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Abstract
A seeing model based on the Tatarskii equation, which uses wind shear and temperature gradient as inputs, has been proposed for Dome A in our previous study. It has been shown that the seeing was correctly reproduced statistically. In this paper, in order to investigate the ability of the Polar version of the Weather Research and Forecasting (PWRF) model to forecast the seeing, we have tried three different PWRF-simulated wind shear and temperature gradient combinations of two levels for the near-ground. The three combinations all give good performance with the correlation coefficients being higher than 0.75, when compared with the seeing measured by the Kunlun Differential Image Motion Monitor at Dome A during the summer of 2019. Besides this, the gradient Richardson number (Ri), which was also simulated by the PWRF model, basically behaved as expected as the Ri is generally large when the measured seeing is small. These results suggest that the PWRF model can be used to forecast the astronomical seeing above Dome A.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
“Seeing” is usually used to quantify the intensity of atmospheric optical turbulence. And the atmospheric optical turbulence is dramatically affecting ground-based optical applications, such as astronomical observation and laser communication. The Antarctic Plateau, which has advantages for astronomical observations compared with the temperate latitude sites for its old, dry and stable air [1], has been found that the optical turbulence is mainly contributed by the boundary layer [2–7].
The boundary-layer atmosphere above Dome A (80.37° S, 77.53° E) in Antarctica seems to be the major region impacting the seeing [8], and stays for the most of the time when temperature inversion (or stable condition) occurs [9,10]. So far, the optical turbulence inside the boundary layer is generally estimated by the Monin-Obukhov similarity (MOS) theory [11–13]. However, the MOS theory becomes unreliable for stable atmospheric conditions [14,15], when the vertical motions are suppressed. Thus, Bolbasova et al. applied machine learning to predict optical turbulence in the stable surface layer [16]. Besides, the Tatarskii equation, which was based on the context of boundary-layer turbulence [17,18], has been recommended to be used for stably stratified conditions [19]. And the seeing could be calculated using the estimated refractive index structure constant ($C_n^2$) by the Tatarskii equation [20,21]. Moreover, the Tatarskii equation has been examined for the boundary layer at Dome A [22] and the Antarctic Taishan Station [23].
Therefore, this study aims to estimate the astronomical seeing using a numerical weather model based on the Tatarskii equation, as simulating seeing would be meaningful for the applications of optical observations. A mesoscale model, namely, the Weather Research and Forecasting (WRF) model, has been used to simulate the surface-layer meteorological parameters of the Antarctic Taishan Station and estimate optical turbulence [24,25], suggesting that the WRF model can simulate the Antarctic near-ground atmosphere.
Herein, the Polar-optimized version of the WRF (Polar WRF; hereinafter, PWRF) model was employed as the numerical weather model to forecast the atmosphere above Dome A. The PWRF model was designed for the Polar regions and optimized for energy balance and heat transfer over ice and snow surface [26–29]. PWRF is now maintained by the Polar Meteorology Group at the Byrd Polar and Climate Research Center (http://polarmet.osu.edu/PWRF/). PWRF appears to simulate the polar regions slightly better than standard WRF [30]. In June 2008, PWRF replaced Polar MM5 as the numerical forecast base model of the Antarctic Mesoscale Prediction System (AMPS; website: https://www2.mmm.ucar.edu/rt/amps/) [31]. Lascaux et al. also simulated the optical turbulence above Dome A using an atmospheric modeling system, they implemented the mesoscale model (Meso-NH) together with the Astro-Meso-NH package, this package was used to calculate optical turbulence [32].
This paper is organized as follows. In Section 2, the experiment and data are briefly described. In Section 3, the PWRF model configurations are introduced. In Section 4, the theories of seeing and gradient Richardson number are given. In Section 5, the methods for applying and evaluating the proposed seeing model are introduced. In Section 6, the PWRF-simulated results are analyzed. Finally, In Section 7, the conclusions are presented.
2. Experiment and data
The astronomical seeing at Dome A has been measured between 25th January 2019 and 11th March 2019, which can be downloaded on the China-VO Paper Data Repository (http://paperdata.china-vo.org/BinMa/DomeA-seeing2019.zip) [8]. The seeing was observed every minute by the KunLun Differential Image Motion Monitor (KL-DIMM [33]) on a tower 8 m high, which was installed by the 35th Chinese National Antarctic Research Expedition during January 2019.
Figure 1 shows the number of data for valid seeing measurements obtained every day during the 2019 summer campaign. In this study, we have tried to use those days’ data that count seeing data over 1000 a day, considering that more complete data of one day is more worth analyzing.
Fig. 1. Number of seeing data per day during the 2019 summer campaign. As for the seeing model, Eq. (4), provided in Section 4.1, only those days that count data over 1000 (black dashed line) were utilized. The blue upward-pointing triangles above bins indicate the seeing values in these days are used to fit the seeing model, while the red downward-pointing triangles are for validating the fitted seeing model.
3. Model configurations
PWRF can simulate the temporal evolution of the three-dimensional atmospheric flow. In this paper, the mainly used prognostic variables of the PWRF model, are the near-ground temperature, wind speed, pressure, etc. The initial and lateral boundary conditions used here are the final analysis products (FNL; website: https://rda.ucar.edu/datasets/ds084.1/) at 0.25°×0.25° horizontal resolution with 6 h intervals. The basic PWRF settings, shown in Table 1, will be discussed in terms of time and space in the following.
Table 1. Basic Parameter Settings
As for time, Polar models require some time to become quasi-steady; a 12 h spin-up time (the first 12 forecast hours were not used) was utilized, as in some polar simulations [28,34]. The integration period was 36 h, then only the PWRF outputs during the last 24 h of each simulation were used since the model was initialized with the first 12 h. Inspired by the AMPS Configuration (https://www2.mmm.ucar.edu/rt/amps/information/information.html), the time step of Domain 1 is set as 60 s, less than the recommended WRF time step (6×$\Delta X$; the unit of $\Delta X$ is km). The parent-to-nest domain time step ratio is 5. The output time interval is 10 min.
Considering space, the vertical level setting in the cold region provided for reference by the WRF user's guide is 44 (https://www2.mmm.ucar.edu/wrf/users/docs/user_guide_v4/contents.html). To better represent the flow structures of the boundary layer, a 2× vertical resolution (87 levels) was employed as inspired by [35]. However, simply increasing the number of eta levels (model vertical coordinate: terrain-following normalized pressure levels) may not significantly increase the boundary-layer mesh density. Then the 2× resolution is defined by interpolating the median values at adjacent levels from the default setting, as shown in Fig. 2. The new interpolated eta levels may be rational as they are generated based on the default eta levels of WRF. The AGL (Above Ground Level) of the same eta level varied slightly during simulation (∼ 1 m near the ground), so the AGL in this paper is the average over all simulations. To achieve a high horizontal resolution, a model domain with three nested levels was constructed referring to [24,32], the nesting ratio is 5, coinciding with the parent-to-nest time step ratio (similar to a previous article [36]). The simulation area center was Dome A.
Fig. 2. Full eta levels (left) and eta levels above 0.945 (right) of the default 44 eta levels (red lines), plus the interpolated 2× resolution 87 eta levels (blue lines). Eta = 1 represents the model surface, eta = 0 represents the model lid.
The WRF physics options, which determine a large number of meteorological parameters (temperature, wind, humidity, pressure, etc.), are mainly referred to the AMPS settings [34], as listed in Table 2. The WRF Single-Moment 5-class (WSM5) scheme can predict cloud water and cloud ice, among others [37]. The Noah Land Surface Model (LSM) accounts for the energy balance and can calculate the skin temperature [38]. The Mellor-Yamada-Janjic (MYJ) Planetary Boundary layer (PBL) is a one-dimensional prognostic turbulent kinetic energy scheme with local vertical mixing [39]. The PWRF model includes some modifications to these WSM5 microphysics, Noah LSM, and MYJ PBL schemes compared with the standard WRF [28]. As suggested by a PWRF simulation [29], both longwave and shortwave radiation schemes employ the rapid radiative transfer model for general circulation models (RRTMG) [40].
Table 2. Main Physics Options
4. Theory
4.1 Seeing model
The seeing is always estimated by integrating the $C_n^2$, which generally represents the intensity of optical turbulence and can be calculated by the Tatarskii equation [17]:
where ${L_0}$ (in m) and M are the outer scale of turbulence and the vertical gradient of the potential refractive index, respectively. The value of ${M^2}$ can be calculated from the pressure P (in hPa) and temperature $T$ (in K) [41]:The KL-DIMM observed seeing (${\varepsilon _{TOT}}$) can be considered to be contributed from the free atmosphere (${\varepsilon _{FA}}$) and the boundary layer (${\varepsilon _{BLA}}$). Similar to [42], we used the median value of free-atmosphere seeing (the median value is 0.31 arcsec at Dome A [8]) as a rough estimation:
According to Eq. (3), the “observed” boundary-layer seeing (${\varepsilon _{BLA}}$) can be calculated by the KL-DIMM observed seeing (${\varepsilon _{TOT}}$).
Considering $L_{0}$ can be regarded as a function of wind shear ($S$ in s−1) and temperature gradient ($G$ in K·m−1) [43,44], and the optical turbulence intensity inside the boundary layer seems to be uniform [7], the boundary-layer seeing model for Dome A thus has been proposed by Ref. [22], written by:
4.2 Gradient Richardson number
The atmospheric stability is closely related to the generation of optical turbulence. The Gradient Richardson number ($Ri$), which is used to indicate the dynamic stability, is also expressed as a function of wind shear and temperature gradient [47]:
The critical Richardson number, $R{i_\textrm{c}}$, is about 0.25 (although reported values have ranged from roughly 0.2 to 1.0). The atmosphere has a rather turbulent flow when $Ri$<$R{i_\textrm{c}}$. This helps us to analyze the boundary-layer stability at Dome A.
In this study, we apply NCL (NCAR Command Language; website: http://www.ncl.ucar.edu/index.shtml) to compute $Ri$ from the PWRF outputs, using the function “rigrad_bruntv_atm” (http://www.ncl.ucar.edu/Document/Functions/Contributed/rigrad_bruntv_atm.shtml).
5. Method for applying and evaluating the proposed seeing model
5.1 Determination of seeing model
PWRF can export many levels of meteorological parameters near the ground. The PWRF-simulated meteorological data are read from the nearest model grid point to Dome A. The wind shear S and temperature gradient G, which are used to estimate the seeing by Eq. (4), were calculated using three different combinations of two eta levels of PWRF near the ground, as shown in Table 3. In Table 3, the subscripts of S and G represent the model levels of PWRF, e.g., ${S_{2\textrm{ - }1}}$ represents the wind shear calculated from the 2nd levels (see Fig. 2; AGL is 30 m) and 1st layers (see Fig. 2; AGL is 10 m) above the model surface. For consistency, the vertical gradient of the potential refractive index M used the same levels of meteorological parameters as S and G. Finally, we found the nearest seeing measurement time (1 minute interval except for missing) to the given Polar WRF output (10 minutes interval) for fitting.
Table 3. The Fitted Undetermined Coefficients for Seeing Model (Eq. (4)) Using Three Different PWRF-simulated S and $G$
The three undetermined coefficients ($A$, B, $C$) in Eq. (4) were determined using linear fitting. The fitting data were obtained from the KL-DIMM observations (seeing) and the PWRF outputs (wind speed, temperature, etc), during these days that are marked by the blue upward-pointing triangles in Fig. 1. Such fitting may only make sense as the KL-DIMM is within the boundary layer, then only the measurements under the condition of ${U_{3.6}}$>19/12 m·s−1 (when ${H_{BLA}}$>0 m) and ${\varepsilon_{TOT}}$>0.31 arcsec (the value of free-atmosphere seeing is assumed to be 0.31 arcsec) were used. Besides, considering the weakness of the Tatarskii equation for a neutral atmospheric condition ($\partial \theta /\partial z$ ∼ 0) [22], the fitting function is divided into two cases ($\partial \theta /\partial z$>0.005 K·m−1 and $\partial \theta /\partial z$<−0.005 K·m−1). Finally, the coefficients ($A$, B, $C$) are fitted, as shown in Table 3.
In Table 3, the coefficients ($A$, B, $C$) show a slight difference between different combinations when $\partial \theta /\partial z$> 0.005 K·m−1, but vary more in the case of $\partial \theta /\partial z$< −0.005 K·m−1. This may be due to the fewer data for $\partial \theta /\partial z$< −0.005 K·m−1 (e.g. in the case of ${S_{4\textrm{ - }1}}$&${G_{4\textrm{ - }1}}$; it counts for about 8%), considering a more robust model needs more fitting data.
Finally, $\varepsilon_{BLA}$ can be estimated by substituting the PWRF outputs into Eq. (4) using the values of A, B, and C in Table 3. Then, ${\varepsilon _{TOT}}$ can be calculated by Eq. (3).
5.1 Statistical operators
To evaluate the statistical reliability of the PWRF model in reconstructing the seeing, the average bias ($Bias$), root mean square error ($RMSE$), and Pearson correlation coefficient ${R_{XY}}$, which have been used for quantitative analysis of the estimated seeing [22], are defined as:
6. Results
To test the PWRF reliability in forecasting the seeing trends above Dome A, other 8 days of the PWRF outputs (28th February to 7th March in 2019; marked by the red downward-pointing triangles in Fig. 1) were used to estimate the total atmosphere seeing values (${\varepsilon _{TOT}}$), based on Eqs. (3) and (4) using the fitted model coefficients shown in Table 3. The model estimated ${\varepsilon _{TOT}}$ would not be smaller than 0.31 arcsec as we assumed ${\varepsilon _{FA}}$=0.31 arcsec in Eq. (3).
Table 4 shows that the three different PWRF-simulated S and G produce a similar performance. This likely is due to the stable boundary-layer atmosphere at Dome A, inside which the temperature and wind structure change slightly with height. This may also suggest that the optical turbulence intensity inside the boundary layer could be uniform.
Table 4. Statistical Analysis for ${\varepsilon _{TOT}}$ When Seeing Model ((Eq. (4))) Using Three Different PWRF-simulated $S$ and $G$
The ${R_{xy}}$ of the 2nd combination (${S_{3\textrm{ - }1}}$&${G_{3\textrm{ - }1}}$) has the maximum value (0.7691). Thus, the temporal evolutions of ${\varepsilon _{TOT}}$ estimated by ${S_{3\textrm{ - }1}}$&${G_{3\textrm{ - }1}}$ using Eqs. (3) and (4) are shown in Figs. 3(d) and 4(d) (4 days per figure). The corresponding temporal evolution of the PWRF-simulated wind speed, temperature, and gradient Gradient number are also displayed. The PWRF model is initialized with a 12 h lead time and integrated continuously for 36 h, only forecasts from the last 24 h from each of the forecasts are joined to make a continuous time series. This could lead to the discontinuities of the PWRF outputs, as shown in Figs. 3(a-c) and 4(a-c).
Fig. 3. The results recorded from 00:00 28th February to 00:00 4th March in 2019 (UTC) . (a) Cross-section pattern of the Wind speed simulated by the PWRF model assuming a zero wind speed at the surface. (b) Cross-section pattern of the temperature simulated by the PWRF model. (c) Cross-section pattern of the gradient Richardson number ($Ri$) simulated by the PWRF model. (d) the black circles represent the ${\varepsilon _{TOT}}$ observed by the KL-DIMM, the red crosses represent the ${\varepsilon _{TOT}}$ estimated by the PWRF model coupled with Eqs. (3) and (4) using ${S_{3\textrm{ - }1}}$&${G_{3\textrm{ - }1}}$.
Fig. 4. Similar to Fig. 3, but for the results recorded from 00:00 4th to 00:00 8th March in 2019 (UTC).
One can see that the simulated wind speed is weak (see green color shown in Figs. 3(a) and 4(a)) when the measured seeing values is small (see Figs. 3(d) and 4(d)), this agrees with measured wind speed [22]. The simulated temperature near the surface is smaller (see blue color shown in Figs. 3(b) and 4(b)) approximately during 12:00∼00:00 (UTC) every day, being consistent with the results from the temperature sensors [22], such diurnal variation is caused by the changing of solar radiation. As shown in Figs. 3(c) and 4(c), the PWRF-simulated gradient Richardson number ($Ri$) are generally large when the seeing is small. There are some values of $Ri$ less than 0.25 near the surface on 28th February and 1st March (see Fig. 3(c)), suggesting the boundary-layer atmosphere tends to be turbulent, and the seeing indeed becomes worse in the two days as shown in Fig. 3(d). Figures 3(d) and 4(d) show that the seeing obtained from the KL-DIMM and the PWRF model agree well in trends. But the seeing are obviously overestimated by the PWRF model on 2nd March when the PWRF-simulated $Ri$ is also small, this suggests such bias may be attributed by the PWRF simulation, rather than our proposed seeing model (i.e., Eq. (4)). Nevertheless, the overall results indicate that the estimated seeing is reliable.
We also can get the PWRF-simulated $Ri$ at high altitude (not shown); the free atmosphere above Dome A shows stable conditions with much larger values of $Ri$ (similar to [48]). This suggests that the atmosphere near the ground is the key regime affecting the observational condition (or seeing) above Dome A.
7. Conclusion
This study proposed a simple method to forecast the seeing based on the PWRF-simulated meteorological parameters (temperature, wind speed, etc.), using the seeing model proposed by our previous study [22]. The seeing model is based on the Tatarskii equation, of which the outer scale is parameterized by the wind shear and temperature gradient. Three different PWRF-simulated wind shear and temperature gradient, which are related to combinations of two eta levels near the ground, have been tried for evaluating the seeing model. All of them give good performance with ${R_{xy}}$ higher than 0.75, when compared with the seeing measured by the KL-DIMM at Dome A during summer 2019. Ideally, one can forecast the summer seeing above Dome A using the fitted seeing model in this study and doesn’t need seeing measurements anymore (provided the atmospheric conditions above at Dome A doesn’t significantly different from the 2019 summer campaign). The proposed method for forecasting seeing might well be used for some other astronomical sites as they generally have something in common (e.g., the atmosphere is stable; their location is on a summit) [45,49].
The gradient Richardson number ($Ri$), which was also simulated by the PWRF model, basically behaves as expected as the PWRF-simulated $Ri$ is generally large when the measured seeing is small, confirming the PWRF model can simulate the atmospheric turbulence above Dome A.
In sum, when dealing with the condition of astronomical observation above Dome A, qualitative analysis can refer to the PWRF-simulated $Ri$ and quantitative analysis can use the PWRF-simulated meteorological parameters coupled with our proposed seeing model [22].
Funding
Foundation of Key Laboratory of Science and Technology Innovation of Chinese Academy of Sciences (CXJJ-19S028); National Natural Science Foundation of China (41576185, 91752103).
Acknowledgments
We wish to thank Bin Ma, Zhaohui Shang, Yi Hu, et al. (from the National Astronomical Observatories, Chinese Academy of Sciences) for their shared data [8].
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are available in Ref. [8].
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