PUBLICATIONS. Journal of Geophysical Research: Earth Surface. Aeolian creeping mass of different grain sizes over sand beds of varying length

Transcription

1 PUBLICATIONS Journal of Geophysical Research: Earth Surface RESEARCH ARTICLE Key Points: Measure creeping mass with different particle sizes and length of sand beds Develop a model of aeolian creeping mass with friction velocity Analyze the effect of particle sizes and length on aeolian creeping mass Correspondence to: H. Cheng, chengh@bnu.edu.cn Citation: Cheng, H., C. Liu, X. Zou, J. Li, J. He, B. Liu, Y. Wu, L. Kang, and Y. Fang (2015), Aeolian creeping mass of different grain sizes over sand beds of varying length, J. Geophys. Res. Earth Surf., 120, , doi:. Received 11 OCT 2014 Accepted 9 JUN 2015 Accepted article online 11 JUN 2015 Published online 27 JUL 2015 Aeolian creeping mass of different grain sizes over sand beds of varying length Hong Cheng 1, Chenchen Liu 1, Xueyong Zou 1, Jifeng Li 1, Jiajia He 1, Bo Liu 1, Yongqiu Wu 1, Liqiang Kang 1, and Yi Fang 1 1 State Key Laboratory of Earth Surface Processes and Resource Ecology, MOE Engineering Center of Desertification and Blown-sand Control, Beijing Normal University, Beijing, China Abstract Creep is an important mode of aeolian sand transport, but it has received little attention in previous studies due to experimental difficulties and insufficient theory. In this study, we conducted 116 groups of experiments with three repeats for each group in a wind tunnel to measure the creeping mass of four different mean grain sizes (152, 257, 321, and 382 μm) over six bed lengths (2.0, 3.5, 5.0, 6.5, 8.0, and 10.0 m) at six different friction velocities (0.23, 0.35, 0.41, 0.47, 0.55, and 0.61 m/s). We attempted to develop a comprehensive model of the aeolian creeping mass by analyzing the effect of wind velocity, the particle size, and the sand bed length based on the experimental data. The primary conclusions are as follows: (1) the complex relationship among the wind velocity, the grain size, the length of the bed, and the surface shape determines sand creep. There was no unified formula to express the effect of particle sizes and the sand bed length on aeolian creeping masses, and their effects appeared to depend on each other and wind velocity, whereas the creeping mass increases with increasing wind velocity for any particle size with any length of sand bed. (2) This paper presented a predicting model to determine the aeolian creeping mass, whose calculating results can match to experimental data with correlation coefficients (R 2 )of0.94orhigher.(3)the effect of grain size on creeping mass can be classified into three categories: the creeping mass increases with increasing grain size, the creeping mass initially decreases and subsequently increases with increasing grain size, and the creeping mass fluctuates with the grain size. (4) The effect of increasing bed length appears to depend on the grain size. For mean grain sizes of 152, 257, and 321 μm, creep initially increases with increasing bed length before decreasing above a certain value, while for a mean grain size of 382 μm, the creeping mass gradually increased with increasing bed length. The results help to elucidate aeolian creep and provide an intense foundation for advanced study American Geophysical Union. All Rights Reserved. 1. Introduction Scientists have studied aeolian sand transport for several decades; however, all studies remain within the framework established by Bagnold [1941]. This framework classified sand grain displacement into three primary modes (creep, saltation, and suspension) according to the characteristics of motion. In contrast to the numerous studies of saltating sand grains [Bagnold, 1941; Shao and Raupach, 1992; Zou et al., 1999, 2007; Dong et al., 2004; Cheng et al., 2006, 2009; Huang et al., 2008] and suspension [Liu, 1960; Pye and Tsoar, 1990], few studies have focused on creeping sand grains. The complex material and energy transfer between sand grains during creep and saltation, together accounting for 95% of the total aeolian sand transport [Bagnold, 1941], are the most important dynamic processes of aeolian sand transport. Bagnold [1941] first studied aeolian creep by using a bed trap with a 1 3 mm entrance in a wind tunnel and noted that the creeping proportion of the total sand flux was 25%. However, the creeping proportion greatly differed in existing literatures, such as 25% [Bagnold, 1941; Willetts and Rice, 1985], 16% [Chepil, 1945], 7 17% [Horikawa, 1960], 20% [Namikas, 2003], 3 6% [Nickling and McKenna Neuman, 1997], 4 29% [Dong et al., 2004], and 19 57% [Cheng et al., 2013]. Furthermore, the creeping proportion decreased with increasing wind velocity [Chepil, 1959; Horikawa, 1960; Wu, 2003]. Although bed traps were regarded as an effective method to measure creeping transport masses, creeping sand grains and saltating sand grains were both captured by bed traps. Thus, Bagnold [1941] overestimated the creeping mass. Bagnold s approach has been repeated in several other studies, using both larger [Wu et al., 2011] and varied [Chepil, 1959; Qi et al., 2001; Wu, 2003] widths of bed traps; however, no accurate creeping mass was reported. Based on the relationship between the aeolian sand flux and height, Dong et al. [2004] thought that the aeolian sand flux CHENG ET AL. AEOLIAN CREEPING MASS 1404

2 at zero height was equal to the creeping mass. In fact, there were many saltating sand grains within the sand flow at zero height. Recently, Zou et al. [1999, 2007] developed a noncontact s optical method to measure the aeolian sand flow using high-speed digital photography. Several scientists measured creeping mass by analyzing pictures taken by high-speed digital photography [Wang et al., 2009; Zhang et al., 2010], but the approach was only suited for aeolian sand flow with a sparse concentration and ignored that there was a thickness Figure 1. A layout scheme of the roughness elements in the wind tunnel. of creeping sand grains [Zhang et al., 2010]. It is generally considered that the above mentioned results from bed trap measurements and model inversion results have overestimated the creeping mass, whereas the optical approach has underestimated the creeping mass. This debate highlights the lack of accurate experimental data regarding creeping mass; thus, it is necessary to carry out additional research regarding creeping sand grains. Although several studies have indicated that the creeping mass varies with grain size [Chepil, 1959; Dong et al., 2004; Wu, 2003], these studies generally relied on qualitative results that lacked accurate experimental data. In particular, the length of the bed used in the experiments has a significant effect on sand movement, as demonstrated by the wind erosion equation (WEQ) [Woodruff and Siddoway, 1965]. In recent decades, many studies have been performed on this problem [Bagnold, 1941; Chepil and Milne, 1941; Chepil, 1946, 1959; Woodruff and Siddoway, 1965; Stout, 1990; Shao and Raupach, 1992; Fryrear and Saleh, 1996; Gillette et al., 1996; Li and Zhang, 2006]; however, there were significant discrepancies in the equilibrium bed length for aeolian sand transport [Bagnold, 1941; Shao and Raupach, 1992; Dong et al., 2004; Li and Zhang, 2006]. For example, Bagnold [1941] determined an equilibrium length of 4 7m, whereas Shao and Raupach [1992] and Li and Zhang [2006] suggested equilibrium lengths in excess of 15 m. Furthermore, previous studies have exclusively focused on saltating sand, with no consideration of the effects of the length of the sand beds on transport mass by creep. In the present study, we carried out detailed wind tunnel experiments for four different grain sizes over six different sand bed lengths at different wind velocities in a wind tunnel to measure the creeping transport mass. Based on experimental data, this paper studied the effect of wind velocity, particle size, and bed length to develop a compressive model of creeping transport masses. Although these results require additional experimental validation, they provide new information for the research of creeping sand grains. 2. Wind Tunnel Experiments We developed our experiment in the wind tunnel at the State Key Laboratory of Earth Surface Processes and Resource Ecology, Beijing Normal University, Beijing, China. The wind tunnel has a total length of m and includes a power section, an inlet section, a transition section, a first diffusion section, a stabilization section, a contraction section, an experimental section, and a second diffusion section. The experimental section was 24 m, and its cross section was a rectangle whose width and height were 3 m and 2 m, respectively. We could continuously change the wind velocities from 1 to 45 m/s, with a turbulence intensity of less than 0.8%. It was necessary to set a series of roughness elements to satisfy the principle of dynamic similarity in wind tunnel testing. For the present study, we used bricks with a length of 0.24 m, a width of m, and a height of 0.05 m as roughness elements. These bricks were arranged with regular spacing at a distance of 3.60 m, decreasing in height from the windward to the leeward side (Figure 1). We used eight rows of roughness elements and adjusted the number of bricks in each row as necessary for the tests in this experiment. For example, for a friction velocity of 0.23 m/s, we placed two layers of bricks in the first row and only one layer in the remaining seven rows; however, for a friction velocity of 0.35 m/s, we used four CHENG ET AL. AEOLIAN CREEPING MASS 1405

3 layers of bricks in the first row; two layers in the second, third, and fourth rows; and one layer in the remaining rows. For friction velocities of 0.41, 0.47, 0.55, and 0.61 m/s, their roughness elements were consistent with our prior research [Cheng et al., 2015]. Sand grains of four different mean particle sizes were sourced from the central areas of the Taklimakan Desert, the Qaidam Basin, the Mu Us Desert, and the Badain Jaran Desert in China. The Figure 2. Particle-size distribution of the experimental sand grains. particle-size analysis was performed in the Key Laboratory of Environmental Change and Natural Disasters, Ministry of Education of China, Beijing Normal University, Beijing, China, using a Malvern Mastersizer 2000 particle-size analyzer. Figure 2 presents the grain-size distributions for the four sand samples. The volume mean diameters of the samples were 152, 257, 321, and 382 μm, with standard deviations of 0.45, 0.57, 0.56, and 0.92 μm, respectively. For each sand sample, we created m thick sand layers, all of which were 1 m wide. We measured the mean wind velocities at eight different heights above the surface (0.02, 0.05, 0.10, 0.20, 0.30, 0.40, 0.50, and 0.70 m) using a pitot tube (Figure 1) [Cheng et al., 2015]. The wind velocities increased logarithmically with increasing height.the relationship between wind velocity and height could be expressed as u = a ln(z)+b. According to Dong et al. s [2004] method, the friction velocity (u * ) is expressed as k a (where k is the von Kármán constant (0.4), whereas a and b are the fitted parameters of the wind profile). In this paper, we used six different friction velocities (0.23, 0.35, 0.41, 0.47, 0.55, and 0.61 m/s). In order to illustrate the main principle of creeping transport mass, this paper assumed several parameters: (1) M i was the mass of the sand grains captured by bed traps (g/s), (2) q c was the creeping transport mass per unit time and unit length (g/(m s)), (3) q s was the mass of saltating sand grains captured by the bed trap per unit time and unit area (g/(m m s)), and (4) L and D i were the entrance length and width of bed trap, respectively. Based on mass conservation, the relationship among M, q 0, and k 0 can be expressed as follows: Equation (1) could also be expressed as M i ¼ q c L þ q s D i L (1) M i =L ¼ q c þ q s D i (2) If M i /L is regarded as q i, equation (2) could also be expressed as q i ¼ q c þ q s D i (3) There are two dependent variables (q c and q s ) in equation (3). In theory, q c and q s could be determined by using the mass of captured sand grains by two different entrance bed traps with the same length and different width (D i ). To reduce the effect of measurement error on q c and q s, six traps with the same length (L = 20 mm) and different width (D i = 1, 2, 3, 4, 5, and 6 mm) were used to measure M i. The six bed traps in a line perpendicular to the wind direction were located at a position 0.10 m away from the edge of the sand bed; their structure is listed in Cheng et al. [2013]. To study the effect of the bed length on the creeping transport mass, this paper measured the creeping transport mass with six bed lengths (2, 3.5, 5.0, 6.5, 8.0, and 10 m). Overall, this paper conducted 116 groups of experiments with three repeats for each group, including four different grain sizes (152, 257, 321, and 382 μm) over six bed lengths at six friction velocities (0.23, 0.35, 0.41, 0.47, 0.55, and 0.61 m/s). Due to the variation in the grain sizes, there were some differences for the range of friction velocities (Table 1). This paper tested six friction velocities (u * = 0.23, 0.35, 0.41, 0.47, 0.55, and 0.61 m/s) for sand with a grain size of 152 μm, five friction velocities (u * = 0.35, 0.41, 0.47, 0.55, and 0.61 m/s) for sand with grain sizes of 257 and 321 μm, and four friction velocities (u * = 0.41, 0.47, 0.55, and 0.61 m/s) for sand with a grain size of 382 μm. CHENG ET AL. AEOLIAN CREEPING MASS 1406

4 Table 1. The Fitting Parameters (q c, q s, and R 2 ) Determined by Equation (3) by Using Experimental Data for Four Mean Grain Sizes Over Sand Beds of Six Different Lengths at Different Friction Velocities Fetch Length (m) Particle Size (μm) Friction Velocity (m/s) q c /q s /R 2 q c /q s /R 2 q c /q s /R 2 q c /q s /R 2 q c /q s /R 2 q c /q s /R /0.22/ /0.18/ /0.25/ /0.24/ /0.22/ /0.04/ /0.55/ /0.45/ /0.43/ /0.42/ /0.49/ /0.63/ /0.88/ /0.66/ /0.93/ /0.72/ /0.94/ /0.89/ /1.79/ /1.77/ /1.49/ /1.62/ /1.49/ /1.10/ /2.59/ /2.71/ /2.15/ /2.01/ /2.36/ /1.38/ /3.88/ /4.21/ /3.03/ /2.79/ /3.37/ /1.59/ /0.65/ /0.72/ /0.64/ /0.69/ /0.70/ /0.52/ /1.18/ /1.24/ /1.16/ /1.52/ /1.46/ /1.17/ /1.87/ /1.93/ /1.86/ /2.19/ /2.18/ /1.89/ /2.70/ /2.81/ /2.78/ /2.79/ /2.85/ /2.69/ /3.54/ /3.87/ /3.90/ /3.30/ /3.48/ /3.57/ /0.37/ /0.34/ /0.33/ /0.11/ /0.02/ /0.0002/ /0.91/ /0.92/ /0.86/ /0.97/ /0.94/ /0.72/ /1.74/ /1.69/ /1.57/ /1.82/ /1.18/ /1.19/ /2.11/ /2.04/ /1.92/ /2.04/ /1.84/ /0.90/ /3.00/ /3.76/ /3.41/ /2.89/ /2.53/ /1.82/ /0.71/ /0.11/ /0.08/ /0.04/ /0.01/ /1.65/ /1.77/ /1.71/ /1.56/ /1.27/ /2.05/ /2.31/ /2.76/ /2.07/ /1.85/ /2.46/ /3.04/ /3.16/ /2.42/ /2.16/ Results and Discussion 3.1. A Transport Model of Creeping Sand Grains Based on the Friction Velocity Aeolian Creeping Mass of Different Particle Sizes With Different Bed Lengths for Different Friction Velocities Figure 3 illustrates the change law of the mass of sand grains captured by bed traps with different entrance widths (X = 1, 2, 3, 4, 5, and 6 mm) for a grain size of 152 μm over sand beds of six different lengths at six different friction velocities. In this case, the horizontal axis is the entrance width of the sand trap, and the vertical axis is the captured mass of sand grains for each bed trap. Positive linear relationships between the entrance width of the bed trap and its captured mass of sand grains were distinctly revealed (Figure 3), and the regression analysis demonstrates that this relationship is significant for all bed lengths because all R 2 values are 0.94 or higher (Figure 3 and Table 1). Furthermore, there were similar results for the other three grain sizes (257, 321, and 382 μm) with the bed lengths and friction velocities (Table 1). These experimental results demonstrated the validity of equation (3). According to equation (3), the intercept (q c ) and slope (q s ) of the linear relationship in Figure 3 describe the creeping and saltating mass of sand grains captured by the bed trap, respectively. The parameters (q c, q s,andr 2 ) for four grain sizes (152, 257, 321, and 382 μm) over sand beds of six different lengths at six different friction velocities are listed in Table 1. For a mean particle size of 152 μm under the action of friction velocities (0.23, 0.35, 0.41, 0.47, 0.55, and 0.61 m/s), the creeping masses in the 10 m long bed were 0.13, 0.33, 1.21, 1.62, 3.50, and 5.34 g/(20 mm min), respectively. In the 8 m long bed, the masses transported by creep increased to 0.17, 0.62, 1.50, 2.13, 3.07, and 4.56 g/(20 mm min) with increasing friction velocity. The masses transported by creep were 0.10, 0.74, 0.97, 2.11, 3.88, and 6.29 g/ (20 mm min), respectively, when the bed length was decreased to 6.5 m, and were 0.14, 0.68, 1.62, 2.73, 5.58, and 9.36 g/(20 mm min), respectively, in the 5 m long bed. In the 3.5 m long sand bed, the masses transported by creep were 0.26, 0.61, 1.20, 3.14, 4.62, and 7.32 g/(20 mm min) and were 0.006, 0.66, 1.59, 2.18, 2.63, and 3.07 g/(20 mm min) when we reduced the bed length to 2 m. Similar relationships between the friction velocity and creeping mass were obtained in all the particle-size categories. These results distinctly indicated that the creeping mass increases quickly with wind velocity, in agreement with prior studies [Bagnold, 1941; Qi et al., 2001; Wang et al., 2009; Zhang et al., 2010; Wu et al., 2011]. Our results show some significant differences compared to the transport rates reported in previous studies. These differences can largely be attributed to the methods used to measure the creeping mass. First, prior CHENG ET AL. AEOLIAN CREEPING MASS 1407

5 Figure 3. The change law of the captured mass of sand grains by a bed trap for four mean grain sizes (152 μm) over sand beds of six different lengths (2.0, 3.5, 5.0, 6.5, 8.0, and 10.0 m) at different friction velocities. use of bed trap resulted in an overestimation of the creep because it did not exclude saltating sand grains [Bagnold, 1941; Chepil, 1959;Qi et al., 2001; Wu, 2003;Wu et al., 2011]. It was also very difficult to assess the true creeping transport rates because there were variable entrance widths of bed traps: 1 3mm [Bagnold, 1941], 20 mm [Wu et al., 2011], or not provided [Qi et al., 2001; Wu, 2003]. For the entrance width of 20 mm, CHENG ET AL. AEOLIAN CREEPING MASS 1408

6 Cheng et al. [2013] thought that the saltating mass captured by the bed trap resulted in estimates times the actual creeping mass. Second, the creeping transport rate estimation by inverting the vertical distribution of the flux of blown sand flow to zero height [Dong et al., 2004] also overestimated the creeping mass because there were plenty of saltating sand grains at zero height. Finally, noncontact s high-speed digital photography was only suited for aeolian sand flow with a sparse concentration [Zhang et al., 2010] and ignored the a thickness of creeping sand grains [Cheng et al., 2013]. As a result, it underestimates the creeping transport rates. For example, the creeping mass with particle sizes of μm wasonly g/(20 mm min) at a friction velocity of 0.60 m/s [Wang et al.,2009],incomparisonwithourresults showing 3.50 g/(20 mm min) for a mean particle size of 152 μm andafrictionvelocityof0.55m/s. Although our method to measure the creeping mass is similar to our prior results [Cheng et al., 2013], key differences remain in the experimental design. For example, there were no roughness elements within the experimental section of the wind tunnel in our prior research, whereas we lifted the baseboard of the experimental section by 0.05 m and altered the roughness elements for different friction velocities. In addition, we measured the wind velocity 8.5 m from the entrance of the experimental section, whereas the wind velocity was measured at 7 m in our prior research. These differences may account for some of the variations in the obtained results. For example, at u * = 0.35 m/s, we measured a creeping mass of 0.62 g/(20 mm min) on an 8 m long bed compared to 0.51 g/(20 mm min) measured in our prior research on a 7.35 m long sand bed A Model of Aeolian Creeping Mass Wind velocity is the driving force of aeolian sand movement. Therefore, the role of the wind velocity on aeolian creep is a key problem for understanding creep. However, little has been reported due to the lack of accurate creep measurements. Qi et al. [2001] and Wu et al. [2011] presented a linear relationship between the creeping mass and the wind velocity but based this relationship on data that overestimated the creeping mass. Recently, Cheng et al. [2013] established a power model (q = u * ) with experimental data, which was interesting in that this model showed a similarity to the cubic relationship between the saltation flux and the friction velocity established in previous studies [Bagnold, 1941; Hsu, 1971]. Figure 4 illustrates the change law of creeping transport masses with friction velocity for a grain size of 152 μm over sand beds of six different lengths. The creep transport increased with friction velocities, and the best fit result was a power function (q = a + bu 3 * ) as determined by the TableCurve software. All correlation coefficients (R 2 ) were 0.94 or higher with similar results for the other three grain sizes (257, 321, and 382 μm) (Table 2). These findings indicated that the creeping mass linearly increased with the cube of the wind velocity (q = a + bu 3 * ). In general, our results are consistent with the results presented by Cheng et al. [2013] and the relationship between the saltation flux and the friction velocity [Bagnold, 1941]. However, Figure 4 also indicated that a linear relationship exists between the creeping transport mass and friction velocity when the bed length is short (2 m). The other three grain sizes (257, 321, and 382 μm) had similar results, and the creep transport seemed to increase linearly with friction velocity over sand beds of five different lengths (3.5, 5.0, 6.5, 8.0, and 10.0 m) for the coarsest sand grains (382 μm) (Table 2), with linear correlations (R 2 ) of 0.98, 0.97, 1.00, 0.96, and Because saltating grains affect the mass transported by creep, reducing the influence of saltation increases the relative importance of the wind velocity in determining the mass transported by creep. The influence of saltation is likely to decrease over short beds, where sand grains with low velocity can saltate off the bed, and in coarse-grained beds, where saltation requires higher wind velocities. In Figure 4 and Table 2, the parameter a waslessthanzeroandtheparameterb was larger than zero for different grain sizes and five or six different bed lengths. In theory, the threshold friction velocity (u *t )wasequalto (a/b) 1/3 by taking the inverse of the relationship between the aeolian sand transport and friction velocities, assuming that no aeolian sand transport occurred (i.e., q = 0). Although the uncertainty of the experimental data would result in errors in the threshold friction velocity, this was an effective method to determine the threshold friction velocity [Qi et al., 2001; Wu et al., 2011; Cheng et al., 2013; Li et al., 2014]. Therefore, the bestwaytogainanaccurate threshold friction velocity is to integrate between theoretical calculation results and experimental data. The classic formula of threshold friction velocity is expressed as follows: rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ρ u *t ¼ A s ρ gd (4) ρ where ρ s is the density of sand grains, ρ is the air density, g is the acceleration of gravity, d is the particle size of sand grains, and A is an empirical coefficient. Although equation (4) was accepted by plenty of scientists CHENG ET AL. AEOLIAN CREEPING MASS 1409

7 Figure 4. The effect of friction velocities on the aeolian creep mass for a grain size of 152 μm over sand beds of six different lengths. [Bagnold, 1941; Chepil, 1946, 1959; Zingg, 1953; Lyles and Krauss, 1971; Chepil and Woodruff, 1963; Wu, 2003], there were very large differences regarding the empirical coefficient (A)[Bagnold, 1941; Chepil, 1946, 1959; Zingg, 1953; Lyles and Krauss, 1971]. For example, the empirical coefficient (A) was equal to 0.1 [Bagnold, 1941], [Chepil, 1946], 0.12 [Zingg, 1953], [Lyles and Krauss, 1971], and [Li et al., 2014]. Wu [2003] CHENG ET AL. AEOLIAN CREEPING MASS 1410

8 Table 2. The Parameters of the Relationship Between Creeping Transport Mass (q c ) and Friction Velocity (u * ) Particle Size (μm) Regression Function Fetch Length (m) a/b/r 2 a/b/r 2 a/b/r 2 a/b/r 2 a/b/r 2 a/b/r q c = a + bu * 0.55/24.57/ /19.83/ /28.39/ /42.57/ /33.23/ /13.65/ q c = a + bu * 1.43/24.98/ /21.89/ /28.29/ /23.48/ /15.47/ /11.99/ q c = a + bu * 1.46/24.85/ /28.14/ /45.72/ /16.74/ /22.72/ /22.61/ q c = a + bu * 2.43/37.69/ /31.12/ /25.33/ /29.20/ /22.10/ q c = a + bu * 12.42/30.41/ /24.78/ /20.41/ /23.71/ /17.30/ summarized the reasons resulting in these different empirical coefficients (A) as the difference in determining the threshold standard, the errors in determining friction velocity based on measuring the result of mean wind velocities and differences in the experimental method. Based on the parameters (a and b) in Figure 4, Table 2, and equation (4), we recorded the empirical coefficient (A) of different bed lengths for different sand grain sizes. In theory, the threshold friction velocity should derive from a short bed length, but the experimental uncertainty would result in errors. Therefore, the threshold friction velocity is equal to the mean value for different bed lengths. Based on the threshold friction velocity and equation (4), we found that the empirical coefficient (A) was equal to 0.15, which was similar to previous results [Li et al., 2014]. Thus, the relationship (q = a + bu 3 * ) between the aeolian creeping mass and friction velocities (Figure 4 and Table 2) was rewritten as follows: q ¼ bu 3 3 u *t ¼ bu u 2 2 *t u þ u *t u þ u *t (5) In addition, the parameter (b) in equation (5) is the fitted result. To satisfy dimensional analysis, this paper introduced a nondimensional parameter (k) and rewrote equation (5) as follows: q ¼ k ρ g u u 2 2 *t u þ u *t u þ u *t (6) where q is the creep mass (kg/(m s)), ρ is the air density (kg/m 3 ), g is the acceleration of gravity (m/s 2 ), and k is a nondimensional parameter whose numerical value is equal to b g 1200ρ. As mentioned in Figure 4 and Table 2, the parameter (b) is related to the grain size and the bed length, so the nondimensional parameter (k) was determined by the grain size and the bed length. There was no obvious relationship among the nondimensional parameter (k), the grain size, and the bed length (Table 3). For example, the nondimensional parameter (k) was 0.09, 0.21, 0.27, 0.18, 0.13, and 0.16 for a mean sand grain diameter of 152 μm with bed lengths of 2.0, 3.5, 5.0, 6.5, 8.0, and 10.0 m, respectively; the nondimensional parameter (k) was 0.08, 0.10, 0.15, 0.18, 0.14, and 0.16 for a mean sand grain diameter of 257 μm with above corresponding bed lengths, respectively; the nondimensional parameter (k) was 0.14, 0.14, 0.11, 0.29, 0.18, and 0.16 for a mean sand grain diameter of 321 μm with above corresponding bed lengths, respectively; and the nondimensional parameter (k) was 0.14, 0.18, 0.16, 0.20, and 0.24 for a mean sand grain diameter of 382 μm with bed lengths of 3.5, 5.0, 6.5, 8.0, and 10.0 m, respectively. This result suggests that other factors affect creeping transport rate. First, sand grains in this paper are not single diameter and have certain frequency distribution (Figure 2), especially for particle size with 382 μm. Under the action of wind, there are very great differences of coarsening level for different positions. For the same wind Table 3. The Nondimensional Parameter (k) in Equation (6) for Different velocity, there is small creeping transport Particle Sizes With Different Bed Length rate in severe coarsening position; The Nondimensional Parameter (k) in Equation (6) Length of Sand there is large creeping transport rate in Bed (m) slightly coarsing position. Second, under the action of wind, the formation and development of ripples would affect creeping transport rate. Third, coarsening bed and ripples result in the adjustment of friction velocity (u * ), which changes with time and space. CHENG ET AL. AEOLIAN CREEPING MASS 1411

9 Figure 5. Comparison of experimental data of the aeolian creeping masses and the predicted results. According to the above processes, there is reason to believe that equation (6) can represent the relationship between aeolian creeping masses and wind velocities, and Figure 5 supports this conclusion. The nondimensional parameter (k) was related to the grain size and the sand length. It was more interesting to describe the parameter (k) as a function of the particle size and bed length. The comprehensive effects of wind velocity, particle size, length of the bed surface, and bed surface change under the action of wind velocity determined whether the parameter (k) was uniformly related to the particle size or the length of the bed surface. Although there were some differences between the calculated creep mass by using equation (7) and the wind tunnel experiment data for the same particle size with different bed surface lengths, these findings seemed to concentrate on the line (y = x). Therefore, this paper colligated the data for each experiment length into a data set for all experiment lengths and found that this relationship is significant for all four sand grain sizes because all correlation coefficients (R 2 ) are 0.94 or higher (Figure 5). The model not only helps to determine creeping transport rate but also helps to assess the contribution of creeping sand grains to the total flux of aeolian sand. However, it is necessary to verify if the model is applicable for conditions in the field. As a whole, equation (6) is similar to the formula of the flux of aeolian sand provided by Kawamura [1951]: ρ q 1 ¼ k 1 g u 2 ρ u *t u þ u *t ¼ k1 g u u 2 2 *t u þ 2u *t u þ u t (7) Except for the numerical value of the constant (k 1 ) in equation (6) and the constant (k 1 ) in equation (7), the difference between equations (6) and (7) is u * t u *. As mentioned above, the constant (k) in equation (6) is a function of the particle size and bed length, while the constant (k 1 ) in equation (7) would change for CHENG ET AL. AEOLIAN CREEPING MASS 1412

10 different sand grain sizes, equaling 2.78 when the sand grain size is 250 μm[kawamura, 1951]. Subsequently, scientists gained different constants (k 1 ), such as 1.0 [Horikawa, 1960; Horikawa et al., 1984], 2.61 [White, 1979; Namikas and Sherman, 1997], and 2.78 [Sherman and Li, 2012]. By comparing the model prediction results [Bagnold, 1937; Kawamura, 1951; Zingg, 1953; Owen, 1964; Kadib, 1965; Hsu, 1971; Lettau and Lettau, 1978; Sørensen, 2004] with field observation data, Sherman and Li [2012] found that the match between Kawamura s [1951] model prediction and the field observation results was very poor and Kawamura s prediction substantially overvalues aeolian transport rates. The main reason was that Sherman and Li [2012] retained Kawamura s original value of Sherman et al. [2013] found by deep analysis that the constant (k 1 ) in equation (7) was equal to 0.70 for the von Kármán constant (0.40). As mentioned above, the gain size affects the constant (k 1 ). For comparison purposes, this paper only discussed sand grains with a particle size of 257 μm. The creeping proportion of sand flow for sand bed lengths of 2.0, 3.5, 5.0, 6.5, 8.0, and 10.0 m ranged from 4.54% to 27.26% with a mean value of 12.73% on the basis of the constant (k 1 = 0.7) in Sherman et al. [2013]. Although these findings were consistent with the 7 17% value [Horikawa, 1960], they were less than most results [Bagnold, 1941; Chepil, 1945; Willetts and Rice, 1985; Nickling and McKenna Neuman, 1997; Namikas, 2003; Dong et al., 2004], which overestimated the aeolian creeping transport mass, as mentioned in section In addition, the explanation of the above differences required us to clarify an accurate value for the constant (k 1 ) in equation (7) and the contribution of creeping sand grains occupied by the flux of aeolian sand. However, the essence of this work was how to determine an accurate flux of aeolian sand flow Effects of Sand Grain Size and Bed Length on Aeolian Creeping Mass Grain size is an important factor affecting mass transported by creep. Several studies have noted a relationship between creep and grain size [Chepil, 1959; Dong et al., 2004; Wu, 2003]; however, few have provided accurate experimental data to quantify the relation. The length of the sand bed also has a significant effect on sand movement and is a key parameter in the wind erosion equation (WEQ) [Woodruff and Siddoway, 1965]. In addition, studies have demonstrated that topographic surface features are important for determining the creeping mass. In general, creep is greater in the trough of a ripple than at the crest. Overall, a complex relationship among the wind velocity, the grain size, the length of the bed, and the surface shape determines the sand creep. In this section, we attempted to isolate the effects of particle size and sand bed length as a first step toward developing a comprehensive model of sand grain creep in the future The Evolution of Aeolian Creeping Masses Along Sand Beds Figure 6 presents the creeping mass for different particle sizes and bed lengths. The effect of increasing bed length appears to depend on the grain size. For mean grain sizes of 152, 257, and 321 μm, creep initially increases with increasing bed length, particularly at higher wind velocities (u * = 0.55 and 0.61 m/s), before decreasing above a certain value. The peak bed length for creeping transport is inconsistent across grain sizes (Figure 6). For example, the peak length for a grain size of 152 μm and friction velocities of 0.55 and 0.61 m/s was 5.0 m compared to 6.5 m for mean grain sizes of 257 and 321 μm. However, these results are generally consistent with the peak bed length of 7 m for saltating grains, as reported by Shao and Raupach [1992]. The general increase in creep with bed length is also consistent with the relationship observed in saltating grains by Bagnold [1941]. Aerodynamic entrainment is particularly important in short beds. However, as the bed length increases, the kinetic energy from collisions becomes increasingly influential to the point where entrained grains are sufficient to modify the airflow near the surface. At this stage, the mean wind speed near the bed is reduced, thereby decreasing the number of collisions and, consequently, the potential for sand transport. In our experiment with a mean grain size of 382 μm, the creeping mass gradually increased with increasing bed length (Figure 6). However, an anomaly occurred at high friction velocities (u * = 0.55 and 0.61 m/s), where the transport mass on the 3.5 m long bed exceeded that on the beds with lengths of 5.0 and 6.5 m. Bagnold [1941] noted that the maximum flux of saltating grains occurs at a considerable distance from the leading edge of the erodible surface when the particles are coarse and exhibit a wide size distribution. Because our experiment with a mean grain size of 382 μm contained the coarsest grains and the widest size distribution (Figure 2), we expected the maximum flux to occur on the longest sand bed. We hypothesize that the wider grain-size distribution resulted in a peak at 3.5 m. CHENG ET AL. AEOLIAN CREEPING MASS 1413

11 Figure 6. The creeping mass for different grain sizes with different lengths of sand beds The Variation in the Aeolian Creeping Mass With the Particle Size of the Sand Grains From Figure 6, it appears that the effect of grain size on creeping mass can be classified into three categories. The first category includes scenarios where the transported mass increases with increasing grain size. For friction velocities of 0.35 and 0.47 m/s, this increase occurred on bed lengths of 2.0, 3.5, and 5.0 m. This can also be observed at a friction velocity of 0.41 m/s on bed lengths of 2.0, 3.5, 5.0, and 8.0 m and on the 3.5 m long bed at a friction velocity of 0.55 m/s. In general, the creeping mass increased with increasing grain size on short beds (2.0, 3.5, and 5.0 m) at low friction velocities (u * = 0.35, 0.41, and 0.47 m/s). For example, at a friction velocity of 0.35 m/s, the creeping masses on the 3.5 m long bed were 0.61, 0.053, and 0.02 g/(20 mm min) for mean grain sizes of 152, 257, and 321 μm, respectively. With a friction velocity of 0.41 m/s, the creeping masses were 1.12, 0. 39, 0.04, and 0.01 g/(20 mm min) for mean grain sizes of 152, 257, 321, and 382 μm. Increasing the friction velocity to 0.47 m/s resulted in transported masses of 3.14, 0.99, 0.08, and 0.02 g/(20 mm min) for mean grain sizes of 152, 257, 321, and 382 μm. Finally, u * = 0.55 m/s, transported masses were 4.62, 1.84, 1.71, and 1.67 g/(20 mm min) for mean grain sizes of 152, 257, 321, and 382 μm. The second category includes situations in which the creeping mass initially decreases and then increases with increasing grain size. This situation occurred on the 10.0 m sand bed at friction velocities of 0.35, 0.41, 0.47, 0.55, and 0.61 m/s; on the 8.0 m bed at 0.35 and 0.47 m/s; on the 6.5 m bed at 0.35 and 0.41 m/s; and on both the 2.0 and 5.0 m beds at 0.61 m/s. For example, on the 10 m bed with u * = 0.35 m/s, the creeping masses were 0.33, 0.08, and 0.10 g/(20 mm min) for mean grain sizes of 152, 257, and 321 μm, respectively. CHENG ET AL. AEOLIAN CREEPING MASS 1414

12 Table 4. The Creeping Proportion to the Total Mass of Sand Grains Captured by the Width of the Bed Trap for Four Different Widths (1, 2, 3, and 20 mm) Particle Size (μm) Friction Wind Velocity (m/s) Width of Bed Trap (mm) With u * = 0.41 m/s, the creeping masses were 1.21, 0.30, 0.20, and 0.38 g/ (20 mm min) for grain sizes of 152, 257, 321, and 382 μm, respectively. Further increasing the friction velocity to 0.47 m/s produced creeping masses of 1.62, 0.88, 0.34, and 1.36 g/(20 mm min) for grain sizes of 152, 257, 321, and 382 μm, respectively. At friction velocities of 0.55 m/s and 0.61 m/s, the equivalent transport masses were 3.50, 2.16, 3.08, and 3.81 g/(20 mm min) and 5.34, 4.84, 4.29, and 6.34 g/(20 mm min) for grain sizes of 152, 257, 321, and 382 μm, respectively. The third category includes scenarios where the creeping mass fluctuates with the grain size. These scenarios generally involve an initial decrease in creep with increasing grain size, followed by an increase and then a final decrease as the particle size further increases. For example, at u * = 0.41 m/s on the 6.5 m long bed, the creeping masses were 0.97, 0.45, 0.91, and 0.03 g/(20 mm min) for mean grain sizes of 152, 257, 321, and 382 μm, respectively. Increasing the friction velocity to 0.47 m/s resulted in creeping masses of 2.11, 1.33, 1.38, and 0.92 g/(20 mm min). At a friction velocity of 0.55 m/s and 0.61 m/s, the creeping masses were 3.88, 2.89, 5.32, and 2.71 g/(20 mm min) and 6.29, 5.37, 8.50, and 4.03 g/(20 mm min) for particle sizes of 152, 257, 321, and 382 μm Contribution of Creeping Grains to the Total Trapped Sand As discussed in section 2, bed traps also capture a significant quantity of saltating grains. Therefore, it is important to assess the creeping proportional of all sand grains captured by the bed trap for different entrance widths. Previous studies have used bed trap widths of 1 3mm [Bagnold, 1941] and 20 mm [Wu et al., 2011]; however, little has been reported regarding the creeping proportional of all sand grains captured by the bed trap. In addition, Bagnold [1941] assumed that a bed length of 4 7mwas sufficient to accurately assess sand transport. In the present study, we assessed the relative importance of creeping grains to the mass captured in the bed traps with sand samples of four different particle sizes (152, 257, 321, and 384 μm) on a 10 m long sand bed. Table 4 presents the creeping proportion of the total mass of sand grains captured by bed traps of four different widths (1, 2, 3, and 20 mm). With a 1 mm wide trap, the creeping proportions ranged from 0.37 to 0.58 for a mean grain size of 152 μm, for 257 μm, for 321 μm, and for 382 μm (Table 4), with mean proportions of 0.49, 0.33, 0.35, and 0.54, respectively. This finding shows that grains transported by creep generally provide less than 50% of the total sand grains captured by a 1 mm wide bed trap. More saltating sand grains were captured with increasing bed trap widths; thus, the creep proportion decreased quickly with increasing trap width. For example, the maximum contribution of creeping grains in a 1 mm wide trap was 0.72; however, this contribution decreased to 0.56, 0.462, and 0.11 with trap widths of 2, 3, and 20 mm, respectively. This result suggests that the mass captured by a bed trap is only representative of creep if the trap is sufficiently narrow. However, Cheng et al. [2013] demonstrated that if the width of the trap is too narrow, then the entrance can become partially or fully buried. To avoid this situation, it is necessary to determine the minimum width of the trap. Therefore, we can conclude that the direct measurement of creep using a bed trap is not possible. The mean grain size appears to affect the creep proportion of the mass captured by a bed trap. At high friction velocities (u * = 0.61 m/s), the creeping proportion in the 1 mm wide bed trap increased with grain size, whereas at lower friction velocities, the creeping proportion initially decreased and then increased CHENG ET AL. AEOLIAN CREEPING MASS 1415

13 with increasing grain size (Table 4). At u * = 0.61 m/s, the creeping proportions were 0.58, 0.58, 0.59, and 0.72 for mean grain sizes of 152, 257, 321, and 382 μm, respectively. At u * = 0.55 m/s, the creeping proportions were 0.58, 0.44, 0.59, and 0.65 for mean grain sizes of 152, 257, 321, and 382 μm. Wind velocity is the primary factor controlling aeolian sand transport and affects the creep proportion of sand grains captured by the bed trap. The creep proportion showed an increase with increasing friction velocities, except when sediments had a mean grain size of 321 μm. There were particularly strong linear relationships (R 2 = 0.99) between the friction velocity and the proportion of creep at mean grain sizes of 257 and 382 μm. This result shows that although numbers of creeping and saltating sand grains increase with increasing friction velocity, the increase is more marked for creeping grains than for saltating grains. 4. Conclusions In this paper, we conducted detailed wind tunnel experiments to measure creep transport for four different mean grain sizes over six bed lengths at various friction velocities. The main conclusions are as follows: (1) this paper developed an aeolian creeping transport mass model q ¼ k ρ g u u 2 2 *t u þ u u *t þ u *t, where ρ is the air density (kg/m 3 ), g is the acceleration of gravity (m/s 2 ), u * is the friction velocity (m/s), u *t is the threshold friction velocity (m/s), and k is a nondimensional parameter. Its results match to experimental data for four different grain sizes with different bed lengths because their correlation coefficients (R 2 ) are 0.94 or higher. (2) The nondimensional parameter (k) in the model did not have a consistent relationship for changing bed length and grain size because of the comprehensive coactions of coarsening bed and ripples and the adjustment of friction velocity. (3) The empirical coefficient (A) in Bagnold s threshold shear velocity equation was equal to 0.15 based on the inversion of the relationship between creep transport rate and friction velocity to a creep transport rate of zero. (4) There was no unified formula to describe the effect of particle size and the bed length on creep transport. The effect of grain size can be classified into three categories: the creep mass increases with increasing grain size, the creep mass initially decreases and then increases with increasing grain size, and the creep mass fluctuates with the grain size. The effect of increasing bed length appears to depend on the grain size. For mean grain sizes of 152, 257, and 321 μm, creep initially increases with increasing bed length before decreasing above a certain value, while for the mean grain size of 382 μm, the creep mass gradually increased with increasing bed length. Acknowledgments This research was funded by the National Natural Science Foundation of China (grant ), by the Key Program of the National Natural Science Foundation of China (grant ), by CERS (CERS-China Equipment and Education Resources System), and by the Program for New Century Excellent Talents in University (NCET ). The authors gratefully acknowledge the Editor, Associate Editor, and three reviewers, whose valuable comments and suggestions led to the improvement of our manuscript. We also thank Tian Jinglu, Du Sisong, Wu Wangyang, Wang Hongying, and Peng Shuai for their assistance during the wind tunnel experiments. Authors state that we shall provide relative data within the manuscript without any restriction to users who are interested in them. References Bagnold, R. A. (1937), The transport of sand by wind, Geogr. J., 89, Bagnold, R. A. (1941), The Physics of Blown Sand and Desert Dunes, Methuen, London. Cheng, H., X. Y. Zou, and C. L. Zhang (2006), Probability distribution functions for the initial liftoff velocities of saltating sand grains in air, J. Geophys. Res., 111, D22205, doi: /2006jd Cheng, H., X. Y. Zou, C. L. Zhang, and Z. J. Quan (2009), Fall velocities of saltating sand grains in air and their distribution laws, Powder Technol., 2009(192), Cheng, H., X. Y. Zou, C. C. Liu, J. J. He, and Y. Q. Wu (2013), Transport mass of creeping sand grains and their movement velocities, J. Geophys. Res. Atmos., 118, , doi: /jgrd Cheng, H., J. J. He, X. Y. Zou, J. F. Li, C. C. Liu, B. Liu, C. L. Zhang, Y. Q. Wu, and L. Q. Kamg (2015), Characteristics of particle size for creeping and saltating sand grains in aeolian transport, Sedimentology, doi: /sed Chepil, W. S. (1945), Dynamics of wind erosion: I. Nature of movement of soil by wind, Soil Sci., 60, Chepil, W. S. (1946), Dynamics of wind erosion: V. Cumulative intensity of soil drifting across eroding field, Soil Sci., 61(3), Chepil, W. S. (1959), Wind erodibility of farm fields, J. Soil Water Conserv., 14, Chepil, W. S., and R. A. Milne (1941), Wind erosion of soils in relation to size and nature of the exposed area, Sci. Agric., 21(7), Chepil, W. S., and N. P. Woodruff (1963), The physics of wind erosion and its control, Adv. Agron., 15, Dong, Z. B., H. T. Wang, X. P. Liu, and X. M. Wang (2004), The blown sand flux over a sandy surface: A wind tunnel investigation on the fetch effect, Geomorphology, 57, Fryrear, D. W., and A. Saleh (1996), Wind erosion: Field length, Soil Sci., 161(6), Gillette, D. A., G. Herbert, P. H. Stockton, and P. R. Owen (1996), Causes of the fetch effect in wind erosion, Earth Surf. Processes Landforms, 21(7), Horikawa, K. (1960), Sand Movement by Wind Action on the Characteristics of Sand Traps, Tech. Mem. 119, U.S.A. Beach Erosion Board, U.S. Army Corps of Eng., Washington, D. C. Horikawa, K., S. Hotta, S. Kubota, and S. Katori (1984), On the sand transport rate by wind on a beach, Coastal Eng. Jpn., 26, Hsu, S. A. (1971), Measurement of shear stress and roughness length on a beach, J. Geophys. Res., 76, , doi: / JC076i036p Huang, N., S. Ren, and X. J. Zheng (2008), Effects of the mid-air collision on sand saltation, Sci. China Ser. G, 51(9), Kadib, A. A. (1965), A function for sand movement by wind, Rep. HEL 2-8, Univ. of Calif. Hydraulics Eng. Lab., UCLA-Berkeley, Berkeley. Kawamura, R. (1951), Study on sand movement by wind, Rep. Inst. Sci. Technol. Univ. Tokyo, 5(3), CHENG ET AL. AEOLIAN CREEPING MASS 1416