject is derived. * apyshao@cityu.edu.hk

2 36 YAPING SHAO AND MASAO MIKAMI Saltation in nature is not homogeneous, but heterogeneous: saltating particles differ in size, follow different trajectories and respond differently to turbulence. Consequently, saltation observed at different heights shows different characteristics (Leys and McTainsh, 1996). This is by itself an interesting aerodynamic problem. It is now known that saltation is a key mechanism for dust emission (Gillette, 1981; Shao et al., 1993) and the prediction of dust emission requires detailed information of saltation. It is hence desirable to develop a general theory for heterogeneous saltation, as homogeneous saltation models are no longer adequate for the purpose. With respect to heterogeneous saltation, a range of questions can be asked. Of particular interest is the variation of particle size distribution (PSD) with height, the entrainment rates for particles in different size ranges and the associated profiles of saltation flux, particle momentum flux and particle concentration (Sorensen, 1985). Anderson and Hallet (1986) developed a model for sediment transport for multiple particles and Anderson (1987) introduced the Lagrangian stochastic technique to sediment transport modelling. These are useful studies, but seem to have not continued since that time. Jensen and Sorensen (1986) proposed a model for heterogeneous saltation and estimated the dislodgement rate and the probability distribution of lift-off velocity for each size class by finding the values of these quantities for which their model exactly predicts the transport profile obtained by Williams (1964). However, until today, the existing theories of saltation remained quite inadequate in describing heterogeneous saltation. In this study, together with Shao (25), we develop a theory for heterogeneous saltation. We hypothesise that the saltations of different sized particles in turbulent flows are dynamically similar and universal similarity functions can be established for representing quantities such as streamwise saltation (mass) flux, particle momentum flux, etc. As described in Shao (25), in establishing the similarity functions, we constructed a numerical model and applied it to simulating saltation under various flow conditions. The similarity functions were then derived from the numerical results. In Shao (25), an outline of the theory, the modelling technique and the profile of saltation flux were considered. Here, we emphasise on the saltation of multi-sized particles and the comparison of the theory with observations. Field observations were carried out at Qira, China (8.8 E, 36.9 N, located at the southern fringe of the Takla Makan Desert) during the Intensive Observation Period (IOP) of the Asian Dust Experiment (ADEC) in April 22 (Mikami et al., 22). In this experiment, measurements of streamwise saltation fluxes of 32 particle size groups were made using a sand particle counter (SPC). Wind speed and other atmospheric variables were also measured. This dataset is used for testing the theory.

3 HETEROGENEOUS SALTATION A Theory for Heterogeneous Saltation 2.1. SALTATION OF UNIFORM PARTICLES First, we consider the saltation of uniform particles of size d. Shao (25) described a saltation model for uniform particles (uniform trajectory is not assumed); the model consists of three components: A Lagrangian stochastic model for particle motion in turbulent flows; A statistical representation of particle lift-off conditions; and A specification of the atmospheric surface-layer wind profile and turbulence on the basis of the Monin Obukhov similarity theory. The model allows the determination of particle concentration, ^cðz; dþ, streamwise particle mass flux, ^qðz; dþ, and particle momentum flux, ^s p ðz; dþ. We express these quantities as (see also Sorensen, 1985, 24), ^cðz; dþ ¼ ^F s ðdþj c ðz; dþ; ð1þ ^qðz; dþ ¼ ^F s ðdþj q ðz; dþ; ð2þ ^s p ðz; dþ ¼ ^F s ðdþj s ðz; dþ; ð3þ where ^F s ðdþ ¼^nðdÞmðdÞ is entrainment particle mass flux (or dislodgement rate), with ^n being the entrainment particle number flux and m particle mass. The J functions are dimensional functions that can be determined through observation or numerical modelling. Note that J q is dimensionless while the other two are not. If saltation is dynamically similar then it is possible to describe the J functions, i.e., the saltation similarity functions, in terms of simple analytical expressions. Shao (25) studied J q using a Lagrangian stochastic model and suggested that J q ¼ A exp zk L k ; ð4þ z where A is a dimensionless coefficient, L z a scaling height and k a quantity that varies between 1 and 2. For saltation in equilibrium, Equation (3) implies that ^F s ðdþ can be expressed as ^F s ðdþ ¼ s s tðdþ J s ð; dþ ; ð5þ where s ¼ qu 2 and s tðdþ ¼qu 2 t ðdþ, with u being friction velocity and u t ðdþ being threshold friction velocity for particle of size d. Alternatively, as can be seen from Equation (2), a global constraint can be applied to ^F s such that

4 362 YAPING SHAO AND MASAO MIKAMI Z 1 ^F s ðdþ ¼ ^QðdÞ J q ðz; dþdz; ð6þ where ^QðdÞ is the streamwise mass transport of particle of size d. Equation (6) is a manifestation of the second hypothesis of Owen (1964) and is an interesting result, since it provides an estimate of dislodgement rate from the streamwise mass transport. Sorensen (24) used the same formula as a basis of his model. We hypothesise that ^QðdÞ can be approximated using the Owen (1964) model, 8 < u u t ðdþ ^QðdÞ ¼ c qu3 : ; u > u t ðdþ g 1 u2 t ðdþ u 2 ð7aþ (7b) where c is a coefficient, q is air density and g is the acceleration due to gravity. Equation (7) (together with Equation (12)) represents a simplification of heterogeneous saltation: embedded in this equation is the assumption that the lift-off of particles is entirely determined by surface shear stress and the impact of saltation splash is neglected. It has been suggested (e.g., Anderson and Haff, 1991) that splash entrainment may play an important role in the process of saltation. Wind-tunnel experiments (e.g. Rice et al., 1995) and numerical simulations (e.g. Shao and Li, 1999; Doorschot and Lehning, 22) have been carried out to study splash entrainment and its significance in the process of saltation. Shao and Li (1999) suggested that a critical friction velocity u c exists. If u < u c, saltation is in the weak saltation mode for which splash entrainment is unimportant; if u > u c, saltation enters the strong saltation mode for which splash entrainment is important. This critical friction velocity is probably around.7.8 m s )1. Under such conditions, Equation (7) becomes less valid. In this study, we assume the correctness of Equation (7) for simplicity, and from Equations (5) and (6) follows another useful relationship, J s ð; dþ ¼ g Z 1 J q dz: ð8þ cu 2.2. SALTATION OF MULTI-SIZED PARTICLES Suppose a soil composed of multi-sized particles is well mixed and the particle-mass size distribution is p M ðdþ. Then the mass fraction of particles in the size range d 1 2 dd is p MðdÞdd and the area fraction these particles occupy on the surface is p A ðdþdd. The particle-area size distribution p A ðdþ is related to p M ðdþ by

5 p A ðdþ ¼ p MðdÞ 1 R d pm ðdþd ln d : The streamwise mass transport for all particle sizes, Q, is Q ¼ Z 1 ^QðdÞp A ðdþdd and the total particle momentum flux at the surface is s p ¼ Z 1 ^s p ð; dþp A ðdþdd: ð9þ ð1þ Note that the above integrations require the knowledge of ^nðdþ. Equations (9) and (1) are hypothetic. It is assumed that saltating particles of different sizes behave independently and aerodynamic forces play a determining role. In reality, saltating particles may interact through two processes, one is collision in air and the other is splash on surface. The former affects particle motion and the latter particle entrainment rate and lift-off conditions. The effect of the first process can be neglected, because the concentration of saltating particles is low. The importance of splash entrainment is being debated, as discussed in Section 2.1. Suppose soil particles are divided into I size bins and the ith bin has a particle size d i, a bin width D i and occupies a surface-area fraction P i (the probability of finding a particle from the ith size group on the surface). Then, we have X I 1 and P i is P i ¼ P i ¼ 1; Z di þd i =2 d i D i =2 p A ðdþdd: HETEROGENEOUS SALTATION 363 Conceptually, we can view the aeolian surface covered by a mixture of particles of different sizes as a well-sorted surface, as illustrated in Figure 1. Consider now the balance of momentum at the surface. Suppose the particle-borne momentum flux (in the vertical direction) due to the saltation of the ith size group is ^s pi. Then, the particle-borne momentum flux at z = due to the saltation of all size groups is s p ¼ XI 1 ^s pi P i : ð11þ

6 364 YAPING SHAO AND MASAO MIKAMI Well Mixed Soil Surface Well Sorted Soil Surface Figure 1. The concept of approximating a well-mixed aeolian surface with a well-sorted one. Similarly to Owen (1964), we hypothesis that ^s pi must satisfy ^s pi ¼ u u ti ð12aþ s s ti ; u > u ti : ð12bþ where s ti ¼ qu 2 ti with u ti being the threshold friction velocity for the ith particle size group. It follows that s p ¼ XK ðs s tk ÞP k ¼ rs XK s tk P k ; ð13þ where K is the number of erodible groups among the I groups for a given u and r ¼ XK P k is the total erodible fraction of the surface. The airborne momentum transfer to the surface (at equilibrium saltation) satisfies s a ¼ð1 rþs þ XK s tk P k : ð14þ 2.3. SALTATION MASS FLUX The entrainment mass flux for the ith particle size group is F s ðd i Þ. Following the configuration depicted by Figure 1, this flux is P i ^F s ðd i Þ. ^F s ðd i Þ must be so large that Equation (12) is satisfied. It follows from Equation (5) that F s ðd i Þ¼ u u ti ð15aþ s s P ti i J s ð;d i Þ ; u > u ti : ð15bþ The streamwise mass flux is

7 HETEROGENEOUS SALTATION 365 qðz; d i Þ¼F s ðd i ÞJ q ðz; d i Þ¼P i ^F s ðd i ÞJ q ðz; d i Þ: Finally, Q i ¼ and Z 1 qðz; d i Þdz ð16þ ð17þ Q ¼ XI 1 Q i : ð18þ Equation (16) is one of the main statements of the theory. The study of Shao (25) suggests that J q may be a universal function of height, depending on d (particle size), z (aerodynamic roughness length), u (friction velocity), etc. This suggestion requires further refinement and verification, but we assume here that J q is known and can be specified according to Shao (25). Note that ^F s can be estimated from either (5) or (6). Thus, if p M ðdþ and hence P i is known, the profile of saltation flux for a given particle-size group, qðz; d i Þ, is known VARIATION OF PARTICLE SIZE DISTRIBUTION WITH HEIGHT We can approximate the PSD at level z as p M ðd i ; zþ ¼ 1 qðz; d i Þ D i qðzþ ; ð19þ where qðzþ is the sum of qðz; d i Þ over all particle size bins, i.e., qðzþ ¼ XI 1 qðz; d i Þ: Equation (16) implies that qðz; d i Þ¼qðz r ; d i Þ J qðz; d i Þ J q ðz r ; d i Þ ; where z r is a reference level. Therefore, it is possible to obtain PSD at any z,if it is known for a given z r, because it follows from p M ðd i ; zþ ¼p M ðd i ; z r Þ qðz; d iþ qðz r Þ ð2þ qðz r ; d i Þ qðzþ that p M ðd i ; zþ ¼cp M ðd i ; z r Þ= X cp M ðd i ; z r Þ; ð21þ where c ¼ J q ðz; d i Þ=J q ðz r ; d i Þ.

8 366 YAPING SHAO AND MASAO MIKAMI 3. Qira Field Observation Field observations were carried out at Qira in China during ADEC-IOP in April 22 (Mikami et al. 22). The directly relevant information to our study is summarised here. Qira (8.8 E, 36.9 N) is located at the southern fringe of the Takla Makan Desert. The elevation of the site is 14 m. The surface at the site, composed of sand dunes with small and large grains, Gobi with small and large stones and farm land, is actively aeolian. Atmospheric quantities, including wind speed, wind direction, air temperature and air humidity, were monitored using automatic weather stations. Surface quantities, such as soil moisture, soil temperature and soil heat fluxes, as well as radiation fluxes, including upward and downward shortwave and longwave radiation components, were measured. Visibilities at two different levels were observed. A SPC was used for measuring the streamwise transport of particles ranging from about 4 to 66 lm. The functioning and calibration of the SPC were described by Yamada et al. (22); it consists of a light source and a detector, and as a particle passes through the sampling area, a decrease in light signal is detected. Larger particles produce stronger decreases than smaller ones. The SPC has a sampling cross-section 25 mm wide and 2 mm high, and thus a sampling area of A = 5 mm 2. The number of particles passing through A is recorded for every 1 min. If the number of particles of size d i passing through A in a time interval T is N i, then the particle mass flux is q i ¼ m in i AT ; where m i is particle mass. The SPC measured N i for 32 particle size bins and the particle sizes are 41.3, 56.6, 71.6, 87.1, 13.4, 12.4, 137.5, 154.3, 171.6, 189., 26., 223.5, 242.1, 26.8, 279.5, 298.5, 318., 338., 358.6, 379.8, 41.1, 422.4, 444.2, 466.2, 488.3, 51.8, 534., 557.8, 582.2, 67.3, and lm. An example of the atmospheric measurements is shown in Figure 2. The atmospheric flow in the region is characterised by a mountain-basin wind pattern. During the daytime (about 93 to 183 LT, LT = UTC + 5:5 h), the prevailing wind is a northerly flow from the desert areas of the Tarim Basin. It is in general warm, dry and turbulent. At nighttime, cooler, more humid and less turbulent south to south-easterly flows from the mountainous regions dominate. During the IOP, starting from 14 April 22, the Qira site experienced a relatively cool and humid period. Strong north-westerly winds exceeding 6 m s 1 (measured at 3:8 m above ground) occurred in the morning before the establishment of the northerly basin flow. Strong winds produced

9 HETEROGENEOUS SALTATION 367 (a) (b) T ( o c) DD ( o ) (c) 3 R (%) Time (day, LT = UTC hr) Figure 2. Atmospheric quantities, including (a) wind speed, (b) wind direction (dashed line) and temperature (solid line) and (c) relative humidity, observed at 3.8 m above ground over a period of seven days. significant wind erosion that lasted several hours on both days. Figure 3 shows a sample of SPC N i measurements for 17 April 22. The 1-min values of N i fluctuate significantly with time, responding apparently to atmospheric turbulence. The values of N i varied from over 4 for 7-lm particles to less than 1 for 4-lm particles. Based on the wind speed and SPC measurements, the threshold wind velocity and the threshold friction velocity can be determined. The former is height dependent. As Figure 4 shows, the threshold wind velocity at heights.38, 1.38 and 3:8 m are approximately 4.2, 5.5 and 6:3 m s 1, respectively. The threshold friction velocity is approximately :39 m s 1 (with roughness length z set to.5 m). These threshold values are similar for both 14 and 17 April, 22. As the saltation sampler was mounted at a rather high level (.3 m), the threshold friction velocity may be a slight overestimate. We now consider the relationship between q and u. To facilitate description, we denote the average of q over 1 min, and the running averages over 1 and 2 min as q 1min, q 1min and q 2min, respectively. Correspondingly, we have u 3 1min, u3 1min and u3 2min for friction velocity cubed. Figure 5 shows

10 368 YAPING SHAO AND MASAO MIKAMI 4 ~7 micron ~1 ~15 N i 2 N i ~2 micron ~3 ~ Time (hr, LT = UTC + 5.5, 17 April 22) 15 Figure 3. SPC measurements of streamwise saltation flux. Thin dashed lines represent the number of particles passing through the SPC sampling area in 1-min intervals, using 7, 1, 15, 2, 3 and 4 l m as examples. The solid curves are 1-min running means. plots of q 1min against u 3 1min, q 1min against u 3 1min, etc. It is seen that q 1min is poorly correlated with u 3 1min. However, as the averaging time increases, better correlations emerge. As the figure shows, q 2min is well related to u 3 2min. Note that for a fixed height (z =.3 m in our case), q does not linearly depend on u 3. Instead, q appears to increase exponentially with u 3. Figure 4. Saltation flux q at.3 m, wind speed at.38, 1.38 and 3.8 m and friction velocity for 14 April 22. The threshold wind velocities at height.38, 1.38 and 3.8 m are 4.2, 5.5 and 6.3 m s 1, respectively. The threshold friction velocity is.39 m s 1, estimated using the logarithmic wind profile with z ¼ :5 m.

11 HETEROGENEOUS SALTATION 369 Figure 5. Relationship between q and u 3 for 14 April 22 (a) and 17 April 22 (b), both q and u 3 are 1-min averages and 1- or 2-min running means. An explanation for this is that for a small u, the saltation layer is shallow as most particles saltate close to the surface and, as u increases, particles increasingly saltate at higher levels.

12 37 YAPING SHAO AND MASAO MIKAMI 4. Comparison of Theory with Measurements The Qira data are used for testing the saltation theory, mainly by examining whether the theory reproduces the observed saltation flux and PSD at a given height. For the Qira experiment, these two quantities were measured only at the.3-m level. The dataset has several other uncertainties and the following manipulations are necessary for the comparison. Aerodynamic roughness length z : we distinguish between the roughness length of the surface z and the effective roughness z e. It is known (e.g. Rasmussen et al., 1996) that z e varies with saltation intensity. In the Qira experiment, wind velocities averaged over 1-min intervals were measured at height.38, 1.38 and 3.8 m; the data are further averaged over 2-min intervals. For neutral cases, the averaged wind speeds are fitted to the logarithmic wind profile to estimate u and z. For other stabilities, stability corrections are made following the Monin Obukhov similarity theory. However, a constant z could not be found, and during times of steady winds, z mostly varied between.1 and.1 m. The range of z values is consistent with the earlier analysis of Mikami et al. (1995). For the model calculations, z is assumed to be.3,.4,.5 and.1 m. We present the results obtained using z =.5 m and discuss the sensitivity of the results to z. Saltating particles transfer momentum to the surface and increase its capacity in absorbing momentum from the mean flow. Hence, the effective roughness length z e is larger than z. A related consequence is that the profile of mean wind in the saltation layer is no longer logarithmic. The issue related to z e has been accounted for in the determination of J q (Shao, 25). In that paper, the wind profile for the saltation layer is specified according to Raupach (1991), rather than the logarithmic assumption. An explicit calculation of z e is not necessary. Streamwise saltation transport Q: Q was not measured, but was estimated using Equation (7). There are uncertainties in the c coefficient. The threshold friction velocity, u t, is set to u at which qðzþ becomes zero. As u is calculated from wind measurements assuming a logarithmic profile, u t is z dependent. For 14 April 22, u t is about.39 m s 1 for z =.5 m. For 17 April 22, u t is.4 m s 1 for z =.4 m. Parent soil particle size distribution: parent soil PSD is obtained through both laboratory analysis and dry sieving. The former is made by using a laser diffractometry (Microtrac FRA, Leed & Northrup Co. Ltd), and requiring the soil sample to be dispersed in solvents such as hexametaphosphate. For the sand dune, the distribution has a distinct mode at 21 lm, while for the Gobi site, the mode is at 144 lm. Soil samples are also analysed using dry sieves with the following mesh sizes: 1, 5, 25, 125, 9, 63, 45 and 32 lm

13 (plus a base). The results are shown in Figure 9 and, as can be seen, the particle size characteristics of the soils at the experiment site can be substantially different. It is now important to determine which particle size data should be used. The Gobi site laser-diffractometry PSD is first used for testing. We have compared the simulated and observed saltation flux q for 14 April 22 and 17 April 22 and found a reasonable agreement between the prediction and observation (not shown). The PSD of the saltating particles at height z can be estimated by calculating the ratio X I p i ¼ q i q i : ð22þ i HETEROGENEOUS SALTATION 371 Using the theory presented in this study, the calculation of p i for any z can be done for any given height. We have done so for z =.1,.2,.5,.1,.2 and.3 m and compared results with the PSD observed at the.3-m level and that of the parent soil (Figure 6). It is seen that for z =.1 m, the predicted PSD is similar to that of the parent soil. For higher levels, the mode of particle size decreases. This result is plausible, since large particles generally fly lower than smaller particles. However, a comparison of the predicted and the observed PSD for z =.3 m shows a substantial discrepancy: the former has a narrow range of particle sizes and a sharper peak than the latter. The PSD observed at z =.3 m is similar to that of the parent soil. This is in conflict with the model concept that large particles saltate at lower levels than smaller particles. In the model, this is determined by the fact that particles lift-off with vertical velocity proportional to ðu u t Þ and that large particles fall faster..4.3 q i /q, obs. z =.3 m Model, z =.1 m.5 m.1 m.3 m Gobi Soil, Laser Diffractometry q i /q d (µm) Figure 6. Simulated PSD for z =.1,.5,.1 and.3 m, compared with that observed at z =.3 m and of the parent soil, using 14 April 22 as an example.

14 372 YAPING SHAO AND MASAO MIKAMI It may be incorrect to use the laser-diffractometry PSD data for Gobi site to represent the parent soil PSD for three reasons. First, Figure 6 shows that the parent soil PSD and the observed PSD at z =.3 m level are almost identical, implying that the aerodynamic process has no effect on particle sorting. This contradicts the saltation theory. Second, the laser diffractometry PSD is obtained using the fully dispersed method, and is not the same as the PSD seen by wind. Third, the surface at the experiment site is a complex mixture of Gobi with small and large stones and sand dunes and, hence, it is not exactly certain what particle size signal is seen by the SPC. Using the theory described in Section 2.4 (e.g., Equation (21)), we can estimate what PSD the parent soil must have so that the PSD at z =.3 m is as observed. This is an inverse calculation and the results are shown in Figure 7. The parent soil PSD determined using the inverse calculation shows a particle size mode at around 2 lm. The results presented below are obtained using the inverse calculated PSD of the parent soil. We then compare the inverse calculated PSD with that of the parent soil analysed using dry sieving. Figure 8 shows the comparison of the simulated and observed saltation flux q for 14 and 17 April 22. In computing q for 14 and 17 April, the c coefficient in Equation (7) is set to 1.5 and 1, respectively. While the Owen (a).2.15 q i /q q i /q (b) Parent Soil (Inverse) Mod., z =.3 m Obs., z =.3 m Mod., z =.1 m Mod., z =.3 m d (µm) Figure 7. (a) Comparison of simulated and observed PSD for z =.3 m, together with the inverse calculated parent soil PSD, for 14 April 22; (b) as (a) but for 17 April 22. In addition, simulated PSD for z =.3 and.1 m are shown.

15 HETEROGENEOUS SALTATION 373 model has been verified on many occasions (e.g., Iversen and Rasmussen, 1999), the c coefficient is known to vary considerably, depending on whether saltation is supply limited. For supply limited saltation, c is smaller. Saltation over a Gobi surface covered by small and large stones is often supply limited. The decrease of c from 14 to 17 April is probably due to the fact that the Gobi surface was stablized by the first erosion event. As can be seen, the theory well reproduced the main features of the observations, apart from minor differences. The simulated evolution of PSD with height shown in Figure 7 now reproduces that observed at the.3-m level. Rasmussen et al. (1985) reported grain characteristics obtained from a field experiment in which saltation was recorded between the surface and.7 m. Their data indicate a weak decrease of the modal size with height. Leys and McTainsh (1996) measured particle size distributions at various levels between the surface and 2 m and found there is a significant decrease of the modal size with height below.5 m. Our model simulation is in qualitative agreement with the observations of Leys and McTainsh (1996). A stringent test to the theory is whether the inverse calculated PSD agrees with the observed parent soil PSD. In Figure 9, the inverse calculated PSD is compared with the dry-sieve PSDs of soil samples collected in the vicinity of the Qira site. For the sand-dune soil sample, the dry-sieve and (a) 2 q mod, c=1.5 q obs.6 u * (b) 2 q mod,, c= Time (LT, Day).4 Figure 8. Simulated and observed time series of saltation flux q together with friction velocity u : (a) for 14 April 22 and (b) for 17 April 22.

16 374 YAPING SHAO AND MASAO MIKAMI.2 Model Inverse Gobi, Dry Sieve PSD Density (µm -1 ).15.1 Farm land, Dry Sieve Sand dune d (µm) Figure 9. A comparison of inverse calculated PSD with dry-sieve PSDs for the Gobi, farmland and sand-dune soils at Qira. laser-differactometry PSD are similar, because sand particles are predominantly individual grains rather than aggregates. The inverse calculated PSD is in good agreement with the sand-dune soil PSD. However, this comparison does not unequivocally verify the model, because the surface at the Qira site is made of a mixture of soils (Gobi, sand dune and farm land) with quite different particle size characteristics, as Figure 9 shows. As the saltation measurements were made directly over the Gobi surface, we have expected the inverse calculated PSD to match the Gobi soil PSD. Unfortunately, this is not the case, as Figure 9 reveals. One possible reason for this discrepancy from expectation is the complex nature of the Gobi surface and the Gobi soil. About 7% of the Gobi surface is covered by small stones, and the model assumptions (e.g. Equations (7) and (12)) may be invalid for such surfaces. The Gobi soil shows strong variations in space and with soil depth and another possible reason is that the Gobi soil sample we analysed is unrepresentative. 5. Sensitivity to z The saltation theory requires friction velocity u, roughness length z and parent soil PSD as input. Embedded in the theory are the assumptions on particle lift-off velocity and its probabilistic distribution. There are

17 HETEROGENEOUS SALTATION 375 uncertainties in these assumptions. For simplicity, we have fixed the parameters associated with these assumptions, e.g., assuming the particle lift-off velocity distribution to be exponential and mean lift-off velocity to be ðu u t Þ. The roughness length turns out to be an important parameter and the quantitative predictions of the theory are sensitive to z. Figure 1 shows the sensitivity test for q with z =.3,.4,.5 and.1 m. Clearly, the magnitude of q varies to a considerable degree. For z =.5 m, the predicted q is in good agreement with the observations; for z =.3 and.4 m, the predictions are too small, while for z =.1 m, the predictions are almost four times too large. The sensitivity of PSD to z is shown in Figure 11, where it is again seen that the uncertainties in z result in quantitative differences in PSD predictions. On the other hand, both figures show that the qualitative features of the predictions are insensitive to z. From the modelling perspective, this sensitivity is understandable. First, q is an exponentially decaying function of z and the decaying speed depends on z, because z is a parameter used to describe both the (logarithmic) wind profile and the structure of turbulence (dissipation rate for turbulent kinetic energy); second, z is used to estimate u, and Q is proportional to u 3 and hence q is also sensitively related to z. As there are insufficient data for further comparison, we cannot conclude whether this sensitivity is due to the deficiency of the theory or a fact. 8 6 q obs z =.3 m z =.4 m z =.5 m z =.1 m q (gm 2 s 1 ) Time (LT, day 22) Figure 1. Sensitivity tests of the predicted saltation flux at z =.3 m to z, for z =.3,.4,.5 and.1 m.

18 376 YAPING SHAO AND MASAO MIKAMI.3.2 Parent Soil (Inverse) Obs, z =.3 m Mod, z =.1 m z =.3 m z =.4 m z =.5 m qi/q d ( µ m) Figure 11. Sensitivity tests of the predicted PSD at z =.3 m to z, for z ¼ :3,.4,.5 and.1 m. 6. Conclusions We have presented a theory for heterogeneous saltation, i.e., the saltation of multi-sized particles, as a further development of the saltation similarity theory described in Shao (25). A surface with a mixture of particles is approximated by a well sorted surface with uniform particles occupying fractions according to PSD. The theory first deals with the saltation of uniform particles. We have proposed the relationships for particle concentration, particle mass flux and particle momentum flux, namely, Equations (1), (2) and (3). The Js are saltation similarity functions, which are universal ; although further verification is required, the study of Shao (25) suggests that J q is an exponential function. Several newly established relationships are presented, e.g., the relationship between particle entrainment rate and momentum flux is given by Equation (5), that between particle entrainment rate and streamwise saltation flux by Equation (6), and that between surface particle momentum flux and streamwise saltation flux by Equation (8). We summarise the theory here. In studying heterogeneous saltation, we divide soil into a number of particle size bins, each with an entrainment mass flux. The theory makes two main statements: (1) for a bin with particle size d i, the streamwise mass flux qðz; d i Þ is given by qðz; d i Þ¼P i ^F s ðd i ÞJ q ðz; d i Þ;

19 HETEROGENEOUS SALTATION 377 and (2) if the particle size distribution at a given z r, p M ðd; z r Þ, is known, then that at level z, p M ðd; zþ, is given by p M ðd; zþ ¼cp M ðd; z r Þ X cpm ðd; z r Þ; where c ¼ J q ðz; dþ=j q ðz r ; dþ. The theory requires as input, (i) parent soil PSD; (ii) friction velocity u and (iii) roughness length z. Parent soil PSD (determined through dry sieving) is used to estimate P i as described in Section 2.2; u is required to evaluate ^F s ðd i Þ as described in Section 2.1; u and z are used to evaluate J q ðz; d i Þ. For the specific forms of J q, see Shao (25). If Q is measured, then it is sensible to partition Q into ^Qðd i Þ as described in Section 6 of Shao (25) and then calculate ^F s ðd i Þ from ^Qðd i Þ. The theory has two main uncertainties that require further examination. One involves the derivation of J q ðz; d i Þ, as already discussed in Section 7 of Shao (25). The other involves the linearization of heterogeneous saltation as reflected in Equations (9) and (1). The validity of these hypotheses awaits confirmation in future with more detailed data. The main achievement of this study is that a plausible framework for studying heterogeneous saltation is established. The relevant Qira field measurements are presented. The SPC measurements show that saltation fluxes fluctuate with time and the high frequency fluctuations are poorly correlated with those of wind speed. However, the low frequency fluctuations of saltation fluxes and wind speed are well correlated. We have attempted to use the Qira data to verify the theory, by examining whether the observed saltation flux and PSD can be reproduced. Our attempt is partially successful in that the observed time series of q is reproduced by the model. We have shown that the size distribution of saltating particles varies with height: at low levels, PSD is similar to that of the parent soil, but as height increases, the mode of PSD shifts to smaller values. This finding is in agreement with the observations of Leys and McTainsh (1996). We have used the model to inversely calculate the parent soil PSD from SPC measurements and compared it with the PSDs of several soil samples collected at the experimental site. The inverse calculated PSD is found to match the PSD of the sand-dune soil sample, but not that of the Gobi soil sample. Since only a limited amount of field data is available to this study, we have not been able to fully explain the implication of this unexpected outcome. The two likely reasons are, (1) the model assumptions are not valid for the Gobi surface, and (2) the Gobi soil sample is unrepresentative of the Gobi surface at large. Additional investigations will be carried out in future to clarify these issues. The q profile is found to be somewhat sensitive to z. This is because q is an exponential function of height and z is a parameter that affects both

20 378 YAPING SHAO AND MASAO MIKAMI the intercept and the scaling height of this function. This sensitivity imposes difficulties in the applications of the theory, if z cannot be estimated accurately. On the other hand, it is encouraging that the qualitative behaviour of the predictions is less sensitive to z. Acknowledgements We wish to thank Dr. M. Ishizuka of Wakayama University, Mr. Y. Yamada of RIKEN, Dr. F. Zeng and Mr. W. Gao of Xinjiang Institute of Ecology and Geography, CAS for their collaboration in the field observation at Qira, China. We are grateful to Dr. K. R. Rasmussen, Dr. M. Sorensen and another referee for their constructive comments. The field experiment is part of the ADEC project sponsored by the Ministry of Education, Culture, Sports, Science and Technology, the Japanese Government. References Anderson, R. S.: 1987, Eolian Sediment Transport as a Stochastic Process: The Effects of a Fluctuating Wind on Particle Trajectories, J. Geology 95, Anderson, R. S. and Haff, P. K.: 1991, Wind Modification and Bed Response During Saltation of Sand in Air, Acta Mech. Suppl. 1, Anderson, R. S. and Hallet, B.: 1986, Sediment Transport by Wind: Toward a General Model, Geol. Soc. Amer. Bull. 97, Bagnold, R. A.: 1941, The Physics of Blown Sand and Desert Dunes, Methuen, 265 pp. Doorschot, J. and Lehning, M.: 22, Equilibrium Saltation: Mass Fluxes, Aerodynamic Entrainment, and Dependence on Grain Properties, Boundary-Layer Meteorol. 14, Gillette, D. A.: 1981, Production of Dust that May be Carried Great Distances, Spec. Pap. Geol. Soc. Amer. 186, Iversen, J. D. and Rasmussen, K. R.: 1999, The Effect of Wind Speed and Bed Slope on Sand Transport, Sedimentology 46, Jensen, J. L. and Sorensen, M.: 1986, Estimation of Some Aeolian Saltation Transport Parameters: A Re-analysis of Williams Data, Sedimentology 33, Leys, J. and McTainsh, G. H.: 1996, Sediment Fluxes and Particle Grain Size Characteristics of Wind Eroded Sediments in South Eastern Australia, Earth Surf. Proc. Landforms 21, Mikami, M., Fujitani, T., and Zhang, X.: 1995, Basic Characteristics of Meteorological Elements and Observed Local Wind Circulation in Taklimakan Desert, China, J. Meteorol. Soc. Japan 73, Mikami, M., Osamu, A., Du, M. Y., Chiba, O., Fujita, K., Hayashi, M., Iwasaka, Y., Kai, K., Masuda, K., Nagai, T., Ootomo, T., Suzuki, J., Uchiyama, A., Yabuki, S., Yamada, Y., Yasui, M., Shi, G. Y., Zhang, X. Y., Shen, Z. B., Wei, W. S., and Zhou, J.: 22, The Impact of Aeolian Dust on Climate: Sino-Japanese Cooperative Project ADEC, J. Arid Land Stud. 11, Owen, R. P.: 1964, Saltation of Uniform Grains in Air, J. Fluid Mech. 2,

21 HETEROGENEOUS SALTATION 379 Rasmussen, K. R., Iversen, J. D., and Rautaheimo, P.: 1996, Saltation and Wind Flow Interaction in a Variable Slope Wind Tunnel, Geomorphology 17, Rasmussen, K. R., Sorensen, M., and Willetts, B. B.: 1985, Measurements of Saltation and Wind Strength on Beaches, in O. E. Barndorff-Nielsen, J. T. Mo ller, K. R. Rasmussen and B. B. Willets (eds.), Proceedings of the International Workshop on the Physics of Blown Sand, No. 8, Dept Theor. Statist., Aarhus University, Denmark, pp Raupach, M. R.: 1991, Saltation Layers, Vegetation Canopies and Roughness Lengths, Acta Mech. Suppl. 1, Rice, M. A., Willetts, B. B., and McEwan, I. K.: 1995, An Experimental Study of Multiple Grain-size Ejecta Produced by Collisions of Saltating Grains with a Flat Bed, Sedimentology 42, Shao, Y.: 25, A Similarity Theory for Saltation and Application to Aeolian Mass Flux, Boundary-Layer Meteorol. 115, Shao, Y. and Li, A.: 1999, Numerical Modelling of Saltation in Atmospheric Surface Layer, Boundary-Layer Meteorol. 91, Shao, Y., Raupach, M. R., and Findlater, P. A.: 1993, The Effect of Saltation Bombardment on the Entrainment of Dust by Wind, J. Geophys. Res. 98, Sorensen, M.: 1985, Estimation of Some Aeolian Saltation Transport Parameters from Transport Rate Profiles, in O. E., Barndorff-Nielsen, J. T., Mo ller, K. R. Rasmussen and B. B. Willets (eds.), Proceedings of the International Workshop on the Physics of Blown Sand, No. 8, Dept Theor. Statist., Aarhus University, Denmark, pp Sorensen, M.: 24, On the Rate of Aeolian Sand Transport, Geomorphology, Williams, G.: 1964, Some Aspects of the Eolian Aaltation Load, Sedimentology 3, Yamada, Y., Mikami, M. and Nagashima, H.: 22, Dust Particle Measuring System for Streamwise Dust Flux, J. Aird Land Stud. 11,