## Abstract

The neutrally stable atmospheric surface layer is used as a physical model of a very high Reynolds number, canonical turbulent boundary layer. Challenges and limitations with this model are addressed in detail, including the inherent thermal stratification, surface roughness and non-stationarity of the atmosphere. Concurrent hot-wire and sonic anemometry data acquired in Utah's western desert provide insight to Reynolds number trends in the axial velocity statistics and spectra.

## 1. Introduction

In the quest to explore the structure of the zero-pressure-gradient turbulent boundary layer, high Reynolds number facilities are required that permit well-resolved measurements over a very large range of turbulent scales. Typical laboratory facilities offer only moderate Reynolds number ranges, while also incurring sensor resolution issues due to both the finite probe size and the physical size constraints on the facilities. Under appropriate conditions, the atmospheric surface layer (ASL), defined as the layer of air in the immediate vicinity of the Earth's surface, *resembles* a very high Reynolds number canonical turbulent boundary layer. The thickness of the ASL eases spatial resolution problems by orders of magnitude, compared with nearly all high Reynolds number facilities built to date. For example, a typical value of the Karman number (*Re*_{τ}) based on the depth of the ASL is *O*(10^{6}), which affords at least one or two orders of magnitude in Reynolds number above current laboratory facilities. As such, the ASL offers a unique opportunity to advance physical models of turbulent boundary layers, as well as address long-standing questions concerning trends in near-surface scaling as the Reynolds number approaches *O*(10^{6}). The nature of the ASL, however, presents other unique difficulties towards this end, including non-stationarity and thermal/roughness effects at the surface.

Owing to diurnal heating/cooling of the Earth–atmosphere system, the ASL alternates between thermally unstable conditions during the day, when the surface temperature exceeds that of the air aloft and buoyancy drives the turbulence, and thermally stable conditions during the night, as the surface temperature becomes cooler than the air aloft and the heat flux reverses sign. Near dawn and dusk, the atmosphere transitions through neutral stability, wherein mechanical shear dominates turbulence production and buoyancy remains negligible. The relative influences of buoyancy and mechanical shear are commonly characterized through the stability parameter, *ζ*(=*z*/*L*), where *z* denotes the height above the surface and *L* represents the Monin–Obukhov length,(1.1)Here, the subscript *o* indicates a surface quantity, the overline represents a suitable time average, a prime indicates a fluctuating quantity with the ‘mean’ removed, *g* denotes the gravitational constant, *κ* represents the von Kármán constant of magnitude approximately 0.4, *w* represents the vertical velocity, represents the friction velocity, with *τ*_{w} and *ρ* denoting the surface shear stress and air density, respectively, and is the virtual potential temperature, where *P* represents the atmospheric pressure, *P*_{0} denotes a reference pressure (typically 1000 mb), *R*_{d}=287 J kg^{−1} K is the gas constant for dry air and is the specific heat of dry air at constant pressure. The virtual temperature is defined as *T*_{v}=*T*(1+0.61*q*), where *q* denotes the specific humidity. Note that *L* remains height independent since equation (1.1) depends only on surface fluxes. The Monin–Obukhov length *L* is often interpreted as the height at which the buoyant energy and shear production terms in the turbulent kinetic energy equation are equal (Brutsaert 1982); below *L*, mechanical shear dominates turbulence production. As *L*→∞ (or, equivalently *ζ*→0), the ASL is said to be neutrally stable. The neutral period is often rather short (of the order of several minutes), thereby posing additional challenges in terms of achieving convergence of statistics during neutral stability.

Aside from the diurnal cycle of the ASL, non-stationary effects also arise from synoptic weather. Owing to these combined effects, simultaneous measurements in *z* are required in order to faithfully capture the instantaneous and time-averaged structure of the turbulence over any given time-interval. The trade-off, compared with traversing a single probe, is the additional cost of experimental complexity. The surface roughness of the ground, typically quantified in terms of the aerodynamic roughness length *z*_{0} and/or the equivalent sand-grain roughness *k*_{s}, also poses challenges in comparing the ASL with smooth-wall laboratory boundary layers. Townsend's (1976) hypothesis, though, suggests that roughness effects are confined to the small scales, such that turbulence measurements obtained in the presence of hydraulically rough or transitionally rough terrain may still be compared with hydraulically smooth results far from the surface.

In this paper, we explore the use of the ASL as a physical model of a very high Reynolds number canonical turbulent boundary layer. The implications of non-stationarity in successfully realizing this physical model are discussed using well-resolved datasets obtained from the Surface Layer Turbulence and Environmental Science Test (SLTEST) site in Utah's western desert. Determination of the neutral period and friction velocity are considered in detail. The axial velocity statistics and spectrum as a function of height are also presented. As such, the present study compliments the rich base of contributions that have emanated from SLTEST over the past decade, including Klewicki *et al.* (1995), Kurien *et al.* (2001), Metzger & Klewicki (2001), Metzger *et al.* (2001, 2003), Hommema & Adrian (2003), Priyadarshana & Klewicki (2004), Kunkel & Marusic (2006), Hutchins & Marusic (submitted) and Metzger (2006).

## 2. Field experiments

Field experiments were conducted at the SLTEST facility located on the southern end of the salt flats of the Great Salt Lake Desert (113°27.5′ W, 40°8.5′ N). This desert consists of a silty-clay playa, devoid of vegetation and extremely flat. Sparse crust imperfections, constituting the actual surface roughness, yield an aerodynamic roughness length between 0.2≤*z*_{0}≤0.5 mm (Metzger 2002), defined such that *κU*/*u*_{τ}=log(*z*/*z*_{0}), where *U* denotes the ‘mean’ horizontal wind speed at height *z*. The observed variation in *z*_{0} at SLTEST appears to be related to the water content in the soil, which may be tied to annual precipitation and regional drought cycles.

Topographical maps (US Geological Survey 1952–1972) with 1 : 24 000 scale indicate that surface elevation varies by less than 1 m over the 13 km fetch just north of the test site. Often during summer evenings, in the absence of synoptic scale weather, the SLTEST site experiences a prevailing northerly approach flow, which runs parallel to isocontour lines. No abrupt changes in surface elevation occur over the area spanning 12 km east–west and extending 100 km north of the site. Over this 100 km northerly fetch, the elevation gradually decreases by a total of 20 m to a height of 1297 m above sea level at the test site.

### (a) Surface layer depth

An important length-scale in wall-bounded flows is the boundary layer thickness, *δ*. For the purposes of comparing with laboratory data, the near-neutral surface layer depth of the ASL (denoted herein as *δ*_{s}) will be considered as an effective boundary layer thickness. In order to estimate *δ*_{s}, profiles of the horizontal wind speed and virtual potential temperature acquired near sunset were compiled using all available SLTEST data. The result of this historical compilation spanning 1996–2004 is shown in figure 1. The radiosonde data in figure 1*d*,*e* are not suitable for obtaining *δ*_{s}, because they were acquired under stable conditions, as evident by the significant increase in the virtual potential temperature, *θ*_{v}, with height. The horizontal dotted lines in the other four cases (*a*–*c*,*f*) represent the best estimate of *δ*_{s}, and correspond approximately to the height at which the gradient of the horizontal wind profile (approximated by Δ*U*/Δ*z*) reaches a minimum. This is similar to the definition used in the laboratory, whereby *δ* represents the height at which the boundary layer velocity reaches 99% of the freestream value. Owing to the ambiguity of an equivalent freestream in the ASL, the aforementioned definition for *δ*_{s} is preferred. An ensemble average of the profiles (*a*–*c*,*f*) yields *δ*_{s}=80±8 m. Interestingly, the mean horizontal wind profile in all four cases (*a*–*c*,*f*) appears to deviate from the expected log-law behaviour at *z*≈0.15*δ*_{s}. This result agrees with both the very high Reynolds number data of McKeon *et al.* (2004) from the Superpipe, which indicate that the log layer does not extend beyond approximately 0.1*δ* over the Reynolds number range 8.6×10^{2}≤*Re*_{τ}≤5.3×10^{5}, and the boundary layer data of Osterlund *et al.* (2000), which indicate a log-layer height of approximately 0.18*δ* for 3×10^{3}≤*Re*_{τ}≤1×10^{4}.

### (b) Experimental set-up

The bulk of the new data presented herein was obtained on 25 May 2005. A photograph of the experimental set-up is shown in figure 2. Thirty-one hot-wire probes, simultaneously sampled at 5 kHz, were used to measure the axial velocity. The hot-wires were spaced approximately logarithmically over a distance 1 mm≤*z*≤4.6 m, using a combination surface-mounted rake, 1 m tower and 5 m telescoping mast. The surface rake housed 15 hot-wires (spanning 1 mm≤*z*≤13 cm) and was mounted directly on a circular disc that rotated inside a polyethylene platform made flush with the desert surface. A 1 m high tower was also mounted to the circular disc and contained eight hot-wires (spanning 18 cm≤*z*≤1 m) collocated with the surface rake wires. Lastly, eight hot-wires (spanning 1.3 m≤*z*≤4.6 m) were mounted on a 5 m rotatable telescoping mast that was located directly behind the polyethylene platform. The hot-wires on the 5 m tower were displaced by approximately 10 cm from the other wires.

All hot-wires were manufactured in-house and connected to constant-temperature anemometers (two AA Labs, 1003 units and one Digital Flow Technologies, PWM–CTA unit). The hot-wires consisted of a *d*_{w}=5 μm diameter tungsten wire, copper plated at the ends with a ℓ_{w}=1 mm active (unplated) region in the centre. The ratio ℓ_{w}/*d*_{w}≈200 as suggested by Ligrani & Bradshaw (1987). The frequency response of the hot-wires was estimated as greater than 10 kHz, based on an internal square wave test. The length of the individual data records was 210 s, limited primarily by the data acquisition system. Between each run, the hot-wire rake, tower and mast were rotated into the mean wind direction, as determined by visually inspecting a cup and vane anemometer at *z*=1 m.

Pre- and post-calibrations of all of the hot-wires were performed prior to and following each evening experiment, i.e. typically around 16.00 LDST (pre) and 23.00 LDST (post), using a custom-designed two-dimensional calibration jet facility (Metzger 2002). In the time period between the pre- and the post-calibrations on 25 May 2005, the ambient temperature *T* dropped by less than 4°C. Temperature and drift compensation were performed using the relation *u*=*αu*_{pre}+(1−*α*)*u*_{post}, where *u* represents the instantaneous axial velocity during any given record and *u*_{pre}, *u*_{post} denote the *u* signals calculated with the pre- and post-calibration coefficients, as obtained from typical power law fits (Bruun 1995) to the calibration data. The parameter *α* represents the fraction of time between the pre- and post-calibrations, i.e. *α*=(*t*_{post}−*t*_{data})/(*t*_{post}−*t*_{pre}), where *t* represents the time stamp of the pre-, post-calibration and data record accordingly. Since the ambient temperature of the ASL decreases approximately linearly during the evening transition through neutral stability, the present method is nearly equivalent to the temperature compensation technique used previously by Metzger & Klewicki (2001) based on the work of Abdel-Rahman *et al.* (1987). Explicit temperature compensation was not performed here due to the uncertainty in measuring the air temperature compared with the time of day (greater than 1°C variation existed between the thermocouple located on the 5 m tower, the thermocouple located in the calibration facility and the sonic anemometers located on the 30 m tower). Furthermore, as shown in figure 10, statistics from the hot-wires agree very well with the sonic anemometers, which do not require day-to-day calibration. This suggests that the present method used to compensate between the pre- and post-hot-wire calibrations is adequate, at least to within the uncertainty indicated by the error bars in figure 10.

Nine sonic anemometers (Cambell Scientific, CSAT3), with a sensing length of 10 cm, provided the three-dimensional velocity and virtual temperature, *T*_{s}(≡*T*_{v}), spanning 1.4 m≤*z*≤26 m. The sonic anemometers were spaced approximately logarithmically on a guyed 30 m tower located approximately 15 m west of the hot-wire experiment, and were simultaneously sampled at 20 Hz. Sonic anemometry data were acquired concurrently (but not synchronized) with that of the hot-wires. In the reference frame fixed to the sonic anemometer, *U*_{x}, *U*_{y} and *U*_{z} denote the velocity components along the north–south, east–west and vertical directions, respectively, with positive *U*_{x} reflecting flow from the north.

## 3. Atmospheric stability

At SLTEST, conditions of neutral stability usually occur diurnally near sunrise and sunset. In order to effectively compare data from canonical laboratory boundary layers, typically designed to be statistically stationary, to that acquired in the ASL, inherently not statistically stationary, care must be taken in identifying the near-neutral time period of the atmosphere, to minimize potential contamination of the turbulence analysis by buoyancy effects. Previous work (Högström 1988) suggests using the criterion |*z*/*L*|<0.1 to designate the ‘near-neutral’ period.

Figure 3 shows the stability parameter (*z*/*L*) as a function of time of day from sonic anemometers at four different heights between 1.4 m≤*z*≤12.5 m. Data acquired at 12.5≤*z*≤26 m displayed very similar behaviour to that at *z*=12.5 m. The vertical dashed lines in figure 3 represent the time period over which |*z*/*L*|≤0.1 for the sonic anemometer located at *z*=1.4 m and, thus, provide a good estimate of near-neutral conditions. Figure 3 also shows that the near-neutral period occurred just following sunset. This behaviour of a relatively rapid transition through neutral stability near sunset characterizes typical summertime observations at SLTEST (Metzger 2002). Since *L* remains essentially independent of height, the time-interval over which |*z*/*L*|≤0.1 narrows slightly as *z* increases. For example, at *z*=1.4 m, near-neutral occurs between 20.55.00 and 21.27.00; whereas, at *z*=26 m, near-neutral occurs between 21.02.20 and 21.17.30. In these data, the mid-point of the near-neutral period also shifts slightly towards earlier times as *z* increases. Note that a sliding window average of 15 min was used in equation (1.1) to calculate *L*; however, window sizes between 5 and 25 min all yielded very similar results.

In order to support the claim that buoyancy effects are negligible during the near-neutral period defined by the criterion |*z*/*L*|≤0.1, cospectra associated with the heat flux, , were calculated over a 20 min time-interval spanning three different stability regimes: (i) 20.40–21.00, slightly unstable, (ii) 21.00–21.20, near-neutral, and (iii) 21.20–21.40, slightly stable. The near-neutral cospectrum was calculated over a 20 min interval centred between the time, where |*z*/*L*|≤0.1. Figure 4 compares the resultant premultiplied cospectra from the sonic anemometer located at *z*=1.4 m, where and *θ*_{v}≡*T*_{s}. The cospectra are plotted versus frequency (as opposed to wavenumber) because changes in stability are due to *temporally* varying processes. Cospectra are normalized such that the area under the curve equals the heat flux measured over the corresponding time period (2.4×10^{−3}, −2.2×10^{−4} and −1.4×10^{−3} m°C s^{−1} for the three cases (i)–(iii), respectively). Note that the near-neutral heat flux is an order of magnitude lower in absolute magnitude than the other two. One of the notable features of the cospectra in figure 4, despite the noise, is a broadband peak in both the slightly unstable and the stable cases. These peaks reside at approximately 0.2 Hz in both cases, and are accentuated by third-order least-squares spline fits to the raw data (shown by the black lines). During the near-neutral period, however, this dominant broadband peak disappears leaving the cospectra with zero average intensity over four decades in frequency, suggesting that buoyancy effects do not contribute significantly to local organized turbulent motions during this time-interval. Similar behaviour was observed for 1.4 m≤*z*≤26 m.

## 4. Friction velocity

The friction velocity, *u*_{τ}, is an important scale in wall-bounded flows, and ideally should be derived from direct measurement of the surface shear stress. In the ASL, however, this often proves too difficult, especially for rough surfaces. Therefore, it is typical to estimate the friction velocity from the momentum flux (or Reynolds shear stress) near the surface, i.e. , where *o* indicates a height in the lower portion of the log layer. In the present study, *u*_{τ} was derived from the sonic anemometer measurements at *z*=1.4 m. Following the convention of Finnigan *et al.* (2003), the momentum flux may be calculated according to(4.1)where the overline denotes an average over some time-interval of interest and ‘∼’ denotes a ‘mean’ quantity, calculated in the present study using a sliding window average, e.g. for a generic quantity *β*,(4.2)Note that the expression on the right-hand side of equation (4.1) represents the total horizontal momentum flux. Alternatively, the friction velocity may be calculated from the axial momentum flux, , where *U*_{r} represents the axial velocity, following a coordinate rotation of *U*_{x} and *U*_{y} into the ‘mean’ horizontal wind direction, i.e. , where *γ*=tan^{−1}(*U*_{y}/*U*_{x}).

Since the ASL is not statistically stationary over any arbitrary time-interval, some ambiguity exists in the definition of ‘mean’, as mathematically expressed through the parameter *τ* in equation (4.2). The characteristic time-scale, *τ*, used in the temporal averaging (or low-pass filtering) operation to determine the ‘mean’ quantities can have a significant effect on the magnitude of the resultant turbulent fluxes (Rannik & Vesala 1999; Dias *et al.* 2004). A critical time-scale, *τ*_{c}, is identified herein that separates the local turbulence in the ASL from the diurnal/mesoscale motions, associated with the periodic heating/cooling of the Earth's surface and changing regional weather patterns. This critical time-scale is then used to remove the ‘mean’ component of the instantaneous signals in order to calculate the turbulent fluxes. The basis for selecting *τ*_{c} stems from the spectral gap, which was first observed by Van der Hoven (1957) and further analysed by Lumley & Panofsky (1964). In the present study, the spectral gap is defined as the intermediate frequency band in the one-dimensional power spectra of the velocity that provides essentially zero contribution to the turbulent kinetic energy. Figure 5 shows the spectral gap observed from the sonic anemometer at *z*=1.4 m during the near-neutral period 20.55–21.17. For completeness, both horizontal velocity components and corresponding momentum fluxes (as referenced to the coordinate system fixed to the sonic anemometer) are presented.

The high-frequency end of the spectral gap sits at approximately *f*_{c}=5×10^{−3} Hz, as indicated by the vertical dashed lines in figure 5, and is consistent between both the power spectrum and the cospectrum. The value of *f*_{c} from these data did not vary appreciably with height for 1.4 m≤*z*≤26 m. The critical averaging time follows as with a corresponding value of *τ*_{c}=3.33 min. For comparison, the spectral gap identified in figure 5 corresponds to an equivalent wavelength of approximately 10*δ*_{s} using an estimate of *δ*_{s}=80 m. As apparent in figure 5, significant energy exists at frequencies of approximately 1×10^{−2} Hz, which corresponds in this case to a wavelength of approximately 6*δ*_{s} (figure 11 presents the same data in a slightly different form). For lower frequencies (*f*<1×10^{−2} Hz) approaching the spectral gap, the energy drops off sharply. Data at lower Reynolds number (Nickels *et al.* 2005) show a much more gradual decrease in the energy at low frequencies (low wavenumbers). The sharp drop-off observed here may be due to Reynolds number effects and/or non-stationary effects due to the relatively short near-neutral period.

The contour plot of the horizontal momentum flux, as calculated from equation (4.1) using *τ*_{c}=3.33 min, is shown in figure 6. The difference between the near-neutral *u*_{τ} obtained from the total momentum flux (4.1) compared with the axial momentum flux is approximately 1.5%, with the total momentum flux yielding consistently higher *u*_{τ} values. Qualitative features of the contour plot, though, remain intact regardless of the method used to calculate . Aside from the obvious non-stationarity of the ASL, figure 6 shows the existence of an approximately constant stress layer between 1.4≤*z*≤8 m during the near-neutral period (21.02–21.18). To highlight this, figure 7 shows the inner normalized Reynolds stress profile as averaged over the near-neutral period only. The error bars represent the variation between individual 3.33 min averaging windows. The superscript + denotes inner normalization by *u*_{τ} and the kinematic viscosity by *ν*, i.e. *y*^{+}=*ku*_{τ}/*ν* and .

## 5. Ramifications of non-stationarity

The ramifications of the non-stationarity of the ASL are multifaceted. One important ramification relates to the difficulty in achieving converged statistics and spectra. For example in the present study, the 15 min duration of the near-neutral period only admits 4.5 independent realizations of the lowest frequency turbulent motions (based on the time-scale of the spectral gap). Kunkel & Marusic (2006) discuss the convergence of velocity statistics in the atmospheric boundary layer. For comparison, the integral time-scale (*T*_{δ}) of the ASL turbulence may be approximated as *T*_{δ}≈*δ*_{s}/*U*_{δ}, where *U*_{δ} denotes the horizontal wind speed at *z*=*δ*_{s}. In the present study, *T*_{δ}≈21.5 s based on an estimate *δ*_{s}=80 m. Interestingly, the time-scale associated with the spectral gap is slightly more than 10 times larger than the integral time-scale, i.e. *τ*_{c}>10*T*_{δ}.

In order to capture all of the turbulent energy (from the dissipation scales up to the spectral gap), the minimum sampling time *T*_{d} of each individual data record should be *T*_{d}≥2*τ*_{c}. The factor of 2 arises from the necessity of using a Hamming window function (or equivalent) on the entire record, prior to calculating the spectra/cospectra. The window function alleviates end effects in the Fourier transform due to the non-periodic, non-stationary nature of the ASL signals (Bendat & Piersol 1986). In the present study, therefore, the lowest frequency that can be accurately measured from the hot-wire spectra is *f*_{min}=9.5×10^{−3} Hz, which is about twice the spectral gap frequency, *f*_{c}=5×10^{−3} Hz. Consequently, care must be taken when interpreting the hot-wire spectra. To highlight this, figure 8 compares the one-dimensional power spectra of *U*_{r} from the sonic anemometer at *z*=3 m for the entire near-neutral period (21.02–21.18) versus a 3.5 min sub-interval (21.08–21.11.30). The main discrepancy between the two occurs in the low-frequency regime, *f*<7×10^{−2} Hz. The short-record spectrum suffers from reduced resolution (in the frequency domain) at the lower frequencies, explaining the attenuation of energy in the short-record spectrum for *f*<7×10^{−2}. Note that the hot-wire spectrum near *z*=3 m exhibited very similar behaviour to that of the short-record sonic anemometer spectrum shown in figure 8.

Besides the friction velocity and the stability parameter, the horizontal wind speed (*U*_{r}) and direction (*γ*) also appear to be non-stationary, especially when viewed over long times. Instantaneous signals of *U*_{r} and *γ* are shown in figure 9. Interestingly, the horizontal wind speed remains approximately constant (stationary) over the near-neutral time period, indicated by the dashed lines in figure 9. The wind direction, however, does experience a slow drift easterly during that same time period. These behaviours are typical in the summertime at SLTEST in the absence of synoptic scale weather. The slow easterly drift of the wind during evening transition likely results from thermal effects, due to the unequal cooling of the mountain ranges (bordering the eastern and western sides of the Great Salt Lake Desert) as the Sun begins to set.

The non-stationarity of the wind direction during near-neutral conditions poses little measurement difficulty for the three-dimensional sonic anemometers. On the contrary, single-element hot-wire probes are limited to measuring the axial velocity component (parallel to the axis of the probe body) and, thus, become contaminated by a mean cross wind (Lekakis 1996). In order to overcome this deficiency, hot-wire data were acquired in the present study over relatively short records (3.5 min). All of the hot-wires were realigned with the mean wind direction, as described in §2*b*, following the conclusion of one run and prior to the start of the subsequent run. The short hot-wire records, though, pose additional complications in terms of the spectral analysis, as discussed below.

The non-stationarity of *u*_{τ} and *U*_{r} may lead to the ‘smearing’ of normalized quantities. For example, given a probe at a fixed height *z*, the inner normalized probe height (*z*^{+}≡*zu*_{τ}/*ν*) decreases proportionally with *u*_{τ}. The potential for ‘smearing’ in wavenumber space (*k*=2*πf*/*U*_{r}) also exists due to temporal variations in *U*_{r}. Over the course of the evening transition through neutral stability, *u*_{τ} and *U*_{r} usually decrease gradually. During the near-neutral period (21.02–21.18) of the 2005 data, *u*_{τ} remained constant at 0.128±0.003 m s^{−1}. Recall that *U*_{r} (figure 9) appeared to level off during that same time period. Based on the behaviour of *U*_{r}, it is believed that *δ*_{s} also remained constant over the near-neutral period; although, further data are needed to verify this. Thus in the present study, ‘smearing’ of the normalized quantities is not expected to be significant. Nevertheless, to avoid any possible effects of ‘smearing’, ensemble averages were performed after normalization, rather than using dimensional quantities.

## 6. Scaling the axial velocity spectra

One of the main motivations for the present study is to investigate the Reynolds number scaling behaviours of turbulent boundary layers. Of particular interest is the low wavenumber form of the turbulent axial velocity spectrum. Kim & Adrian (1999) identified energetic wavelengths, which they called very large-scale motions (VLSMs), of the order of 10 times the radius in pipe flow at Reynolds numbers based on bulk velocity and pipe diameter *Re*_{D}≤115×10^{3}. The VLSMs were attributed to the streamwise alignment of packets of hairpin vortices. These wavelengths were subsequently also observed by Morrison *et al.* (2004) in streamwise spectra from the Princeton Superpipe at Reynolds numbers *Re*_{D}≤5.7×10^{6}. It was noted that these large wavelengths become increasingly dominant as the Reynolds number increases, such that they eventually contribute more than half of the energy in the streamwise velocity fluctuation. In boundary layers, Hutchins & Marusic (submitted) documented the existence of ‘superstructures’, long wavelength streaks of low and high streamwise momentum that have a large vertical extent and exert an imprint on the near-surface region of the flow. These large structures and the associated energy content have strong implications for spectral self-similarity, particularly the so-called ‘*k*^{−1}’ scaling (where *k* denotes the wavenumber) arising from a self-similar eddy structure with length-scale *k*, see Perry *et al.* (1986).

For the present 2005 data, *Re*_{τ}=7.8×10^{5} based on the estimate *δ*_{s}=80 m. Sensor spatial resolution (ℓ^{+}=ℓ*u*_{τ}/*ν*, where ℓ denotes the sensing length) of the hot-wires and sonic anemometers was ℓ^{+}≈7 and ℓ^{+}≈700, respectively. For comparison, the highest Reynolds number achieved in the well-resolved LDV measurements of DeGraaff & Eaton (2000), having comparable spatial resolution as the hot-wires in the present study, was *Re*_{τ}=1×10^{4}. This range in *Re*_{τ}, over two orders of magnitude between the laboratory and the ASL data, coupled with the very good spatiotemporal resolution of the sensors are important towards discerning Reynolds number trends in the structure of turbulent boundary layers.

The inner normalized mean (*U*^{+}) and r.m.s. (*u*′^{+}) axial velocity profiles from a compilation of laboratory (Klewicki & Falco 1990; DeGraaff & Eaton 2000) and SLTEST (Folz 1997; Metzger & Klewicki 2001; Metzger *et al.* 2001) data are shown in figure 10. The solid black symbols represent an ensemble average over the near-neutral period (21.02–21.17) of the present 2005 hot-wire and sonic anemometry data, which agree very well. The corresponding error bars indicate one standard deviation in the ensemble average. The 2005 data are unique in that all of the measurements were obtained concurrently, with sufficient range in height to capture the entire log layer (as described below). This is important in order to accurately discern wall-normal variations in the turbulence. Traversing a probe in the vertical direction (as typically done in laboratory boundary layers) is not as suitable in the near-surface ASL, due to the non-stationarity of the flow, and leads to further uncertainties in the statistics.

The 2005 and 2003 data display characteristics of a transitionally rough boundary layer, as evident by the downward shift of the *U*^{+} profile. Log laws, with coefficients from Coles (1969) and Osterlund *et al.* (2000), are drawn for comparison in the *U*^{+} profile, in addition to lines of constant inner normalized sand-grain roughness (), as taken from Schlichting & Gersten (2000). For the case of 2005, based on the values of *U*^{+} between 30≤*z*^{+}≤200, which appear to follow the conventional log law. Above *z*^{+}≈200, however, the 2005 *U*^{+} profile exhibits a much steeper slope than that expected with the traditional range of the von Kármán constant, *κ*=0.38–0.41. In fact, this behaviour is observed in all of the SLTEST data, except for the data of Folz (1997) wherein the change in slope occurs at *z*^{+}≈55. The 2005 data also appear to depart from logarithmic behaviour near *z*^{+}=1×10^{5}, and thus exhibit a relatively short log layer compared with other SLTEST data (Metzger *et al.* 2001). Unfortunately, velocity measurements above *z*^{+}>1.5×10^{5} were not available in 2005, leading to the ambiguity of the actual surface layer depth.

Stability effects on the slope of *U*^{+} in the logarithmic layer are usually accounted for through the similarity function *ϕ*(*ζ*), such that using the semi-empirical forms of *ϕ*(*ζ*) given by Andreas *et al.* (2004), the apparent value of the log-layer slope (uncorrected for *ϕ*) lies in the range 0.32<*κ*/*ϕ*<0.46 for |*ζ*|≤0.05. This may explain the observed trends in the *U*^{+} profile from the SLTEST data for *z*^{+}>200, as shown in figure 10. Of particular interest is that for *z*≤200, below many meteorological measurement heights, data appear to follow traditional log-law behaviour (in terms of *κ*≈0.4). This may reflect the fact that very near-surface turbulence is less susceptible to stability effects, because for small *z*, the stability parameter remains near-neutral for a longer time period.

Surface roughness effects in 2005 also lead to a substantial reduction in the peak value of *u*′^{+} near *z*^{+}=15. The study of Flack *et al.* (2005) found that the roughness influence on *u*′^{+} extended from . The dotted black line in figure 10 indicates the upper limit of for the 2005 data. It remains to be known whether roughness effects extend to higher *z*^{+} with increasing *Re*_{τ}. The apparent lack of a plateau region in the present 2005 data between 200≤*z*^{+}≤1000 may be due to roughness effects and/or insufficient resolution of the low-frequency motions, as described in §5. The black dashed line indicates the similarity formulation of Marusic & Kunkel (2003), assuming *δ*_{s}=80 m. This result is relatively insensitive to *δ*_{s}. For example, ±20 m change in *δ*_{s} leads to an approximate 2% variation in the value of *u*′^{+} at *z*^{+}=1×10^{4}, as predicted by the similarity formulation of Marusic & Kunkel (2003). The 2005 r.m.s. values from the sonic anemometers are slightly lower than the 2005 hot-wire measurements at similar *z*^{+}. This is attributed to unresolved energy at high frequencies in the spectra from the sonic anemometers, due to insufficient spatiotemporal resolution. The increased uncertainty in the r.m.s. measurements (compared with the mean) reaffirms the difficulty in achieving converged statistics in the near-neutral ASL.

A composite contour plot of the axial velocity power spectra from the hot-wires and sonic anemometers during the near-neutral period (21.02–21.17) is shown in figure 11. The spectra from the sonic anemometers extend from 1×10^{4}<*z*^{+}<2×10^{5} and are visible in the upper right-hand portion of the spectra. Distinctions in the temporal resolution and sampling times of the two different sensor types are evident, particularly the coarse nature of the hot-wire spectra at large wavelengths (due to the 3.5 min sampling time of the individual runs). Spectra from both sensor types lack convergence as a result of the relatively short duration of the near-neutral period, and, hence, are noisy. Nevertheless, several features are noteworthy. First of all, compared with the smooth-wall case (Metzger & Klewicki 2001; Metzger 2006), a reduction in energy at all wavelengths for *z*^{+}<200 is apparent in the present 2005 spectra, due to surface roughness effects. Energy shifts to larger normalized wavelengths with increasing distance from the wall, and an energy peak corresponding to *λ*_{x}∼*z* exists, in agreement with wall-normal scaling of the energy containing motions (Perry & Abell 1977). A second region of relatively high energy emerges farther from the wall starting at *z*^{+}=*O*(10^{3}), with a peak value similar to the lower wavelength peak. Near *z*/*δ*_{s}=0.1, the peaks lie at *λ*_{s}/*δ*_{s}≈1 and *λ*_{x}/*δ*_{s}≈5, and are separated by an extended wavelength band containing much lower energy. The high energy peak observed at *λ*_{s}/*δ*_{s}≈5 extends from z/*δ*_{s}≈0.2 all the way down to *z*^{+}=300. This latter feature qualitatively agrees with the low Reynolds number results of Hutchins & Marusic (submitted) (their fig. 11), which supports the existence of a broadband ‘footprint’ in spectral space, supposedly due to meandering superstructures that physically reside in the log layer. The extent of this phenomenon is clearly demonstrated in the present study due to both the high Reynolds number and the wide range of simultaneous wall-normal measurements.

## 7. Conclusions

The ASL is used as a physical model for a very high Reynolds number, canonical turbulent boundary layer in order to ascertain Reynolds number effects on axial velocity statistics and spectra. Challenges in realizing this model are presented, with emphasis on thermal stability, non-stationarity and surface roughness. Only a short-time window exists (the near-neutral period) wherein thermal effects become sufficiently small for the physical model to be valid. The 2005 sonic anemometry data at SLTEST indicate that the magnitude of the axial velocity and friction velocity remain almost constant (stationary) during the near-neutral period, defined by |*z*/*L*|≤0.1; whereas, the direction of the axial velocity slowly drifts easterly during the same time period. The friction velocity is determined from the surface momentum flux, which involves calculating the covariance between the ‘fluctuations’ in *w* and *θ*_{v}. A time-scale, based on the inverse of the spectral gap frequency, is used in the mean removal operation defining these fluctuations. Selection of an appropriate averaging time is considered critical for obtaining an accurate estimate of *u*_{τ}.

Near the surface (*z*^{+}<1×10^{3}), roughness and insufficient resolution of the low-frequency (low wavelength) motions cause an attenuation in the axial velocity fluctuations. The exact wall-normal extent of roughness effects at these high Reynolds numbers, however, requires additional investigation. Further into the log layer, the effect of buoyancy (expressed in terms of *L*) becomes apparent, particularly with regard to observations of a steeper slope in the inner normalized mean profile. The influence of *L* is more subtle in the axial velocity spectrum and on the overall turbulence structure. For example, it remains to be determined whether long structures that originate non-locally in physical space (i.e. formed prior to the onset of neutral stability) are advected over the test site during the near-neutral period and what effect they might have on the local turbulence. This latter point speaks to the fact that the ASL, in contrast to laboratory boundary layers, develops both spatially (downwind fetch) as well as temporally.

The composite picture of the axial velocity spectrum, at the high Reynolds number and over the wide wall-normal range of the present study, demonstrates similar features to the lower Reynolds number, laboratory boundary layer results of Hutchins & Marusic (submitted). Evidence exists of an extended spatial region (from *z*^{+}≈300 to *z*/*δ*_{s}≈0.2) influenced by VLSMs. Future work will exploit the nature of the simultaneous measurements in 2005 at SLTEST to further explore the scaling behaviours of high Reynolds number turbulence.

## Acknowledgments

This work was funded by the Office of Naval Research. B.J.M. is grateful for the support of a Royal Society Dorothy Hodgkin Fellowship during the field experiments. Field support from Dugway Proving Ground is also appreciated. The authors further benefited from the help of Drs I. Marusic and E. Pardyjak.

## Footnotes

One contribution of 14 to a Theme Issue ‘Scaling and structure in high Reynolds number wall-bounded flows’.

- © 2007 The Royal Society