1. Introduction

The stable nocturnal boundary layer (NBL) is characterized by a rich collection of waves, interfaces, turbulence, and intermittent processes. The Cooperative Atmosphere–Surface Exchange Study-1999 (CASES-99) was focused on the atmospheric processes of the nocturnal boundary layer near Leon, Kansas (50 km), east of Wichita, Kansas, during the month of October (Poulos et al. 2002). Turbulence in the stable boundary layer and the lower free troposphere have been studied using instrumented towers (Garratt 1981; Nieuwstadt 1984a,b; Gossard et al. 1984; Cuijpers and Kohsiek 1989; Smedman 1988; Smedman et al. 1995), instrumented aircraft (Mahrt 1985; Mahrt and Gamage 1987; Muschinski and Wode 1998), acoustic sounders (Gossard et al. 1984; Smedman 1988), and the Frequency-Modulated Continuous Wave (FMCW) radar, which produces high spatial resolution information on the refractive index structure constant, that is, the level of refractive turbulence (Richter 1969; Gossard et al. 1984; Eaton et al. 1995). Each measurement has its own advantages and disadvantages. However, reliable high-resolution in situ data are critical to observe and validate the many examples of intermittent turbulent processes in the NBL.

One of the most striking examples of intermittent turbulence and waves in the NBL was produced by the FMCW measurements of Gossard et al. (1984) and Eaton et al. (1995). These high-resolution measurements of signal power are proportional to the refractive index structure constant C²_n if the turbulence has a well-defined inertial range over the sensing volume of the radar at the Bragg wavelength. Gossard et al. (1984) determined that a smoothly varying interface of refractive index can produce detectable radar backscatter, which could be incorrectly interpreted as a region of high C²_n from Bragg scattering. The in situ turbulence measurements of the tethered lifting system can answer these questions (Balsley et al. 2003).

Another important aspect of stable nighttime turbulence is the “global intermittency” (Mahrt 1989), that is, variations in space and time of the finescale turbulence quantities, such as the energy dissipation rate ϵ and the temperature structure constant C²_T. Accurate in situ measurements of the spatial and temporal variability of ϵ are essential for valid comparison with other remote sensing techniques for estimating ϵ, such as Doppler radar (Cohn 1995; Jacoby-Koaly et al. 2002) and Doppler lidar (Frehlich et al. 1998). The large-scale variations of global intermittency also affect the interpretation of the small-scale statistics of turbulence (Sreenivasan and Stolovitzky 1996; Alisse and Sidi 2000; Danaila et al. 2002). In addition, similarity properties of structure constants in stable conditions can produce estimates of temperature and momentum flux, as well as the local Obukhov length (Wyngaard and Kosovic 1994). Muschinski and Lenschow (2001) have produced an overview of the various research needs in the area of finescale turbulence. Some of these needs have been addressed by the CASES-99 campaign (Poulos et al. 2002).

In situ measurements of the small-scale turbulence is typically performed with fixed sensors on towers or on moving platforms such as aircraft and helicopter (Muschinski and Wode 1998). Both measurements are limited in their altitude coverage and aircraft provide only a single measurement. In addition, measurements from a fast-moving platform require higher bandwidth and lower noise than the same measurements from a slow-moving platform to sample a given regime of spatial scales (Muschinski et al. 2001).

The tethered lifting system (TLS) was developed by the Cooperative Institute for Research in the Environmental Sciences (CIRES) at the University of Colorado for a variety of atmospheric measurements (Balsley et al. 1998, 2003; Muschinski et al. 2001). The relatively stable windspeed in the nocturnal jet is ideally suited for the operation of the kite system while low windspeed conditions is ideal for the tethered blimp. A vertical array of turbulence sensors was specifically designed for the CASES-99 campaign to produce finescale temperature and velocity measurements at a fixed altitude and also to produce profiles of atmospheric quantities up to an altitude of 2 km (see Fig. 1). The temperature measurements were produced with a low-frequency response solid-state temperature sensor and a high-frequency response fine cold-wire sensor. The velocity measurements consisted of a sensitive Pitot-tube velocity sensor vaned into the wind and a high-frequency response fine hot-wire sensor. The fine-wire sensors were chosen to provide the most robust mechanical lifetime, that is, a tungsten wire with a diameter of 5 μm, a length of 1.5 mm, and a typical resistance of 4Ω. The diameter of the cylindrical sensor package is 10 cm and the length of the supports for the hot-wire/cold-wire sensor is 40.6 cm to reduce the effects of flow distortion (Miller et al. 1999; see Fig. 1). The data was recorded with 12 bits on onboard compact flash disks and downloaded to a PC after each flight. During the campaign, there were 11 flights of 3–10-h duration. For each flight, the high-frequency turbulence data consisted of many records of 180 s of continuous data separated by a 9-s gap to write the data to the compact flash disk. To provide accurate measurements of the finescale fluctuations, each signal was split into a low-frequency component and an amplified high-frequency component, which is ideally suited for the analysis of small-scale turbulence. Note that triple hot-film anemometers were deployed at a surface tower during the CASES-99 campaign (Skelly et al. 2002).

Accurate calibration of the fine-wire sensors is difficult in the field because the changing atmospheric conditions alter the properties of the wire. This is especially critical for the calibration of the hot-wire sensors. Therefore, calibrations were performed with simultaneous independent low-frequency measurements of temperature and velocity. The low-frequency temperature sensor (see Fig. 1) is located 33 cm from the cold-wire sensor (15.2 cm vertical and 29.2 cm horizontal) and the Pitot tube is located 22.5 cm from the hot-wire sensor (6.4 cm vertical and 21.6 cm horizontal). In past campaigns (Muschinski et al. 2001), the high-frequency signal was calibrated assuming a calibration constant for the fluctuations based on a linear approximation to the low-frequency calibration data. Here, we propose a more accurate calibration that merges the low- and high-frequency information into a reconstructed signal.

The calibration procedure will be discussed as well as methods for estimating the temperature and velocity structure constants that describe the small-scale turbulence. The structure constants are attractive because they have good statistical properties, that is, many independent spectral estimates can be produced with short lengths of data, which produces accurate estimates (Smalikho 1997). However, local isotropy and Taylor's frozen hypothesis is required for accurate estimates. These assumptions will be investigated in detail to evaluate the performance of the TLS for high-resolution turbulence measurements.

2. Merging the low-frequency and high-frequency signals

For both the temperature and velocity, the total signal x(t) consists of a low-frequency component y(t) and a high-frequency component z(t). The total signal is recovered using a merging algorithm in the frequency domain. The Fourier transform X(f) of time series x(t) is given by

View Expanded

with an inverse Fourier transform

View Expanded

Similarly, the Fourier transform X_HF(f) of the high-frequency signal is

X_HFfH_HFfXf(4)

where H_HF(f) is the transfer function of the high-pass filter given by

View Expanded

where G_HF is the gain of the high-pass circuit compared with the low-frequency signal component and τ_HF = 4.99 s for the temperature and velocity electronic circuit.

The sampling interval Δ_LF for the low-frequency signal is an integral multiple of the sampling interval Δ_HF for the high-frequency signal, which produces equal frequency resolution when the total observation times are equal; that is, T = Δ_LFM_LF = Δ_HFM_HF, where M_LF and M_HF are the number of points for the low-frequency and high-frequency data, respectively. Typically, Δ_LF = 1.0 s, Δ_HF = 0.005 s, and M_HF = 200 M_LF.

The complete Fourier transform X(f) is produced by merging the corrected Fourier transforms from the two signals; that is,

View Expanded

and the merging frequency f₁ is chosen as the center of the overlap region. The merging frequency f₁ was typically chosen as 0.01 Hz to emphasize the higher-quality high-frequency signal, which was digitized with more significant bits. The reconstructed signal x(t) is produced by the inverse Fourier transform. Accurate merging requires that the spectral signal power is much larger than the spectral noise power in the overlap region around the frequency f₁. This merging techniques was applied to the temperature and velocity measurements.

3. Temperature measurements

The high-frequency temperature measurements were produced by operating the tungsten wire with a constant current of 2 mA. The voltage across the tungsten wire was amplified and split into two output voltage signals: a low-frequency signal υ_LF(t) and a high-frequency signal υ_HF(t). The high-frequency signal had a high-pass filter with a time constant of 4.99 s and a low-pass filter with a 3-dB bandwidth of 488 Hz. The low-frequency signal υ_LF(t) and the solid-state external temperature signal V_ext(t) (Analog Devices AD22100) were sampled simultaneously at 1 Hz. The solid-state sensor was calibrated in the National Center for Atmospheric Research (NCAR) calibration facility (see Fig. 2) and had excellent linearity and accuracy over the typical temperature range experienced during CASES-99. The largest deviation from the best-fit straight line is 0.0223 K. A radiation shield was placed around the solid-state temperature sensor to improve the absolute accuracy of the temperature measurements.

a. Spectral calibration method

The calibration constant G_CW can be estimated by relating the external temperature signal to the low-frequency voltage signal in the low-frequency region. The Fourier transform T_ext(f) of the low-frequency temperature signal is given by Eq. (3):

T_extfH_LFextfXf(8)

where

View Expanded

and τ_LFext = 25 s was determined by the time constant of the external temperature signal as the sensor passed through sharp temperature interfaces (e.g., Fig. 4).

The Fourier transform of the low-frequency cold-wire signal υ_LF(t) is given by Eq. (3):

View Expanded

where K_CW is the sensitivity of the cold-wire and,

View Expanded

where R_LF = 100.0 and τ_LF = 0.1 sec. The calibration constant G_CW for the high-frequency signal is given by

View Expanded

where G_HF = 705.334 is the gain of the high-pass circuit compared with the low-frequency signal υ_LF.

Solving Eqs. (8) and (10) for X(f) and equating the results produces

View Expanded

which can be written in spectral form as

S^1/2_LFfK_CWS^1/2_LFCWf(14)

where

View Expanded

is the spectrum of the temperature signal from the solid-state sensor, and

View Expanded

is the spectrum of the temperature signal from the cold-wire signal. The slope of S^1/2_LF(f) versus S^1/2_LFCW(f) is K_CW [note that S^1/2_LF(f) and S^1/2_LFCW(f) have a correlation coefficient of unity if the additive noise is negligible]. The error in the best-fit slope provides an estimate for the error in the calibration constant G_CW. The results of this analysis is shown in Fig. 3 and G_CW = 0.5887 ± 0.0026, that is, an accuracy of 0.44%. The traditional method of calibration based on the average external temperature and the average of the low-frequency voltage signal produced G_CW = 0.5776 ± 0.0058, which has less accuracy than the spectral method.

The calibrated high-frequency temperature signal x_HF(t) is given by Eq. (7). This signal is merged with the calibrated low-frequency external temperature signal T_ext(t) using the algorithm in section 2. Results of the merging are shown in Fig. 4 for a slow transect through a sharp temperature gradient (Balsley et al. 2003). The absolute accuracy of all the temperature data is better than 0.5 K and the accuracy of all the calibration constants G_CW is better than 2%, based on the error in the best-fit slope of the calibration curves (see Fig. 3).

4. Velocity measurements

Hot-wire (HW) velocity measurements were produced to provide accurate measurements of the finescale turbulence. The tungsten wire was oriented along the vertical axis and operated with an overheat ratio of 1.8. The output voltage was split into two signals: a low-frequency signal x_LF(t) and a high-frequency signal x_HF(t) (note that the same notation was used for the temperature measurements). The total voltage signal x(t) was produced with the merging algorithm of section 2. The low-frequency transfer function Eq. (3) is

View Expanded

and τ_LF = 0.1 s.

The high-frequency transfer function is given by Eq. (5), where G_HF = 8.7406 and τ_HF = 4.99 s based on the electronic circuit.

The calibrated velocity U(t) was produced by applying the modified King's law to the reconstructed hot-wire signal x(t) (Bruun 1995; Hugo et al. 1999); that is,

xt²T_wTcdρⁿUtⁿ(18)

where T_w − T is the temperature difference of the hot-wire; ρ is the density of air; and c, d, and n are empirically determined parameters. The calibrated hot-wire velocity is then

View Expanded

The calibration of fluctuating velocity in Muschinski et al. (2001) used the slope of the King's law curve instead of the reconstructed voltage signal.

The constants of King's law (c, d, n, T_w) were determined by minimizing the mean-square error between the Pitot-tube velocity U_pitot and the predicted hot-wire velocity U_hw using the low-frequency signal y(t) in King's law Eq. (19). The time intervals for a King's law fit was typically longer than 6 min and shorter than 30 min. An example of the King's law fit is shown in Fig. 6 for 12 min of data. The scatter in the fit is from four main sources: the difference in the bandwidth of the two measurements, the 22.5-cm separation between the two measurements, errors in the angle of the vane, and the additive noise in the sensors.

The final calibrated velocity U(t) was produced from the merged hot-wire voltage signal x(t) (see section 2) using Eq. (19) (see Fig. 4). An example of the corrected spectra of the low-frequency and high-frequency signals is shown in Fig. 7.

5. Velocity spectra

The hot-wire velocity is essentially a measurement of the horizontal velocity V_H(t), which can be written as

View Expanded

where U(t) is the longitudinal velocity, υ(t) is the transverse velocity, U₀ is the local mean velocity, and u(t) is the fluctuations in the longitudinal velocity. The one-dimensional spectrum S_u(f) of the longitudinal velocity fluctuations is an important statistic for the velocity field. If the last term of Eq. (21) can be neglected, then the hot-wire velocity is the longitudinal velocity U(t). This is an excellent approximation especially for the calculation of higher-order statistics such as the spectrum S_u(f).

The fast Fourier transform (FFT) estimate of the raw spectrum of the longitudinal velocity u(t) = u(mΔ_t) is

View Expanded

which is normalized such that the integral of S_u(f) over positive frequency is the variance of u(t). The sampling interval Δ_t = Δ_HF = 0.005 s, and the frequency resolution Δ_f = (MΔ_t)⁻¹. There have been many models proposed for S_u(f) under the assumption of homogeneous, isotropic turbulence (Monin and Yaglom 1975; Azizyan et al. 1989) and Taylor's frozen hypothesis, which is well satisfied for short time intervals (Hill 1996). For stably stratified mixing layers, the assumption of isotropy is valid for estimates of energy dissipation rate when the buoyancy Reynolds number is larger than 100 (Smyth and Moum 2000). We assume the theoretical model from Azizyan et al. (1989), which was derived assuming a Gaussian cutoff function for the high-wavenumber region of the three-dimensional spectrum and can be written as

View Expanded

where U is the mean velocity over the measurement time, ϵ is the energy dissipation rate, β = 2.955 18,

ην^3/4ϵ^1/4(24)

is the Kolmogorov microscale, ν is the kinematic viscosity, and

View Expanded

where

View Expanded

is the incomplete Gamma function.

The kinematic viscosity is given by (NOAA 1976)

View Expanded

where A = 1.458 × 10⁻⁶ kg s⁻¹ m⁻¹ K^−1/2, T is temperature in Kelvin, ρ is the density of air, and S = 110 K is the Sutherland constant. To facilitate fitting to the data we write the spectral model as

View Expanded

where S_N is the spectral noise floor and

x₀U^2/3ϵ^2/3(29)

The spectral model is a function of x₀, ν, S_N, and U, or equivalently ϵ, η, S_N, and U. The maximum likelihood estimate of ϵ is given by

ϵx₀^3/2U,(30)

where x₀ is the maximum likelihood estimate of the spectral level (Smalikho 1997; Ruddick et al. 2000).

6. Temperature spectra

High-resolution measurements of the finescale temperature fluctuations were produced using the calibrated high-frequency signal x_HF(t) because the analog high-pass filter performs as an excellent detrending filter. The FFT estimate of the raw spectrum of the temperature fluctuations x_HF(t) = x_HF(mΔ_t) is

View Expanded

which is normalized such that the integral of S_T(f) over positive frequency is the variance of x_HF(t). The theoretical model for the one-dimensional temperature spectrum assuming locally homogeneous isotropic turbulence is produced from the three-dimensional model given by Eqs. (20), (23), and (49) of Frehlich (1992) following the analysis of Azizyan et al. (1989). The resulting spectrum can be written as

View Expanded

where

l₀χ^3/4ϵ^1/4(33)

is the temperature inner scale [defined as the intercept of the square-law dependence r² and inertial range dependence r^2/3; Tatarskii (1961)], χ is the thermal diffusivity of air, U is the mean velocity over the measurement time, and

View Expanded

describes the Hill bump (Hill 1978). The thermal diffusivity of air is given by (NOAA 1976)

View Expanded

where T is average temperature in Kelvin, ρ is the density of air, and C_P = 1004 J kg⁻¹ K⁻¹ is the specific heat of air at constant pressure. A useful approximation is

View Expanded

The inner-scale l₀ is calculated from the best-fit ϵ and then C²_T is estimated from the best fit to the temperature spectrum. The spectral model is written as

S_Tfx₀f^−5/3hπl₀fUS_N(37)

where

x₀U^2/3C²_T(38)

and the maximum likelihood estimate of C²_T is

View Expanded

where x₀ is the maximum likelihood estimate of the spectral level (Smalikho 1997; Ruddick et al. 2000). Examples of velocity and temperature spectra from 1 s of data for very weak turbulence are shown in Figs. 8 and 9, as well as the estimates for ϵ and C²_T. The scatter reflects the statistical properties of the raw spectral estimates. Approximately N = 50 spectral points above the noise floor are used for each estimate of x₀. Therefore, the estimation error for the estimates of x₀ (C²_T and ϵ^2/3) is approximately 1/N or 15% (Smalikho 1997). The spectral estimates at high frequency (50–100 Hz) are dominated by aliasing and the noise floor of the sensor. These 1-s estimates of C²_T and ϵ provide useful high-resolution measurements of the finescale turbulence structure in the atmospheric boundary layer and troposphere (Balsley et al. 2003). The fine-wire sensors are ideally suited for measurements in very low turbulence. During CASES-99, values of ϵ less than 10⁻⁷ m² s⁻³ and C²_T less than 10⁻⁶ K² m^−2/3 were observed. The spectrum in Fig. 8 would have ϵ = 10⁻⁷ m² s⁻³ if the spectral level was reduced by a factor of 10.536. Similarly, the spectrum in Fig. 9 would have C²_T = 10⁻⁶ K² m^−2/3 if the spectral level was reduced by a factor of 65.6. In both cases, there is sufficient spectral signal above the noise floor to produce an estimate. The accuracy of these estimates can be improved by computing spectra over longer time intervals (Muschinski et al. 2001).

7. Summary and discussion

Accurate finescale measurements of temperature and velocity are feasible for a tethered lifting system (TLS) using cold-wire and hot-wire sensors. The calibration of the cold-wire temperature sensor is produced using the fluctuations of a calibrated low-frequency solid-state temperature sensor. The absolute accuracy of the temperature measurements are typically better than 1 K. This limit was established by the drifting in the solid-state sensor with time. However, the accuracy of temperature fluctuations is typically much better because the gain of the solid-state sensor is very linear and stable with time. The accuracy of the slope of the temperature calibration is better than 2%. The hot-wire velocity sensor is calibrated by a modified King's law and a low-frequency measurement of the wind speed produced from a nearby Pitot tube vaned into the wind. The absolute accuracy is typically better than 1 m s⁻¹ and the accuracy of the slope of the calibration curve is better than 5%. Careful calibrations are required because the parameters of the modified King's law change with conditions.

The TLS is ideally suited for high-resolution measurements of finescale turbulence using multiple vertically spaced sensors. Accurate measurements of energy dissipation rate ϵ and temperature structure constant C²_T are possible with 1 s of data (see Figs. 8 and 9). The accuracy of Taylor's frozen hypothesis is very good over these short time intervals. The assumption that the hot-wire velocity is the longitudinal velocity U(t) is valid when the local mean wind speed U₀ is large compared with the transverse velocity υ(t) [see Eq. (21)]. This is an excellent assumption for the low turbulence levels in the nocturnal jet and stable boundary layer. In addition, the errors from the transverse velocity are reduced for covariance and spectral estimates because they are second-order quantities. The random variations of ϵ and C²_T (global intermittency) are clearly observed (Balsley et al. 2003) since the accuracy of the estimates for the spectral level in the inertial range (ϵ^2/3 and C²_T) is typically 15% with 1 s of data. The assumption of local isotropy is required to estimate ϵ and C²_T. This is a good assumption for high Reynold's number flows and a local description of turbulence described by Kolmogorov's refined similarity hypothesis (Kolmogorov 1962; Smyth and Moum 2000), that is, sufficiently small observation intervals (Sreenivasan and Stolovitzky 1996; Monin and Yaglom 1975).

Results from the TLS finescale turbulence measurements in the NBL are presented in Balsley et al. (2003). These include the observation of very large temperature gradients (28 K m⁻¹) over a few centimeters in altitude, large changes of turbulence intensity C²_T over a few meters in altitude, and very low levels of C²_T ≈ 10⁻⁵ K² m^−2/3 extending over 60 m in altitude. The high-resolution TLS measurements are ideally suited for the statistical analysis of the variability of ϵ and C²_T, as well as the spatial statistics of temperature and velocity in very low turbulence regions and regions of very high Reynolds number such as the middle of the convective mixed layer. Another important area of future work is the measurement of fluxes using improved platform angle and position sensors. New light-weight sensors of trace gases will permit accurate measurement of important fluxes for climate change and transport.

Recent advances in electronic circuits and flash memory will permit even higher data acquisition rates to fully resolve the dissipation range of turbulence. In addition, multiple hot-wire probes will produce measurements of all the components of the velocity vector. This will provide new information on the anisotropy of turbulence for many atmospheric phenomena at all altitudes from the surface to the free troposphere. The low effective mean velocity U of the TLS (2–25 m s⁻¹) system compared with instrumented aircraft (100–200 m s⁻¹) permits more accurate measurements of the small-scale turbulence (Muschinski et al. 2001). Aircraft measurements are preferable when a large region of the atmosphere must be sampled for statistical accuracy, for example, with flux measurements. The TLS, on the other hand, has the advantage of multiple sensors, which provide instantaneous information in the vertical that is difficult to produce with aircraft measurements.