A description and some results of the calculated aeolian sand transport including the turbulence intensity and the autocorrelation of the wind field in the transport equation’s are presented in this study. The simulated transport related to Sørensen (2004) transport equation varies by more than 100% for one and the same friction velocity and one and the same sand type near the thresholds by variation of turbulence
intensity and/or the autocorrelation of the wind. Published results of sand transport measurements are imperfect and not comparable without information about the characteristics of the turbulent wind. An indication of friction velocity, threshold friction velocity and measured transport must be supplement by the specification of the turbulence intensity and the autocorrelation of the wind velocity

Schönfeldt, H.-J.

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Turbulence and aeolian sand transport  H.-J. Schönfeldt  Summary A description and some results of the calculated aeolian sand transport including the turbulence intensity and the autocorrelation of the wind field in the transport equation’s are presented in this study. The simulated transport related to Sørensen (2004) transport equation varies by more than 100% for one and the same friction velocity and one and the same sand type near the thresholds by variation of turbulence intensity and/or the autocorrelation of the wind. Published results of sand transport measurements are imperfect and not comparable without information about the characteristics of the turbulent wind. An indication of friction velocity, threshold friction velocity and measured transport must be supplement by the specification of the turbulence intensity and the autocorrelation of the wind velocity.   Zusammenfassung Es wird beschrieben, wie die Turbulenzintensität und die Autokorrelation des Windfeldes in Transportgleichungen des aeolischen Sandtransports berücksichtigt werden kann und es werden einige Ergebnisse vorgestellt. Der simulierte Transport angewandt auf die Sørensen (2004) Transportgleichung variiert um 100 % für ein und dieselbe Schubspannungsgeschwindigkeit und den gleichen Sandtyp in Schwellennähe bei Variation der Turbulenzintensität und/oder der Autokorrelation des Windes. Ohne Informationen über die Eigenschaften des turbulenten Windes sind veröffentlichte Ergebnisse von Sandtransportmessungen unvollständig und  nicht vergleichbar. Die Angabe der Schubspannungsgeschwindigkeit, Schwellenschubspannungs-geschwindigkeit und der gemessene Transport müssen durch die Angabe der Turbulenzintensität und Autokorrelation der Windgeschwindigkeit ergänzt werden.  Introduction Knowledge about the airflow patterns near the surface and the resulting transport of sediment can enhance our understanding of desert sand movement and dust production. Furthermore, aeolian sediment transport is important in redistributing plant nutrients, determining vegetation patterns, and contributing to the change in vegetation type (Schlesinger et al. 1990). A number of equations have been proposed linking horizontal sand fluxes with wind velocities (Bagnold 1941; Zingg 1953; Williams 1964; Kawamura 1964; Owen 1964; Gillette and Goodwin 1974; Gillette 1979; Lettau and Lettau 1978; White 1979; Sørensen 1985; Gillette and Stockton 1989; Leys and Raupach 1991; Shao et al. 1993; Stout and Zobeck 1997; Zheng et al. 2003, 2006; Stout 2004; Leenders et al. 2005). Figure 1 shows data points of measured transport and the calculated results of three transport equations which all meet the measured points more or less good. Bowker et al. (2007) compared the sand flux of model predictions with field measurements and then assess the sensitivity of the simulations to several aspects such as the formulation of the sand flux equation and the specific value of the threshold velocity, tu* . They used the transport equations of Kawamura (1964)/White (1979) (Eq. 1) and Owen (1964) (Eq. 2), with Q the transport rate, *u  the friction velocity and A, A1 constants. Kawamura/White’s equation has a correction Wiss. Mitteil. Inst. f. Meteorol. Univ. Leipzig Band 48(2011)103term reciprocally proportional to the friction velocity but the difference between the measured and simulated data are nevertheless significant as shown in the Table 1.   +−=**2*2*3*1 11uuuuuAQ tt Kawamura (1964), White (1979)   (1) −= 2*2*3*1uuuAQ t  Owen (1964)      (2)  Eq. (1) and Eq. (2) underestimate the measurement during the storm-event 1 in 2003 (see Table 1) and on the storm 5 in 2003 the equations overestimate the measurements up to 100%. The storm 1 in 2004 is not predictable with this method. Problematic in the prediction equations is the threshold friction velocity. Stout (1998) experimentally and Schönfeldt (2003) theoretically found that the averaging time of wind speed measurement affects the observed threshold. Measurements of this threshold friction velocity have not solved the problem (Bowker et al., 2007).  The object of this study is to examine the influence of the turbulence intensity on the sand transport. Generally the turbulence is parameterized by the friction velocity, a quantity derived from the law of the wall of fluid dynamics. The law of the wall describes the alteration   Figure 1. Mass transport rate as a function of friction  velocity (Greeley and Iversen, 1985).  Wiss. Mitteil. Inst. f. Meteorol. Univ. Leipzig Band 48(2011)104of the wind speed with distance or height above the ground surface. Spectral parameters and the relation of standard deviation to mean wind speed (turbulence intensity) are not incorporated in this law. In the following the importance of these parameters on the sand transport is demonstrated.  Table 1. Total accumulated sand flux for each of the eight storms (Bowker et al., 2007).     Methods The threshold friction velocity in Equation (1) and (2) is a constant; therefore the normalized saltation velocity V is introduced as the horizontal velocity at the height z divided by the impact threshold velocity at the same height z. Assuming a logarithmic wind profile (according to the law of the wall) then the mean of the normalized saltation velocity is the friction velocity *u  divided by the impact threshold friction velocity tu* . The normalized saltation velocity V is similar statistically distributed as the velocity u. It can be Gaussian distributed or otherwise. tt uuVzuzuV**)()(=→=        (3)  In order to describe gusts in the turbulent time series of wind, the turbulence intensity I is used. The turbulence intensity I is defined as the standard deviation of the wind speed, σ divided by the mean of the wind speed. Assigned to the normalized saltation velocity V it follows: VI Vσ=.           (4) Wiss. Mitteil. Inst. f. Meteorol. Univ. Leipzig Band 48(2011)105 Figure 2. The probability density of exceeding the normalized saltation threshold velocity Vt in dependence of the turbulence intensity (I = standard deviation of the wind velocity divided by the mean of the wind velocity). The mean of the Weibull- distributed wind velocity is always 8.0=V .   Figure 2 shows different Weibull distributions of wind velocity with the same mean and three different turbulence intensities. The threshold is per definition one. The mean wind speed is chosen to 8.0=V . This wind speed will not give any transport using such prediction equations as Eq. (2). A wind time series with a turbulence intensity of 0.12 provides some transport and wind with a turbulence intensity of 0.2 results in a rather considerable amount of transport. Note that the mean velocity is constant and therefore *u  is always the same. Every transport equations ignore the gusts and will give one and the same value for the sand transport. All transport equations such as Equation (2) provide for *u  < tu*  ( 1<V ) a negative transport and must actually written as  ( ) ( )23*11 −−−= VAuVHQ               (5)   with H(x) as Heaviside function (H(x) = 0 for x < 0 else H(x) = 1). Introducing a low frequency turbulence and gusts, the Heaviside function in Equation (5) is transferred from a Heaviside function, H(V-1), to a distribution function, F(V,I) which describes the probability of saltation. Eq. (5) becomes then  ( ) ( )23*1, −−= VAuIVFQ .         (6) Wiss. Mitteil. Inst. f. Meteorol. Univ. Leipzig Band 48(2011)106Schönfeldt (2003) examined the resulting transport determined after Eq. (6). Using Weibull and Gaussian distributions for the horizontal wind speed, Schönfeldt (2003) found that the averaging time, the variance and the form of horizontal wind speed distribution affects the observed threshold. Furthermore, calculations reveal that the sediment transport rate and its increase with the friction velocity above the apparent threshold depends on the variance and the form of horizontal wind speed distribution. The proportionality between streamwise sand flux and friction velocity cubed is a result of the distribution function of the wind strength and the average procedure.  . Figure 3a. Heaviside function H(V-1).        Figure 3b. Probability of saltation F(V,I).  Bagnold (1941) showed that there are two thresholds for saltation: the fluid threshold, which is defined as the speed at which particles start moving due to the forces of wind only, and the impact threshold, which is the speed at which the combined action of wind forces and saltation impacts can just sustain movement, or alternatively, the speed at which the energy received by the average saltating grains becomes equal to that lost (by impact) so that motion is sustained. The impact threshold is smaller than the fluid threshold. After Bagnold the impact threshold is 0.8 of the fluid threshold and therefore the normalized saltation fluid threshold is Vf = 1.25 with Vi = 1.0, the impact threshold. In the case of intermittent saltation a transport equation must take into account this circumstance. Intermittent saltation is characterized by periods with wind speed falling below that necessary threshold for grain movement, producing a momentary calm followed immediately by gusts. In Figure 4 the explained process is illustrated. There are two thresholds, the impact threshold Vi = 1.0 and the fluid threshold Vf = 1.25. The saltation process begins when the fluid threshold is overcome (red points) and stops, when the normalized wind velocity falls below the impact threshold (blue points). The saltation process depends on how frequently the wind speed exceeds the fluid threshold and how long the wind speed remains over the impact threshold. Therefore saltation also depends on spectral parameters of the wind velocity time series. The simplest parameter to describe a spectrum is the value of the autocorrelation function at time lag ∆t. In the following an interval for the time series of ∆t = 1 second is used.    Wiss. Mitteil. Inst. f. Meteorol. Univ. Leipzig Band 48(2011)107Figure 4. An example of modelled time series of the normalized saltation velocity (Schönfeldt, 2004). After Bagnold (1941) the saltation starts at the red points and stops at the blue points.  From Fig. 4 one can see that the saltation process has a characteristic hysteresis curve and is strongly nonlinear. In a transport equation as Sørensen (2004) recommends for sand of size > 170 µm the non-written Heaviside function in the equation should be replaced by the probability of saltation F(V,I,r1) (see Fig. (3a,b)) and the normalized saltation velocity V must be replaced by its expected value during saltation, where V during saltation is denoted by Vs.  { } { } −−+=→−−+=ssstt VVVrIVFugQVVVugQ 1),,(1 213*23* βεγβεγερβγβγρ            (7)  In Eq. (7) and (8) the expected value of the variable during saltation is denoted by ε{variables}, β and γ are constants for a fixed grain size (see also Sørensen, 2004). Other transport equations, which are valid for constant velocities as e.g. derived by a wind tunnel experiment, must be transformed by the following rules: { } { } ),,()1(;11;; 122 rIVFVHVVVVVV s →−→→→ εεε (8) An analytic solution for the quantities in Eq. (8) as a function of the normalized saltation velocity (or friction velocity) is difficult because of hysteresis in the process. We solved the problem with synthetic time series. The simplest form is the first order Markov chain. A Markov chain is a sequence of random values whose probabilities at a time interval k + 1 depend upon the value of the number at the previous time k. A time step of one second was used. The controlling factor in a Markov chain is the transition probability. The first order Markov chain is described by Eq. (9), where ε is a random noise. Wiss. Mitteil. Inst. f. Meteorol. Univ. Leipzig Band 48(2011)108ε+−=−+ )(11 VVrVV kk         (9) Using r1 as the autocorrelation of V(t) with a time lag of one second in Eq. (9), the resulting time series has also this autocorrelation r1. Epsilon can be Gaussian, Weibull or otherwise distributed. In the following the Weibull distribution is used for the noise. Gaussian distributed time series give in principle the same result.  Results A table of excepted values for the quantities in Eq. (7) as well as for the distribution function F (V, I, r1) is numerically calculated using turbulence intensities from 0.1 to 0.3, values of autocorrelation from 0.5 to 0.9 and normalized saltation velocities from 0.8 to 4.0. The resulting values are fitted whereas the distribution function of the saltation probability is somewhat complex. The fitted distribution function, F (V, I, r1) has a form like a Gumbel distribution with an additional quadratic and cubic term in V.  ( )( )3132121111101 ),(),(),(),(),(),,( VrIfVrIfVrIfVrIfrIfExpExprIVF ++++−=   (10)  The coefficients fi are polynomial functions of the turbulence intensity I and r1. The dependency from the autocorrelation is simply linear. The polynomial functions  fi can be expressed as  ( ) 131321211103032020100 rcfcfcffcfcfcfff iiiiiiiii +++++++= ,  (11)  There are 32 coefficients to describe the probability of saltation for given V, I, r1 values. The probability of saltation as function of the normalized saltation velocity V is shown in Fig. 5 with the turbulence intensities from 0.1 to 0.2 and the autocorrelation r1   from 0.5 to 0.9 as parameters The expected values of 1/Vs, Vs and Vs2 are polynomial functions of the turbulence intensity I and r1.  { } { } +++=+++=+++= VaVaVaaVVVaVaVaaVVVaVaVaaVV sss 3222212022131221110030220100 ,,11εεε            (12)  ( ) 131321211103032020100 rcacacaacacacaaa ijijijijijijijijij +++++++=  (13)  The so corrected equation of Sørensen (2004) has 128 coefficients. The implementation of these coefficients in a simple FORTRAN program can be downloaded over the website http://www.uni-leipzig.de/~meteo/de/HANS/hans.htm. This program is written for homogeneous sand of size 242 µm and can be simply adapted to any transport equation by interchanging the prefactors of the expected values (β, γ) in equation (7) or by introducing other linear combinations of the expected values.   The results for the turbulent sand transport using an adapted transport equation compared with the equation by Sørensen (2004)  are shown in Fig. 6. It demonstrates that there are combinations of turbulence intensity I and autocorrelation r1 parameters Wiss. Mitteil. Inst. f. Meteorol. Univ. Leipzig Band 48(2011)109 Figure 5. The probability of saltation as function of the normalized saltation velocity V with the turbulence intensity I and the autocorrelation r1 as parameters.  Figure 6. The sand transport Q as function of the normalized saltation velocity V with the turbulence intensity I and the autocorrelation r1 as parameters.  Wiss. Mitteil. Inst. f. Meteorol. Univ. Leipzig Band 48(2011)110 Figure 7. The sand transport Q related to the Sørensen (2004) transport as function of the normalized saltation velocity V with the turbulence intensity I and the autocorrelation r1 as parameters.   with lower as well as higher transports as predicted by Sørensen (2004). Figure 7 shows the normalized transport and makes clear that the transport near the thresholds can differ by more than 100% for one and the same friction velocity and one and the same sand type. Therefore, measurements as done by Bowker et al. (2007) are imperfect without information about the characteristics of the wind velocity time series (I, r1).  References Bagnold, R.A., 1941: The Physics of Blown Sand and Desert Dunes. Methuen, London, 256 pp. Bowker, G.E., Gillette, D.A., Bergametti, G., Marticorena, B. and Heist, D.K., 2007: Sand flux simulations at a small scale over a heterogeneous mesquite area of the northern Chihuahuan Desert. Journal of Applied Meteorology and Climatology 46(9), 1410-1422. 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Lettau, K., Lettau, H., 1978: Experimental and micrometeorological field studies of dune migration. Exploring the World’s Driest Climate, K. Lettau and H. Lettau, Eds., University of Wisconsin—Madison IES Rep. 101, 110–147. Leys, J. F., Raupach, M. R., 1991: Soil flux measurements using a portable wind erosion tunnel. Aust. J. Soil Res., 29, 533–552. Owen, R.P., 1964: Saltation of uniform grains in air. J. Fluid. Mech., 20, 225-242. Schlesinger, W.H., Reynolds, J.F., Cunningham, G.L., Huenneke, L.F., Jarrell, W.M.,  Virginia R.A., Whitford W. G., 1990: Biological feedbacks in global desertification. Science, 247,1043–1048.  Schönfeldt, H.-J., 2003: Remarks on the definition and estimation of the aeolian erosion threshold friction velocity. Meteorol. Z., 12, 137-142. Schönfeldt, H.-J., 2004: Establishing the threshold for intermittent aeolian sediment transport. Meteorol. Z., 13, 3, 437-444. Shao, Y, Raupach, M. R., Findlater, P. A., 1993: Effect of saltation bombardment on the entrainment of dust by wind. J. Geophys. Res., 98, 12, 719–12 726. Sørensen, M., 1985: Estimation of some Aeolian saltation transport parameters from transport rate profiles. Proc. Int. Workshop on the Physics of Blown Sand, Vol. 1, Aarhus, Denmark, University of Aarhus, 141–190. Sørensen, M., 2004: On the rate of aeolian sand transport. Geomorphology, 59, 53-62  Stout, J. E, Zobeck, T. M., 1997: Intermittent saltation. Sedimentology, 44, 959–970. Stout, J.E., 1998: Effect of averaging time on the apparent threshold for aeolian transport. Journal of Arid Environments, 39, 395-401 Stout, J. E., 2004: A method for establishing the critical threshold for aeolian transport in the field. Earth Surf. Processes Landforms, 29, 1195–1207. White, B., 1979: Soil transport by winds on Mars. - J. Geophys. Res., 84, 4643-4651 Williams, G., 1964: Some aspects of the eolian saltation load. Sedimentology, 3, 257–287. Zheng, X. J., Huang, N., Zhou, Y.-H., 2003: Laboratory measurement of electrification of wind-blown sands and simulation of its effect on sand saltation movement. J. Geophys. Res., 108, 4322, doi:10.1029/2002JD002572. Zheng, X. J., Huang , N., Zhou, Y., 2006: The effect of electrostatic force on the evolution of sand saltation cloud. Eur. Phys. J. E, 19, 129–138. Zingg, A. W., 1953: Wind tunnel studies of the movement of sedimentary material. Proc. Fifth Hydraulics Conf., Bulletin 34, Iowa City, IA, Institute of Hydraulics, 111–135.  Address of the Author Hans-Jürgen Schönfeldt (hans@uni-leipzig.de) Universität Leipzig Institut für Meteorologie Stephanstraße 3 04103 Leipzig Wiss. Mitteil. Inst. f. Meteorol. Univ. Leipzig Band 48(2011)112

Turbulence and aeolian sand transport

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Turbulence and aeolian sand transport

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