4.1. Comparison With Theory: Turbulence With Enhanced Schmidt Number
 We now turn to a detailed comparison of volume reflectivities observed at the two radar frequencies (and Bragg wavelengths). In an initial attempt we have tested whether there is any direct and obvious correlation between η at the two Bragg wavelengths of 2.8 and 30 cm. For this direct comparison, the measurements were put on an identical time grid of 1 min. Subsequently, times and altitudes were identified during which PMSE occurred simultaneously in both radar observations. A scatterplot of corresponding reflectivities is presented in Figure 9. At first glance, there is no obvious correlation between the data.
Figure 9. Scatterplot of the reflectivities of ESR and SSR for all simultaneous observations during June 2006. The red solid line indicates a reflectivity ratio of 816, i.e., all points where η(SSR) = (kESR/kSSR)3 · η(ESR).
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 The question, however, is what we may expect to find based on the above-mentioned PMSE theory according to which PMSE is caused by a combination of neutral air turbulence and an enhanced Schmidt number (reduced diffusivity) of electron density fluctuations (= fluctuations of the radio refractive index). For looking into this, we follow the formulation of Rapp et al.  which is based on the classical formulation for pure turbulent scatter by Hocking  and using the results of Batchelor  for the case of tracers with Schmidt numbers larger than 1. According to this work, the theoretical expression for the radar volume reflectivity for the case of the turbulent scatter aided by an enhanced Schmidt number is given by
where ε is the turbulent energy dissipation rate, ν is the kinematic viscosity, N is the buoyancy frequency, k = 4π/λ is the Bragg wave number of the radar, is the Kolmogorow microscale, re is the classical electron radius, and Sc = ν/De is the Schmidt number introduced in section 1. is the reduced potential refractive index gradient, i.e., which depends on the electron number density ne, the buoyancy frequency, and the density scale height Hn. Finally, fα, q, Ri and Prt are all “constants” (i.e., they are constant for a given event) derived from either theory or by comparison with observations (see Appendix A of Rapp et al.  for more details).
 In Figure 10 we have plotted various theoretical curves η(k) for different combinations of the electron density and electron density gradient and the Schmidt number Sc. In addition, the two vertical bars indicate the Bragg wave numbers of the SSR (k = 2.24 m−1) and the ESR (k = 20.93 m−1). Figure 10 shows two interesting things: if the Schmidt number is large, then the ratio between η(SSR) and η(ESR) should be a constant value; that is, the curves of log(η(k)) are parallel to each other. Going back to equation (3) we see that in this case η(k) k−3; that is, for large Sc, both η values fall into the viscous-convective subrange such that the ratio η(SSR)/η(ESR) is given by (kESR/kSSR)3 = (500 MHz/53.5 MHz)3 = 816. If, however, the Schmidt number is not very large, then η(SSR) would fall into a part of η(k) where the latter is dominated by the k−3 dependence whereas η(ESR) would already fall into the part where η(k) is dominated by the exponential term. In consequence, the ratio η(SSR)/η(ESR) should be larger than the minimum value of (kESR/kSSR)3 = (500 MHz/53.5 MHz)3 = 816 in those cases (indicated by the red line in Figure 9).
Figure 10. Calculated volume reflectivities for turbulent backscatter for a turbulent energy dissipation rate of 0.1 W/kg and electron number density Ne and its gradient dNe/dz indicated at the top of the plot. The solid, dashed, and dash-dotted lines were calculated for Schmidt numbers Sc = 650, 2600, and 41,600, respectively (corresponding to radii rA = 10, 20, and 80 nm, respectively). The two vertical bars indicate the Bragg wavelengths of the SSR (left bar) and the ESR (right bar). The black horizontal line indicates the volume reflectivity due to incoherent scatter for an electron density of 1 × 1010 m−3 for comparison.
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 In order to investigate this further, we next show histograms of the ratio η(SSR)/η(ESR) in Figure 11. Figure 11 demonstrates that the ratios vary between ∼400 and 100,000. Importantly, however, more than 94% are larger than a value of 816, i.e., the ratio of the radar frequencies to the third power (see above). This means that the large majority of our data is consistent with the expectations based on the here considered theory. The remaining 6% which lie to the left of 816 can likely be explained by the uncertainty of the calibrations (see above) as well as by the fact that the radar volume of the ESR is significantly smaller (beam width 1.2°; see Table 1) than the radar volume of the SSR (beam width 5°; see Table 1). This means that an incomplete filling of the latter by scatterers will result in an underestimate of the “real” volume reflectivity. Importantly, we can find independent support for this explanation by means of a similar analysis of PMSE observations with the EISCAT VHF and UHF radars in Tromsø. From measurements conducted in the years 2004, 2005 and 2006 we could identify a total of 296 1 min samples of simultaneous PMSE observations with these two radars operating at 224 MHz and 930 MHz (Bragg wavelengths of 67 cm and 16 cm, respectively). Importantly, unlike in the case of the SSR and ESR, the beam widths and hence observing volumes of these two radars are near identical such that one would expect to find an even larger percentage in agreement with theory, provided that the latter is correct. Indeed, this analysis yielded the result that 99% of derived volume reflectivity ratios were larger than the corresponding ratio of the radar frequencies to the third power. A detailed description of the latter analysis of the EISCAT VHF and UHF measurements will be published in the near future (M. Rapp et al., manuscript in preparation, 2010). We note, however, that we can certainly not exclude that some other, yet unidentified physical process, is responsible for the above-described discrepancy for 6% of the presented data.
Figure 11. (top) Distribution of the ratio of the volume reflectivities observed with the SSR and ESR based on a total of ∼300 simultaneous and common volume observations with a integration of one minute each. (bottom) The same data shown as the cumulative percentage of the ratio, i.e., indicating which part of the observations showed a reflectivity ratio larger than the value indicated on the x axis.
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 We note that the here demonstrated consistency between theoretical expectations and observations allows us to disregard any theory which predicts a wave number dependence which is less steep than k−3. However, it certainly does not allow us to disregard other potential theories with functional dependencies steeper than k−3. In order to distinguish such different possible functional dependencies, observations at more than two frequencies would be required. Hence, our analysis can certainly not prove that the assumed turbulence theory is correct. Nevertheless, we consider the consistency demonstrated above as sufficient motivation to proceed further and derive microphysical parameters from our data which may then be compared to independent observations in order to further corroborate or falsify our assumptions.
4.2. Microphysical Properties Derived From Signal Ratios
 We now proceed and apply the turbulence with large Schmidt number theory to the observations in order to derive microphysical parameters from the radar measurements. Using equation (3) we see that the ratio η(SSR)/η(ESR) can be written as
Note that this equation only contains two remaining unknowns, i.e., the Schmidt number Sc and the Kolmogorow microscale ηK, while all other, partly largely uncertain, factors contributing to the expression for the volume reflectivity in equation (3) have canceled. The ηK can be easily calculated using the energy dissipation rates estimated from the ESR spectral widths (see section 3.2) and an estimate of the kinematic viscosity using Sutherland's formula and densities and temperatures from the MSIS climatology [Picone et al., 2002]. As a result, we may solve equation (4) for the Schmidt number Sc and obtain
Taking further into account that the Schmidt number can be expressed in terms of the properties of the charged ice particles, corresponding ice particle radii can be derived from the following relation:
where the radius rA is in nm [Cho et al., 1992; Lübken et al., 1998; Rapp and Lübken, 2003]. Note that equation (6) assumes that collisions between the ice particles with the neutral gas, and hence the corresponding diffusion, can be modeled as hard sphere collisions. The latter was shown to be a reasonable assumption for particle radii larger than ∼0.5 nm by Cho et al. .
 All simultaneous measurements with the SSR and ESR were analyzed for Sc and rA resulting finally in histograms of the Schmidt number and ice particle radii shown in Figure 12. Figure 12 shows that Schmidt numbers vary between a few hundred and ∼32,000 with corresponding radii between 10 and 70 nm. Interestingly, we note that the Schmidt number distribution appears to show two distinct peaks. We will show below, that these two peaks correspond to different particle populations below and above ∼85 km altitude.
Figure 12. (top) Schmidt numbers and (bottom) radii of the charged particles derived from the ratio of the volume reflectivities between the SSR and ESR PMSE which occurred simultaneously for the observations in June 2006.
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 Note that similarly large Schmidt numbers with maximum values of 4000 have recently been independently reported based on direct in situ observations of the spectral content of small-scale fluctuations in the neutral gas and in charged particles under PMSE conditions [Strelnikov et al., 2009]. Importantly, independent observations on the same sounding rocket proved the presence of large ice particles [Megner et al., 2009] and allowed determination of corresponding particle radii in the range of 20–30 nm [Rapp et al., 2009], i.e., in good agreement with equation (6).
 In order to further assess the accuracy of the derived ice particle radii, in Figure 13 (top) we have plotted contour lines of particle radii as a function of turbulent energy dissipation rates ε and ratios η(SSR)/η(ESR) according to equations (6) and (5). In the same plot we have marked combinations of these two quantities derived from our simultaneous SSR and ESR PMSE observations. This comparison confirms that our measurements correspond to particle radii between ∼10–70 nm. Furthermore, it also shows that this method to derive particle radii becomes more and more inaccurate as it approaches the theoretical lower limit of η(SSR)/η(ESR) corresponding to the largest particle radii. The reason for this behavior can be easily understood by going back to Figure 10: As the ratio η(SSR)/η(ESR) approaches its smallest possible theoretical value of 816, a further increase of the particle radius and hence the Schmidt number will hardly lead to any further change in η(SSR)/η(ESR) because for large Schmidt number it is asymptotically given by ratio of the Bragg wave numbers to the third power. As a next step, Figure 13 (bottom) shows relative errors of the radii assuming relative errors in ε and η(SSR)/η(ESR) of 10% and 50%, respectively. We note that the 10% error estimate for the accuracy of ε only reflects the uncertainty of deriving the spectral width from measured Doppler spectra and does not take into account possible systematic problems with Equation 7 as suggested on the basis of the direct numerical simulations presented by Gibson-Wilde et al. . These authors showed that for the case of a simulated Kelvin-Helmholtz instability ε derived on the basis of equation (7) and using a constant background buoyancy frequency resulted in an underestimate of the real energy dissipation rate by a factor of five which would lead to a corresponding systematic shift of derived particle radii to smaller values. Whether or not this factor does apply to our observations is difficult to judge since no information is available to us about the type of instability which led to the observed turbulence events. Furthermore, we note that the only so far available direct comparison of ε measurements based on the spectral width method with a well established in situ technique showed overall very good quantitative agreement [Engler et al., 2005, Figure 9] such that our choice to only consider random errors and ignore potential systematic errors appears to be justified.
Figure 13. (top) Contour lines of particle radii (in nanometers) as a function of turbulent energy dissipation rates ε and ratios η(SSR)/η(ESR) according to equations (6) and (5). The grey diamonds indicate combinations of these two quantities derived from the simultaneous SSR and ESR PMSE observations. (bottom) Relative error (in percent) of the derived particle radius as a function of ε and η(SSR)/η(ESR) assuming relative errors of 10% in ε and 50% in η(SSR)/η(ESR). Again, grey diamonds indicate combinations of ε and η(SSR)/η(ESR) derived from the simultaneous SSR and ESR PMSE observations.
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 Note further that also other potential systematic effects like an incomplete filling of the radar beam in the case of the SSR, or a systematic overestimate or underestimate of the SSR antenna gain cannot be excluded and would bias the results shown above to one direction. However, since we are so far incapable of quantifying these effects, they will not be considered any further.
 In any case, using the above-stated assumptions, Figure 13 (bottom) reveals that derived particle radii have typical uncertainties of 30% and less except for those cases where η(SSR)/η(ESR) is close to its theoretical lower limit as already mentioned above. We also note that smaller radii (<20 nm) have even much smaller uncertainties of less than 15%.
 In order to further judge whether these distributions are geophysically reasonable or not we have finally compared our results to independent observations of ice particle radii in mesospheric ice clouds. A survey of available measurements quickly shows that there is currently only one instrument which has sufficient sensitivity to detect ice particles with radii down to as small values as 10 nm. This is the Solar Occultation For Ice Experiment (SOFIE) on board the Aeronomy of Ice in the Mesosphere (AIM) spacecraft [Gordley et al., 2009]. SOFIE measures vertical profiles of limb path atmospheric transmission within 16 spectral bands between 0.29 and 5.32 μm wavelength. SOFIE observes about 15 sunsets in the Southern Hemisphere and 15 sunrises in the Northern Hemisphere each day. Measurement latitude coverage ranges from about 65° to 80° north or south. SOFIE measurements are used to retrieve extinctions of mesospheric ice clouds at eleven wavelengths from 0.330 to 5.01 μm. In addition to temperature and the abundance of gaseous species, mesospheric ice clouds are measured by monitoring the attenuation of solar radiation using broadband radiometers. The SOFIE field of view is about 1.5 km vertical and about 4.3 km horizontal. Detectors are sampled at 20 Hz which corresponds to ∼145 m vertical spacing. The sample volume length, as defined by the line-of-sight entrance and exit of a spherical shell with vertical thickness of the FOV, is ∼290 km. Details of the method to derive microphysical parameters including effective radii (which are independent of the assumption of a specific particle size distribution) from SOFIE observations have recently been presented by Hervig et al. . Here, we compare estimates of effective radii obtained during the entire northern summer season 2007 (data version V1.022) to our own results.
 Finally, we also compare the radar results to observations with the ALOMAR Rayleigh/Mie/Raman (RMR) lidar located at 69°N from June 2006. The ALOMAR RMR lidar measures relative density profiles and particle (aerosol) properties in the stratosphere and mesosphere and has been described in detail by von Zahn et al. . A recent review of corresponding results including a detailed description of the method to derive ice particle parameters from observations at three wavelengths is presented by Baumgarten et al. . For the current purpose, suitable three-wavelength lidar data were integrated for 14 min and then analyzed for particle number densities, mean radii and widths of a Gaussian particle size distribution. For comparison to SOFIE and radar data, these parameters were then converted to effective radii as described by Hervig et al. .
 Before going into the details, a few caveats should be mentioned: first we note that we compare measurements from different years and also from different latitudes: while our radar measurements were done in June 2006, the AIM satellite was only launched on 25 April 2007 so that the 2007 summer season is the first available data set. Concerning latitudinal coverage, we note that the bulk of the SOFIE observations is from 68°N and the lidar observations are from 69°N and not from 78°N as in the case of our radar measurements. Furthermore, SOFIE observations were taken between 2200 and 2300 LST and lidar data were gathered between 2300 and 0300 LST, whereas the radar observations were taken between 1000 and 1400 LST such that potential tidal differences in particle properties might occur. Based on the multiyear lidar statistics of Fiedler et al.  this effect is, however, expected to be small since both local time intervals correspond to comparable values in the observed semidiurnal variation of brightness values (see their Figure 3). Then one must realize that SOFIE observations are proportional to the cube of particle radius (i.e., absorption in the Rayleigh regime) whereas the Schmidt number depends quadratically on the particle radius because a hard sphere collision model is here assumed. Finally, one must also note that there are certainly many occasions where PMSE (and hence mesospheric ice particles) was only observed with the SSR and not with the ESR. Hence, our data set of combined SSR/ESR observations represents in itself a biased data set; that is, it is not representative of an average state of the atmosphere but rather of a situation which allows PMSE to occur at the rather large frequency of 500 MHz. As described in detail above, these favorable conditions depend on the turbulence activity, the background ionization, and the size of the ice particles involved.
 Having all these caveats in mind, we now compare radii derived from our radar observations to radii derived from the SOFIE occultations and ALOMAR RMR lidar observations at three wavelengths in Figure 14. For this comparison, we have divided the data again into a subset for altitudes from 81 to 85 km and from 85 to 88 km. Focusing first on the radar results, we see that radii above 85 km are on average significantly smaller (median value of ∼20 nm) than below 85 km (median value of ∼35 nm) which is in line with our current microphysical understanding of these ice clouds which assumes that the particles nucleate at the mesopause and then grow and sediment to lower altitudes [e.g., Rapp and Thomas, 2006]. Coming now to the comparison to the SOFIE data, we note that the overall agreement between radar and optical measurements is actually very good: Above 85 km, the agreement is actually remarkably good with both data sets peaking at around 15 nm and showing a tail down to values of about 40 nm. Below 85 km, both satellite, radar, and lidar data show a rather broad distribution with median values of 28 nm in the case of the SOFIE measurements, 30 nm in the case of the lidar data, and 35 nm in the case of the radar observations. Taking into account the very different statistics of the three data sets (see caption of Figure 14) as well as the known fact that ice particle radii tend to increase toward the pole [Karlsson and Rapp, 2006] this difference can likely be explained by different latitudes at which radar (78°N) and satellite observations (68°N) were taken. In summary, taking all the caveats mentioned above into account, we consider this an excellent agreement and very strong support that the radii retrieved from the radar observations are meaningful.
Figure 14. Distribution of ice particle radii derived from radar (dark blue) and optical observations from the SOFIE instrument on the AIM satellite (red) and from the ALOMAR RMR lidar (light blue) for altitudes (bottom) below and (top) above 85 km. Please note the very different statistics of the different data sets: below 85 km, histograms are based on 174 values for the radar, 88 for the lidar, and 6945 values for SOFIE; above 85 km, there are 155 radar values, no lidar values, and 834 values from the SOFIE instrument.
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 Finally, we present full altitude profiles of ice particle radii for one given event together with the mean of all observations in Figure 15. These results demonstrate that both in a given event as well as in the statistical average, the radii show an increase with decreasing altitude as suggested by independent observations [e.g., von Savigny et al., 2005; Hervig et al., 2009; Rapp et al., 2009] and our current microphysical understanding [e.g., Rapp and Thomas, 2006].
Figure 15. (left) Individual altitude profile of particle radii obtained from PMSE observations on 30 June between 0900 and 1000 UT. (right) Average altitude profile of particle radii from entire data set. In both plots, horizontal bars indicate the standard deviation of the radius values.
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