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  • View in gallery

    Vertical profiles of Cm in algebraic stress models and vertical shear of potential temperature observed by rawinsonde.

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    Atmospheric conditions at 1200 LST 4 Oct 2000. Horizontal bars indicate root-mean-square fluctuating velocities during 2 h (1100–1300 LST) at typical heights. Horizontal lines in T, RH, and mixing ratio plots indicate the region of radar range coverage.

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    Power spectra of horizontal wind velocity Su and cospectrum of horizontal and vertical wind velocity Cuw from 0000 to 2359 LST 4 Oct 2000. The thick solid line represents z = 6 km; the thick dashed line, z = 9 km; the thin solid line, z = 15 km; and the thin dashed line, z = 18 km.

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    Vertical profiles of σ, N, ε, and Kh estimated by the spectral width method. The thick solid line represents Kh = 0.1σ2/N; the thick dashed line, ε; the thin solid line, σ; and the thin dashed line, N.

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    Vertical profiles of wind shear, stability parameters, and Km measured directly by the MU radar.

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    Eddy diffusivity for momentum measured directly and its comparison with values estimated using turbulence models. Only the spectral width method estimates eddy diffusivity for heat. The solid lines of the profiles become thinner when Km and Kh contain large uncertainty due to spectral broadening effects.

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    Stability dependence of (a) Kh/Km and (b) Km/Kmo. Solid lines in (a) and (b) represent Eqs. (32) and (33), respectively. Height of the observed atmospheric layers is 4–8 km for (a) and 4–18 km for (b), in which circular and triangular symbols indicate 4–8 and 14–18 km, respectively. Open and solid symbols in (b) are for δ = 2000 and 4000 m, respectively.

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Eddy Diffusivities for Momentum and Heat in the Upper Troposphere and Lower Stratosphere Measured by MU Radar and RASS, and a Comparison of Turbulence Model Predictions

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  • 1 Department of Environmental and Life Engineering, Toyohashi Institute of Technology, Toyohashi, Japan
  • | 2 Institute of Behavioral Sciences, Shinjuku, Tokyo, Japan
  • | 3 Meteorological Research Institute, Tsukuba, Ibaraki, Japan
  • | 4 Disaster Prevention Research Institute, Kyoto University, Kyoto, Japan
  • | 5 Research Institute for Sustainable Humanosphere, Kyoto University, Kyoto, Japan
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Abstract

Recently, middle- and upper-atmosphere Doppler radar (MU radar) has enabled the measurement of middle-atmosphere turbulence from radar backscatter Doppler spectra. In this work, eddy diffusivities for momentum Km in the upper troposphere and lower stratosphere during clear-air conditions were derived from direct measurements of the Reynolds stress and vertical gradient of mean wind velocity measured by MU radar. Eddy diffusivity for heat Kh below 8 km was determined from measurements of temperature fluctuations by the Radio Acoustic Sounding System (RASS) attached to the MU radar. The eddy diffusivity for momentum was on the order of 10 m2 s−1 in the upper troposphere and decreased gradually in the stratosphere by an order of magnitude or more. The eddy diffusivity for heat was almost of the same order of magnitude as Km.

Estimates of eddy diffusivity from the radar echo power spectral width give fairly good values compared with the direct measurement of Km. Applicability of three turbulence models—the spectral width method, the k–ε model modified for stratified flows, and the algebraic stress model—were also examined, using radar observation values of turbulent kinetic energy k and turbulent energy dissipation rate ε together with atmospheric stability observations from rawinsonde data. It is concluded that the algebraic stress model shows the best fit with the direct measurement of Km, even in the free atmosphere above the atmospheric boundary layer once k and ε values are obtained from observations or a model.

Corresponding author address: Hiromasa Ueda, Toyohashi Institute of Technology, Toyohashi, Aichi 441-8580, Japan. E-mail: ueda@acap.asia

Abstract

Recently, middle- and upper-atmosphere Doppler radar (MU radar) has enabled the measurement of middle-atmosphere turbulence from radar backscatter Doppler spectra. In this work, eddy diffusivities for momentum Km in the upper troposphere and lower stratosphere during clear-air conditions were derived from direct measurements of the Reynolds stress and vertical gradient of mean wind velocity measured by MU radar. Eddy diffusivity for heat Kh below 8 km was determined from measurements of temperature fluctuations by the Radio Acoustic Sounding System (RASS) attached to the MU radar. The eddy diffusivity for momentum was on the order of 10 m2 s−1 in the upper troposphere and decreased gradually in the stratosphere by an order of magnitude or more. The eddy diffusivity for heat was almost of the same order of magnitude as Km.

Estimates of eddy diffusivity from the radar echo power spectral width give fairly good values compared with the direct measurement of Km. Applicability of three turbulence models—the spectral width method, the k–ε model modified for stratified flows, and the algebraic stress model—were also examined, using radar observation values of turbulent kinetic energy k and turbulent energy dissipation rate ε together with atmospheric stability observations from rawinsonde data. It is concluded that the algebraic stress model shows the best fit with the direct measurement of Km, even in the free atmosphere above the atmospheric boundary layer once k and ε values are obtained from observations or a model.

Corresponding author address: Hiromasa Ueda, Toyohashi Institute of Technology, Toyohashi, Aichi 441-8580, Japan. E-mail: ueda@acap.asia

1. Introduction

Diffusion processes in the free atmosphere play an important role in the transport of momentum, heat, and mass on global and regional scales, although the eddy diffusivity there is much smaller than that in the atmospheric boundary layer. In particular, diffusion processes of minor constituents in the upper troposphere and lower stratosphere are essential to global warming, stratospheric ozone depletion, and transboundary air pollution problems because they govern the exchange of mass between the troposphere and stratosphere.

In the upper troposphere and lower stratosphere, air is usually stably stratified, and internal gravity waves induced by boundary layer flow and geography are predominant. Turbulence eddies in these layers are generated intermittently and sporadically when gravity wave breaking and shear instability occur. These turbulence eddies transport heat and mass, and then they are partly destructed by buoyancy and viscous forces. Thus, turbulent motions and diffusion processes in these layers are complicated and not yet well understood. In practice, in many meteorological models, as well as in chemical transport models of minor constituents, eddy diffusivity in the upper troposphere and lower stratosphere has been simply assumed to be an “appropriate” minimum value or is predicted by the Smagorinsky model (Smagorinsky 1963), a type of mixing length subgrid-scale model with a length scale equal to the geometric mean of the horizontal and vertical grid sizes.

Observationally, direct measurements have been carried out on turbulence in the upper troposphere and in the middle atmosphere by means of balloons, aircraft, and rockets. However, the measurements are sporadic in time, making it difficult to extract general features of turbulence. Moreover, measurements of eddy diffusivity have been mainly based on dispersion experiments involving tracers and atmospheric constituents. However, such measurements are limited mainly to the eddy diffusivity in the horizontal spanwise direction, which is significantly different from that in the vertical direction during stably stratified conditions.

Recently, a middle and upper atmosphere very high-frequency (VHF) Doppler radar (hereafter MU radar) technique has been developed that allows us to make continuous measurements of wind velocity in the upper troposphere and middle atmosphere. From the radar echo Doppler shift, wind velocity and its fluctuations in the line-of-sight direction can be determined. Extensive studies have been done on gravity waves and turbulence. One of the striking results is the fact that over most of the frequency f range, the energy spectrum of velocity fluctuations has an f−5/3 power-law dependence (Carter and Balsley 1982). On the assumption of inertial subrange turbulence with the same f−5/3 power-law energy spectrum, the echo power spectral width σ, which represents the variance of wind velocity fluctuation inside the radar scattering volume, has been correlated with the turbulence dissipation rate ε (Lilly et al. 1974). In addition, eddy diffusivity has been estimated by assuming that dissipation of turbulent energy balances production of turbulent energy by wind shear and destruction by buoyancy under the critical condition of stable stratification. This estimation method, called the spectral width model, was first applied by Sato and Woodman (1982) to the tropical lower stratosphere and was developed comprehensively by Hocking (1983, 1985, 1986, 1988). Applying this method, Fukao et al. (1986, 1994) made systematic observations of eddy diffusivity in the upper troposphere and middle atmosphere and determined the vertical distribution and seasonal variability of eddy diffusivity.

However, many assumptions are involved in the derivation of eddy diffusivity by the spectral width method, including inertial subrange turbulence, a balance between turbulent energy production and dissipation, and stable stratification. Since the recent development of MU radar technology has made it possible to determine reliable values for Reynolds stress and wind shear, the eddy diffusivity can be obtained by inserting directly measured values of Reynolds stress and wind shear in its definition equation. Thus, it is worthwhile to evaluate the eddy diffusivity from the spectral width method by comparing the eddy diffusivity calculated from directly measured values of Reynolds stress and wind shear. Hereafter in this paper, the latter is referred to as directly measured eddy diffusivity.

The purpose of this work was to make direct measurements of eddy diffusivity and to examine estimates from the spectral width method and existing turbulence closure models. Direct measurements of eddy diffusivity for momentum Km in the upper troposphere and in the lower stratosphere under clear-air conditions were made by the MU radar. Eddy diffusivity for heat Kh in the upper troposphere was measured directly by the Radio Acoustic Sounding System (RASS) operated in conjunction with the MU radar (Furumoto and Tsuda 2001). Furthermore, using observed values of turbulent kinetic energy k and the turbulent energy dissipation rate from the MU radar, we evaluated eddy diffusivity values estimated by the spectral width method as well as by existing turbulence closure models. We then compared the estimated and directly measured Km values to evaluate the applicability of turbulence closure models. Observations made by the MU radar and RASS, as well as data processing, are described in section 2. Estimation methods for determining eddy diffusivity by the spectral width method and existing turbulence closure models are introduced in section 3. We compare directly measured eddy diffusivities with the estimates and discuss the applicability of the spectral width method and turbulence closure models. In addition, on the basis of directly measured eddy diffusivity data, we discuss about stability dependence of Km and the turbulent Prandtl number Km/Kh in section 4. Conclusions are presented in section 5.

2. Observations and data processing

a. Wind velocity and temperature measurements

The MU radar is a VHF-band (46.5 MHz) Doppler radar installed at Shigaraki (34.85°N, 136.10°E), about 50 km southeast of Kyoto, Japan. It is an active phased-array system composed of 475 Yagi antennas with nominal peak and average transmitted powers of 1 MW and 50 kW, respectively, and a receiver dynamic range of 70 dB (Fukao et al. 1985a,b, 1990). It observes atmospheric echoes from altitude ranges of 1–25 km (day and night) and 55–90 km (daytime only). This radar can be steered to any position within 30° of the zenith in each interpulse period (400 μs), which enables us to observe Doppler velocities in five line-of-sight directions within 2 ms (minimum).

In the present work, three velocity components—the eastward, northward, and upward directions—were measured every 2 ms by the MU radar at a zenith angle of 10°. The height resolution was 150 m, and the beamwidth (half power for the full array) was 3.6°.

It is practical for the MU radar to measure instantaneous values of velocity components within 2 ms, but the values are averaged over the measuring volume (e.g., 150 m vertically and 630 m horizontally at 10-km altitude). The mean velocity components U, V, and W, and their fluctuations u, υ, and w in the eastward, northward, and upward directions (x, y, and z, respectively), were calculated from the velocity data series from the MU radar. Here, 2-h averaging was selected for the mean values, except for analyses of the energy spectrum and cospectrum of velocity fluctuations presented in section 4b.

RASS can continuously monitor temperature in the troposphere. It consists of four ground-based loudspeakers attached to the MU radar (Tsuda et al. 1994). Acoustic pulses are transmitted upward from the speakers and produce refraction-index fluctuations at acoustic wave fronts. The MU radar receives echo signals scattered by the acoustic wave fronts and determines Doppler shifts from the transmitted signals that correspond to their propagation speed. The speed cs observed by the MU radar is the apparent acoustic speed and is the sum of the true acoustic speed ca and the background wind velocity υr:
e1
Atmospheric temperature T can be derived as
e2
and K is determined by
e3
where γ is the ratio of specific heat, R* is the universal gas constant, and Md is the mean molecular weight.
In a moist atmosphere, sound waves propagate slightly faster than in a dry atmosphere. Note that K varies depending on the humidity, and T in Eq. (2) is replaced by the atmospheric virtual temperature Tυ, defined as
e4
Thus, the specific humidity q (kg kg−1) is a necessary parameter for RASS temperature measurements, but in the present case, where the humidity in the upper troposphere and lower stratosphere is very low, the contribution of q can be neglected.

This system can measure the temperature every 2.7 min. Height resolution is same as for the MU radar (150 m). Mean potential temperature Θ and its fluctuations θ are calculated using 2-h averaging. Rawinsondes were launched to collect vertical profile measurements of temperature and relative humidity at the MU radar and RASS site.

b. Eddy diffusivities for momentum and for heat

Eddy diffusivity for momentum was calculated according to the equation
e5
where the subscript h denotes the horizontal component and the overbar indicates a time-averaged value. The presence of clouds was judged from relative humidity from rawinsonde measurement and image of a video camera, and cloudy conditions were excluded from data analysis.
For the calculation of eddy diffusivity for heat we used RASS data. After calculating the mean potential temperature Θ and its fluctuation θ using 2-h averaging, we obtained Kh from the definition equation
e6

c. Brunt–Väisälä frequency and gradient Richardson number

The mean temperature field was represented in terms of the Brunt–Väisälä frequency N,
e7
and the gradient Richardson number Ri was calculated according to the equation
e8
where g is the gravitational acceleration.

While data from the MU radar and RASS are indispensable for determining Km and Kh, and they are better than those from rawinsonde measurement even for calculating N and Ri and correlating the resulted Km and Kh by them, nonetheless the temperature measurement by RASS was limited up to a level of 8 km, although the velocity measurement by MU radar was 18 km. For this reason, we decided to use rawinsonde temperature data for calculating N, after comparing the mean temperature profiles from rawinsonde measurement with those by RASS and confirming the representativeness of the former profiles. Then, together with the mean velocity profiles from MU radar, we calculated Ri for correlating Km and Kh by Ri up to a level of 18 km. It is considered to be legitimate because the observation of Km and Kh are based on the mean and turbulent quantities averaged over 2 h and so it is regarded as a measurement on the mesoscale comparable to the flight distance of the rawinsonde.

d. Turbulent kinetic energy k and turbulent energy dissipation rate ε

The turbulent kinetic energy was calculated according to the equation
e9
The turbulent energy dissipation rate was estimated from the Doppler spectrum of the radar echo.
The Doppler spectrum represents the probability density distribution of the velocity associated with turbulent eddies of spatial scales ranging from half the radar wavelength (3 m) up to the radar scattering volume. Its vertical extent is 150 m, and its horizontal extent is about 600 m in the upper troposphere and the lower stratosphere. Therefore, the MU radar can provide turbulence characteristics at scales that extend approximately over the entire inertial subrange of three-dimensional atmospheric turbulence. The nominal beamwidth 1/e is 3.6°, corresponding to horizontal diameters of less than 630 m at altitudes of about 10 km. We can convert the half-power half-width σ of the Doppler velocity spectrum to the variance of the radial (line-of-sight) velocity (where ur is the deviation from the mean radial velocity) inside the measuring volume using the simple relationship
e10
as described by Hocking (1983, 1985).
To extract turbulence parameters from σ, we postulated that the observed signal extends over the entire inertial subrange and obeys the Kolmogorov power law for the power spectrum of the scalar wavenumber κ. We may then state (Frisch and Clifford 1974; Bohne 1982) that
e11
where α (~1.5) is the Kolmogorov constant, ε is the turbulence dissipation rate, is the lowest wavenumber of the inertial subrange (the highest wavenumber of the buoyancy subrange), and κν is the highest wavenumber of the inertial subrange. Performing the integration on the right-hand side of Eq. (11) and noting that ε is constant in the inertial subrange, we obtain
e12
where
e13
If κνκB, then Cα−3/2 ≈ 0.5. In the present study, we use C ≈ 0.4, following Weinstock (1981), who selected this value on the basis of observational data. Using Eq. (10), we may extract ε from σ through
e14
if N is known from an appropriate temperature observation.

e. Spectral broadening effects on σ

When we allow for a finite beamwidth and finite sampling duration, radial velocities are not uniform in the measuring volume, so that, for example, the experimental value of spectral width σobs includes several experimental contaminations that are not directly produced by turbulence. Atlas et al. (1969), Frisch and Clifford (1974), and Hocking (1983, 1985, 1986, 1988) showed that spectral broadening effects result from 1) beam broadening resulting from different velocities in the line-of-sight directions within a finite beamwidth σbeam, 2) wind shear in the radial (approximately vertical) direction σshear, and 3) transient atmospheric motion within measuring time σtrans. Following the formulation by Hocking (1985, 1986), data corrections of these three effects were made for the specification of the MU radar (Fukao et al. 1986, 1994).

For common MU radar observations made at a sufficiently small zenith angle, beam broadening is the most serious contamination. According to in situ measurements in the lower stratosphere, turbulence has a typical maximum fluctuating velocity of 1 m s−1 (Barat 1975, 1982; Yamanaka et al. 1985). We found that the Doppler spectral width resulting from beam broadening exceeds 1 m s−1 when horizontal wind speed exceeds 40 m s−1. Therefore, σ directly produced by turbulence cannot be extracted from the spectral width when horizontal wind speed is greater than 40 m s−1. Thus, in the present work, observation data collected when wind speeds were greater than 40 m s−1 were excluded from analysis. Such wind speeds appear frequently near the tropopause jet stream in winter. However, beam-broadening effects were not significant in the observation data during other seasons or at other altitudes.

3. Estimation methods of eddy diffusivity

a. Spectral width method

From MU radar signals, Lilly et al. (1974), Hocking (1983), and Fukao et al. (1986, 1994) estimated eddy diffusivity in the upper troposphere and lower stratosphere. They assumed a local equilibrium in turbulence—that is, that energy production resulting from wind shear P and buoyancy G was balanced by ε,
e15
where Rf is the flux Richardson number,
e16
Substituting these expressions into Eq. (6), they obtained
e17
Adopting an Rf value of ¼ for strongly stratified conditions (Lilly et al. 1974) and using Eqs. (7) and (14), they then obtained
e18
Strictly speaking, Eq. (18) represents the eddy diffusivity for heat. In principle, because diffusions of vector and scalar quantities are considered to be substantially different in turbulent flows, Kh may be different from Km under stably stratified conditions (Ueda et al. 1981).

b. The k–ε model modified for stratified flows

In the modified version of the kε model, the conservation equations of turbulent energy k and turbulent energy dissipation ε are written as
e19
e20
e21
where D denotes the substantial derivative, and σk and σε are the turbulent Prandtl numbers for k and ε.
In the modified kε model (Launder and Spalding 1974), the coefficients take the values
e22
Thus, the eddy diffusivity for momentum in this model is given as
e21a
In this analysis, values of turbulent kinetic energy k and turbulent energy dissipation rate ε observed by the MU radar are used.

c. Algebraic stress model

Second-order turbulence closures can take into account anisotropic features of turbulence caused by buoyancy in stratified flows. Following Rodi (1976), we derived the algebraic stress models (Uno et al. 1989). Transport equations for Reynolds stresses , turbulent heat fluxes , and temperature variance θ2 are
e23
e24
and
e25
where β is the volumetric expansion coefficient (β = 1/Θ for gases). The subscripts i, j, and k denote direction, and the summation convention is applied; Pij and Gij are generations of resulting from shear and buoyancy, respectively, and ϕij is the pressure strain term. Following Gibson and Launder (1978), transport Eqs. (23)(25) were closed by means of assumptions for certain terms, including the pressure strain term.
Assuming that advection and diffusion processes in turbulent flows show behavior similar to that of k, then, for example, the transport equation for Reynolds stresses becomes
e26
and the following algebraic equation is obtained:
e27
where δij is the Kronekar delta. Similar assumptions were made for transport Eqs. (24) and (25) for turbulent heat fluxes and temperature variance, resulting in algebraic equations similar to Eq. (27). After some complicated algebra, Km is given as
e28
where
e29
eq1

The model constants have been determined from various laboratory experimental data as C1 = 1.804, C2 = 0.594, C3 = 0.5, = 0.599, = 0.307, C1t = 2.916, C2t = 0.448, C3t = 0.33, = 0.604, and R = 0.7. In this analysis, observed values from MU radar data are used for turbulent kinetic energy and the turbulent energy dissipation rate. The stability dependence takes Cm into account. Figure 1 describes the vertical profiles of Cm and vertical shear of potential temperature observed by the rawinsonde. The value of Cm ranged from 0.06 to 0.12, depending on variations in stratification stability.

Fig. 1.
Fig. 1.

Vertical profiles of Cm in algebraic stress models and vertical shear of potential temperature observed by rawinsonde.

Citation: Journal of the Atmospheric Sciences 69, 1; 10.1175/JAS-D-11-023.1

4. Results and discussion

a. Atmospheric conditions

Clear days were selected for analysis from observation periods in 1999 and 2000 based on the humidity data from rawinsonde measurement and imagery of a video camera. These days were assumed not to be affected by cloud activity. Mean velocity components and their fluctuations were obtained from the MU radar. Mean potential temperature and its fluctuation were obtained from RASS. The eddy diffusivities and turbulent kinetic energy were calculated from their respective definitions, Eqs. (5), (6), and (9). The turbulent energy dissipation rate was estimated from Eq. (12). The mean temperature field was represented in terms of the Brunt–Väisälä frequency and gradient Richardson number defined by Eqs. (7) and (8), with the potential temperature gradient calculated from rawinsonde data.

The selected dataset showed that the amplitude of the vertical velocity in the upper troposphere and lower stratosphere was less than 3 m s−1, and that its average was less than 0.3 m s−1. Horizontal velocity increased with altitude and attained a westerly wind maximum at a level ranging from 10 to 15 km. Large wind shear occurred within the 5-km layers above and below this level. The selected wind profiles had a maximum speed of about 40 m s−1, except for a speed of 78 m s−1 at 1300–1500 local standard time (LST) 19 February 1999. As discussed in section 2e, Uh data greater than 40 m s−1 were eliminated from analysis because of large spectral broadening effects.

Typical atmospheric conditions are illustrated in Fig. 2. The vertical profile of temperature did not change so much in all dataset selected for analysis, and the minimum temperature generally occurred at about 17 km, although it occurred at 15 km on 19 February 1999. The tropopause is located a little lower than this level. In the lower stratosphere, potential temperature increased monotonically, indicating strongly stable stratification. Echo intensity averaged over five beam directions was weak, and relative humidity was low above 10 km. However, the 18 May 1999 dataset showed relatively high relative humidity (RH > 50%), with high echo intensity below 10 km. In the 5 October 2000 dataset, a thin layer at 7 km had a RH of 75% and high echo intensity. These data are supposed to be influenced by clouds and constrained convection at the cloud top and are excluded from the discussion.

Fig. 2.
Fig. 2.

Atmospheric conditions at 1200 LST 4 Oct 2000. Horizontal bars indicate root-mean-square fluctuating velocities during 2 h (1100–1300 LST) at typical heights. Horizontal lines in T, RH, and mixing ratio plots indicate the region of radar range coverage.

Citation: Journal of the Atmospheric Sciences 69, 1; 10.1175/JAS-D-11-023.1

b. Wind velocity spectra

To examine how far MU radar data covered the power spectrum range of turbulence, we calculated the power spectra of horizontal wind velocity Su and cospectra of horizontal and vertical wind velocities Cuw (Fig. 3). The power spectra of the horizontal velocity and cospectra of horizontal and vertical velocities were derived from data from 0000 to 2359 LST 4 October 2000. These spectra and cospectra were smoothed in the frequency range using a repeated triangular filter. Su remained almost constant in the lower frequency range, but they began to decrease at the frequency value f ~ 3–5 × 10−4 s−1. After removing the aliasing effect caused by digital data sampling in the frequency range higher than 2.1 × 10−3 s−1, the decreasing rate of Su was approximated by the power of f. This decreasing rate extended over the frequency range for about two orders of magnitude. This tendency is similar to the Kolmogorov power law in the inertial subrange of turbulence. The power spectra of the vertical wind velocity Sw also showed a similar decreasing tendency of f−5/3 in the higher frequency range (not shown).

Fig. 3.
Fig. 3.

Power spectra of horizontal wind velocity Su and cospectrum of horizontal and vertical wind velocity Cuw from 0000 to 2359 LST 4 Oct 2000. The thick solid line represents z = 6 km; the thick dashed line, z = 9 km; the thin solid line, z = 15 km; and the thin dashed line, z = 18 km.

Citation: Journal of the Atmospheric Sciences 69, 1; 10.1175/JAS-D-11-023.1

Figure 3 shows the contribution of the frequency range to the Reynolds stress. The observed frequency range covers almost all contributions to the Reynolds stress at all levels. For this 1-day spectrum analysis, 2-min averaged velocity fluctuation data were used. Therefore, the power spectra and cospectra in the frequency range higher than 4 × 10−3 s−1 were not included in the diagrams. However, it may be assumed that the observed wind velocity fluctuations include almost all components of the velocity fluctuations caused by energy-containing eddies, as is also the case for the temperature fluctuations. Thus, we conclude that the observed Reynolds stress and turbulent heat flux can be used to estimate eddy diffusivities for momentum and heat.

c. Profiles of turbulence quantities and atmospheric parameters

Vertical profiles of the turbulence quantities and atmospheric parameters are presented in Figs. 4 and 5. The Brunt–Väisälä frequency increased gradually in the range of N ~ 10−2 s−1, and σ and ε showed similar profiles when plotted on a logarithmic scale. Because ε is estimated from Eq. (12), the ε variation is about twice the σ variation. They have peaks in the subtropical westerly jet region near the tropopause and then decrease in the stratosphere. Other atmospheric parameters presented in Fig. 5 were vertically smoothed using a moving average. In the subtropical jet region, wind shear varied dramatically with altitude and changed sign near the axis of the jet, where the absolute value of the wind shear |∂Uh/∂z| was at a minimum. The vertically smoothed profile of Ri increased about one order of magnitude in the stratosphere.

Fig. 4.
Fig. 4.

Vertical profiles of σ, N, ε, and Kh estimated by the spectral width method. The thick solid line represents Kh = 0.1σ2/N; the thick dashed line, ε; the thin solid line, σ; and the thin dashed line, N.

Citation: Journal of the Atmospheric Sciences 69, 1; 10.1175/JAS-D-11-023.1

Fig. 5.
Fig. 5.

Vertical profiles of wind shear, stability parameters, and Km measured directly by the MU radar.

Citation: Journal of the Atmospheric Sciences 69, 1; 10.1175/JAS-D-11-023.1

Results of direct observation of eddy diffusivity are shown in Fig. 5. Eddy diffusivity values were about 10 m2 s−1 in the upper troposphere. In the region above the axis of the subtropical jets, eddy diffusivity values decreased by one or two orders of magnitude, except on 19 February 1999, when an exceptionally strong winter jet with a speed of 78 m s−1 was observed.

The spectrum width method has been applied in previous work regarding radar observations, as described in section 3a. Using observed values for σ and ε and applying the spectrum width method, we estimated eddy diffusivity (Fig. 4). Eddy diffusivity values were about 10 m2 s−1 in the upper troposphere, had a small peak in the subtropical jet region, and decreased gradually by a factor of 10 or more in the lower stratosphere, where values were on the order of 1 m2 s−1.

d. Comparison of turbulence models in eddy diffusivity prediction

Eddy diffusivity for momentum was estimated by the k–ε model and algebraic stress model and compared with the observed one in Fig. 6. Since Kh can be assumed almost equal to Km, as will be shown in the next section, eddy diffusivity profile estimated by the spectral width method is also shown in Fig. 6. The estimated eddy diffusivities from all three models showed good agreement with those observed directly. Moreover, they showed similar vertical profiles, except on 19 February 1999.

Fig. 6.
Fig. 6.

Eddy diffusivity for momentum measured directly and its comparison with values estimated using turbulence models. Only the spectral width method estimates eddy diffusivity for heat. The solid lines of the profiles become thinner when Km and Kh contain large uncertainty due to spectral broadening effects.

Citation: Journal of the Atmospheric Sciences 69, 1; 10.1175/JAS-D-11-023.1

On 19 February 1999, the spectrum width method predicted eddy diffusivity one order larger than the observed Km in the jet stream layer at 7–12 km. This discrepancy was caused by insufficient correction for Doppler spectrum broadening at wind speeds associating with large wind shear, which are much larger than the limiting velocity of 40 m s−1 for correction of spectrum broadening. A similar deviation resulting from strong winds was also seen on 5 October 2000.

The kε model modified for stratified flows gave a relatively poor estimation of Km. In this model, Km is assumed to be k2/ε multiplied by a constant (0.09), as shown in Eq. (21′).

The algebraic stress model is a modification of the k–ε model, as far as these eddy diffusivity expressions are concerned. The algebraic stress model is a simplified model derived from transport equations of and . By assuming that the sum of their advection and diffusion terms behaves in the similar manner to that of the k equation, we derive a set of algebraic equations for turbulent stresses and heat flux components with two unknown parameters k and ε. These k and ε are obtained by solving the respective differential equations, similar to the k–ε model. However, the kε model takes into account the stratification effects in the production (or destruction) terms of the k and ε equations, but it does not in the expression of the eddy diffusivity. In contrast, in the algebraic stress model the derived expression of Km becomes the same in form as that for the kε model—that is, Km is equal to k2 multiplied by a constant Cm, which is dependent, however, on the stratification, as shown in Eq. (28). In that sense the algebraic stress model is a modification of the kε model, and as seen in Fig. 6 the vertical profiles of Km predicted by the k–ε model and algebraic stress model are in good agreement with each other, insofar as the same values of k and ε are used in both models. They can predict well the directly observed Km but the result of the algebraic stress model is superior to that of the kε model. In addition, comparison of the eddy diffusivity profiles from all three turbulence models with the directly observed ones indicates that the algebraic stress model gives the best prediction.

e. Stability dependence of eddy diffusivities for momentum and heat

Eddy diffusivity estimated by the spectral width method showed good agreement with the directly observed values for Km. However, it is important to understand that this method estimates Kh but not Km.

In a previous paper (Ueda et al. 1981), we used laboratory experiments and meteorological tower experiments to show that both Km and Kh decrease with stability and are correlated with the local stability parameters Ri and Rf. In addition, we stressed the substantially different stability dependence between turbulent transfer mechanisms in the surface layer and in the layer above it in the atmospheric boundary layer, particularly in the stability dependence of the Kh/Km ratio. The eddy diffusivities Km and Kh were well described by the following empirical formulas:
e30
e31
where Rf = (Kh/Km)Ri, Kmo is the eddy diffusivity for momentum under neutral stratification condition, and Rfcrit is the critical flux Richardson number. In the latter formula, the ratio of Kh/Km decreases with stability and attains a value of 0.1 at Ri = 0.6. However, this formulation was valid for the outer region of the turbulent boundary layer. Turbulence eddies produced mainly in the surface layer are transported into this outer region, and they are deformed by buoyancy and partly transformed into internal gravity waves. Since only the heat transfer is strongly prohibited in the wavy motions, consequently, Kh becomes much smaller than Km in a large Ri number range.
The stability dependence of the Kh/Km ratio is presented in Fig. 7. Surprisingly, the Kh values do not differ much from the Km values and Kh = O(Km) where Kh is slightly smaller than Km. The average Kh/Km ratio is 0.80, if two low values in Fig. 7a are excluded. Because there are many error sources, including instrumental ones from the MU radar, RASS, and rawinsonde, and data processing error sources such as the calculation of vertical gradients of mean velocity and temperature, it is difficult to estimate experimental error. The above two low Kh/Km values in Fig. 7a may also be supposed to be obtained in the transient atmosphere to the outer region of the boundary layer. Allowing for such experimental error and taking into account the Kh/Km in the outer region of turbulent boundary layer being by one or two orders of magnitude smaller than the observed Kh/Km, the Kh/Km ratio is assumed to be almost equal to one; that is,
e32
Fig. 7.
Fig. 7.

Stability dependence of (a) Kh/Km and (b) Km/Kmo. Solid lines in (a) and (b) represent Eqs. (32) and (33), respectively. Height of the observed atmospheric layers is 4–8 km for (a) and 4–18 km for (b), in which circular and triangular symbols indicate 4–8 and 14–18 km, respectively. Open and solid symbols in (b) are for δ = 2000 and 4000 m, respectively.

Citation: Journal of the Atmospheric Sciences 69, 1; 10.1175/JAS-D-11-023.1

The relationship in Eq. (32) is the same as that for the atmospheric surface layer (e.g., Fukui et al. 1983). In the surface layer, turbulence is considered to be produced intermittently and sporadically by shear instability (Kim et al. 1971; Ueda and Hinze 1975). Such events, which are referred to as “bursting,” are the main mechanism for turbulence production in conditions of neutral and stable stratification (Ueda et al. 1981). By the bursting motion, heat is transferred vertically, together with momentum. Thus, Kh is considered almost equal to Km. Similarly, in the upper troposphere and lower stratosphere turbulent eddies are produced by shear instability and wave breaking. By these motions, momentum and heat are considered to be transferred simultaneously in the same manner. After these events, turbulent eddies are quickly destructed by buoyancy and do not contribute significantly to vertical diffusion of heat and momentum. Therefore, it is reasonable to assume Kh ~ Km in the upper troposphere and lower stratosphere. As seen in Fig. 7, direct observations by the MU radar and RASS verify this equality within the experimental uncertainty of our work.

However, in the outer region of the turbulent boundary layer, turbulence eddies transported mainly from the atmospheric surface layer are deformed by buoyancy and partly transformed to internal gravity waves. Since only the vertical transport of heat is strongly prohibited in the wavy motion, it causes a substantial difference between eddy diffusivity for momentum and that for heat.

Wind tunnel experiments (e.g., Klebanoff 1954; Townsend 1951) suggest that Km in the outer regions of the turbulent boundary layer during conditions of neutral stability may be represented by Kmo = 0.0023Uδ, where U is the wind speed at the outer edge of the boundary layer and δ is the boundary layer thickness. This relationship may also be applied to the turbulent–nonturbulent intermittent layer immediately outside the outer edge of the boundary layer. In this study, this relationship is assumed to be applicable to the upper troposphere and lower stratosphere when those layers are neutrally stable. To estimate Kmo, an average wind speed from 2 to 4 km was used for U, and values of 2 and 4 km were used for δ. Results are presented in Fig. 7. The observed Km is one or two orders of magnitude smaller than Kmo.

The stability dependence of the Km/Kmo ratio was examined, assuming similar mechanisms in the surface layer and in upper troposphere and lower stratosphere as discussed above. The shear function of momentum ϕm (i.e., the reciprocal of the Km/Kmo ratio) is described by
e33
where Kmo is the eddy diffusivity for momentum during conditions of neutral stability,
e34
L is the Monin–Obukhov length, and u* is the friction velocity. Substituting Eq. (34) into Eq. (33), we get
e35
In Eq. (35), the critical value of flux Richardson number in the extreme of stable stratification is , although this value is slightly smaller than the value of ¼ adopted in section 3a.
When shear instability or wave breaking occurs during conditions of extremely strong stratification and hot or cold turbulent eddies are produced, they begin to move up and down and oscillate at the buoyancy Brunt–Väisälä frequency. As a hot eddy descends or a cold eddy ascends, a countergradient diffusion of heat occurs. This is also true for the eddy diffusion of momentum. Thus, as the eddies decay, gradient and countergradient diffusions, and therefore negative and positive fluxes, occur alternately (Komori and Nagata 1996; Hanazaki and Hunt 2004). Even when averaged over long time intervals, negative eddy diffusivity is evident during conditions of strong stratification (Komori et al. 1983). To correlate the observed Km/Kmo, the following empirical equation (Ueda et al. 1981) is applied, taking into account that there exist large Ri numbers in the upper troposphere and lower stratosphere under extremely stable stratifications and that it approaches asymptotically to Eq. (35) at low Ri numbers:
e36
In Fig. 7, the observed Km/Kmo ratio is compared with calculations using Eq. (36). The Km/Kmo ratio was well described by Eq. (36). Better correlation of observation data by Eq. (36) is obtained if δ is assumed to be equal to 4 km in Kmo = 0.0023Uδ. This result is not surprising because the upper part of the atmospheric boundary layer is stably stratified as a result of longwave radiation, and the stable stratification shortens the “apparent” boundary layer thickness. Therefore, δ can be assumed to be larger than the apparent thickness. Moreover, it is important to note that the Km/Kmo ratio is proportional to Ri−1 for a large range of Ri values.

5. Conclusions and perspectives

To understand the details of diffusion processes in clear air above the atmospheric boundary layer in which turbulence is produced by internal gravity wave breaking or by shear instability during conditions of stable stratification, measurements of wind velocity, temperature, and turbulence parameters were made using the MU radar and RASS.

Direct measurements of eddy diffusivity in the upper troposphere and lower stratosphere were made during conditions of clear air. The eddy diffusivities for momentum Km and heat Kh were obtained from the definitions and , where the mean velocity components U, V, and W and their fluctuations u, υ, and w were measured by the MU radar; potential temperature Θ and temperature fluctuation θ were measured by RASS; and the subscript h denotes the horizontal component. The temperature measurement by RASS was limited up to a level of 8 km, while the velocity measurement by MU radar was 18 km. Thus, for calculating the Brunt–Väisälä frequency N, here we used rawinsonde temperature data, after comparing the mean temperature profiles from rawinsonde measurement with those by RASS and confirming the representativeness of the former profiles. Then, together with the mean velocity profiles from MU radar, we calculated Ri for correlating Km and Kh by Ri up to a level of 18 km.

The Km values were on the order of 10 m2 s−1 in the upper troposphere and decreased gradually in the stratosphere by one order or more of magnitude to an altitude of 18 km. Although direct measurements of Kh were limited to the altitudes below 8 km, observed values of Kh seemed almost equal to the values of Km. This is a remarkable contrast to in the outer region of the boundary layer where the ratio Kh/Km is 0.1–0.02 in such strong stratification conditions (e.g., Ueda et al. 1981).

The relation Kh ~ Km can also be applied in the atmospheric surface layer (e.g., Fukui et al. 1983). The similarity is understood since in both of these two layers turbulence is produced locally and intermittently by the so-called bursting due to the shear instability. By such bursting motions, heat is considered to be transported together with momentum in the same manner. After these events turbulent eddies are quickly destructed and do not contribute significantly to diffusion of momentum and heat. It is a great contrast to conditions in the outer region of turbulent boundary layer. In the outer region of turbulent boundary layer, turbulent eddies produced mainly in the surface layer are deformed by buoyancy force, and in stable stratification conditions they are depressed vertically and elongated horizontally and are partly transformed to internal gravity waves. Since only the transfer of heat is strongly prohibited, similarity between momentum and heat transports is destructed, and in strongly stable stratification conditions Kh becomes by one or two orders smaller than Km, in the Ri range observed (Ueda et al. 1981).

Moreover, the stability dependence of Km was examined with regard to the ratio Km/Kmo, assuming that the eddy diffusivity Kmo during conditions of neutral stability can be expressed as Kmo = 0.0023Uδ, where U is the wind speed at the outer edge of the boundary layer and δ is the boundary layer thickness. The observed Km was one or two orders of magnitude smaller than Kmo. Assuming similar mechanisms, P + G = ε both in the surface layer and in the upper troposphere and lower stratosphere, and furthermore assuming that the equality KhKm can be applied in the upper troposphere and lower stratosphere in the same way as in the surface layer, the stability dependence of Km was obtained as Km/Kmo = 1/(1 + 4.7Ri). This relation represented the observed stability dependence fairly well. Moreover, it was proportional to Ri−1 in the range of Ri > 1.

To evaluate the applicability of turbulence models to the upper troposphere and lower stratosphere, model comparisons were made against directly observed eddy diffusivities for momentum and heat. The eddy diffusivity formula K = 0.1σ2/N proposed by Fukao et al. (1986, 1994) from the spectrum width method, where σ is the half-power half-width of the radar Doppler velocity spectrum, agreed fairly well with the directly measured values of Km. Using observed values of turbulent kinetic energy k and the turbulent energy dissipation rate ε, Km was estimated by the k–ε model modified for stratified flows (Launder and Spalding 1974) and the algebraic stress model (Uno et al. 1989). The algebraic stress model showed the best fit with direct measurements of both Km and Kh. Thus, we conclude that the algebraic stress model is applicable for the prediction of eddy diffusivity and its stability dependence even in the clear free atmosphere above the atmospheric boundary layer, once k and ε values have been observed directly or predicted.

Because of the rapid advance of MU radar and RASS technologies, it has been possible to observe three-dimensional images of internal gravity waves and turbulence. Thus, the next step is to obtain detailed observations of the three-dimensional structures of the production of turbulence from gravity wave breaking and shear instability, and also the reorganization of turbulent motion by buoyancy forces into internal gravity waves.

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