Statistical behaviors of different-sized grains lifting off in stochastic collisions between mixed sand grains and the bed in aeolian saltation

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1 JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 112,, doi: /2006jd007888, 2007 Statistical behaviors of different-sized grains lifting off in stochastic collisions between mixed sand grains and the bed in aeolian saltation Wan-Qing Li 1,2,3 and You-He Zhou 1,2 Received 4 August 2006; revised 1 May 2007; accepted 3 August 2007; published 27 November [1] Multiple-size splash models are derived from the simulation results of mixed grainbed impact process of windblown sand flow based on the Particles Dynamics Method (PDM) and parallel algorithm. Unlike previous studies, a probability density distribution of sand diameter is considered in the present study, in which a two-dimensional mixed sand bed is generated by a random method. After the diameter distribution of incident grains is divided into n subregions of representative diameters, first, the collisions are simulated out for each representative diameter of incident grains with an incident velocity impacting the mixed sand bed, to which the information of grains experiencing saltation liftoff may be gained at the end of collisions. After that, a statistical approach is proposed to obtain the average values of velocity and number of the ejecting and rebounding grains as well as the probability density function (PDF) of ejection particle sizes. The results confirm earlier findings that the ratio between outgoing and incoming speed remains about 60% and the ejection angles are typically between However, other properties of grains experiencing saltation liftoff depend not only upon incident velocity as previously argued, but also much upon grain size of incident grains. We found that rebound angle decreases and death rate of incident particles increases with incident grain size exponentially, and ejection speed and number increase logarithmically with both speed and diameter of incident grains. In addition, the PDF of initial diameter is also presented for the first time. These results well agree with the multiple-size measurement data. Citation: Li, W.-Q., and Y.-H. Zhou (2007), Statistical behaviors of different-sized grains lifting off in stochastic collisions between mixed sand grains and the bed in aeolian saltation, J. Geophys. Res., 112,, doi: /2006jd Introduction [2] As an essential way to clarify the wind erosion of soil and sand desert formations research on mechanisms of windblown sand movements has been intensified [e.g., Bagnold, 1941; Owen, 1964]. Except for sand/dust storms in which the suspension is dominant, the saltation movement plays a central role in the wind erosion process, along with the creeping motion [e.g., Shao, 2000]. A great deal of effort has been devoted to understanding saltation by analytical and numerical models, for instance, Anderson and Haff [1988, 1991] and McEwan and Willetts [1991, 1993]. More recently, valuable insights into the process of saltation have been provided based on the knowledge of initial speed and angle of liftoff grains by Zheng et al. They argued that the stratification pattern of mass flux may be used to explain the sand cut phenomena reasonably [Zheng et al., 2004a; Zheng, 2006], to demonstrate that 1 Key Laboratory of Mechanics on Western Disaster and Environment, Ministry of Education of China, Lanzhou, China. 2 Also at Department of Mechanics, Lanzhou University, Lanzhou, China. 3 Now at Department of Computer and Software, Hangzhou Dianzi University, Hangzhou, China. Copyright 2007 by the American Geophysical Union /07/2006JD the electric field plays an important role in sand flux [Yue and Zheng, 2006; Zheng et al., 2006a], and to provide theoretical predictions [Zheng et al., 2004b]. Especially, the probability density of angular velocity was asserted for the first time [Zheng et al., 2006b], and aeolian sand ripples were simulated successfully and they agree well with the natural ones whether in shape or propagation speed [Bo et al., 2006]. Although these splash functions stem from very limited amounts of empirical data, they do enable realistic modeling of the process by treating the outcome of collision statistically in terms of the number, speed and angle of ejecta. There remain, however, important aspects of the process that need to be and can be amplified or verified by mixed size grain-bed impact simulation, such as the grain size parameter. [3] Saltation implies different-size sand particles moving away off and close by the sand bed driven by wind. Because of gravity, some particles which are defined as incident grains will drop down and collide with the bed to exchange mass, momentum and energy. If the incident grain takes off from the bed after collision, it is referred to as a rebound grain, while the others lifted off on the surface are called ejection grains, as shown in Figure 1. Generally, not all incident particles will rebound; a few of them will be captured by the bed. We define death rate of incident particles as h dr which means the percentage of those nonrebounding, where dr is the abbreviation of death rate. 1of9

2 measurement of Rice et al. [1995], are displayed to show the efficiency of the simulations. Figure 1. Schematic diagram of a grain-bed collision. The grain in the saltation layer impacts the bed with an incident speed V in, diameter D in and angle from horizontal q in.it rebounds from the bed at V rb and q rb ; meanwhile, N ej ejected grains leave the bed at speed V ej, diameter D ej and angle q ej. The superscripts in and ej are the abbreviation of rebound and ejection, respectively. At the end of each collision, the more energy incident grains have, more particles will be splashed away from the bed with higher liftoff speed. It is obvious that the motions of sand particles in saltation depend not only on the liftoff speed, but also strongly on grain size. A complete saltation model requires knowledge of the behavior of each grain size in the population. The knowledge of particle size in grainbed impact process may be very important to understand aeolian sand saltation and is needed for prediction of the evolution of a mixed grain cloud, along with the consequent modification of the bed in terms of texture and bed forms. Until recently, the role of grain size in grain-bed collision had been even less well studied, despite the fact that the grain-bed interaction plays a crucial role in saltation. Progress has been made toward a multiple-grain-size model. Haff and Anderson [1993] used computer simulation to incorporate two grain sizes in which they produced splash functions for coarse grains (0.32 mm) impacting a bed consisting only of fine grains (0.23 mm), and for fine grains impacting a coarse bed. Anderson and Bunas [1993] developed a cellular automaton model of a two-grain-size (again 0.32 and 0.23 mm) sand bed, which was bombarded by saltating grains from both of these size categories. In experiments Rice et al. [1995] adopted three ranges of size ( , and mm) particles to observe the behaviors of ejecta in which the grains of each range are dyed different color. Color high-speed film was therefore used so that three variously colored size fractions of the sand population could be more recognized on film than before [Willetts and Rice, 1989]. Recently, Shao et al. found that soil particle size distribution is one of the determined parameters for saltation mass flux profile [Shao, 2005; Shao and Mikami, 2005]. [4] Here, a statistical approach of liftoff velocity of the random collisions between incident grains and the sandy bed with mixed grain diameters is proposed on the basis of the PDM simulations. The quantitative results for a case study of the sandy bed with the Gaussian distribution of mixed diameter of volume average of 321 mm in the region of ( mm), to which the average diameter is nearly same as that of the multiple-size grains employed in the 2. Theoretical Model [5] The sophisticated PDM (also called Discrete Element Method DEM) has been used for many years in the study of collisions between particles, including both rotational and transitional motion, by solution of the Newtonian equations of particle motion [e.g., Haff and Anderson, 1993; Zhou et al., 2006]. If the total force exerting on one particle is known as F, then the path of this particle can be tracked by equation (1) F ¼ m d2 x dt 2 where m is its mass and x its position vector. As shown in Figure 2, grain i comes into contact with particle 1 and particle 2 at velocity V i and angular velocity w i simultaneously. Besides gravitational force, the total force of particle i inserted includes the contact forces, namely normal forces F ni1 (compressive), F ni2 caused by the stiffness of the particles between them, tangential forces F ti1, F ti2 provided by the surface friction due to relative movement, and torques M i1, M i2 caused by rotation. Although the wind turbulence influences the movement of grains in the saltation layer, its effect can usually be neglected in the study of collision since the timescale of the collision is so short compared to that for a load like aerodynamic force. For simplicity we restrict our study to two dimensions, all forces lie in x y plane, and angular velocities and torques are taken positive for anticlockwise rotations in subsequent discussions. More detailed discussions of the theoretical model are given by Zhou et al. [2006] Normal Forces of Contact [6] The simplest model of contacting grains exhibiting finite compressibility is the linear spring model. When two grains are in contact, a repulsive force proportional to the amount of overlap of the two particles is invoked. The Figure 2. Schematic view of forces acted on particle i by gravity and by particles 1 and 2. Symbols with bold are vector quantities. ð1þ 2of9

3 overlap is defined as the difference between the sum of the radii of the two particles and the distance between their centers of mass, and the force is proportional to this overlap through the rigidity constant a. To reflect that the collisions between particles are usually inelastic, the normal contact force produced by particle j, of radius R j, on particle i, of radius R i, includes a damping term, as F n ij 8 < ad ij b i dij _ d ij 0 ¼ : 0 d ij < 0 where d ij is the amount of overlap between particles i and j at an instant during contact and can be written as ð2þ vectors by w i and w j, we get the expression of v tij in the form v tij ¼ v j v i tij tij þ w i R i þ w j R j ð7þ in which R i and R j are the relative vectors from the mass centers of grains i and j to their contact point, respectively Equations of Motion [8] Here, the two-dimensional Cartesian coordinate system of xoy with unit vectors of i and j is employed. Once the resultant force F i exerted on each grain i is known, we can describe the motion of the grains by the following equations m i d 2 x i =dt 2 ¼ F xi ¼ X F n ij þ X! F t ij i ð8þ j6¼i j6¼i in which d ij ¼ R i þ R j r ij ð3þ m i d 2 y i =dt 2 ¼ F yi ¼ X j6¼i F n ij þ X! F t ij j ð9þ j6¼i r ij ¼ r j r i and r ij ¼ r j r i ð4þ I i d 2 q i =dt 2 ¼ M i ¼ X r ij; con F t ij þ F n ij k ð10þ j6¼i Here, r i and r j indicate the position vectors of mass center of the grains, b i is a damping constant and d _ ij represents the rate of change of the distance between centers of the colliding particles, or the differentiation of d ij with respect to time variable t. The minus sign reflects the fact that the damping force always opposes the (relative) motion. The damping is relevant to the inelastic part of collision; hence the constant factor b i is related to the coefficient of restitution e. By means of the analysis of Haff and Anderson [1993], we have e ¼ e pg i=w 1i pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where g i = b i /2~m i, w 1i = w 2 0i pffiffiffiffiffiffiffiffiffiffi g2 i, w 0i = a=~m i, and ~m i = m i m j /(m i + m j ) is the reduced mass. Once the coefficient of restitution e is specified, the constant factor b i is obtained by equation (5) Tangential Forces of Contact [7] Because of existence of friction, there is a tangential force on the contacting point of surface during the collision, which yields rotation of particles in movement except for the translation. Considering first the case in which the surface of one grain is in contact with a second one, a mutual slipping at the point of contact, i.e., the relative tangential velocity v t,ij between grains j and i at the contact point is nonzero. It has been known that the simplest model employed for such a slipping contact is the Coulomb friction model in which the magnitude of the tangential friction force F t is assumed to be proportional to the magnitude of the instantaneous normal force jf n j, through a friction coefficient m, and its direction is opposite to the slip. For particle i, thus, we can write F t ij ¼ mf n ij tij where t ij is a unit vector parallel to v tij. Considering rotation of the grains, denoting their angular velocity ð5þ ð6þ Here, x i and y i are the coordinates of the mass center of grain i, I i is the inertial moment of grain i about the axis passing through its mass center, and the summation extends over all particles j contacted to the grain i, k = i j and r ij,con is a relative vector running from the center of mass of grain i to the point of contact with particle j. [9] To pursue the movement of each grain, the Runge- Kutta method of order 4 is employed to perform the time integration of the dynamic equations (8) (10) with an initial state, where the velocity components of the grain are calculated by v xi = d xi /dt, v yi = dy i /dt, and w i = dq i /dt. 3. Arithmetic Approach [10] Before the simulation of the mixed size grain-bed impact process, a mixed target grain bed should be prepared. Numerous measurements [e.g., Bagnold, 1941; Willetts and Rice, 1989; Rice et al., 1995] of the diameter distribution of natural sand grains implied that the distribution is approximately Gaussian. In order to consist with nature and compare with experimental data, in present study we choose Gaussian distribution f = f(d) in which D means the diameter of a grain. The unit of diameter D is m and its average value nearly equal to the literature of Rice et al. [1995], as shown in Figure 3. N 0 grains will be used to comprise a mixed sand bed, and the maximum and minimum diameters of the distribution are D max and D min, respectively. Divide the region of diameter distribution into n equal subregions. In each subregion (D i, D i+1 ), we select a representative diameter named by ~D i 2(D i, D i+1 ). Then the number of the grains in (D i, D i+1 )isn i =int(n 0 f(~d i )DD), where int is the function of integer. By means of the program of random number generator [Knuth, 1981], N i uniform random numbers can be generated with magnitude in the region of (D i, D i+1 ). While rearranging these random numbers N = Pn N i randomly, a stochastic sequence is i¼1 prepared. On the basis of the particle cooling technique [Haff and Anderson, 1993], as Figure 4a shows, we impel 3of9

4 [12] Because of ejectors from the mixed size grain bed, it is not difficult for us to give the PDF of liftoff diameter D ej. We denote the same liftoff diameter subregion as the target bed with different subscript (D k, D k+1 )(k =0,1,2,, n), and select the same representative diameter ~D ej k = ~D k 2 (D k, D k+1 ). Let N ej l (D in i, V in j ) present the total number of grains undergoing aeolian liftoff by saltation in lth sample for one ej RI, and N l,k (~D ej k, D in i, V in j ) means the number of grains experiencing saltation liftoff in the subregion of (D k, D k+1 ), then the probability of grains undergoing aeolian liftoff by saltation in (D k, D k+1 ) is defined by P l (~D ej k, D in i, V in j ) = N ej ~D ej l;kð k ; Din i ; Vj in Þ. For L samples the average value of probability can be expressed as P(~D ej k, D in i, V in j )= 1 P L N ej D in l ð i ; Vj in Þ L P l (~D ej k, D in i, l¼1 V in j ). Considering the total collisions of the mixed sand grains with a specific incident speed, then, we get, Figure 3. Diameter distribution of a mixed sand bed with A the Gaussian distribution of P(D) =y 0 + pffiffiffiffiffi these N particles to drop down in a periodic box one after another with a random velocity and then control their movement and collision together by using PDM. Because no energy will be input after all grains dropped in the box, the total energy of N particles will dissipate to zero owing to inelastic collision as time goes on. Then, the target bed is achieved, as Figure 4b shows. [11] It is difficult to measure the diameter distribution of sand grains in the saltation layer. For one location, the sand grains in the saltation layer should be from the bed of the same location, their diameter distribution being similar to that of the sand bed. For simplicity we suppose the incident group has the same distribution as the target bed. We defined, as Figure 1 shows, the process of one incident grain bombarding the surface of mixed grain bed with diameter D i in = ~D i, V j in and q in as one representative impact (RI). L impact points (samples) on the surface of bed will be chosen for each RI. If the incident grain does not rebound in L* samples, the death rate of incident particles can be described as h dr = L*/L 100. Denoting random physical qualities (e.g., q rb, h dr, V rb /V in, q ej, V ej, N ej ) of liftoff particles by S l in the lth sample, it may be related to conditions of incident grains as S l = S l (D i in, V j in, q in ). Since the features of grain-bed impact are only slightly dependent on the narrow range (8 15 ) of incident angle q in, which cover the range of expected impact angles in aeolian saltation [e.g., Bagnold, 1941; Anderson and Hallet, 1986; Greeley and Iversen, 1985; Anderson and Haff, 1988, 1991; Zhou et al., 2006], we neglect its effect and choose one medium value of (8 15 ), i.e., 11.5, as incident angle in the following simulation. By averaging S l for all samples L of one RI, we get the value S(D i in, V j in ) as follows: S D in i ; Vj in ¼ 1 X L L l¼1 S l s p=2 D in i ; Vj in 2 D m e ð Þ2 =s 2 where y 0 = , A = 0.98, m = 2.97 and s = The unit of diameter D is m. The diameter of volume average is about 321 mm. ð11þ P ~D ej k ; V j in ¼ Z Dmax D min ¼ Xn i¼1 P ~D ej k ; Din i ; Vj in f D in i dd in P ~D ej k ; Din i ; Vj in f D in i DD ð12þ The probability density is also presented as, p ~D ej k ; V j in ¼ P ~D ej k ; V j in DD 4. Simulation Results and Discussions ð13þ [13] Because of some f(~d i ) not being great enough to make N i =int(n 0 f (~D i )DD) larger than integer 1, the region of mm is chosen in this paper. Moreover V j in is Figure 4. Before generation of the mixed sand bed, some grains are fixed on the bottom (see the biggest black grains) and the periodic boundary is employed at the edges. The PDF of diameter distribution is shown in Figure 3 with N = 900. (a) Particles are dropped in the box. (b) Target bed is prepared. 4of9

5 4.2. Rebound Particles [15] Figure 6a shows the variation of rebound angle with speed and diameter of incident grains. From Figure 6a, we find that rebound angles are not typically as Anderson and Haff [1988] reported or in the work by Anderson and Haff [1991], but decay rapidly from 90 to 20 with incident particle sizes increasing. On the contrary, incident speed hardly influences rebound angles which also agrees well with earlier arguments [e.g., Anderson and Haff, 1991; Zhou et al., 2006]. Therefore averaging the results of Figure 6a along incident speed, the relationship between rebound angles and incident particle diameters is plotted in Figure 6b. Using an exponential function as equation (14) expressed, q rb D in ¼ 161:46 e 2:50 Din þ 0:15 ð14þ the curve can be well fitted with correlation coefficient R 2 = Measured data [Rice et al., 1995] were somewhat Figure 5. For different samples of each RI, (a) curves of ejection angle q ej versus incident particle size and (b) curves of ejection number versus ejection grain size. selected in the region of 18 m/s which cover the range of expected impact speeds in aeolian saltation [Anderson and Haff, 1991; Zhou et al., 2006; Xie et al., 2006] Stability Analysis [14] Before we display those statistical results of physical quantities S which we concentrate in the theoretical studies of windblown sand saltation to show the effect of particle size, we should check the stability of statistical approach in our simulation since S depends on the number of samples to each collision with a representative diameter of incident grains. According to statistical theory, we know that the statistical quantities are steady when the number of samples is large enough. When the samples are taken as 20, 50, 80, and 110, the numerical results of ejection angle and PDF of liftoff diameter are plotted in Figure 5. It can be found that the relative errors for all statistical quantities are all within 2% between the results of the samples taken as 80 and 110, and the maximum relative error occurs for the ejection angle when the incident particle size is 140 mm. Thus the statistical results of the stochastic collisions are reliable when the samples are taken as 110. In the following simulation results for the mixed sand grain-bed collisions, to each representative collision there are 110 samples. Figure 6. (a) Mesh of rebound angle q rb varies with incident particle size and incident speed, compared with those experimental data of Rice et al. [1995] in which incident grains are indicated as follows: solid circle, coarse grains (D mm); solid triangle, medium grains (D mm); solid square, fine grains (D mm). (b) Specifications of Figure 6a. 5of9

6 Figure 7. Ratio of rebound speed and incident speed V rb / V in varies with grain size and speed of incident particle, compared with those experimental data of Rice et al. [1995] in which incident grains are indicated as follows: solid circle, coarse grains (D mm); solid triangle, medium grains (D mm); solid square, fine grains (D mm). smaller than predicted, perhaps because of the different detection methods. It is implied that small particles tend to rebound vertically and more horizontal energy is converted to vertical component than with larger grains. [16] Incident particle size effects rebound angle heavily and rebound speed weakly according to Figure 7. The ratio of in and out speed of incident particles is about 60%, consistent with previous literature [e.g., Anderson and Haff, 1988, 1991; Rice et al., 1995]. [17] In windblown sand some particles deposit on the bed, meanwhile some others lift off from the bed. When the particle number captured by bed equals that of liftoff, a steady state is achieved [e.g., Anderson and Haff, 1988; McEwan and Willetts, 1991, 1993; Shao and Raupach, 1992]. According to earlier literature [e.g., Mitha et al., 1986; Anderson and Haff, 1988, 1991] there are 5 6% of incident grains missing. This percentage has been defined as death rate of incident particles h dr in this paper. It increases exponentially with incident particle size from 5% to 25%, as Figure 8a shows. Only when diameter of incident grains is in 120 mm 160 mm, the rate h dr agrees with 5 6%. It implied that larger incident grains do not rebound easily. Because h dr does not vary with incident speed evidently, averaging the rate along with incident speed, the curve is plotted in Figure 8b. It is well fitted by exponential function of equation (15) with correlation coefficient R 2 = Figure 8. (a) Mesh of death rate of incident particles h dr varies with incident particle size and incident speed. (b) Specifications of Figure 8a. h dr D in D ¼ 0:064 e in 3:49 0:036 ð15þ 4.3. Ejection Particles [18] From Figure 9 we find that ejection angle does not vary apparently with diameter and speed of incident grains, and it is typically in the region of This can be compared with the results of Werner s experiments [Werner, 1987], which showed mean ejection angles ranging from Figure 9. Ejection angle q ej varies with incident speed and particle size, compared with those experimental data of Rice et al. [1995] in which incident grains are indicated as follows: solid circle, coarse grains (D mm); solid triangle, medium grains (D mm); solid square, fine grains (D mm). 6of9

7 logarithmically with momentum of incident grains. Using equations (17) (18), V ej M in ¼ 0:43 þ 0:077 ln M in þ 0:029 ð17þ N ej M in ¼ 17:96 þ 18:05 ln M in þ 0:37 ð18þ These two curves can be well fitted with correlation coefficient of and , respectively. Comparing with the experimental data of Rice et al. [1995], our results are in good agreement with them PDF of Liftoff Particle Size [19] Particle size is important to particle motion, influencing both particle liftoff and particle response time [Shao, 2005]. After grains impact a natural sand bed in aeolian sand saltation, some grains on the bed surface will be ejected with different diameter, in addition to various Figure 10. (a) Mesh of ejection speed V ej varies with incident particle size and incident speed, compared with those experimental data of Rice et al. [1995] in which incident grains are indicated as follows: solid circle, coarse grains (D mm); solid triangle, medium grains (D mm); solid square, fine grains (D mm). (b) Ejection speed as a function of incident momentum. about 56 to 73. Willetts and Rice [1989] measured mean ejection angles of about 55. Rice et al. [1995] observed mean ejection angles in the region of Otherwise, speed and number of ejecta increase with both incident grain size and speed increasing, as Figures 10a and 11a show, supplementing the limited knowledge [e.g., Mitha et al., 1986; Anderson and Haff, 1988, 1991] that these two physical quantities only depend on incident speed. Using momentum of incident grains, the effect of size and speed can be formulated as equation (16) expresses, M in ¼ mv ¼ 4p D in 3 r 3 2 p V in ð16þ where r p = 2500 kg m 3 is the density of sand grains. The functions of ejection speed and number with momentum are plotted in Figures 10b and 11b, respectively. From Figures 10b and 11b we find that speed and number of grains undergoing aeolian liftoff by saltation all increase Figure 11. (a) Ejection number of per collision N ej varies with incident particle size and incident speed, compared with those experimental data of Rice et al. [1995] in which incident grains are indicated as follows: solid circle, coarse grains (D mm); solid triangle, medium grains (D mm); solid square, fine grains (D mm). (b) Ejection number as a function of incident momentum. 7of9

8 Experimental data of Rice et al. [1995] is on both sides of the curve of liftoff. 5. Conclusions [20] In order to provide insight on the collision mechanism in windblown sand saltation, on the basis of PDM simulations of the stochastic mixed diameter grain-bed impact, a statistical approach to predict the features of liftoff diameter of grains is proposed in this paper. The obtained numerical results of a case study exhibit that in addition to incident speed, the behaviors of grains experiencing saltation liftoff also depend much on diameter of incident grains. With incident particle size increasing, rebound angle decays exponentially and the percentage of nonrebounding grains h dr increases exponentially, and rebound speed is about 60% of incident one. Except that ejection angle is almost independent of incident particle size, speed and number of grains experiencing saltation liftoff relate to both speed and diameter of incident grains, and increase logarithmically with incident momentum M in. Finally the PDF of liftoff diameters is for the first time reported in this paper. The results of this paper well agree with the experimental data of the multiple-size sand grains, which suggests to us that the simulation model for natural sand grain-bed collisions in a windblown saltation layer should be conducted by the model of mixed sand grains. Figure 12. (a) PDFs of liftoff diameter for different incident speed, compared with those experimental data of Rice et al. [1995] in which incident grains are indicated as follows: solid circle, coarse grains (D mm); solid triangle, medium grains (D mm); solid square, fine grains (D mm). (b) Specifications of Figure 12a. velocities. Regrettably, the PDF of liftoff grain size is very difficult to detect by experiments [e.g., Williams, 1964; Willetts and Rice, 1989; Rice et al., 1995] and uniform size grain-bed impact simulation [e.g., Werner and Haff, 1988; Anderson and Haff, 1991]. Herein we get the PDF of liftoff particle size and relation with diameter distribution of initial sand bed for the first time. From Figure 12a we find that diameter distributions of liftoff particles at different incident speed are nearly the same. Therefore averaging the results along incident speed, the PDF of liftoff size is plotted in Figure 12b. Using the similar diameter PDF of target bed, the bins are well fitted by equation (19) ej p ~D ej 1:13 k ¼ 0:024 þ p s ffiffiffiffiffiffiffiffi ð~d e 2 k m Þ 2 s 2 p=2 ð19þ with s = 1.70, m = 2.81 and R 2 = Comparing with the target bed, one finds that diameter distribution of liftoff grains moves slightly toward smaller values of diameter. [21] Acknowledgments. This research was supported by a grant of the Key Project of the National Natural Science Foundation of China ( ), the National Key Basic Research and Development Foundation of the Ministry of Science and Technology of China (2003CB716707), the National Natural Science Foundation of China for Abroad Outstanding Young Scientists, the Key Fund of Sciences and Technologies of the Ministry of Education of China, and the Science Fund of the Ministry of Education of China for Ph.D. Program. The computations of this research were performed in the Center of High Capability Parallel Computation of Department of Mechanics, Lanzhou University. The authors would like to express their sincere appreciation for these means of support. References Anderson, R. S., and K. L. Bunas (1993), Grain size segregation and stratigraphy in Aeolian ripples modeled with a cellular automaton, Nature, 365, Anderson, R. S., and P. K. Haff (1988), Simulation of aeolian saltation, Science, 241, Anderson, R. S., and P. K. Haff (1991), Wind modification and bed response during saltation of sand in air, Acta Mech. Suppl., 1, Anderson, R. S., and B. Hallet (1986), Sediment transport by wind: Toward a general model, Geol. Soc. Am. Bull., 97, Bagnold, R. A. (1941), The Physics of Blown Sand and Desert Dunes, Methuen, New York. Bo, T. L., L. Xie, and X. J. Zheng (2006), Numerical approach to wind ripple in desert, Int. J. Nonlinear Sci. Numer. Simul., 8, Greeley, R., and J. D. Iversen (1985), Wind as a Geologic Process, Cambridge Univ. Press, Cambridge, U. K. Haff, P. K., and R. S. Anderson (1993), Grain scale simulations of loose sedimentary beds: The example of particle-bed impact in aeolian saltation, Sedimentology, 40, Knuth, D. E. (1981), Seminumerical Algorithms, The Art of Computer Programming, 2nd ed., vol. 2, p. 116, Addison-Wesley, Boston, Mass. McEwan, I. K., and B. B. Willetts (1991), Numerical model of the saltation cloud, Acta Mech. Suppl., 1, McEwan, I. K., and B. B. Willetts (1993), Adaptation of the near surface wind to the development of sand transport, J. Fluid Mech., 252, Mitha, S., M. Q. Tran, B. T. Werner, and P. K. Haff (1986), The grain-bed impact process in Aeolian saltation, Acta Mech., 63, Owen, P. R. (1964), Saltation of uniform particles in air, J. Fluid Mech., 20, of9

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