Analysis of velocity profile measurements from wind-tunnel experiments with saltation

Geomorphology 59 (2004) 81 – 98 www.elsevier.com/locate/geomorph

Analysis of velocity profile measurements from wind-tunnel experiments with saltation B.O. Bauer a,*, C.A. Houser b, W.G. Nickling c a

Department of Geography, University of Southern California, Los Angeles, CA 90089-0255, USA Physical Sciences, University of Toronto at Scarborough, West Hill, Ontario, Canada M1C 1A4 c Wind Erosion Laboratory, Department of Geography, University of Guelph, Guelph, Ontario, Canada N1G 2W1 b

Accepted 16 July 2003

Abstract Investigations of wind-field modification due to the presence of saltating sediments have relied heavily on wind tunnels, which are known to impose geometric constraints on full boundary layer development. There remains great uncertainty as to which portion of the vertical wind-speed profile to analyze when deriving estimates of shear velocity or surface roughness length because the lower sections are modified to varying degree by saltation, whereas the upper segments may be altered by artificially induced wake-like effects. Thus, it is not obvious which of several alternative velocity-profile parameterizations (e.g., Law of the Wall, Velocity Defect Law, Wake Law) should be employed under such circumstances. A series of experimental wind-tunnel runs was conducted across a range of wind speed using fine- and coarse-grained sand to collect high-quality, fine-resolution data within and above the saltation layer using thermal anemometry and ruggedized probes. After each run, the rippled bottom was fixed with fine mist, and the experiment repeated without saltation. The measured wind-speed profiles were analyzed using six different approaches to derive estimates of shear velocity and roughness length. The results were compared to parameter estimates derived directly from sediment transport rate measurements, and on this basis, it is suggested that one of the six approaches is more robust than the others. Specifically, the best estimate of shear velocity during saltation is provided by the logarithmic law applied to the profile data within about 0.05 m of the bottom, despite the fact that this near-surface region is where profile modification by saltating sediments is most pronounced. Uncertainty remains as to whether this conclusion can be generalized to field situations because progressive downwind adjustments in the interrelationship between the saltation layer and the wind field are anticipated in wind tunnels, thereby confounding most analyses based on equilibrium assumptions. D 2003 Elsevier B.V. All rights reserved. Keywords: Aeolian geomorphology; Wind-speed profiles; Shear velocity; Logarithmic law; Wake law; Sediment transport; Equilibrium; Overshoot

1. Introduction

* Corresponding author. Tel.: +1-213-740-0050; fax: +1-213740-0056. E-mail address: [email protected] (B.O. Bauer). 0169-555X/$ - see front matter D 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.geomorph.2003.09.008

Sophisticated high-speed instrumentation for measuring the near-surface stress distribution within aeolian saltation layers has become available only recently. Heavy reliance was therefore placed on the

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precise measurement and analysis of vertical windspeed profiles to draw inferences about stress distributions in the plane perpendicular to the flow. In particular, the slope of the wind-speed profile was used to estimate shear velocity, which in turn, was used to predict sediment transport rate (e.g., Bagnold, 1941; Kawamura, 1950). Unfortunately, several practical and theoretical impediments preclude an easy and universally applicable solution to the problem of sediment transport prediction following this approach. Wind-speed profiles are rarely ideal in the sense that they consistently display log-linear behavior across the entire depth of flow. In saltation systems, for example, sand grains participate in momentum exchange processes with the fluid (e.g., Owen, 1964; Anderson and Haff, 1991), and some modification of the near-surface velocity and stress profiles is expected. In the absence of detailed measurements in such sediment-laden flow regions, there remains considerable uncertainty about the exact nature of these modifications as well as their influence on flow conditions in the upper, sedimentfree region. As an expedient, aeolian geomorphologists have generally assumed that a saltation layer manifests itself simply as a ‘surface’ of enhanced roughness (Owen, 1964) that, in turn, alters the stress – strain relationship in the flow field immediately above, not unlike any other bottom roughness element. Thus, intense sediment transport rate within the saltation layer implies large ‘effective’ surface roughness, which leads to a steep wind-speed profile (i.e., large shear velocity). In order to examine the validity of this heuristic model relating sediment transport rate to shear velocity above the saltation layer, increasing effort over the past decade has been directed at probing and modeling the internal dynamics of near-surface regions within saltating systems (e.g., Anderson and Haff, 1991; McEwan and Willetts, 1991, 1993; McEwan, 1993; McKenna Neuman and Nickling, 1994; Spies et al., 1995; McKenna Neuman and Maljaars, 1997; Butterfield, 1999; Spies et al., 2000a,b). Wind tunnels have proven indispensable in this regard, but they are also plagued by a host of constraints that may cause significant departures from natural flow systems in the field (Owen and Gillette, 1985). Because of restricted length scales, the full spectrum of turbulent motions found in atmospheric boundary layers cannot be reproduced in its entirety. Moreover, bottom boundary layers are not free to grow vertically in unrestricted

fashion because of sidewall and ceiling effects. In most wind tunnels, the entire boundary layer thickness is less than about 0.2 –0.3 m, whereas the constant stress region, if it exists, may be of only centimeter scale. In comparison, a typical saltation layer is of order 0.05– 0.10 m, and it may therefore occupy the entire inner layer as well as part of the overlap region (i.e., lower 10– 30% of the total boundary layer). To complicate matters, there exist large numbers of velocity-profile parameterizations that are potentially applicable to a given flow situation or profile segment (e.g., Law of the Wall, Velocity Defect Law, Law of the Wake, Power Law), so it is not always apparent which is most applicable at the outset (McKenna Neuman and Nickling, 1994; Spies et al., 1995; McKenna Neuman and Maljaars, 1997). Shear velocity estimates from a single wind-speed profile can differ as a consequence of model choice alone. The analytical challenge facing the aeolian researcher is to determine which of several alternative velocity-profile parameterizations is most appropriate for characterizing wind-tunnel boundary layers given that (a) the inner layer and overlap region are very small, (b) the saltation layer may extend above the overlap region, (c) flow in the saltation layer is altered due to the presence of the grains, and (d) flow above the saltation layer may be susceptible to wake deviations (Coles, 1956) arising from geometrical constraints within the experimental apparatus (e.g., Owen and Gillette, 1985; Spies et al., 1995; McKenna Neuman and Maljaars, 1997). The remainder of this paper will (1) briefly introduce a variety of alternative velocity-profile and surface roughness – length parameterizations; (2) describe a series of wind-tunnel experiments designed to yield high-quality data on the vertical wind-speed distribution above a rippled sand surface across a range of free-stream wind speeds, including both sediment-free and sediment-laden conditions; (3) present and analyze the experimental results; and (4) draw tentative conclusions about best practices for analyzing wind-speed profiles in the context of aeolian geomorphology.

2. Velocity profile parameterizations Complete descriptions of the vertical distribution of mean horizontal velocity, U(z), in clean-air turbu-

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lent boundary layers are generally derived by a classical asymptotic matching procedure that requires the inner layer profile (Law of the Wall) to be matched to the outer layer profile (Velocity Defect Law) in an overlap region or ‘inertial sublayer’ (Tennekes and Lumley, 1972; Raupach et al., 1991). In the inner layer, the relevant velocity scale is the shear (or friction) velocity, u*, whereas the relevant length scale for a smooth-wall boundary is the viscous length, m/ u*, where m is the kinematic viscosity. Prandtl (1932) proposed the following similarity relationship,
zu  U ðzÞ  ¼f u m

ð1Þ

which is expected to be universal (i.e., independent of external conditions) for inner layer flow over a smooth boundary. For a rough-wall boundary, the functional relationship in parentheses may also incorporate additional length scales such as the roughness height, k, and any other lengths necessary to define the geometry of the bottom roughness elements. In the outer layer, the velocity distribution is governed by Ul, the free-stream velocity, and d, the boundary-layer thickness. Flow in the outer layer is described by the Velocity Defect Law,
z U ðzÞ  Ul ¼f d u

ð2Þ

and unlike Eq. (1), the functional relationship should vary with the specifics of the flow field (i.e., it is not universal). Matching of the inner and outer layer profiles in the overlap region (or constant stress layer) leads to the well-known logarithmic law, U ðzÞ 1
zu  ¼ ln þB u j m

ð3Þ

where B is an integration constant that depends on surface roughness and shear velocity. In many earth science applications, the logarithmic law takes the following form:   U ðzÞ 1 z ¼ ln ð4Þ u j zo where j is the von Karman constant (taken as 0.41), z is height above the bed, and zo is roughness length or the height above the bed at which flow velocity tends to zero.

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Despite widespread acknowledgement that sediment contamination in the boundary layer alters the fluid dynamics of the inner layer, aeolian geomorphologists have liberally applied Eq. (4) to saltation systems. This was rationalized by Owen (1964), among others, who argued that the saltation layer acts only to enhance the total roughness felt by the velocity profile. If measurements are derived only from the region above the saltation layer, the velocity profile should be log-linear. Bagnold (1941), in fact, used similar reasoning when he proposed his expression for the velocity profile above the saltation layer: u
z  þ UV ð5Þ U ðzÞ ¼ ln kV j where k V is the height of the focal point ( f 0.003 m) and UV is the average wind velocity at the focal point, or alternatively, the sediment threshold velocity ( f 2.5 m s 1). Owen (1964) suggested that the velocity profile above the saltation layer should be parameterized as   u 2gz U ðzÞ ¼ ln þ DV ð6Þ u2 j where g is gravitational acceleration and D Vf 9.7. In many natural flow systems, the outer region profile deviates slightly from the logarithmic law. In order to account for this deviation, Coles (1956) proposed the Law of the Wake,   
 U ðzÞ 1 z z ln ¼ ð7Þ Pw u j zo d where P is a profile parameter that depends on the distribution of stress in the boundary layer. The wake function takes the following form,
z
p z  w ¼ 2sin2 ð8Þ d 2 d although various trigonometric identities are often invoked (e.g., 2sin2(Ap/2) = 1  cos(Ap)) leading to what appear to be different forms of the Law of the Wake. In all cases, when z/d = 1, then w(z/d) = 2, and when z/d = 0, then w(z/d) = 0, demonstrating that the Law of the Wake reduces to the logarithmic law close to the bottom, as desired. A velocity-defect form of the Law of the Wake can be derived by observing that

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U(z) = Ul when z = d, substituting in Eq. (7), and subtracting the resultant expression from Eq. (7) to produce
z
p z i Ul  U ðzÞ 1 h ln ¼ þ 2P þ 2Psin2 u j d 2 d ð9Þ This expression applies to the entire boundary layer above zu*/m = 30, presuming that P is constant. A value of P = 0.55 for clean-air flows with no downwind pressure gradients has been widely adopted, but when aeolian sand transport is active, the value of P changes. Janin and Cermak (1988), for example, suggest that P is functionally dependent on u*, with values of P increasing from 0.5 and eventually leveling off at 1.8 for large u*. It also appears that there may be some dependency on the type of wind-tunnel facility being used. Janin and Cermak (1988) used a very large wind tunnel with adjustable ceiling to compensate for longitudinal pressure gradients. In contrast, Spies et al. (1995) used a very small, fixed-ceiling wind tunnel and found that P ranged between 0.25 and 0.45. McKenna Neuman and Maljaars (1997) calculated values of P less than about 0.1 for their mediumsized, fixed-ceiling, wind tunnel, which incorporates a suction system that induces a negative pressure gradient. For open-channel (water) flow systems, Wang et al. (2001) conclude that P can be quite variable, sometimes assuming negative values.

3. Surface roughness parameterizations Nikuradse (1933) showed that for dynamically fully rough flows over a fixed sand surface, the roughness length is constant and given by zo c k/ 30, where k is some linear measure of the bottom roughness elements. It is conventional to approximate k by either 2D50 or D90, where D refers to grain diameter and the numerical subscript refers to the percentage of the population that is finer than the indicated size (Julien, 1995). However, a constant value for roughness length across all flow conditions and turbulence levels is clearly untenable. Charnock (1955) demonstrated that the roughness length over a

deformable water surface varies in proportion to shear velocity, zo ¼ a

u2 g

ð10Þ

where a is a dimensionless constant that ranges widely depending on environmental conditions. Charnock’s relationship has been applied to aeolian saltation (Owen, 1964), but results for the value of a are somewhat conflicting. Rasmussen (1989) suggested a = 0.01, which generally agrees with a value of 0.0091 suggested by Sherman (1992). Rasmussen et al. (1996) concluded that a should lie between 0.01 and 0.06. A modified form of the Charnock (1955) relationship for aeolian systems was eventually proposed by Sherman (1992). This expression for surface roughness includes threshold offsets, and it has the following form, zo 

2D50 ðu  ut Þ2 ¼a 30 g

ð11Þ

where u*t is the threshold shear velocity for the sediment size under investigation. In this case, the proportionality constant, a, becomes 0.0252. Various alternative expressions for roughness length in aeolian systems can be obtained by rearranging Eqs. (3), (5), and (6), and noting the similarity in form to Eq. (4). Respectively, these expressions are m ðjBÞ e u

ð12aÞ

zo ¼ kVeð u Þ

ð12bÞ

u2 ð jDV e u Þ 2g

ð12cÞ

zo ¼

jU V

zo ¼

In all cases, roughness length is functionally dependent on shear velocity, although this is not always explicit in the original form of the velocity profile expression. The practicing aeolian geomorphologist is therefore presented with a large variety of options as to which velocity-profile or roughness – length expression provides the most realistic and complete description of measured data.

B.O. Bauer et al. / Geomorphology 59 (2004) 81–98

4. Experimental design A series of wind-tunnel experiments was conducted to measure the vertical wind-speed profile throughout the entire boundary layer across a range of free-stream wind speeds. The experiments were carried out in the University of Guelph recirculating wind tunnel, which has a working section that is 8.0 m long, 0.92 m wide, and 0.76 m high. The working section is constructed of 19.2-mm (0.75 in.) plywood and is fitted with 12.7-mm (0.5-in.)-thick Plexiglas access windows down its entire length. Air flow in the tunnel is generated by a 0.91-m (36-in.) axial fan powered by a computer-controlled variable-speed 35hp electric motor. Air is moved by the fan through an expansion section, through two sets of turning vanes into a stilling section, followed by a honeycomb screen and three fine screens, and then it enters the working section through a fiberglass contraction bell (2.8:1 reduction). At the entrance to the working section, the air is ‘tripped’ over a dense staggered array (0.75  0.75 m) of vertically oriented 19.1  19.1-mm (0.75  0.75-in.) dowels to induce fully turbulent flow and to enhance boundarylayer development. Boundary layer thickness over the smooth plywood working section 6.8 m downwind of the tripping section is approximately 0.25 m deep. To simulate an upwind sand source, sediment was fed into the working section at a known rate using a hopper system that has a feed auger controlled by a variable speed DC electric motor. Sand was fed onto the bed upwind of the working section by a series of five tubes (20 mm ID) that extended down from the ceiling to 100 mm above the bed. After sediment passed the 8-m working section, it was removed in a catcher/stilling well section located far upwind of the fan in the closed-loop circuit. Electronic sensors measuring wind speed, sediment feed, and transport rate were wired to A-to-D boards housed in a personal computer and controlled by proprietary software. Experiments were conducted with fine-grained (D50 = 0.19 mm) and coarse-grained (D 50 = 0.27 mm) sand consisting predominantly of quartz (McKenna Neuman and Nickling, 1994). For each run, the sand bed was flattened by running a straightedged bar along the tops of two metal side rails fixed

85

to either side of the tunnel. The fan motor was set to a constant frequency, the sediment-feed system was then set to a supply rate that was sufficient to preclude sediment buildup or erosion beneath the hopper, and the transport system was given ample time for an equilibrium surface to establish itself (with low-amplitude ripples in many cases). Sediment transport rate was measured using a Guelph – Trent wedge-type trap (Nickling and McKenna Neuman, 1997) mounted directly above a hole cut into the wind-tunnel bottom at the downwind (measurement) section of the tunnel. Sediment captured by the trap orifice was funneled through the hole and onto a high-precision electronic balance connected to a computer-controlled data-acquisition system. Wind speed was measured using a high-speed thermal anemometry system (TSI, IFA 300) and a stainless-steel hot-film probe (TSI, Model 1266). The probe was fixed to a precision rack-and-pinion mount that was located in the center of the tunnel immediately in front of the wedge trap. The probe was lowered toward the sand surface in approximately 36 uneven steps (from 0.36 to 0.005 m with tighter spacing closer to the surface) and then raised back up to the free-stream core by reversing the steps. A pilot tube connected to a precision manometer remained in a fixed position in the free-stream core to ensure flow steadiness during each run. After a vertical windspeed profile was measured in its entirety, the fan and sediment-feed systems were turned off, the wind tunnel windows were opened, the entire sand surface was sprayed with water mist to ‘fix’ the surface, and an identical experiment was conducted on the stationary surface without sediment transport. Such paired runs were conducted across a range of free-stream velocities for both the fine-grained and coarse-grained sediment mixtures. Ripple dimensions were estimated using a ruler and by averaging across the test section of the wind tunnel at the conclusion of every pair of runs.

5. General results Table 1 provides summary data for each of the paired (dry, wet) data runs measured during the experiments. For both the fine-grained and coarsegrained sediment runs, wind speed varied from just

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B.O. Bauer et al. / Geomorphology 59 (2004) 81–98

Table 1 General wind, sediment, and bed characteristics for all experimental runs Fan (Hz) F1

9.5

F2

11.5

F3

12.75

F4a

14

F5

15.25

F6

16

F7

17.75

C1

15

C2

15.75

C3

17

C4

18.2

C5

19.75

C6

20.75

C7

21.75

dry wet dry wet dry wet dry wet dry wet dry wet dry wet dry wet dry wet dry wet dry wet dry wet dry wet dry wet

Ul (m s 1)

qs (kg m 1 s 1)

5.01 5.13 6.23 6.26 7.15 7.12 7.94 7.91 8.64 8.59 8.78 8.67 10.04 9.90 8.14 8.19 8.78 8.74 9.58 9.59 9.94 9.94 11.05 10.91 11.47 11.36 12.03 11.75

0.004387

Bed state

0.0043933

R (3.2)

0.0145669

R (4.1)

0.0139346

R (3.9)

0.0181356

R (4.4)

0.0242998

R (4.5)

0.0365168

UPB

0.0111984

R (4.3)

0.0170113

R (4.8)

0.0287854

R (4.6)

0.036764

R (5.8)

0.0377707

UPB

0.0493823

UPB

0.0404701

UPB

Runs F1 – F7 used fine-grained sand, whereas runs C1 – C7 used coarse-grained sands. ‘Wet’ indicates that the bed surface was moistened with mist to prevent sediment transport, whereas ‘dry’ indicates an unrestrained surface with active saltation. Ul is freestream wind speed, qs is sediment transport rate, R indicates a rippled bed state, UPB indicates upper plane bed or washed out ripples, and numbers in parentheses give ripple wavelength in centimeters. a Run F4 had unusual traits likely due to experimental error—it was eliminated from further consideration.

above the critical threshold of motion (for the slowest speed), through the ripple regime (medium speed), and almost to upper-plane bed (for the fastest speed). In general, ripple wavelength was slightly longer for faster wind speed and for coarse sediments. Ripple height for the fine-grained runs (F1 – F7) ranged between 1.5 and 2 mm, whereas for the coarse-grained runs (C1 –C7) they were slightly smaller (1– 2 mm) with no discernable dependency on wind speed in either case. Thus, for most runs with a rippled bottom,

the surface roughness was geometrically similar. It should be noted that the Froude Number criterion (Fr < 20; where Fr = U2/gH and H is wind-tunnel height) recommended by Owen and Gillette (1985) for free development of the boundary layer is violated at wind speeds beyond 12.2 m s 1 in the Guelph wind tunnel. Fig. 1 shows selected wind-speed profiles with and without saltation for both fine-grained and coarsegrained runs. With few exceptions, the ‘no transport’ (wet) profiles are log-linear, except in the uppermost flow region where wind-tunnel constraints are apparent. In contrast, the profiles ‘with saltation’ (dry) tended to be convex upward, especially in the lower flow region, which is consistent with prior modeling results (Anderson and Haff, 1988; McEwan and Willetts, 1991) and with the observations of many other researchers (e.g., Gerety, 1985; Sullivan and Greeley, 1993; McKenna Neuman and Nickling, 1994; Butterfield, 1999). The uppermost flow regions of the ‘with saltation’ runs were also influenced by wind-tunnel constraints. Examination of the windspeed time series from these upper regions demonstrated that the turbulence signatures were distinctly different from those lower in the profile. On this basis, the boundary layer thickness was assessed to be between 0.23 and 0.26 m deep for all runs. Data from higher in the profile were eliminated from subsequent analysis. The outer flow regions of the ‘with saltation’ profiles (roughly between about 0.06 and 0.24 m) were log-linear with slight convex curvature, whereas below about 0.06 m, the profile became progressively more convex (i.e., less steep when wind speed is plotted on the graph ordinate). Comparison of the ‘no transport’ and ‘with saltation’ cases (Fig. 1) clearly shows the influence of sediment transport on the wind-speed profile. Whereas the near-surface velocities in the ‘no transport’ cases increase in proportion to increases in free-stream velocity, the presence of a saltation layer tends to constrain near-surface velocities. The range in wind speed measured at 0.005 m above the bed for the suite of ‘no transport’ cases is 2– 8 m s 1, but for the ‘with saltation’ cases, it is only 2 –4 m s 1. Thus, when saltation is active, the free-stream core of the wind tunnel forces an over-steepened velocity profile, which is likely an artifact of the apparatus but nevertheless reaffirms the idea that the presence of a

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87

Fig. 1. Selected wind-speed profiles with and without sediment transport. Solid symbols refer to fine-grained sediment runs, whereas open symbols refer to coarse-grained sediment runs.

saltation layer manifests itself as an enhanced surface roughness to which the overlying wind field must adapt (e.g., Owen, 1964).

6. Shear velocity estimation Given that wind-speed profiles in wind tunnels have near-surface layers that are modified due to the presence of sediments, middle sections that are artificially over-steepened, and upper regions that contain wake-like deviations, there can be considerable uncertainty regarding the best methodology for profile analysis. In order to determine the implications for estimating shear velocity, the velocity profile laws discussed in Section 2 were fitted to the wind-speed data. In an initial series of tests, the logarithmic law (Eq. (4)) was fitted to each of the wind-speed profiles using all the data within the boundary layer (i.e., below 0.245 m). A second series of tests incorporated the velocity-defect form of Coles’ Law of the Wake (Eq. (8)). Since the Law of the Wake applies strictly to outer flow regions, the data below 0.045 m (i.e., lower 15 –20% of total boundary layer depth) were eliminated from these regressions. The inner-layer data were employed, nevertheless, to calculate values of P for individual profiles following the methods outlined in Julien (1995, p. 103). Variable values of P generated in this manner were subsequently used in

the regressions to calculate u* from the Law of the Wake. Table 2 gives the numerical results from the curve-fitting analyses for all data runs, whereas Fig. 2 presents selected examples. The profiles were satisfactorily linearized by either the logarithmic law or the Law of the Wake, especially for the ‘no transport’ cases (open symbols in Fig. 2). Shear velocity (u*) for the fine-grained ‘no transport’ runs ranged between 0.267 and 0.439 m s 1 for the logarithmic law and 0.250– 0.482 m s  1 for the wake law. For the coarse-grained ‘no transport’ runs, the range was 0.438 – 0.526 m s 1 for the logarithmic law and 0.442 –0.450 m s 1 for the wake law. These lawdependent differences are small in comparison to the results for the ‘with saltation’ profiles (solid symbols in Fig. 2). Shear velocity (u*) for the fine-grained ‘with saltation’ runs ranged between 0.278 and 0.827 m s 1 for the logarithmic law but only 0.261 – 0.651 m s 1 for the wake law. Similarly, for the coarsegrained ‘with saltation’ runs, the range was 0.487 – 0.914 m s 1 for the logarithmic law but only 0.501 – 0.684 m s 1 for the wake law. Thus, the wake law (with variable P) leads to estimates of u* that are generally smaller than those derived from the logarithmic law, especially when sediment transport rate is intense or when wind speed is large. Several researchers have advocated using profile data from above the saltation layer only (e.g., Bagnold, 1941; Gerety, 1985). This practice has been

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Table 2 Shear velocity (u*) estimates for all experimental runs based on (1) the logarithmic law using velocity profile data measured at heights less than 0.245 m and (2) the law of the wake incorporating variable values of the wake parameter, P Fan (Hz) F1

9.5

F2

11.5

F3

12.75

F4a

14

F5

15.25

F6

16

F7

17.75

C1

15

C2

15.75

C3

17

C4

18.2

C5

19.75

C6

20.75

C7

21.75

dry wet dry wet dry wet dry wet dry wet dry wet dry wet dry wet dry wet dry wet dry wet dry wet dry wet dry wet

U* (m s1) [z<0.245m]

j

U* (m s1) [j]

0.278 0.267 0.389 0.326 0.476 0.347 0.530 0.339 0.642 0.410 0.641 0.407 0.827 0.439 0.487 0.438 0.577 0.450 0.657 0.493 0.701 0.484 0.809 0.478 0.846 0.504 0.914 0.526

0.133 0.186 0.305  0.012 0.282 0.004 0.399 0.209 0.464  0.033 0.462 0.068 0.425  0.007 0.092 0.001 0.248  0.181 0.472 0.122 0.540 0.065 0.571 0.145 0.535 0.160 0.505 0.127

0.261 0.250 0.332 0.330 0.378 0.329 0.385 0.277 0.481 0.450 0.494 0.393 0.651 0.482 0.501 0.442 0.520 0.583 0.478 0.478 0.492 0.471 0.567 0.449 0.623 0.436 0.686 0.450

a Run F4 had unusual traits likely due to experimental error—it was eliminated from further consideration.

widely adopted in field experiments where the large size of cup anemometers precludes direct measurements within the saltation layer. Accordingly, a third series of log-linear regressions was conducted on the middle section of the profiles, above the region of intense saltation but well below the top of the boundary layer (i.e., from 0.06 – 0.18 m inclusive). An alternative strategy was recommended by McEwan (1993) who notes that many wind profiles contain a distinctive kink that is usually quite close to the surface. The presence of a kink is indicative of a local maximum in the force per unit volume exerted by grains on the wind. McEwan (1993, p. 155) suggests that measurements be acquired above the kink but still in the saltation layer. Although there is little evidence

of a near-surface kink in our profiles, a fourth series of log-linear regressions was conducted using the data below 0.045 m, which are presumably all within the saltation layer. Fig. 3 compares u* estimates from all four regression approaches (z < 0.245 m; wake law with variable P; 0.055 < z < 0.185 m; and z < 0.045 m) plotted as a function of mean wind speed at the top of the boundary layer. For the ‘no transport’ runs, these different parameterizations produced only minor, nonsystematic variations in the estimates of u*. It would be difficult to recommend one method over another. The absolute variation among the four parameterizations for the ‘no transport’ runs is greatest for the coarse-grained surface (because shear velocities are greater), but these differences are relatively small in comparison to the variation evident among the ‘with saltation’ runs. In both the fine-grained and coarse-grained cases, the smallest estimates of u* were given by the logarithmic law using only the inner layer data (z < 0.045 m), which produced values in the range of 0.248 –0.633 m s 1 for the fine-grained runs and 0.457 – 0.675 m s 1 for the coarse-grained runs. The wake law (with variable P) produced results that were virtually identical to the inner-layer method. The logarithmic law using data from the entire profile (z < 0.245 m) gave intermediate u* estimates (see Table 2), while the largest estimates of u* were given by the logarithmic law using data from the middle section of the profiles (0.055 < z < 0.185 m). These latter estimates are somewhat larger than typically found in field experiments, ranging from 0.320 to 1.030 m s 1 for the fine-grained ‘with saltation’ runs and from 0.590 to 1.140 m s 1 for the coarse-grained ‘with saltation’ runs. Such large values suggest that this region of the wind tunnel is a zone of pronounced stress and intense vertical momentum transfer. It is the manifestation of an artificially over-steepened profile induced by the geometric constraints imposed by wind-tunnel size. Given this uncertainty regarding the ‘best’ value of u*, three more series of tests were conducted. The fifth involved fitting the Law of the Wake to the profile data using a fixed clean-air value of P = 0.55, as did White and Mounla (1991) and Arnold (2002). This approach produced u* estimates that were consistently (and unrealistically) smaller

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89

Fig. 2. Selected wind-speed profiles plotted according to the logarithmic law (Eq. (4); upper panel of each pairing) and the velocity-defect form of the Law of the Wake (Eq. (9); lower panel of each pairing). Note that the logarithmic law profiles include all data within the boundary layer (z < 0.245 m), whereas the Law of the Wake profiles do not include near-surface data (z < 0.045 m). Slight curvature is evident in the lowermost portions of the high-intensity ‘with saltation’ runs.

than any other method, especially for the ‘no transport’ runs. During intense sediment transport, the agreement between the variable P and fixed P approaches improved. Table 2 shows that the variable values of P generally increase with sediment transport intensity. For runs C4 –C7, the value of P was actually very close to the clean-air value of

0.55. In no case, however, did the value of P exceed 0.6, which is the value implicit to Owen and Gillette’s (1985) method for analyzing wakecorrected profiles. Notwithstanding the careful measurements of Janin and Cermak (1988) who reported very large values of P, it is noteworthy that a value of P greater than the clean-air value of

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Fig. 3. Variation of shear velocity (with and without sediment transport) as a function of free-stream wind speed. Four different approaches were used to estimate shear velocity including Law of the Wake regression on the outer-layer data with variable C, and log-linear regression on (i) the full boundary-layer data set (z < 0.245 m), (ii) the near-surface segment (z < 0.045 m), and (iii) the middle segment (0.055 < z < 0.185 m).

0.55 would lead to even smaller u* estimates in our wind-tunnel tests. Therefore, adopting a constant a priori value of P>0.55 seems a risky strategy that cannot be recommended. It is interesting to note that the value of P for the ‘no transport’ runs varied between  0.2 and + 0.2 with no sense of a trend (Table 2). This implies that there is no consistent wake-like deviation in these ‘no transport’ velocity profiles, making the law of the wake superfluous for the fixed-surface runs. A value of P = 0 (i.e., no wake deviation) is generally expected for flows in pipes and open channels, whereas a value of P = 0.55 is expected for clean-air flows across a flat plate. In combination, these results suggest that the Guelph wind tunnel behaves essentially like a pipe when clean air moves through it,

whereas a wake-like deviation appears when saltating sediments are introduced into the flow. A sixth series of tests was conducted using the smooth-surface version of the logarithmic law (Eq. (3)) rather than its general form (Eq. (4)). Explicit formulations for zo (Eqs. (12a) – (12c)) demonstrate that roughness length and shear velocity are interdependent quantities. Often there is uncertainty in knowing the exact location of the bottom boundary within geometrically complex surfaces (Bauer et al., 1992), especially under field conditions. To account for this, several researchers have advocated the use of a displacement height in reformulating Eq. (4) (e.g., Jackson, 1981). Unfortunately, this procedure may lead to unrealistic results (Raupach et al., 1991; Sullivan and Greeley, 1993), and it introduces yet

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another unconstrained parameter into the analysis. The alternative approach adopted in this study was to use Eq. (3) directly by regressing U/u* against z+ = u*z/m. Since shear velocity (an unknown quantity) appears in both sides of the equation, standard regression procedures cannot be applied to the data directly. A computer algorithm was written to iteratively adjust u* and B until the sum of squared deviations from a straight-line fit was minimized for each profile. Fig. 4 shows that the wind-speed profiles were adequately linearized by this smoothsurface parameterization, although minor deviations are still evident in the near-surface region for the ‘with transport’ cases. The slopes of the best-fit lines are constant (proportional to 1/j) and independent of u*. A large degree of separation between paired lines indicates a large difference in shear velocity and surface roughness between the ‘no transport’ and

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‘with saltation’ profiles. The higher the placement of the line on the graph, the greater the vertical intercept (B) and the smaller the dynamic roughness (Eq. (12a)). Estimates of u* derived from this approach were surprisingly similar (yet not identical) to those derived from regressions using Eq. (4), and therefore they are not reported here. Presumably, this is a consequence of the inherent log-linearity of these particular wind-tunnel profiles—more striking differences are anticipated with highly nonlinear profiles. Support for this conjecture derives from observations that the greatest deviations between u* estimates from Eq. (3) versus Eq. (4) were most evident in the large wind-speed, coarse-grained, ‘with transport’ cases (i.e., the most nonlinear profiles). A seventh and final series of tests was conducted by back-calculating shear velocity on the basis of measured sediment transport rate (Table 1). Bagnold

Fig. 4. Selected wind-speed profiles on a Clauser plot. Regression lines show specific instances of Eq. (3) that correspond to parameter values (u*,B) for which the data are best linearized according to a least-squares minimization rule. Slight curvature is evident in the lowermost portions of the high-intensity ‘with saltation’ runs.

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(1941) proposed a relatively simple transport formula that has been widely adopted by aeolian researchers,  0:5 d q 3 u q¼C D g 

ð13Þ

where q is sediment (mass) transport rate per unit width (kg m 1 s 1), d is grain size (mm), D is standard grain size (0.25 mm), q is air density (taken as 1.14 kg m 3), g is gravitational acceleration (9.81 m s 2), and C is an empirical constant (equivalent to 1.5 for uniformly sorted sediments). This expression is easily rearranged to isolate u*. All other variables were measured during the wind-

tunnel experiments, thereby allowing explicit solution for u*. The results from the back-calculation procedure are plotted in Fig. 5 alongside estimates of u* obtained from log-linear regression using the full boundary-layer data set (z < 0.245 m) and from the inner layer (z < 0.045 m). Using only the inner layer data in the regression produces u* estimates that conform reasonably well to the back-calculated values from Bagnold’s model. This implies that the Law of the Wake procedure (with variable P) should also provide similar levels of agreement (see Fig. 3). Using a fixed, clean-air value of P = 0.55 will vastly underestimate u*, whereas all other approaches (including log-linear regression using

Fig. 5. Comparison of shear velocity estimates (with and without sediment transport) derived from profile analysis versus those derived from direct measurements of sediment transport rate, as a function of free-stream wind speed. Results from two of the approaches shown in Fig. 3 (i.e., log-linear regression on the full boundary-layer data set (z < 0.245 m) and on the near-surface segment (z < 0.045 m)) are reproduced. Large diamonds show u* estimates derived from Bagnold’s model.

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the full boundary-layer data set) will overestimate u*, especially when transport rate is large. In Fig. 5, the back-calculated shear velocities from the ‘with transport’ cases have also been plotted relative to the ‘no transport’ u* estimates for comparison purposes only. McKenna Neuman and Nickling (1994) speculated that the use of fixed-surface values of u* might contribute to greater experimental consistency in flux model development, but the results here suggest that such fixed-surface (‘no transport’) values for u* only lead to large underestimates of the measured transport rate. This is consistent with the assertion that the saltation layer (rather than the bedform ripple geometry) dominates the effective surface roughness to which the wind adapts.

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7. Roughness length estimation Estimates of roughness length for all wind-speed profiles were calculated directly from the various best fit regression parameters generated for estimating shear velocity. Fig. 6 shows roughness length plotted as a function of shear velocity derived from the logarithmic law (Eq. (4)) with all data in the boundary layer (z < 0.245 m) and with only the nearsurface data (z < 0.045 m). Not shown on these diagrams are the results of regressions using Eq. (4) but using the data from the middle section of the profiles (0.055 < z < 0.185 m). This approach produced roughness length and shear velocity estimates that were larger (by a factor of about two) than the other methods. Also not shown are the results of

Fig. 6. Variation in surface roughness length as a function of shear velocity where both parameter pairs were estimated using two different approaches (log-linear regression on the full boundary-layer data set (z < 0.245 m) and on the near-surface segment (z < 0.045 m)). The sequence of lines shows the trends predicted by the models of Bagnold, Owen, Charnock, and Sherman. See text for explanation. Note different vertical axes.

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applying Eqs. (3) and (12a) to the full boundarylayer data set because this approach produced results that were similar to those from Eq. (4). Regressions using only the near-surface data set produced roughness length estimates (as well as shear velocity estimates) that were invariably smaller than those derived using the complete boundarylayer data set. As mentioned previously, because of convex profile curvature within the saltation layer, the near-surface segment is typically of shallower gradient than the outer layer slope, so this result is not surprising. Estimates of roughness length for the ‘no transport’ cases are generally very small (note different scales on axes), and surface roughness generally decreases with increasing shear velocity. Indeed, the smallest roughness lengths occurred during runs with greatest shear velocity, most likely because the surface configuration was approaching upper plane bed (see Table 1). Thus, any enhanced surface roughness that is registered in the velocity profile when saltation is active is primarily due to the presence and action of the saltation cloud rather than to bedform-induced roughness, which actually becomes smaller as transport intensifies. For reference, a series of curves showing the expected relationship given by the methods of Charnock (Eq. (10)), Sherman (Eq. (11)), Bagnold (Eq. (12b)), and Owen (Eq. (12c)) are shown on Fig. 6. In each case, standard values of the various empirical constants were used—that is, Charnock’s a = 0.0091, Sherman’s a = 0.0252 (with u*t = 0.208 m s 1 for D 5 0 = 0.00019 m; and u * t = 0.239 m s  1 for D50 = 0.00025 m), Bagnold’s focal point (0.003 m, 2.5 m s 1), and Owen’s DV = 9.7. These models and empirical constants are not strictly applicable to the ‘no transport’ case, but they are nevertheless plotted for reference. None of the models accommodate the expected decrease in surface roughness associated with bedform flattening that takes place at increasing shear velocity during intense transport. For the finegrained, ‘with saltation’ runs, all models under-predict surface roughness, especially when shear velocity is large and transport is intense. For the coarse-grained, ‘with saltation’ runs, Owen’s model again leads to under-predictions. The other three models provide varying levels of prediction depending on the magnitude of shear velocity and also on which data set is used in the regression procedure. In all cases, better

fits could have been obtained by changing the empirical constants, but this is an unsatisfying requirement. Overall, Sherman’s modified form of the Charnock relationship (Eq. (11)) appears to provide the best results both in terms of accuracy and curve shape. The predictive power of this model might be improved further if some measure of ripple geometry could be incorporated into the threshold offset rather than simple grain roughness (2D50/30) alone.

8. Discussion Several studies have examined wind-tunnel velocity profiles in the presence of saltation (e.g., Gerety, 1985; Owen and Gillette, 1985; Janin and Cermak, 1988; Spies et al., 1995; McKenna Neuman and Maljaars, 1997). A large number of additional studies have either measured or numerically simulated the vertical profile of sediment mass flux (e.g., Bagnold, 1941; Anderson and Haff, 1991; McEwan and Willetts, 1991, 1993; McEwan, 1993; McKenna Neuman and Nickling, 1994; Butterfield, 1999; Rasmussen and Sørensen, 1999; Namikas, 1999; Spies et al., 2000a,b; Dong et al., 2002; Ni et al., 2002) in attempts to decipher more fully the relationship between grainborne and fluid-borne stress distributions within the near-surface layer. Their purpose, ultimately, is to construct robust sediment transport models that can be used to predict mass flux (difficult to measure) on the basis of easily measurable wind-field variables. However, the results from these many studies are divergent and inconclusive. Under ideal conditions above a flat surface, the clean-air velocity profile should be log-linear throughout most of the lower flow region. During saltation, however, the velocity profile close to the surface is typically segmented with a distinct kink or with smooth curvature. Gerety (1985) analyzed a large number of profiles from her wind-tunnel experiments, as well as those of several others, and noted that many wind-speed profiles have two distinct parts: a shallow-gradient segment near the bed and a steep-gradient segment above. These segments are joined by a transition in slope that Gerety (1985, p. 287) suggests is similar to Bagnold’s kink. The analytical work of McEwan (1993) demonstrates that a wind-speed profile with a distinct kink implies a

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well-defined height maximum in momentum extraction by grains from the wind. However, such a maximum may not occur when there is a wide distribution of particle trajectories in the saltation layer, and therefore profile curvature is produced. Thus, both types of wind-speed profile (kinked or curved) are expected with saltation depending on the distribution of particle trajectories. A kink or slope transition in the wind-speed profile, if it exists, should be located very close to the surface (e.g., Bagnold, 1941; McEwan, 1993), and widespread observations indicate that approximately 50 –80% of sediment flux occurs within 0.02 m of the bed (e.g., Gerety, 1985; McKenna Neuman and Nickling, 1994; Butterfield, 1999). Below the kink, the influence of grains in modifying the wind field can be large, and therefore, the local effective (fluid) shear velocity should change progressively with height above the bed. This implies that a constant stress region should not exist below the kink, and therefore, the simple log-linear model is strictly not valid. Above the kink, the influence of grains decays rapidly with height. Gerety (1985, p. 295) states that Although saltating grains (and perhaps suspended grains) are present above 3 cm, their concentration and their velocities are low enough that they do not contribute significantly to momentum at that level. If this explanation is correct, then the wind velocity profiles above 2 cm should show a constant-stress layer described by the logarithmic law and a wake region above it (Owen and Gillette, 1985). McEwan (1993) concurs and asserts that velocity measurements above the kink, but still within the saltation layer, should provide effective fluid stress estimates that are consistent with total stress. Although this reasoning seems to provide a robust methodology for estimating u* (especially values pertinent to the prediction of sediment transport), a number of uncertainties remain. Most measured profiles from aeolian experiments (even those in wind tunnels) include few data from below about 0.02 m, making it all but impossible to discern a distinctive, near-surface kink as described by Bagnold (1941) and McEwan (1993). Indeed, many wind-speed profiles do not display a kink at all, but

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rather, a region of curvature that extends from the surface up to about 0.05 –0.1 m above the bed (e.g., Gerety, 1985; Spies et al., 1995; Butterfield, 1999). This can be seen clearly in the ‘with saltation’ profiles shown in Fig. 1. Because pronounced curvature does not exist in the ‘no transport’ profiles, and given that the only difference between the data runs was the presence of sediment, such curvature is unambiguously due to saltation. Interestingly, Spies et al. (1995) observed that both their sediment-laden and sedimentfree profiles were curved, and therefore, they concluded that profile curvature in their wind tunnel was not only due to saltation exclusively but also due to wake departures forced by the small dimensions of the tunnel. These ambiguities pose severe practical challenges for the aeolian researcher when faced with analyzing wind-speed profiles. In the absence of very detailed information on the near-surface wind and sediment flux profiles, it is difficult to decide whether a distinct kink exists, and if so, at what height it is located. Ideally, a kink or slope transition appears very close to the surface, and the region immediately above the kink is log-linear over a considerable thickness. In this case, log-linear regression (following Eq. (4)) using the profile segment above the kink should provide robust estimates of u*, as recommended by McEwan (1993). Data from below the kink should not be used in the regression because grainborne stress progressively alters the fluid-borne stress in this region, and therefore, effective shear velocity will vary with height (i.e., the constant stress assumption is invalid). If the upper parts of the profile (i.e., far above the saltation layer) deviate substantially from a log-linear trend, as is likely in wind tunnels, then a choice must be made about whether to use only data from a short segment above the kink (thereby reducing statistical significance) or to apply an alternative model such as the Law of the Wake to take into account externally imposed flow deviations. Either approach retains unknown degrees of uncertainty in the estimate of u*. Spies et al. (1995, p. 520) conclude that ‘in many cases a wake correction is required,’ but they also acknowledge the difficulties in determining a robust value for P. More importantly, the Law of the Wake applies specifically to deviations in the outer part of the profile, yet it is the inner part of the profile that is

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altered when saltation is active. Thus, even if the Law of the Wake provides a more defensible estimate of u* for the outer flow region than the logarithmic law, there remains considerable uncertainty about how such an outer-layer estimate of shear velocity relates to the stress distribution within and directly above the saltation layer. McKenna Neuman and Maljaars (1997) concluded that Coles’ wake function has limited utility for profile analysis in their wind tunnel and simply ascribed the departure from a logarithmic law in the upper flow region to an inherent adjustment to the ‘effective roughness’ produced by the saltation layer. The study results presented in this paper argue for adopting an approach that employs only those measurements from below about 0.05 m, despite the fact that wind-field modification by saltating sediments is pronounced throughout this near-surface layer. Fig. 5 shows that u* estimates so produced are in reasonable agreement with back-calculated values from the Bagnold model (Eq. (13)) using direct sediment transport rate measurements as input. The accompanying surface roughness length estimates are also consistent with those reported in the literature and with predictions based on the models of Charnock (Eq. (10)) and Sherman (Eq. (11)), as shown in Fig. 6. In contrast, when data from the entire boundary layer (z < 0.245 m) are used in a log-linear regression, the values of u* and zo are consistently larger than those using only the near-surface data, therefore leading to transport rate over-prediction. Fig. 5 shows that the discrepancy increases with increasing wind speed (and intensified transport) presumably because the profile segment above the saltation layer becomes unnaturally and progressively over-steepened. This tendency devalues the use of the full boundary-layer profile in log-linear regressions to estimate u* for the purposes of predicting sediment transport in wind tunnels, especially at high wind speeds when the Froude Number criterion is violated. When only the data from the middle section (0.055 < z < 0.185 m) of the profiles were used in the log-linear regression, exceptionally large values of u* and zo were produced. This over-steepened profile segment represents a zone of transition or coupling between the near-surface grain-modified flow within the saltation layer and the free-stream, grain-free flow in the central core of the wind tunnel

(which is mechanically forced by the fan and is integral to preserving mass continuity in this closedcircuit wind tunnel). The existence of such an intense region of shear has been recognized by others, and it raises several interesting concerns regarding the correct interpretation of results derived from windtunnel experiments. Specifically, the issue of disequilibrium looms large. Recent studies by Spies et al. (2000a) and Arnold (2002) demonstrate that the wind field experiences a pronounced ‘shock’ at the leading edge of the tunnel working section due to the sudden introduction of sediments into the flow field. These particles, whether entrained from the bed or introduced via a mechanical feed system, extract considerable momentum from the wind. As the perturbation propagates downwind, the saltation layer and the boundary layer undergo mutual adjustments that eventually lead toward an equilibrium interaction between the transport rate, the effective surface roughness, and the shear velocity. Prior to attainment of this equilibrium state, however, sediment flux and the shear stress overshoot their equilibrium values (Shao and Raupauch, 1992). This overshoot phenomenon occurs in both time (e.g., McEwan and Willetts, 1993) and space (e.g., Spies et al., 2000a; Arnold, 2002). Relaxation times following overshoot are on the order of several seconds and relaxation lengths are on the order of meters to tens of meters. Even under moderate wind speeds, these relaxation scales exceed those of most aeolian transport wind tunnels, implying that the majority of wind-tunnel measurements (including ours) are potentially representative of disequilibrium states, despite temporal steadiness at any fixed location. Since the criterion used to judge superiority of one analytical approach over another (i.e., back-calculated shear velocities based on Bagnold’s model) is inherently predicated on an equilibrium presumption, the recommendation to use only data within 0.05 m of the bed in log-linear regressions to estimate u* should be accepted with due caution. The possibility exists that the intensely sheared layer above 0.05 m may contain the most interesting dynamics relevant to understanding the mysteries of aeolian saltation. A well-designed field experiment examining the spatial evolution of the near-surface boundary layer (within and immediately above the saltation layer) would be most revealing in this regard.

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9. Summary and conclusions Recent numerical and theoretical studies have provided insight into the profiles of fluid velocity, mass flux, and fluid shear within saltation layers, but there is as yet insufficient empirical information to fully validate these models (Arnold, 2002). In particular, one would like to know how a mixed population of saltating sediments interacts with the near-surface wind field under variable conditions and how this interaction influences the shape of the velocity profile as well as the associated stresses and turbulence characteristics within and immediately above the saltation layer. Although several wind-tunnel studies have been directed specifically at this problem, considerable uncertainty remains because the restricted geometry of wind tunnels imposes artificial constraints on boundary layer development. In consequence, the upper and middle parts of wind-tunnel boundary layers typically show some departure from log-linear behavior that is not exclusively attributable to saltation effects propagated upward from below. The question that arises, then, is ‘which, if any, of several alternative approaches to analyzing velocity profiles provides the best estimates of u* and zo?’ This study suggests that the most robust estimates of u* and zo are provided when the standard form of the logarithmic law (Eq. (4)) is applied to the nearsurface data (below 0.05 m) only. Presumably, this is because the zone where grain-borne stress is large was of limited vertical extent in our experiments (i.e., less than 0.01 m), and therefore, most of the data in the near-surface profiles were nominally above the region in which effective shear velocity changes substantially with height. In contrast, the wind-speed profile between 0.05 and 0.18 m was characteristically oversteepened, in large part due to mechanical forcing of the free stream in the core of the wind tunnel. It is probable that such profile over-steepening is related to the overshoot phenomenon described by Shao and Raupauch (1992) and therefore dynamically linked to the spatial evolution of the boundary layer. This implies non-homogeneity and disequilibrium in all but the longest wind tunnels. This study also suggests that the Law of the Wake may be theoretically and empirically inadequate for use in wind-tunnel studies involving saltation. Ironically, applying the Law of the Wake to the outer layer

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data yielded u* estimates that were very close to those obtained from the recommended approach of applying log-linear regression to near-surface data only. The process of calculating variable values of P for each profile via regressions through the near-surface data prior to applying the Law of the Wake to the outerprofile data may have been critical in this regard. In general, however, it is not safe to assume that a constant value for P (i.e., the clean-air value of 0.55) in the Law of the Wake will yield reasonable results during saltation, even if such a parameterization leads to ideal linearization of the profile data. The extent to which any of these wind-tunnel results are applicable to equilibrium saltation systems in natural environments remains to be investigated. Acknowledgements We would like to thank Mr. Mario Finoro who provided outstanding technical and logistical support during the project. The senior author acknowledges the hospitality of the Department of Geography, University of Guelph during his visit. Graeme Butterfield provided a substantive and insightful review, as did Ian McEwan who pointed out the potential relevance of the overshoot phenomenon. Financial support was provided in part from NSF grant SBR-9511529 (BOB) and NSERC Operating grant 7427-02 (WGN). This is Geomorphic Processes Laboratory (GPL) Contribution 42. References Anderson, R.S., Haff, P.K., 1988. Simulation of eolian saltation. Science 241, 820 – 823. Anderson, R.S., Haff, P.K., 1991. Wind modification and bed response during saltation of sand in air. Acta Mechanica, Supplementum 1, 21 – 51. Arnold, S., 2002. Development of the saltation system under controlled environmental conditions. Earth Surface Processes and Landforms 27, 817 – 829. Bagnold, R.A., 1941. The Physics of Blown Sand and Desert Dunes. William Morrow and Company, New York. Bauer, B.O., Sherman, D.J., Wolcott, J.F., 1992. Sources of uncertainty in shear stress and roughness length estimates from velocity profiles. Professional Geographer 44 (4), 453 – 464. Butterfield, G., 1999. Near-bed mass flux profiles in aeolian sand transport: high-resolution measurements in a wind tunnel. Earth Surface Processes and Landforms 24, 393 – 412.

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Charnock, H., 1955. Wind stress on a water surface. Quarterly Journal of the Royal Meteorological Society 81, 639 – 640. Coles, D., 1956. The law of the wake in the turbulent boundary layer. Journal of Fluid Mechanics 1, 191 – 226. Dong, Z., Liu, X., Wang, H., Zhao, A., Wang, X., 2002. The flux profile of a blowing sand cloud: a wind tunnel investigation. Geomorphology 49, 219 – 230. Gerety, K.M., 1985. Problems with determination of U* from wind velocity profiles measured in experiments with saltation. In: Barndorff-Nielsen, O.E., Moller, J.T., Rasmussen, K.R., Willetts, B.B. (Eds.), Proceedings, International Workshop on Physics of Blown Sand. Institute of Mathematics Memoir, vol. 8. University of Aarhus, Denmark, pp. 271 – 300. Jackson, P., 1981. On the displacement height in the logarithmic velocity profile. Journal of Fluid Mechanics 111, 15 – 25. Janin, L.G., Cermak, J.E., 1988. Sediment-laden velocity profiles developed in a long boundary-layer wind tunnel. Journal of Wind Engineering and Industrial Aerodynamics 28, 159 – 168. Julien, P.Y., 1995. Erosion and Sedimentation. Cambridge Univ. Press, Cambridge, UK. Kawamura, R., 1950. Study on Sand Movement by Wind. Report of the Institute of Science and Technology, University of Tokyo, 5(3/4), 95-112 (translated from Japanese, Institute of Engineering Research Technical Report HEL 2-8, Hydraulic Engineering Laboratory, University of California, Berkeley, 1964, pp. 1 – 38). McEwan, I.K., 1993. Bagnold’s kink: a physical feature of a wind velocity profile modified by blown sand? Earth Surface Processes and Landforms 18, 145 – 156. McEwan, I.K., Willetts, B.B., 1991. Numerical model of the saltation cloud. Acta Mechanica Supplementum 1, 53 – 66. McEwan, I.K., Willetts, B.B., 1993. Adaption of the near-surface wind to the development of sand transport. Journal of Fluid Mechanics 252, 99 – 115. McKenna Neuman, C., Maljaars, M., 1997. Wind tunnel measurement of boundary-layer response to sediment transport. Boundary-Layer Meteorology 84, 67 – 83. McKenna Neuman, C., Nickling, W.G., 1994. Momentum extraction with saltation: implications for experimental evaluation of wind profile parameters. Boundary-Layer Meteorology 68, 35 – 50. Namikas, S.L., 1999. Aeolian saltation: field measurements and numerical simulations. Doctoral Dissertation, Department of Geography, University of Southern California, Los Angeles, California. Ni, J.R., Li, Z.S., Mendoza, C., 2002. Vertical profiles of aeolian sand mass flux. Geomorphology 49, 205 – 218. Nickling, W.G., McKenna Neuman, C., 1997. Wind tunnel evaluation of a wedge-shaped aeolian sediment trap. Geomorphology 18, 333 – 345. Nikuradse, J., 1933. Stromungsgesetze in rauhen Rohren: VDI Forschungsheft 361 (translation by National Advisory Committee

on Aeronautics, Technical Memorandum No. 1292, Washington, DC, 1950). Owen, P.R., 1964. Saltation of uniform grains in air. Journal of Fluid Mechanics 20 (2), 225 – 242. Owen, P., Gillette, D., 1985. Wind tunnel constraints on saltation. In: Barndorff-Nielsen, O.E., Moller, J.T., Rasmussen, K.R., Willetts, B.B. (Eds.), Proceedings, International Workshop on Physics of Blown Sand. Institute of Mathematics Memoir, vol. 8. Department of Theoretical Physics, University of Aarhus, Denmark, pp. 253 – 269. Prandtl, L., 1932. Zur turbulenten Stro¨mung in Rohren und la¨ngs Platten. Ergebnisse der Aerodynamischen Versuchsanstalt zu Go¨ttingen 4, 18 – 29. Rasmussen, K.R., 1989. Some aspects of flow over coastal dunes. Proceedings of the Royal Society of Edinburgh 96B, 129 – 147. Rasmussen, K.R., Sørensen, M., 1999. Aeolian mass transport near the saltation threshold. Earth Surface Processes and Landforms 24, 413 – 422. Rasmussen, K.R., Iversen, J.D., Rautaheimo, P., 1996. Saltation and wind-flow interaction in a variable slope wind tunnel. Geomorphology 17, 19 – 28. Raupach, M.R., Antonia, R.A., Rajagopalan, S., 1991. Rough-wall turbulent boundary layers. Applied Mechanics Review 44 (1), 1 – 25. Shao, Y., Raupauch, M.R., 1992. The overshoot and equilibration of saltation. Journal of Geophysical Research 97 (D18), 20559 – 20564. Sherman, D.J., 1992. An equilibrium relationship for shear velocity and apparent roughness length in aeolian saltation. Geomorphology 5, 419 – 431. Spies, P.-J., McEwan, I.K., Butterfield, G.R., 1995. On wind velocity profile measurements taken in wind tunnels with saltating grains. Sedimentology 42, 515 – 521. Spies, P.-J., McEwan, I.K., Butterfield, G.R., 2000a. Equilibration of saltation. Earth Surface Processes and Landforms 25, 437 – 453. Spies, P.-J., McEwan, I.K., Butterfield, G.R., 2000b. One-dimensional transitional behaviour in saltation. Earth Surface Processes and Landforms 25, 505 – 518. Sullivan, R., Greeley, R., 1993. Comparison of aerodynamic roughness measured in a field experiment and in a wind tunnel. Journal of Wind Engineering and Industrial Aerodynamics 48, 25 – 50. Tennekes, H., Lumley, J.L., 1972. A First Course in Turbulence. The MIT Press, Cambridge, MA. Wang, X., Wang, Z.-Y., Yu, M., Li, D., 2001. Velocity profile of sediment suspensions and comparison of log law and wake law. Journal of Hydraulic Research 39, 211 – 217. White, B.R., Mounla, H., 1991. An experimental study of Froude number effect on wind tunnel saltation. Acta Mechanica, Supplementum 1, 145 – 157.