## 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 *K _{m}* in the upper troposphere and in the lower stratosphere under clear-air conditions were made by the MU radar. Eddy diffusivity for heat

*K*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

_{h}*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

*K*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

_{m}*K*and the turbulent Prandtl number

_{m}*K*/

_{m}*K*in section 4. Conclusions are presented in section 5.

_{h}## 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 *μ*s), 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.

**c**

*observed by the MU radar is the apparent acoustic speed and is the sum of the true acoustic speed*

_{s}**c**

*and the background wind velocity*

_{a}**υ**

*:Atmospheric temperature*

_{r}*T*can be derived asand

*K*is determined bywhere

*γ*is the ratio of specific heat,

*R*

^{*}is the universal gas constant, and

*M*is the mean molecular weight.

_{d}*K*varies depending on the humidity, and

*T*in Eq. (2) is replaced by the atmospheric virtual temperature

*T*, defined asThus, the specific humidity

_{υ}*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

*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.

*θ*using 2-h averaging, we obtained

*K*from the definition equation

_{h}### c. Brunt–Väisälä frequency and gradient Richardson number

*N*,and the gradient Richardson number Ri was calculated according to the equationwhere

*g*is the gravitational acceleration.

While data from the MU radar and RASS are indispensable for determining *K _{m}* and

*K*, and they are better than those from rawinsonde measurement even for calculating

_{h}*N*and Ri and correlating the resulted

*K*and

_{m}*K*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

_{h}*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

*K*and

_{m}*K*by Ri up to a level of 18 km. It is considered to be legitimate because the observation of

_{h}*K*and

_{m}*K*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.

_{h}### d. Turbulent kinetic energy k and turbulent energy dissipation rate ε

*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

*u*is the deviation from the mean radial velocity) inside the measuring volume using the simple relationshipas described by Hocking (1983, 1985).

_{r}*σ*, we postulated that the observed signal extends over the entire inertial subrange and obeys the Kolmogorov

*κ*. We may then state (Frisch and Clifford 1974; Bohne 1982) thatwhere

*α*(~1.5) is the Kolmogorov constant,

*ε*is the turbulence dissipation rate,

*κ*is the highest wavenumber of the inertial subrange. Performing the integration on the right-hand side of Eq. (11) and noting that

_{ν}*ε*is constant in the inertial subrange, we obtainwhereIf

*κ*≫

_{ν}*κ*, then

_{B}*C*≈

*α*

^{−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*throughif

*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 *σ*_{obs} 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 *σ*_{beam}, 2) wind shear in the radial (approximately vertical) direction *σ*_{shear}, and 3) transient atmospheric motion within measuring time *σ*_{trans}. 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

*P*and buoyancy

*G*was balanced by

*ε*,where Rf is the flux Richardson number,Substituting these expressions into Eq. (6), they obtainedAdopting an Rf value of ¼ for strongly stratified conditions (Lilly et al. 1974) and using Eqs. (7) and (14), they then obtainedStrictly 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,

*K*may be different from

_{h}*K*under stably stratified conditions (Ueda et al. 1981).

_{m}### b. The k–ε model modified for stratified flows

*k*–

*ε*model, the conservation equations of turbulent energy

*k*and turbulent energy dissipation

*ε*are written aswhere

*D*denotes the substantial derivative, and

*σ*and

_{k}*σ*are the turbulent Prandtl numbers for

_{ε}*k*and

*ε*.

*k*–

*ε*model (Launder and Spalding 1974), the coefficients take the valuesThus, the eddy diffusivity for momentum in this model is given asIn 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

*θ*

^{2}areandwhere

*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;

*P*and

_{ij}*G*are generations of

_{ij}*ϕ*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.

_{ij}*k*, then, for example, the transport equation for Reynolds stresses becomesand the following algebraic equation is obtained:where

*δ*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,

_{ij}*K*is given aswhere

_{m}The model constants have been determined from various laboratory experimental data as *C*_{1} = 1.804, *C*_{2} = 0.594, *C*_{3} = 0.5, *C*_{1t} = 2.916, *C*_{2t} = 0.448, *C*_{3t} = 0.33, *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 *C _{m}* into account. Figure 1 describes the vertical profiles of

*C*and vertical shear of potential temperature observed by the rawinsonde. The value of

_{m}*C*ranged from 0.06 to 0.12, depending on variations in stratification stability.

_{m}## 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, *U _{h}* 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.

### 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 *S _{u}* and cospectra of horizontal and vertical wind velocities

*C*(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.

_{uw}*S*remained almost constant in the lower frequency range, but they began to decrease at the frequency value

_{u}*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

*S*was approximated by the

_{u}*f*. This decreasing rate extended over the frequency range for about two orders of magnitude. This tendency is similar to the Kolmogorov

*S*also showed a similar decreasing tendency of

_{w}*f*

^{−5/3}in the higher frequency range (not shown).

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 |∂*U _{h}*/∂

*z*| was at a minimum. The vertically smoothed profile of Ri increased about one order of magnitude in the stratosphere.

Results of direct observation of eddy diffusivity are shown in Fig. 5. Eddy diffusivity values were about 10 m^{2} 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 m^{2} 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 m^{2} 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 *K _{h}* can be assumed almost equal to

*K*, 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.

_{m}On 19 February 1999, the spectrum width method predicted eddy diffusivity one order larger than the observed *K _{m}* 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 *k*–*ε* model modified for stratified flows gave a relatively poor estimation of *K _{m}*. In this model,

*K*is assumed to be

_{m}*k*/

^{2}*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 *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 *k*–*ε* 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 *K _{m}* becomes the same in form as that for the

*k*–

*ε*model—that is,

*K*is equal to

_{m}*k*multiplied by a constant

^{2}/ε*C*, which is dependent, however, on the stratification, as shown in Eq. (28). In that sense the algebraic stress model is a modification of the

_{m}*k*–

*ε*model, and as seen in Fig. 6 the vertical profiles of

*K*predicted by the

_{m}*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

*K*but the result of the algebraic stress model is superior to that of the

_{m}*k*–

*ε*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 *K _{m}*. However, it is important to understand that this method estimates

*K*but not

_{h}*K*.

_{m}*K*and

_{m}*K*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

_{h}*K*/

_{h}*K*ratio. The eddy diffusivities

_{m}*K*and

_{m}*K*were well described by the following empirical formulas:where Rf = (

_{h}*K*/

_{h}*K*)Ri,

_{m}*K*

_{mo}is the eddy diffusivity for momentum under neutral stratification condition, and Rf

_{crit}is the critical flux Richardson number. In the latter formula, the ratio of

*K*/

_{h}*K*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,

_{m}*K*becomes much smaller than

_{h}*K*in a large Ri number range.

_{m}*K*/

_{h}*K*ratio is presented in Fig. 7. Surprisingly, the

_{m}*K*values do not differ much from the

_{h}*K*values and

_{m}*K*=

_{h}*O*(

*K*) where

_{m}*K*is slightly smaller than

_{h}*K*. The average

_{m}*K*/

_{h}*K*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

_{m}*K*/

_{h}*K*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

_{m}*K*/

_{h}*K*in the outer region of turbulent boundary layer being by one or two orders of magnitude smaller than the observed

_{m}*K*/

_{h}*K*, the

_{m}*K*/

_{h}*K*ratio is assumed to be almost equal to one; that is,

_{m}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, *K _{h}* is considered almost equal to

*K*. 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

_{m}*K*~

_{h}*K*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.

_{m}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 *K _{m}* in the outer regions of the turbulent boundary layer during conditions of neutral stability may be represented by

*K*

_{mo}= 0.0023

*U*

_{∞}

*δ*, 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

*K*

_{mo}, 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

*K*is one or two orders of magnitude smaller than

_{m}*K*

_{mo}.

*K*/

_{m}*K*

_{mo}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

*ϕ*(i.e., the reciprocal of the

_{m}*K*/

_{m}*K*

_{mo}ratio) is described bywhere

*K*

_{mo}is the eddy diffusivity for momentum during conditions of neutral stability,

*L*is the Monin–Obukhov length, and

*u*is the friction velocity. Substituting Eq. (34) into Eq. (33), we getIn Eq. (35), the critical value of flux Richardson number in the extreme of stable stratification is

_{*}*K*

_{m}/K_{mo}, 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:In Fig. 7, the observed

*K*/

_{m}*K*

_{mo}ratio is compared with calculations using Eq. (36). The

*K*/

_{m}*K*

_{mo}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

*K*

_{mo}= 0.0023

*U*

_{∞}

*δ*. 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

*K*/

_{m}*K*

_{mo}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 *K _{m}* and heat

*K*were obtained from the definitions

_{h}*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

*K*and

_{m}*K*by Ri up to a level of 18 km.

_{h}The *K _{m}* values were on the order of 10 m

^{2}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

*K*were limited to the altitudes below 8 km, observed values of

_{h}*K*seemed almost equal to the values of

_{h}*K*. This is a remarkable contrast to in the outer region of the boundary layer where the ratio

_{m}*K*/

_{h}*K*is 0.1–0.02 in such strong stratification conditions (e.g., Ueda et al. 1981).

_{m}The relation *K _{h}* ~

*K*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

_{m}*K*becomes by one or two orders smaller than

_{h}*K*, in the Ri range observed (Ueda et al. 1981).

_{m}Moreover, the stability dependence of *K _{m}* was examined with regard to the ratio

*K*/

_{m}*K*

_{mo}, assuming that the eddy diffusivity

*K*

_{mo}during conditions of neutral stability can be expressed as

*K*

_{mo}= 0.0023

*U*

_{∞}

*δ*, where

*U*

_{∞}is the wind speed at the outer edge of the boundary layer and

*δ*is the boundary layer thickness. The observed

*K*was one or two orders of magnitude smaller than

_{m}*K*

_{mo}. 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

*K*≈

_{h}*K*can be applied in the upper troposphere and lower stratosphere in the same way as in the surface layer, the stability dependence of

_{m}*K*was obtained as

_{m}*K*/

_{m}*K*

_{mo}= 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* = 0.1*σ*^{2}/*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 *K _{m}*. Using observed values of turbulent kinetic energy

*k*and the turbulent energy dissipation rate

*ε*,

*K*was estimated by the

_{m}*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

*K*and

_{m}*K*. 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

_{h}*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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