Long-duration meteor trains have fascinated observers for many years. The great Leonid meteor storms of 1866–1888 were the first to spark organized scientific study on the subject, but despite years of study, more than a century later, persistent trains remain for the most part a mystery. Over the last few years, however, the heightened Leonid activity has fueled considerable research efforts, much of it dealing with persistent trains. Some of the results of a comprehensive study of persistent trains conducted at the Starfire Optical Range (SOR) on Kirtland Air Force Base, New Mexico, during the 1998 and 1999 Leonid showers are reported here. For the first time the time evolution of persistent trains is used to determine the eddy diffusion coefficient at mesospheric heights. In three of the four trains studied, portions of the train exhibited molecular diffusion while the remainder of the train, as well as the entire fourth train, exhibited eddy diffusion. The eddy diffusion coefficients were several hundred m2 s−1, two orders of magnitude higher than the molecular rates. The sodium density in the train was sufficient to use it as a passive scalar tracer of turbulent fluctuations. The spectra are well modeled by the Heisenberg turbulence model, and the values found for the energy dissipation rate are in agreement with the eddy diffusion coefficient estimates. The gradient Richardson number and Brunt-Väisällä frequency were determined from lidar measurements and indicated regions of convective and dynamic instability.
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 In addition to their inherent mystery and beauty, persistent meteor trains have historically proven to be useful tools for the study of the mesosphere. The motion and distortion of such visible markers gave some of the first evidence for winds and waves in the upper atmosphere and, indeed, that the atmosphere even extended to such heights. Wind measurements based on persistent train observations were published by Olivier , and similar, more recent measurements by Drummond et al. . In this paper, a study of mesospheric turbulence properties using persistent train lidar data and video footage is described. A primary result of this turbulence study points to the potential for the use of large aperture resonance lidars such as that used at SOR as a tool for studying turbulence using the Na layer as a passive tracer.
 The data presented here were obtained during two campaigns in which a sodium resonance lidar was fielded at the Starfire Optical Range (SOR), a facility of the Air Force Research Laboratory, Directed Energy Directorate, on Kirtland Air Force Base near Albuquerque, New Mexico. Other results have been published in a series of papers [Chu et al., 2000; Grime et al., 2000; Kelley et al., 2000; Drummond et al., 2001; Kruschwitz et al., 2001], and only a brief description of the system will be provided here. The 3.5 m diameter, fully steerable telescope was used to collect the scattered light from sodium atoms left over in the wake of the meteors. Lingering trails were first identified by eye, and within one or two minutes, captured in the 5° field of view of a bore-sighted camera used by the telescope operator. The trails could then be tracked and illuminated by the laser. In the remarkable event studied here, the sodium density was measured over a 4 km height range with 24 m resolution and exhibited a sodium density 20 times the background sodium density at the same height.
Figure 1 shows four of the trails observed during the campaigns. The large-scale distortion of the originally straight injection of ablated material is due to the presence of internal waves in the region. The event shown in the top left panel has been studied extensively [Drummond et al., 2001]. The wave was shown to be an inertio-gravity wave with a 20-hour period. A rotation of the wind vector with height created the nearly circular pattern seen in the image, which gives rise to the event being dubbed the Diamond Ring. A considerable shear in the wind is generated by the wave. The ‘diamond’ is caused by a considerable broadening of the trail diameter above about 97 km. Below that height the trail is much narrower and has the curious double trail signature commonly observed in persistent trails [Trowbridge, 1907; Olivier, 1957; Drummond et al., 2001]. Such dissimilar behavior as a function of height is commonly observed. It is argued below that this transition marks a boundary between zones of molecular and turbulent diffusion.
 The observations discussed here were made near the turbopause, the statistical boundary between the fully mixed lower atmosphere and the upper region in which atoms and molecules are separated according to their mass. In this context it is of interest to note that the apparently turbulent portion of the Diamond Ring is above the region in which the trail appears quite smooth and limited in horizontal extent. This shows that the turbopause is ill-defined in a localized measurement.
 Man-made trails, using sodium, lithium, and trimethyl aluminum (TMA) released from rockets, have been used for decades to measure upper atmospheric winds. They also have been used to study turbulence in a manner similar to the methods used in the current study [Zimmerman and Champion, 1963; Justus, 1966; Zimmerman and Trowbridge, 1973]. Our results are, in fact, quite similar to those from man-made trails and also those which employed expanding clouds of chaff. We are able to add two new elements, however, which make these results unique, both related to the lidar observations. Just before and after the lingering trail observations in 1999 the lidar provided vertical profiles of the wind and temperature in the ambient atmosphere. This allows us to deduce both the Brunt-Väisällä frequency and the Richardson number, key parameters in assessing the stability of the atmosphere. Gardner et al.  have published an extensive set of measurements of these parameters using the SOR measurements. Second, we have been able to treat the sodium density measured over the 4 km height range as a passive scalar mixed by turbulence to estimate the energy dissipation rate, ε. This can then be used to estimate the eddy diffusion coefficient and compare it with the observed spreading of the train.
 We comment now on the possible role of initial conditions on the sodium atoms. Their collision time with the ambient gas is less than a millisecond. At the time of the observations reported here, many tens of thousands of collisions have occurred and there is little doubt that the sodium atoms have taken up the ambient flow. But could the sodium somehow have retained a memory of the initial event which is, of course, quite violent? The non-turbulent portions of the trail, which are clearly visible in the figures below, are well matched to Gaussian functions [Drummond et al., 2001; Kruschwitz et al., 2001], look smooth to the eye, and, when fit to a classical diffusion model, yield classical rates [Grime et al., 2000]. These results all argue against memory of the initial structure. The portion of the trails we regard as being driven by neutral turbulence are much more extended perpendicular to the original and contain structure of a size that could not be a memory of structure in a trail which initially was, at most, a few meters across. Along the trail we turn to meteor radar results. Within seconds, meteor echoes return from lengths of trail which are smooth for a distance of at least a Fresnel length, some 200 m. This well-established result argues for a rapid destruction of any initial structure.
2. Molecular and Turbulent Diffusion Estimates
Grime et al.  used lidar observations of lingering trails in the 1998 Leonid shower to estimate a diffusion coefficient from the expansion rate of the sodium trails. Their estimates ranged from 5–10 m2 s−1, in good agreement with values of the molecular diffusivity. The trail dimensions were small and therefore similar to the size of the individual components of the double trails in Figure 1. Kruschwitz et al.  used a chemistry/diffusion model to study the light emission from lingering trails and also found reasonable agreement with the size of the individual trails in the lower portion of the Diamond Ring as deduced from photographs. Drummond (personal communication, 2001) investigated the puffy portions of that same trail and estimated that diffusion coefficients nearly two orders of magnitude larger were needed to explain their large dimensions. Taken together, these studies encouraged a more in-depth study of the train evolution reported here.
 Both Grime et al.  and Drummond assumed an expansion of the trail radius determined by the classic t1/2 dependence predicted for classical molecular diffusion. This approach is certainly valid for the former case and reasonable as well for the latter study since only an estimate was needed due to the very large value found; clearly this region of the trail exhibited eddy diffusion. Here we study the development of one of the trails in more detail using methods developed theoretically by Batchelor  and applied to the study of man-made trails by previous authors [e.g., Zimmerman and Champion, 1963; Justus, 1966; Zimmerman and Trowbridge, 1973].
 A Xybion image-intensified video camera fitted with a 5° lens was also fielded during the 1999 Leonids. An observer steered the camera to any persistent trains that were observed, usually beginning to record footage within about 30 s of the associated fireball. In this way the camera captured excellent footage of the two trains observed during the 1999 Leonid observations, the Puff Daddy and the French Curve. In what follows, the Puff Daddy video footage is analyzed to study the train's turbulent motions. This analysis is similar to work using cesium and trimethyl aluminum releases to measure winds and turbulent diffusion [Zimmerman and Champion, 1963; Justus, 1966; Zimmerman and Trowbridge, 1973]. Figure 2 shows an image of a chemical release and a plot of the mean square radius versus time for the cloud studied by Zimmerman and Trowbridge . It is important to note that r2 is proportional to t3 and not t1 as predicted for classical diffusion. This is exactly the prediction made by Batchelor  when the size of a cloud has expanded into the inertial subrange of a turbulent background medium.
 The video frames used here were first digitized with a resolution of 704 × 480 pixels. Because the individual frames are quite noisy, the averages of 10 frames, corresponding to 1/3 s, were used for the analysis. Changes occurring in such a short time period should not have a significant effect on the results. Figure 3 shows a series of the images analyzed, each one spaced by 12 s. The Na laser is clearly visible in several of the images. The locations labeled A and B have been analyzed to study the spreading of the train as a function of time. This is done by fitting a Gaussian curve to the background-subtracted train luminosity profile, and then measuring the width of the Gaussian.
 The results are shown in Figure 4, which shows the square of the width of the train versus time since ablation on logarithmic axes. Gaps in the data occur when either the camera was moved or the Na laser obstructed the region being analyzed. Trend lines are also plotted and the slopes labeled. For both locations, at early times the spreading exhibits close to a t.5 dependence, which is anomalously slow (t1 would be expected for simple diffusion). At later times, though, both exhibit a t3 dependence, consistent with turbulent transport on length scales within the inertial subrange. Location A appears to have a transitional section obeying a t1.5 dependence, while in location B the transition to t3 is very abrupt.
 Following Batchelor  we note that when the cloud is small, with the range of sizes encompassed by the diffusive subrange, its motion is determined by molecular diffusion. In other words, the square of the cloud's radius increases linearly with time, i.e.,
where D is the molecular diffusivity. Eventually, the cloud expands to a sufficiently large size for turbulent motions to start to spread it. The turbulent eddies that begin to affect the spreading are assumed to be within the size range of the inertial subrange. It seems reasonable then to assume that the motion is dependent only upon ε, the turbulent energy dissipation rate (measured in units of energy mass−1 s−1), in addition, of course, to the time t, and the initial radius of the cloud r0. Using dimensional arguments, then, the rate of change of the radius of the cloud can thus be expressed as:
where G is a universal function of its dimensionless argument. At early times the initial distribution of particles, r0, is important because it determines the range of eddy sizes which can affect the spreading. Further analysis, which gives G a functional form, eventually yields [Batchelor, 1950]:
When enough time has passed that the size of the cloud is much greater than r0, the initial size is no longer relevant to the motion. Thus the function G in (2) reduces to a constant:
where c is of order unity. At still later times, when the cloud size is beyond the upper limit of the inertial subrange, the increase in the size of the cloud is expected to follow a law typical of a diffusive process:
By looking at the cubic regions, an independent estimate of ε can be obtained from the relation given above in 4, where, as mentioned previously, c is a constant of order unity, taken to be 0.6 here following Lübken . Fitting a cubic curve, then, to the late section of the data therefore yields ε. For A and B the calculated values of ε were found to be about 0.17 W kg−1 and 0.15 W kg−1. These values are quite reasonable and in agreement with values published by Lübken .
3. Turbulent Mixing of a Passive Scalar
 One effective method for studying turbulence is to evaluate its effects on a conservative and passive tracer. Turbulent velocity fluctuations in turn produce fluctuations of the tracer. The tracer is conservative if the only time dependence is due to the turbulent fluctuations. It is considered passive if it does not itself influence the turbulent flow. Relative neutral density fluctuations are an example of a passive scalar; temperature fluctuations are another example. In the following section Na density fluctuations will be treated as a conservative passive scalar and used to analyze turbulence affecting one of the observed meteor trains.
 The theory of turbulent mixing of a passive scalar has been treated rather extensively by past authors [e.g., Heisenberg, 1948]. The method of studying turbulence through passive scalars typically involves investigating the power spectrum of the turbulence-induced fluctuations. The work presented in this paper concerns turbulence on scales within the inertial subrange and the viscous subrange. In the inertial subrange of turbulence the range of eddy sizes is too large for molecular dissipation to be of importance and too small for buoyant forces to have an influence. In this range, energy cascades from large eddies to smaller ones with essentially no losses. This cascade continues until the energy is passed to eddies of a scale small enough for viscosity to be important, i.e., into the viscous subrange. Kolmogorov determined that in the inertial subrange, therefore, the energy in eddies of a particular length scale depends only on the size of the eddy and on the rate of energy dissipation ε. This is typically expressed in Fourier space as
where k denotes the wave number. In the viscous subrange, a steeper spectral dependence is found:
Figure 5 shows a lidar profile from the Puff Daddy meteor train showing a significant enhancement in the Na levels over an approximately 4 km expanse in altitude; the lower panel shows a close-up of the meteor's Na enhancement. Clearly, the Na trail has a much different look than that which is usually seen by the lidar, typically just a single Gaussian-shaped spike. The profile, in fact, bears a striking resemblance to the neutral density turbulence fluctuations shown by Lübken [1992, 1993, 1997]. In those publications, Lübken gleaned much useful information on mesospheric turbulence from the neutral density turbulent fluctuations. An analysis similar to Lübken's is made here for the Na fluctuations measured by the Na lidar in the Puff Daddy meteor train.
 The section of the Na profile between 96 and 100 km is first broken up into three separate sections. The profiles were detrended by fitting a spline curve over 300 m intervals of the three sections. Figure 6 shows the individual sections and the spline fit adopted for each. Taking these fits to constitute a reasonable background distribution, the relative density fluctuations were obtained by subtracting the fits from the data and then dividing by the fit.
 A standard FFT routine was then used to obtain the spectrum of the residuals. The spectra are shown in Figure 7; the fourth spectrum is the average of the other three. The spectra exhibit characteristics typical of the spectrum of a turbulence-mixed passive scalar; there is an inertial subrange, turning into a viscous subrange at shorter wavelengths. Following Lübken , a theoretical turbulent spectrum was fit to the calculated spectra. The theoretical model used is that of Heisenberg , which predicts the following power spectrum:
The spectrum includes the well-known k−5/3 dependence predicted by Kolmogorov theory in the inertial subrange (k < k0), as well as an extrapolation into the viscous-diffusive subrange (k > k0), exhibiting a k−7 dependence. The wave number at which the transition between the two subranges occurs is denoted by k0 in equation (8). and a are constants, and ε is the turbulent energy dissipation rate, a quantity of some interest. The only concern here is with determining k0, which can be obtained simply from the fitted spectrum. The fitted spectra are also shown in Figure 7. The values for k0 from the three spectra are 4.5 ± 0.6 × 10−3 m−1, 4.8 ± 0.6 × 10−3 m−1, 6.4 ± 1.4 × 10−3 m−1, and k0 = 5.4 ± 0.4 × 10−3 m−1 for the averaged spectrum.
 The parameter k0 is related to the Kolmogorov microscale η through the following relationship
where the factor of 9.9 is calculated from theory [Lübken, 1993]. The Kolmogorov microscale is a rough measure of the size of the smallest eddies, where the turbulent energy is dissipated by viscosity. It is related to ε as follows:
where ν is the kinematic viscosity. Thus, since k0 is known, ε can be calculated,
assuming, of course, that ν is known or can be calculated. Since no measurements of the kinematic viscosity were made on that night and in the region of the Puff Daddy train, an estimate must be used. Using
where ρ is the atmospheric mass density, here taken to be 4 × 10−7 g cm−3, and μ is the dynamic viscosity, given by
with β = 1.458 × 10−6 kg (s × m × K1/2), S = 110.4 K, and T the temperature of 195 K, which was obtained from the lidar measurements, we find ν = 33 m2 s−1. The values of ε calculated for the three sections of data from equation (11) were between 0.2 and 0.5 W kg−1; the ε from the averaged profile was 0.3 ± 0.1 W kg−1. These values are somewhat higher than estimated above, but not unreasonably so.
 It is unusual and somewhat unexpected that the Na lidar appears to have been able to resolve turbulent fluctuations. The signal-to-noise ratio achievable by the lidar is on the order of 100, and slightly higher in the regions of the meteor trail Na enhancement. Typical relative neutral density fluctuations, however, are on the order of a few percent, so it would not seem the lidar has sufficient sensitivity to detect them. Indeed, attempts to detect turbulent fluctuations in the Na density outside of the region of the meteor's Na trail were unsuccessful. Within the trail, though, the magnitude of the fluctuations appears to be on the order of 50 percent of the background (or more precisely, the assumed background), much larger than usual neutral density fluctuations. It seems, then, that there is something particular about the Na enhancement in the wake of the meteor trail that makes it a particularly strong tracer of turbulent fluctuations.
 This is most likely due to the fact that the Na enhancement changes quite rapidly with distance. The magnitude of the Na density fluctuations, δN, are related to the neutral density fluctuations, δn, by
where n and N represent the unperturbed neutral and Na density, respectively. F is defined as
where HNa is the scale height of the Na density, Hp is the pressure scale height, Hn is the neutral density scale height, and γ = cp/cv = 1.45 [Lübken, 1997]. If Hn and Hp are taken to be approximately 6 km, then for the factor F to be at least 0.1, which would explain the size of the relative density fluctuations seen in the Na profiles, HNa would need to be at most 1 km. This is certainly consistent with the lidar profile shown in Figure 5.
 Both the trail expansion rates and the sodium density fluctuations indicate that the Puff Daddy lingering trail was at least partially embedded in a region of turbulence and eddy diffusion. We now investigate the stability of the atmosphere as deduced from the lidar observations. The lidar-based measurements of the mesospheric temperatures and winds allow the stability of the atmosphere to be studied. In Figures 8a and 8b, the temperature and wind measurements for the period of time just before the Puff Daddy train occurred are shown. The data here have been smoothed over a 500 m interval. Knowledge of the temperature structure as a function of altitude can be used to calculate the Brunt-Väisällä frequency, given by the following formula
Figure 8c shows ωB2 as a function of altitude calculated from the data in a. There is a region a few kilometers in thickness where the frequency is negative. This is indicative of convective instability. The wind data obtained from the lidar allows the calculation of the gradient Richardson number, Ri. This is given by
where ū is the mean horizontal wind. Ri values of less than 1/4 indicate regions of dynamic instability necessary for the creation of turbulence. Figure 8d shows Ri as a function of altitude based on the wind data and the calculated ωB2 shown in b and c. Clearly, there are regions in the height range of the meteor trains which are dynamically unstable.
 Based on the foregoing calculations we can estimate the turbulent diffusion coefficient, K. It is related to ε and the Brunt-Väisällä frequency by
where C is a constant of order unity generally taken to be between 0.2 and 1 [Hocking, 1987]. This gives K ∼ 500 m2 s −1.
 Larsen has performed an extensive study of rocket TMA trails and concluded that both the winds and the wind shears in the mesosphere and lower thermosphere are quite high. Gardner et al.  have also found regions of convective and dynamic instability in this height range. It seems that the extended and puffy nature of lingering trails are visual manifestations of these properties.
 The Leonid campaigns offered a rather unexpected opportunity to study mesospheric turbulence in addition to the stated goal of studying the causes and the nature of persistent trains. The large aperture associated with the University of Illinois Na lidar provided sufficient sensitivity to study turbulent fluctuations in the Na trail left by a bright Leonid fireball. This highlights the hitherto unexplored potential of similar large aperture resonance lidars to study mesospheric turbulence using only the ambient, naturally occurring metal layers. Additionally, the long-lived luminosity of the persistent trains allowed them to be used to study turbulent transport. Similar measurements have been done using chemical releases [Zimmerman and Champion, 1963; Justus, 1966; Zimmerman and Trowbridge, 1973], but this appears to be the first time persistent trains have been used for this purpose.
 Research at Cornell was supported by the Atmospheric Science Section of the National Science Foundation under CEDAR grant ATM-9714736 and Aeronomy grant ATM-0000196. Work by the University of Illinois lidar group was supported by the National Science Foundation. T. J. K. was supported in part by NSF CEDAR grant ATM-0086385. We extend our appreciation to the Starfire Optical Range staff for their arduous support during these observations and especially thank Bob Fugate, Bill Lowrey, and Ray Ruane for facilitating our research at the SOR.