Mesospheric radar echoes in midlatitudes, which are generated by refractive index irregularities at half the radar wavelength, are considered to be caused by electron density irregularities induced by neutral turbulence. In this paper we construct a model to evaluate intensities of VHF mesospheric echoes due to electron irregularities with the help of an empirical model profile of mean electron densities and temperatures. We simulate the electron density deviations in a parcel displaced by neutral turbulence, the intensity of which is evaluated from the Doppler spectral width measured with the mesosphere-stratosphere-troposphere radar. Our results show that the estimated reflectivity increases up to the height range of 70–80 km mainly because of the increase of the electron density gradients and decreases above 80 km because of the suppression of turbulence at half the radar wavelength by increasing molecular viscosity. Here 70–80 km is the critical height for the mesospheric echoes where the inner scale of inertial turbulence is equal to half the radar wavelength. Our model reproduces well the characteristics of observed echo intensities. Particularly mesospheric echoes are most frequently found around and just below the critical height, which is much higher than those estimated in the earlier studies.
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 Coherent backscattered signals in VHF wavelength from the mesosphere were first investigated by Woodman and Guillén using the Jicamarca radar in Peru. Mesospheric observations with the middle and upper atmosphere (MU) radar (Shigaraki, Japan; 34°51′N, 136°6′E) have been carried out by Radio Science Center at Kyoto University since 1986. One of the results revealed with the MU radar is that midlatitude mesospheric echoes have seasonal oscillation with maxima in the summer at the altitude range of 70–85 km and that the interannual variations seem to be dependent on solar activity [Kubo et al., 1997].
 Concerning the observational information about echoing mechanism, Fukao et al. [1979, 1980] first showed with the Jicamarca radar that echoes received from around 70 km were enhanced in the vertical direction compared with those in the off-vertical directions, which suggests that echoes are produced by Fresnel reflection due to horizontally stratified structures in the refractive index. Above the altitude of 70 km, however, enhancements of echo scattering in the vertical direction are less pronounced than those below 70 km, echoes being inclined to be isotropic. Similar characteristics are also found with the MU radar [Kubo et al., 1997]. On the one hand, Yamamoto et al. showed with the MU radar observations that descending echo layers in winter occurred at the region where the horizontal wind shear was strong because of amplified mesospheric waves. On the other hand, Muraoka et al.  suggested that echoes were correlated to gradient of electron densities induced by gravity waves.
 In previous theoretical studies, Gage and Balsley  have developed a model of radio wave scattering in the troposphere, stratosphere, and mesosphere. The radio wave is scattered by refractive index structures with a scale of half the radar wavelength (≈ 3.23 m in the case of the MU radar). Refractive index fluctuations have been attributed to spatial irregularities of radar sensitive materials induced by turbulence in the atmosphere. Because of the increase of molecular viscosity with height, the inner scale of inertial turbulence is supposed to increase with height in the mesospheric region, and half the radar wavelength falls into the viscous subrange, causing echoes to be reduced severely. However, a quantitative comparison of echo powers in the mesospheric echoes between observations and theories has not been made yet. There is some advantage to study quantitatively echoing mechanism in the mesosphere rather than in the lower atmosphere. The electron distribution consists of bound electrons in the stratosphere and troposphere and free electrons in the mesosphere. Whereas it is difficult to detect detailed structures of neutral density profiles, electron densities and their irregularities can generally be detected with in situ measurements.
 We provide, in the present paper, the first quantitative approach to the backscattered echo powers from the mesosphere by estimating the strength of turbulence from the observed Doppler spectral width. Our major assumption in the following model is that irregularities in electrons are made in the moving parcel by neutral turbulence.
2. Estimation of Scattered Power in the Mesosphere
 We begin with the equation of the refractive index nin the atmosphere at VHF bands, as shown by Gage and Balsley ,
where P is the total atmospheric pressure (mbar), e is the water vapor pressure (mbar), Tis the temperature (K), Neis the electron density (m−3), and Nc is the characteristic plasma density (m−3) for the radar frequency. The first and the second terms on the right-hand side of (1) correspond to the contributions of bound electrons inherent in the density fluctuations of dry air and water vapor, respectively, which are negligible at mesospheric heights. The third term, the contribution of electrons, dominates above the altitude of 60 km. The refractive index fluctuations δnin the mesopause region are then related to the fluctuations of electron density by the expression
On the other hand, radar volume reflectivity ηturb (radar scattering cross section per unit volume) induced by homogeneous isotropic turbulence is defined by Balsley and Gage  as
where K is the three-dimensional (3-D) wave number vector (double the radar wave number vector) and Φn(K) is the 3-D spatial spectral density of refractive index fluctuations defined by [Tatarskii, 1971]
where the mean square of the refractive index fluctuation is obtained by averaging (δn)2 over some extended volume, such as the radar volume V as follows:
The 3-D spatial spectrum is related to the 1-D wave number spectrum in case of isotropy as follows:
where Fn(k) = 4πk2Φn(k) and kis the 1-D wave number. So it can be rewritten as
 In the theory of isotropic turbulence the power spectral index of velocity fluctuations in wave number space is equal to −5/3 in the inertial subrange, where kinetic energy is transferred from low to high wave numbers without loss, whereas it is equal to −7 in the viscous subrange, where turbulence decays because of molecular viscosity [Heisenberg, 1948]. According to the in situ rocket observations by Røyrvik and Smith  the spectra of electron density fluctuations have been shown to be the same indices, −5/3 and −7, as those of the velocity fluctuations. Taking account of (2), the fluctuations of refractive index in the mesosphere are proportional to the electron density fluctuations. Then we define the power spectrum density of refractive index similarly to the definition in Heisenberg's theory. We describe the spectral density of the refractive index as
where kℓ is the wave number for the scale ℓ and ℓ is the inner scale of isotropic turbulence, which is the boundary between the inertial and viscous subranges. Cf is a constant, which we evaluate in the following paragraph.
 The outer scale of the inertial subrange of isotropic turbulence L, which is the boundary between the inertial subrange and the energy source region, can be defined by requiring [Silverman, 1956]
where kL= 2π/L. Then by integrating (9), we obtain
 Characteristic parameters of turbulence L and ℓ in (12)are evaluated from the Doppler spectral width σwobserved with the radar system. Schlegel et al.  assumed that σwcorresponds to the deviation of air velocity at half the radar wavelength. On the other hand, Hocking assumed that σwis related to the root-mean-square velocity of the largest eddies of isotropic turbulence. In this paper we follow the theory by Hocking . Then Lis estimated as L ≈ σw/fB, where fB is the Brunt Väisälä frequency, which is the measure of the lowest frequency of turbulent motions at the scale of L. The turbulent energy dissipation rate ε has been estimated as follows [see Hocking, 1985]:
 To evaluate δn in the mesosphere, we consider irregularities in electron density induced by turbulence. We assume that the mean distribution of free electrons is homogeneous in the horizontal plane but stratified in height. The inhomogeneous distribution is assumed to be created by vertical displacements of air parcels in turbulent motions. The lifetime of free electrons against recombination is estimated as more than 1000 s in the mesosphere [Brasseur and Solomon, 1986], whereas that of turbulent air parcels is considered to be less than the Brunt Väisälä period (300 s around the mesopause). Electrons in turbulent air parcels do not diffuse due to the electron recombination during a period of a vertical oscillation, resulting in the inhomogeneous distribution of electron density. The variations of electron density δNe in the mean density due to the displacement δz of an atmospheric parcel induced by turbulent motions are tentatively described as
Following Hocking , we assume that the Doppler spectral width in the vertical beam is equal to the deviation of the vertical turbulent velocity wsuch that
Further, we assume that the mean-square displacement of an air parcel in the turbulent field is of the order of L, such that
where a is a constant with an order of 1. Then we can obtain the equation for electron density fluctuations and refractive index variance as follows:
 Thus with the use of Doppler width we can evaluate the variances of refractive index fluctuations in the mesosphere. In the succeeding analyses we will take a= 1.
 Finally, we take account of adiabatic processes in an air parcel moving due to turbulence. In spite of (16) the electron density deviation δNe in the atmosphere is given by
where Hpis the scale height of atmospheric pressure [see Thrane and Grandal, 1981; Sugiyama, 1988]. It should be noted that the fluctuation of electron density is nearly 8 times greater than that of the neutral density in the case that the scale height of electrons He ≡ −(1/)(∂/∂z)−1= −Hp. Using (21) instead of (16) and substituting (20) and (21)into (12) yields
for the inertial and viscous subranges, respectively.
3. Results of the Modeled Reflectivity
 In order to estimate radar reflectivity from (22)we need not only the values of spectral width but also the mean profiles of temperature and electron density. In this study, we use the data sets of the MU radar observations for spectral width of the VHF echoes, the Mass Spectrometer Incoherent Scatter Extension (MSISE90) model [Hedin, 1991] for atmospheric temperature and pressure and the international reference ionosphere (IRI95) model [Danilov and Smirnova, 1995] for electron density.
 The diagrams of Figure 1 show an example of profiles of the temperature T, the reduced viscosity coefficient ν, the electron density gradient ∂/∂z, and the Doppler spectral width σw (from MU radar observations). Each profile shows the hourly averages for a specific date and time in October 1989. It should be noted that the model profiles of T, ν, and ∂/∂z are acquired with a time resolution of 1 hour whereas σw are obtained with a time resolution 3 min.
Figure 2 shows vertical profiles of the inner scale ℓ and the outer scale L of turbulence evaluated by using (15)and (14) with the parameters shown in Figure 1. The inner scale increases exponentially with height because of the increase of the molecular viscosity. The outer scale is of the same order of magnitude in the height range of 60–100 km only with a slight change in accordance with the variation of the Doppler spectral width. When compared with the previous estimates of Hocking and of Fukao et al. , the inner scale of ours exhibits smaller values in the altitude range of 60–90 km. Though the difference between our estimates and the previous ones is only a factor of 2–4 around 70 km, it produces more than a 10 km difference in the critical altitude for half the radar wavelength to be just within the inertial subrange, i.e., λ/2 ≈ ℓ. In our case, the critical altitude extends up to 75 km, to which the magnitude of radar reflectivity is very sensitive, as suggested in (22).
 Let us consider (22)in order to see the dominant terms contributing to the variations in the height profile of echo intensity. The left-hand-side diagram of Figure 3 shows three profiles of individual terms in (22), i.e., (λ/2/ℓ)16/3 (term a), L4/3(term b), and ∂/∂z (term c). The dynamic ranges of terms a and c are much greater than that of term b. The large variations in terms a and c strongly affect radar reflectivity profiles, whereas the outer scale, which is the function of the Doppler spectral width σw, slightly affects them. The resultant radar reflectivity using (22) is indicated by the curve in the right-hand diagram of Figure 3. Our radar reflectivity increases with altitudes up to 72 km, where λ/2 ≈ ℓ. The profile of Doppler spectral width σwin Figure 1 also shows a peak around 72 km, but note that the contribution of the change of the spectral widths to the reflectivity is quite small. We also see that the viscosity to determine ℓ plays the major role in controlling the reflectivity at the altitudes above 75 km, where the reflectivity is decreasing in spite of the increase in electron density gradient.
 In the time-height diagram of Figure 4we show a typical example of the relations between the turbulence inner scale and the radar reflectivity. The thick white solid curves indicate the critical altitude, i.e., λ/2 ≈ ℓ. The height range below these curves is for half the radar wavelength to be within the inertial subrange of turbulence. The strong mesospheric echoes are generally observed below these boundaries. However, below 65 km radar echoes are scarce, which is because electron density is less abundant at lower altitudes than above. In the height range above the thick white curves, half the radar wavelength corresponds to the viscous subrange of turbulence. In this region, mesospheric echoes are observed only scarcely, but they are strong if they occur. In Figure 4, strong echoes around 80 km endure only ∼1 hour after 1500 LT. These echoes may occur because of temporal enhancements of turbulence by breaking of gravity waves or because of induced enhancement of electron density gradient by upward propagating gravity waves whose amplitudes grow with height. In usual cases, mesospheric echoes have peak intensities just below the critical altitude where the inner scale of turbulence equals half the radar wavelength.
4. Radar Reflectivity From Observed Echo Powers
 To compare the results of our model with observations, we derive radar volume reflectivity from signal-to-noise ratio S/N of the MU radar. The MU radar does not have a self-calibration system for signal intensities. We apply the method of intensity calibration with the help of galactic emissions developed by Kubo et al.. In our case in the mesosphere the radar reflectivity per unit volume is given by [VanZandt et al., 1978]
where the parameters used in (23)are described in Table 1. The noise power N is defined by
where NGis the galactic noise level and Nrx is the internal thermal noise of the radar system. From the large amounts of data that have been accumulated in the MU radar, operated with the variability of gains in the receiver system, we have found the radar system noise Nrxto be constant for more than 10 years and much smaller than galactic noise, i.e., NG ≫ Nrx as shown in Table 1. So we obtain
where 〈N〉 is an annual mean noise level and 〈TG〉 is the annual mean temperature of galactic emission from above the MU radar. As a result we obtain from (25)
5. Comparison of the Model Results With Observations
 In this section we compare the model results with observed profiles of radar reflectivity, especially of diurnal and seasonal variations, in order to see characteristics of echoes in the mesosphere.
Figure 5a shows an example of the radar volume reflectivity obtained from echoes observed with the MU radar in 17 October 1989, and Figure 5b shows that calculated from our model. In Figure 5, white curves show the most frequently occurring profiles of reflectivity. Below 70 km the reflectivity increases with height, which is well reproduced in our model. The altitudes of the maximum occurrence of echoes in the two profiles coincide with each other as well as the profiles themselves. Above 70 km both observed and calculated reflectivities decrease, but some disagreements are noticeable in this region: The calculated reflectivity is enhanced around 87 km owing to a large gradient of mean electron density; around 83 km and above 90 km the model underestimates the observed reflectivity, which also occurs at other local times and in other seasons. In Figure 5 we apply logarithmic means of reflectivity because they are more similar to the magnitudes of reflectivity with most frequent occurrence than linear means.
Figure 6 shows examples of diurnal-height variations of observed and calculated radar reflectivities in July and October. Most of the observed mesospheric echoes appear during 800–1600 LT in the height range of 70–80 km, with variable magnitudes in months. The calculated reflectivity reproduces the general features of diurnal occurrence of observed mesospheric echoes, which stresses that echo variations are mainly due to the diurnal changes in electron density. Observations show some detailed diurnal variabilities in seasons. Particularly in July, strong echoes are observed in the height range of 70–85 km, and they descend in 3 hours, which our calculated model could not reproduce.
Figure 7 shows seasonal-height variations of measured and calculated radar reflectivity. Means of reflectivities at each height in each month are obtained from 1986 to 1997 in logarithmic average. The diagram of calculated values exhibits the following similarities to that of the measured ones: (1) Mesospheric echoes tend to occur at a height range of 70–80 km in every season, (2) mesospheric echoes have an annual oscillation with a maximum from May till August, and (3) mesospheric echoes have a second maximum in autumn just above 70 km.
 Consequently, the general agreement outlined above indicates that (22)can be useful, and we consider that the scenario of reflectivity by electron irregularities induced by the parcel movement in neutral turbulence is verified. However, some discrepancies still remain. The calculated radar reflectivity is not consistent with the observations around 90 km, especially in summer, when the model indicates a reflectivity enhancement. The model results at 70–80 km in the winter months show smaller values than those observed. One of the main reasons for these discrepancies must be the lack of knowledge of simultaneous profiles of electron densities and temperatures. In particular, local and instantaneous fluctuations of electron density and temperature induced by amplified gravity wave (especially with a period around 1 hour) cannot be included in the model we used. We also note that mesospheric echoes, except for some stratified layers, are very weak at most altitudes where the Doppler width is difficult to define. In our observations, noise levels in reflectivity are around 10−18 m−1.
 In this paper we have proposed a quantitative model for interpreting VHF echo powers from the mesosphere caused by irregularities in electron distributions which are induced by neutral turbulence. For applying our model we used Doppler spectral widths of the radar echoes that give us information about the intensity of turbulence. Mean profiles of electron densities and temperatures were given by empirical models. First we estimated the boundary between the inertial and viscous subranges of isotropic turbulence by using Doppler spectral widths and temperatures as shown in Figure 2. We have evaluated squared fluctuations in radio refractive index not only in the inertial subrange but also in the viscous subrange under the assumption that electron densities are perturbed in a parcel forced to move by neutral turbulence. As shown in Figure 3, half the radar wavelength lies in the inner scale up to the altitude of 70–75 km, where the vertical profile of radar reflectivity is influenced mainly by the profile of electron density gradient. In the altitudes above 75 km the estimated radar reflectivity decays drastically because above 90 km the inner scale of turbulence is so large that the radar wavelength lies far from the inertial subrange and turbulence with the scale of λ/2 does not contribute to the scattering of radio wave.
 Before the first observations of the mesospheric echoes with the Jicamarca radar [Woodman and Guillén, 1974], VHF radar sounding of the mesosphere was believed to be unsuccessful because the critical altitude for half the radar wavelength to lie within the inertial subrange of turbulence is 60 km at the highest. With the help of successive MU radar observations, according to our new considerations we found that the critical altitude is 70–75 km. The critical altitude is mainly controlled by the molecular viscosity or neutral number density. Then the generation of mesospheric echoes in the height range 70–80 km in all months is well reproduced by our model. As shown in Figure 4, the observed echo layers close to the critical altitude continue more than 3 hours, but the echoes above 80 km are short-lived (less than 1 hour). Above 80 km it may be possible for echoes to occur because of sharp gradients of electron density and/or strong turbulence; the latter may lift the critical altitude up to 90 km. On the other hand, as strong turbulence is forbidden in the thermosphere, the highest altitude of the observed mesospheric echo may imply the altitude of the mesopause. According to the lidar measurements given by Senft et al. , the summer mesopause locates around the height 86 km, which has also been found as the highest altitude for the summer echoes in the MU radar [Kubo et al., 1997].
 It should be noted that observed values of spectral width σ in the vertical beam include components which are not produced by turbulence. Among the components a so-called “beam-broadening effect” is the first-order contribution, in which background horizontal winds, even if they are uniform, expand Doppler spectral width with σbeamdue to a finite width of the radar beam. Then the net spectral width produced by turbulence σturb is written as σturb2= σ2 − σbeam2. However, σbeam is smaller than 1 m/s with the MU radar in the case of such a strong wind as 100 m/s. The beam-broadening effect does not affect the results of our analysis as the mesospheric echoes in the MU radar are only identified with σ larger than ∼1 m/s because echoes are too weak in other cases.
 We consider that we are successful in reproducing general characteristics of the mesospheric echoes with our model, but we must be cautious in interpreting our results because of the uncertainties that lie in the turbulence theory, such as L, ℓ, and the magnitude of the displacement of an air parcel. Further development of the investigations is necessary in the quantitative evaluation of absolute reflectivity.
 Finally, we estimate wavelength dependence of volume reflectivity in our model. In the right-hand panel of Figure 8, for three radar wavelengths (46.5, 25, and 5 MHz) we show efficiency factors contributing reflectivity due to the competition between the radar wavelength and the inner scale according to (22). In the lower altitudes the efficiency factors equal unity because half the radar wavelength lies in the inertial range of turbulence. The right-hand diagram in Figure 8 shows the model results for echo intensities. Mesospheric echoes are strong around 70 km for 46.5 MHz. For 25 MHz the profile exhibits two peaks at 70 and 90 km. For 5 MHz a strongly dominant peak is indicated at 90 km. These results may explain the different heights of maximum mesospheric echo powers observed with radars operating at VHF and HF. Figure 8 suggests that mesospheric sounding with the lower frequency will obtain more information at the upper mesosphere.
 We have evaluated quantitatively radar volume reflectivity in the mesosphere according to a scheme of electron irregularities induced by neutral turbulence. The results show that the maximum scattered power occurs around 70–75 km by the competition between the increase of the electron density and damping of turbulent intensity at the Bragg wavelength owing to the increase of molecular viscosity with height. Comparisons of the estimated radar reflectivity from our model with observations generally show similar height variations, especially below 75 km. We failed to reproduce the infrequent and short-lived but strong radar reflectivity observed at 80–95 km, which is probably because of the lack of knowledge about simultaneous and detailed profiles of electron densities and temperatures. Though the absolute values of radar reflectivity shown in Figure 5 are roughly of the same order of magnitude, further research is required to reduce ambiguity, especially using the galactic noise level calibration.
 The authors thank Y. Muraoka for helpful discussions. The MU radar belongs to and is operated by the Radio Science Center for Space and Atmosphere, Kyoto University. The first author was supported by a grant of the Japan Society for the Promotion of Science (JSPS) under the Research Fellowships for Young Scientists.