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Optical Turbulence Characteristics at the Special Astrophysical Observatory Site

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04 November 2024

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Abstract
In this paper, we used meteorological data to simulate vertical profiles of optical turbulence at the Special Astrophysical Observatory (Russia, 43°40′19 ′′ N 41°26′23′′E, 2100 m a.s.l.), site of the 6-m Big Telescope Alt-azimuthal. The vertical profiles of optical turbulence are calculated using ERA-5 reanalysis data. These profiles are corrected using DIMM measurements as well as estimations of atmospheric boundary layer heights. We may note that the method basically reconstructs the most important features of the shape of the measured profile under clear sky. Atmospheric turbulent layers were identified, and the strength of optical turbulence in these layers was estimated. The model hourly values of seeing corresponding to the obtained vertical profiles range from 0.85 to 2.10 arcsec, values of the isoplanatic angle vary in the range from 1 to 3.0 arcsec (at λ = 500 nm). These estimations are close to the measured median of seeing (1.21 arcsec).
Keywords: 
Subject: 
Physical Sciences  -   Astronomy and Astrophysics

1. Introduction

Ground-based astronomical telescopes operating in different ranges of the electromagnetic spectrum are significantly limited by the Earth’s atmosphere. For optical telescopes, the key atmospheric characteristics are the statistics of cloudiness and optical turbulence [1]. The efficiency of millimeter and submillimeter telescopes depends on the atmospheric transparency in the radio range, which is related to the statistics of the water vapor content in the atmospheric column [2,3,4]. Development of ground-based network of astronomical telescopes requires a detailed knowledge about the meteorological and optical characteristics within troposphere and lower stratosphere.
Turbulent fluctuations in air refractive index are the main reason of wavefront distortions and image scintillation within the plane of telescope aperture. For measurement and compensation the wavefront distortion caused by the turbulent mixing of air with different temperature in the atmosphere in real time adaptive optics systems are used. To obtain diffraction limited images and optimize adaptive optics systems, astronomers should know vertical profiles of optical turbulence and wind speed. These systems are sensitive to vertical distribution of atmospheric characteristics including optical turbulence strength. Among the atmospheric optical turbulence measuring instruments, we can note the Generalized DIMM (GDIMM) with a 3-hole aperture mask [5], the Hartmann DIMM (H-DIMM) with a Hartmann mask with a larger number of sub–apertures [6] and the Shack-Hartmann Image Motion Monitor (SHIMM), employing a Shack–Hartmann wavefront sensor (SHWFS) instead of an aperture mask with isolated sub-apertures [7], the Multi-Aperture Scintillation Sensor (MASS) and the Generalized-Scintillation Detection and Ranging (Generalized-SCIDAR) instrument [8,9,10]. Despite the high prospects for the development of the measuring base, data collection campaigns are typically space and time-limited.
Modeling atmospheric processes can provide qualitatively new information about the structure of small-scale turbulence and on influence of mesoscale and large-scale atmospheric processes on optical turbulence on different time and spatial scales. The key problem in simulation of optical turbulence is the physics of lower atmospheric turbulent layer [9,11,12]. Within this layer, turbulence is often non-uniform and anisotropic [13,14,15]. Vertical changes of the optical turbulence strength depend on the large-scale atmospheric disturbances, the impact of local and non-local turbulent eddies, mesoscale eddy structures formed within the atmospheric boundary layer, gravity waves, the state of atmospheric stratification (in the field of humidity and air temperature). Also, a key issue is the influence of the energy transfer and enstrophy in the energy spectrum of atmospheric turbulence over a wide range of spatial and temporal scales on optical turbulence intensity [16,17]. In this paper, the goal is to determine the characteristic features in the changes of the vertical profiles of optical turbulence above Special Astrophysical Observatory (SAO) (Russia, 43°40′19 ″ N 41°26′23″ E, 2100 m a.s.l.). The geographical position of SAO is shown in Figure 1.
Herein, the focus of our research is on the lower part of the atmosphere, the so-called atmospheric boundary layer. The most difficult thing in this layer is to correctly parameterize the optical turbulence. Over the years, a certain amount of atmospheric optical turbulence models has been developed [18,19,20,21,22,23].For simulation of optical turbulence ERA-5 reanalysis data and different approaches are often used [24,25]. In our opinion, the models focus more on taking into account the altitude dependencies of the turbulence strength, but do not take into account the changes in the altitude of the atmospheric boundary layer from day to night and vice versa. Generally, the application of existing methods and approaches for simulation of optical atmospheric turbulence is a non-trivial task. Formulas for calculation of optical turbulence characteristics often include difficult-to-determine atmospheric characteristics [26]. One of these characteristics is the outer scale of turbulence. Another approach is to parameterize optical turbulence through the kinetic energy of atmospheric motions. In our opinion, the significant drawback of these approaches is the fact that the vertical profiles of optical turbulence are not uniquely related to the kinetic energy of mean motion and the kinetic energy of turbulence.

2. Data

In this study, for simulation of optical turbulence we use the global European Centre for Medium-Range Weather Forecasts atmospheric (ERA-5) reanalysis data at different pressure levels [27]. The reanalysis contains hourly values of meteorological characteristics from 1940 to the present. Data are available for a regular lat-lon grid of 0.25 degrees for the reanalysis, vertical resolution is 37 pressure levels from surface to 1 hPa. In this study, in particular, optical turbulence characteristics at the SAO site were derived from hourly data on wind speed components and air temperature available at pressure levels ranging from its surface value to 7 hPa. For estimation of optical turbulence parameterization coefficients, we considered the period from 25 August 2024 to 30 August 2024. During this time, direct optical measurements of phase distortion characteristics were performed using differential image motion monitor (DIMM) at the SAO site (Figure 2). This mobile monitor is described in [28]. It is a modification of the classic recorder of image motion [29].

3. Research Results

3.1. Results of Optical Turbulence Measurements

Adaptation of optical turbulence models and calculation algorithms based on meteorological data obtained over a specific site is a critical step in estimating the seeing parameter. This parameter is independent of the telescope that is observing through the atmosphere. A similar characteristic to seeing is the full width at half maximum (FWHM) of long exposure stellar images. FWHM is a characteristic of the images obtained in the focal plane of an instrument mounted on a telescope observing through the turbulent atmosphere. Under conditions of moderate and strong turbulent fluctuations along line of sight seeing and FWHM should be close in value. We have used a standard and widely spread theory for calculating the seeing parameter from the measured angles of arrival variance. In particular, the seeing values ( β parameter) were estimated using formulas:
β = 0.98 λ r 0 ,
σ α 2 = K i λ 2 r 0 − 5 / 3 D − 1 / 3 ,
where λ is the light wavelength (500 nm), D is the telescope diameter, r 0 is the Fried parameter, K l is the proportionality coefficient.
Generally speaking, Differential Image Motion Monitor shown in figure is sensitive to phase and amplitude of a light wave propagating through the turbulent atmosphere. By obtaining information about phase and amplitude fluctuations, we are able to interpret them in terms of the optical turbulence characteristics corresponding to both the total atmosphere and the free atmosphere. Using a method similar to MASS (Multi-Aperture Scintillation Sensor)-DIMM technique [30], we measured the seeing in the free atmosphere (above 500 m) and, taking into account DIMM data, estimate the ground-layer seeing. The key relations for the separate assessment of the optical turbulence strength are:
s = C n g 2 ∫ 0 z 0 W ( D , z ) d z + C n f 2 ∫ z 0 Z m a x W ( D , z ) d z ∫ 0 Z m a x C n 2 ( z ) d z = C n g 2 z 0 + C n f 2 ( Z m a x − z 0 ) ,
where s is the scintillation index, C n g 2 is the structure constant of the air refractive index fluctuations within the ground layer, C n f 2 is the value of C n 2 within the free atmosphere, z 0 = 500 m, Z m a x =30000 m, W(D,z) is the weighting function. Analysis of these equations shows that the atmosphere is divided into two layers, for the center of which the turbulence strength is estimated. Taking into account the fact that the ratio of weighting functions between the ground layer and the free atmosphere is small, estimations of the strength of optical turbulence above 500 m can be obtained directly, based on the analysis of time series of the scintillation index.The details of the measurement theory are described in paper [31].
Figure 3 show night-time changes of seeing at the SAO site. The median night-time seeing is 1.42 arc sec at 27 August, 2024. The median value of seeing at 28 August, 2024 is smaller, it is equal to 1.12 arc sec, respectively. The seeing values in the free atmosphere change in a narrow range, from 0.25 to 0.61 arc sec. These estimations obtained for different nights form the basis for the refinement of the algorithm for calculation of optical turbulence characteristic using meteorological quantities.

3.2. Scheme for Parameterization of Optical Turbulence

Optical turbulence characteristics, including vertical profiles of its strength C n 2 , can be found using available meteorological information within a selected region or at a given site. This approach has been used to describe the characteristics of the atmosphere above many astronomical observatories and is essentially standard [32,33]. The essence of the method is based on the generation of small-scale optical turbulence in areas with high vertical wind speed gradients. In particular, the structure constant of the air refractive index fluctuations C n 2 is estimated using the following relation [34]:
C n 2 ( z ) = A m α t L 0 ( z ) 4 / 3 M 2 ( z ) ,
where α t is the numerical constant which equal to 2.8. As in paper [34], in which Masciadri Elena et al. applied a technique for calibration of C n 2 values, integrated over defined height slabs, at the San Pedro Martir site, we calculated a special coefficient A m for SAO. The coefficient A m is determined by comparing measured and calculated values of the total seeing. The parameter M ( z ) is the vertical gradient of air refractive index, L 0 is the outer scale of turbulence. The parameter M ( z ) is calculated using [34]:
M ( z ) = − 79 · 10 − 6 P ( z ) T ( z ) ∂ l n θ ( z ) ∂ z ,
where P is the atmospheric pressure, T is the air temperature. θ is the air potential air temperature [34]:
θ ( z ) = T ( z ) 1000 P ( z ) 0.286 .
However, the "meteorological" approach has limitations, which are primarily related to the type of turbulence parameterization and the high variability of the parameterization coefficients.
The key parameter is the outer scale of atmospheric turbulence L 0 [35,36]. In general, vertical changes of the outer scale L 0 can be determined through vertical gradients of the horizontal component of the wind velocity S [34]:
L 0 ( z ) 4 / 3 = 0 . 1 4 / 3 · 10 a + b · S ( z ) , troposphere 0 . 1 4 / 3 · 10 c + d · S ( z ) , stratosphere ,
where
S ( z ) = ∂ u ( z ) ∂ z 2 + ∂ v ( z ) ∂ z 2 0.5 ,
u and v are the horizontal components of wind velocity. Under conditions of weak turbulence the coefficient a=1.64, b=42, c=0.506 and d=50 [37]. The very high sensitivity of the seeing value to some background value of the outer scale, as well as its component associated with the vertical gradient of wind speed, requires a fairly accurate simulation of the vertical profile of L 0 . Use of standard coefficients a=1.64, b=42, c=0.506 and d=50 gives mediocre results. We recommend refining the parameterization coefficients a=1.64 and b=42 for each month, separately for night and day, by comparing the model results with optical observations of the parameter seeing and measured values C n 2 ( z * ) within the surface layer of atmosphere. Moreover, we believe that optical turbulence modeling must take into account variations in the height of the atmospheric boundary layer, which changes both from season to season and from day to night. Below we discuss the changes of atmospheric boundary layer heights as well as the results of modeling the vertical profiles of optical turbulence taking into account variations in the height of the atmospheric boundary layer.

3.3. Atmospheric Boundary Heights Within SAO Region

The boundary layer height (BLH) determines the thickness of the lower atmospheric layer with the greatest turbulence intensity. Above this height, the values of the structural constant C n 2 decrease to the level of free atmosphere turbulence. Taking into account that typical values of BLH in mid-latitudes are 700-1000 m during the day and 30-200 m at night, BLH has a significant effect on the shape of the optical turbulence profiles and, ultimately, on the parameter seeing.
A convenient method for estimating the height of boundary layer is based on the calculation the bulk Richardson number R i . This method assumes that the boundary layer height is the height at which R i reaches a threshold value R i c [38]. However, in calculation of vertical profiles of R i , the problem of dividing by zero or by values very close to zero often arises.
In order to reduce the number of errors and avoid overestimating the shear production in generation of turbulence, Vogelezang and Holtslag have proposed a method for calculation of R i for different height levels in the atmosphere [39]. According to [39] the value of bulk Richardson number can be defined as follows:
R i = g / θ v s ( θ v z − θ v s ) ( z − z s ) ( u z − u s ) 2 + ( v z − v s ) 2 + 100 u * 2 ,
where z s and z are the heights of the lower and upper levels, θ v z , θ v s are values of virtual potential temperature at the heights of z and z s , respectively. u and v are the horizontal components of wind velocity. The advantage of using this expression is that it takes into account the effect of friction velocity u * on the Richardson number.
To illustrate how we estimated the heights of the boundary layer, we provided Figure 4 which demonstrate vertical profiles of the bulk Richardson number at the Divnoe. This radiosonde station is one of the closest measuring locations to the Observatory.
Figure 4a shows profiles that describe some characteristic changes in R i with height above the ground during daytime and nighttime. The height of atmospheric boundary layer is estimated as some height level for which R i reaches its critical value. Within unstable boundary layer optimal values of R i c range from 0.33 to 0.39. When the boundary layer stability increases the optimal R i c value decreases (under the stable stratification R i c is equal to 0.2–0.25) [39]. Practically, the BL heights are most easily estimated for unstable thermal stratification, when the Richardson numbers in the lower atmospheric layer are negative (for example, profiles shown by orange line in Figure 4a and red line in Figure 4b). The greatest uncertainty corresponds to atmospheric situations when the values of R i are close to zero in the lower part of BL (blue line in Figure 4b). Uniform distributions of wind speed with height and the limitation of vertical resolution can make it difficult to determine the height level where the Richardson numbers reach R i c . As our results show, taking into account the non-zero friction velocity makes it possible to correct the profile shape and significantly improve the conditions for determining the height of BL. Distributions of u * in the lower atmospheric layer derived from ERA-5 reanalysis data for 2012– 2024 within the SAO region are shown in Figure A1. The model values of u * are averaged using available estimations of this characteristics from ERA-5 database. These distributions reflect the intensity of turbulent vertical flows of impulse in the lower layer of the atmosphere. To a first approximation, the obtained estimates of u * can be used for correction of the BL height.
We should note that determination of BLH is complicated by a number of factors. First, the algorithm for estimation of this height is based on the threshold Richardson number and it is sensitive to vertical resolution. As a rule, at night, the boundary layer has a small thickness. At the same time, individual vertical profiles of the Richardson number demonstrate a complex nature. As the analysis of vertical profiles for the Divnoe station shows, at heights of several hundred meters above the earth’s surface, a layer with an increase in the Richardson number with height is formed, and a thin layer with increased values of the Richardson number near the ground is not reproduced. As a result, the height of the atmospheric boundary layer is artificially inflated. In the case when the Richardson number values were not determined in the lower layer of the atmosphere, we used:
- the value determined by the nearest four values of Ri in the time series;
-the value determined for conditions of non-zero friction velocity;
- the value based on the vertical profile of the Richardson number. Changes in the Richardson number between two height levels were considered linear. The Richardson number was calculated using a linear regression of the vertical profile, from the higher layer to the lower one.
Figure 5 show night-time changes of atmospheric boundary layer height at the Divnoe station site. Analysis of Figure 5 shows that the heights of the atmospheric boundary layer in the reanalysis are significantly overestimated for both January and July. Comparison of the heights showed that the greatest deviations are observed in winter. Presumably, this is due to significant deformations of the vertical profiles of wind speed and air temperature in the cold period and large-scale atmospheric disturbances that are most pronounced in the winter season. In addition, the range of changes in the heights of the atmospheric boundary layer in the cold season is significantly wider. In winter, the ERA-5 reanalysis derived value of BLH can be higher by 500–1000 m in comparison with value corresponding to radiosonde station.

3.4. Vertical Profiles of Optical Turbulence at the SAO

Using the ERA-5 reanalysis data and above mentioned formulas, vertical profiles of optical turbulence were obtained. Below, vertical profiles for two time periods are discussed. The first period (1 June 2024 - 15 September) is chosen because it corresponds to the time interval for which meteorological fields are modeled using the WRF model. The second period (January, 2023 - December, 2023) is chosen arbitrarily, it covers all seasons of the year and various atmospheric conditions. Also, a period when the measurements with DIMM were performed is considered separately. Figure 6a and Figure 7a show vertical profiles of optical turbulence at the SAO for different nights. Figures with the letter b) correspond to the profiles of dimensionless optical turbulence. Normalization was carried out by dividing C n 2 profile by the profile of optical turbulence in the first period of measurements.
It is necessary to emphasize that for calculations, the values of the structural constant C n 2 ( z = 0 ) at the lower height level have been obtained taking into account the data of sonic anemometer [40,41]. The sonic anemometer was mounted on the 20m meteorological tower. Preliminary, the values of C n 2 ( z = 0 ) derived from the reanalysis data were corrected by adjusting the model median to its measured value (for the period from 25 October 2012 to 1 November 2012). Using the coefficients estimated for the period as reference values, vertical profiles of C n ( z ) have been calculated for another time intervals (Figure 6 and Figure 7).
Taking into account the height of BL occurs by specifying an additional node between pressure levels (height levels) in the atmosphere. The structural constants of air refractive index turbulent fluctuations for this node is determined as the average between C n 2 values corresponding to the higher and lower pressure levels. Another important step in calculating vertical profiles of optical turbulence is the use of the weighted vertical profile of outer scale of turbulence for a observation period. On the one hand, such profile of outer scale introduces certain errors associated with the influence of various atmospheric situations on a profile C n 2 . On the other hand, the weighted profile of the outer scale is sufficiently smooth, without large fluctuations. This ensures the receipt of a stable statistical vertical profile for a certain time interval. The calculated vertical profiles of optical turbulence are corrected taking into account the data of surface mast measurements of turbulent characteristics and variations in the integral seeing. Due to the lack of complex long-term measurements of optical and micrometeorological (turbulent) characteristics, the verification of vertical profiles was performed based on the analysis of the range of changes in the calculated values of seeing and the isoplanatic angle. The reanalysis data allow us to estimate the vertical profiles of optical turbulence and analyze their deformations based on long time series. In particular, analyzing the vertical profiles of optical turbulence on a long time scale, it can be noted that the characteristic range of changes in the values of the seeing parameter is from 0.60 to 2.40 arc sec (the light wavelength is 500 nm, Figure 9). Model values of the isoplanatic angle θ 0 change from 1.0 to 3.0 arc sec (Figure 10). These estimations are in agreement with typical values of seeing and θ 0 [42].
Analysis of the vertical profiles shows that turbulence is mainly concentrated in a narrow atmospheric layer near the earth’s surface. However, in our opinion, the presented profiles are significantly deformed in the troposphere. It is possible to identify, although weak, individual layers of turbulence throughout the troposphere. This indicates that the atmosphere above the boundary layer is characterized by a high degree of disturbance and, possibly, vorticity. Moreover, we also see that the strength of optical turbulence in individual atmospheric layers changes several times during the night. The highest changes are observed in the surface layer of the atmosphere and at altitudes of 6200 and 10700 m. Surprisingly, the ratios of the structural constants C n 2 are close to each other for these atmospheric layers. The largest night to night changes in C n 2 correspond to heights of 2300 m and 7200 m (large-scale atmospheric flow) and 11700 m (level of jet stream). Figure 8 shows vertical profile of median values of C n 2 at the SAO. The model median of seeing is ∼ 1.21 arc sec. The first quartile corresponds to 0.75 arc sec, the third quartile is 2.31 arc sec (for the period 1 June–15 September, 2024). The range of changes in seeing is in good agreement with the measured data [43]. For the period 1 January 2023–31 December 2023 the first quartile of seeing is 0.78 arc sec, the third quartile is 1.71 arc sec. The values of the isoplanatic angle vary in the range from 1.0 to 3.0 arc sec (at λ = 500 nm).
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Figure 8. Vertical profile of optical turbulence at the SAO site, 1 June–15 September, 2024. The shading corresponds to the interval between the first and the third quartiles.
Figure 8. Vertical profile of optical turbulence at the SAO site, 1 June–15 September, 2024. The shading corresponds to the interval between the first and the third quartiles.
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Figure 9. Histogram of seeing at SAO, calculated at the wavelength 500 nm, January–December, 2023.
Figure 9. Histogram of seeing at SAO, calculated at the wavelength 500 nm, January–December, 2023.
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Figure 10. Histogram of isoplanatic angle at SAO, calculated at the wavelength 500 nm, January–December, 2023.
Figure 10. Histogram of isoplanatic angle at SAO, calculated at the wavelength 500 nm, January–December, 2023.
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4. Discussion

In this paper, we pay attention to a number of factors that lead to changes in thes of field wind speed and air temperature, as well as variations in the intensity of turbulence. These include, first of all:
-deformation of air flows by mountain obstacles and the occurrence of rotor movements on their leeward side, producing an increase in the friction velocity and small-scale turbulence;
- interaction of air masses with different characteristics and the formation of atmospheric fronts. These zones are most often associated with cyclonic weather, cloudiness and are only partially taken into account. In particular, vertical profiles of optical turbulence were obtained with a total cloudiness of less than 0.5;
- friction of the air flow against the earth’s surface and the formation of a wind speed profile with large vertical gradients in its lower part. This factor is taken into account by correcting and analyzing the vertical profiles of the Richardson number.
We present the results of studies on optical turbulence in the atmospheric boundary layer and free atmosphere, up to a height of 30 km. A modified method for estimating vertical profiles of optical turbulence is proposed. The method directly takes into account variations in the height of the atmospheric boundary layer that depend on the friction velocity. The latter allows us to minimize the number of atmospheric situations with overestimation of turbulence generation under the influence of vertical shears of wind speed. We examine the location of the Special Astrophysical Observatory and analyze the peculiarities in vertical structure of atmospheric optical turbulence (We do not know the previously obtained vertical profiles of optical turbulence above SAO). For a better understanding of conditions for the occurrence of atmospheric disturbances within the SAO region, below we discuss some features of atmospheric processes.
Atmospheric flow structure in the Caucasus region depends mainly on circulation regime and features of the interaction of large-scale pressure perturbations with the underlying surface. In our opinion, complex relief and the character of the interaction of large-scale air movements with the underlying surface play a significant role in generation of mesoscale vortex structures and formation/dissipation of optical turbulence, but not only in the surface layer. For example, we have previously shown that mountain (gravity) waves are often formed above the Observatory, which are the source of collapsing mesoscale atmospheric inhomogeneities [44].
Let’s consider the characteristic relief within the Observatory area.
In particular, from the south, the SAO site is isolated from Transcaucasia by the Main Caucasian Ridge (MCR), with a height of 2 to 3 km. The MCR is an obstacle for weather fronts (and large scale turbulent structures). Due to this Transcaucasia has a milder climate than the North Caucasus. To the north of the SAO, the mountains descend to the North-Jurassic intermountain depression, which is located parallel to the Great Caucasus Range, along the settlements of Kurdzhinovo - Pregradnaya - Zelenchukskaya - Karachayevsk. The SAO area is located in a zone of a fairly sharp relief, up to 3000 meters above sea level. The location of river valleys in the SAO area is almost meridional. The relief features determine the development of air subsidence processes: in the intermountain North Jurassic Depression; in the Arkhyz valley (where the Abishira-Akhuba ridge prevents the intrusion of air masses from the north); and in other isolated valleys. Probably, some decrease in the intensity of optical turbulence is associated with the downward movements. The relief stimulates the development of ascending flows on the windward slopes of the mountains (since the average angle of inclination of the mountain surface is close to the angle of inclination of the warm frontal section, -0.02o). Because of this feature, in the intrusions of frontal sections from the north and northwest into the SAO area, the mountain relief stimulates more powerful processes of cloud development and precipitation formation. In this case, we believe that we can expect an increase in optical turbulence. Frontal sections coming from the south and southwest, on the contrary, tend to weaken. The location of river valleys and spurs, close to the meridional, forms meridional air flows and prevents the movement of air masses along the latitudes.
The results of simulation make it possible to to believe that the repeatability of low seeing is quite high. Despite this, the number of atmospheric situations with high values of seeing is also significant. We associate these situations and the amplification of atmospheric optical turbulence, largely, with large-scale atmospheric pressure disturbances. The presence of additional local peaks in the optical turbulence profiles is confirmed by analysis of measured optical turbulence profiles for the Terskol Astronomical Observatory (TAO) [?]. Analyzing optical measurements (at TAO), it is shown that seeing higher than 1.5 arc sec are often associated with the development of optical turbulence within the surface layer and an additional turbulent layer at heights of 2–2.5 km (Figure A2).
The description of the terrain and the features of large-scale air circulation are important for further research into the influence of large-scale and mesoscale air flows on small-scale turbulence.

5. Conclusions

In this study we applied a modified method that uses meteorological characteristics at pressure levels extracted from the ERA5 reanalysis to estimate vertical profiles of optical turbulence at the Special Astrophysical Observatory site. The method differs from its analogues by simultaneously taking into account the measured values of the total seeing and the variations of atmospheric boundary layer height. The approach itself, based on the gradients of wind speed and the outer scale of turbulence, is necessarily not new. However, its adaptation for each observatory is individual. At the same time, we used the measurement data (at two heights) to calibrate the calculated data. We analyzed the spread of the Fried parameter, seeing and the outer scale of turbulence for different height levels. As a result, the method does not give non-physical ranges of changes in the parameters under consideration. Moreover, if for many observatories in the world the optical turbulence profiles are obtained and known, then for SAO there are simply no such data. We also added a point on estimating the height of the atmospheric boundary layer, using data on typical measured and model values of the friction velocity and the structural constant of turbulent fluctuations of the refractive index of air in the surface layer of the atmosphere. This is the novelty of the research.
To improve calculations, a special parameterization scheme of C n 2 within the surface layer based on a comparison of model results with estimates of C n 2 obtained by processing mast measurements (sonic anemometer) is used. Vertical profiles of optical turbulence at the SAO site were obtained. The range of amplitude changes of C n 2 at different height levels is in good agreement with the measurement results widely available in the literature. Moreover, the typical values of the isoplanatic angle (500 nm) corresponding to the obtained vertical profiles vary in the range from 1 to 3.0 arcsec. This is comparable to reported values [42], which supports the assumption that the obtained C n 2 profiles have a vertical distribution that is consistent with measured profiles. Thus, we should note that the method presumably reconstructs the most important features of the shape of the measured profile under clear sky. But, we should pay more attention to the reconstruction of vertical profiles under very good and very bad turbulence conditions. Under these conditions, the parameterization coefficients will change. Analysis of vertical profiles shows that optical turbulence over the SAO is highly non-uniform with height above the ground. Over time, the structure of optical turbulence changes significantly in different atmospheric layers. Orography appears to influence optical turbulence throughout the troposphere, with largest changes of turbulence strength corresponding to the levels of the leading flow and the jet stream in the upper troposphere. In the future, we would like to accumulate richer samples of measurements carried out with DIMM and sonic anemometers in order to be able to improve the proposed method for description of different optical turbulence. The obtained results can be used for determination of vertical turbulence profiles for the period from 1940 to the present.
The novelty is that we are trying to create a method for describing optical turbulence above the observatory. The approach itself, based on the gradients of wind speed and the outer scale of turbulence, is necessarily not new. However, its adaptation for each observatory is individual. At the same time, we used the measurement data (at two heights) to calibrate the calculated data. We analyzed the spread of the Fried parameter, seeing and the outer scale of turbulence for different height levels. As a result, the method does not give non-physical ranges of changes in the parameters under consideration. Moreover, if for many observatories in the world the optical turbulence profiles are obtained and known, then for SAO there are simply no such data. We also added a point on estimating the height of the atmospheric boundary layer, using data on typical measured and model values of the friction velocity and the structural constant of turbulent fluctuations of the refractive index of air in the surface layer of the atmosphere. This is the novelty of the research.

Author Contributions

A.Y.S. and P.G.K. developed the methodology and performed the investigation; S.A.P., E.A.K. and L.A.B. performed measurements using DIMM; X.Q. and A.V.P. were engaged in data analysis. All authors have read and agreed to the published version of the manuscript.

Funding

The study was funded by RSF grant No. 24-72-10043.

Data Availability Statement

Data used are available on request from the corresponding author.

Conflicts of Interest

The author declares no conflict of interest.

Appendix A

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Figure A1. Distributions of u * in the lower atmospheric layer for 2012– 2024 within the SAO region (a) December- February and (b) June-August.
Figure A1. Distributions of u * in the lower atmospheric layer for 2012– 2024 within the SAO region (a) December- February and (b) June-August.
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Figure A2. Typical vertical distribution of optical turbulence for medium and high values of seeing obtained from the image scintillation, Terskol Peak Observatory.
Figure A2. Typical vertical distribution of optical turbulence for medium and high values of seeing obtained from the image scintillation, Terskol Peak Observatory.
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Figure 1. Geographical position of SAO. The figure also shows the sites where measurements are planned or carried out.
Figure 1. Geographical position of SAO. The figure also shows the sites where measurements are planned or carried out.
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Figure 2. Differential image motion monitor (DIMM).
Figure 2. Differential image motion monitor (DIMM).
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Figure 3. Night-time changes of seeing at the SAO. For brevity, the parameter seeing is designated as β (Local time). Black lines correspond to variations of total β . The changes of β within the free atmosphere (above 500 m) are shown by red lines.
Figure 3. Night-time changes of seeing at the SAO. For brevity, the parameter seeing is designated as β (Local time). Black lines correspond to variations of total β . The changes of β within the free atmosphere (above 500 m) are shown by red lines.
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Figure 4. Vertical profiles of the bulk Richardson number at the Divnoe. The grey narrow vertical bars show the ranges of changes in the critical Richardson number. The horizontal lines are estimations of atmospheric boundary layer heights.
Figure 4. Vertical profiles of the bulk Richardson number at the Divnoe. The grey narrow vertical bars show the ranges of changes in the critical Richardson number. The horizontal lines are estimations of atmospheric boundary layer heights.
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Figure 5. Night-time changes of atmospheric boundary layer height at the Divnoe station site (00 UTC). Blue lines correspond to HBL values estimated from ERA-5 reanalysis data. Heights of BL determined by the threshold value of R i (radiosondes) are shown by orange lines.
Figure 5. Night-time changes of atmospheric boundary layer height at the Divnoe station site (00 UTC). Blue lines correspond to HBL values estimated from ERA-5 reanalysis data. Heights of BL determined by the threshold value of R i (radiosondes) are shown by orange lines.
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Figure 6. Night-time vertical profiles of optical turbulence at the SAO, 27/08/2024.
Figure 6. Night-time vertical profiles of optical turbulence at the SAO, 27/08/2024.
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Figure 7. Night-time vertical profiles of optical turbulence at the SAO, 28/08/2024-29/08/2024.
Figure 7. Night-time vertical profiles of optical turbulence at the SAO, 28/08/2024-29/08/2024.
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