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Abstract
Astronomical seeing parameters calculated based on the Kolmogorov turbulence model cannot fully evaluate the effect of the natural convection (NC) above a solar telescope mirror on the image quality, as the convective air motions and temperature variations of the NC are significantly different from the Kolmogorov turbulence. In this work, a new method based on the transient behaviors and frequency characteristics of NC-related wavefront error (WFE) are investigated in detail and used to evaluate the image quality degradation caused by a heated telescope mirror, aiming to make up for the deficiency of astronomical seeing parameters with the conventional method in evaluating the image quality degradation. Transient computational fluid dynamics (CFD) simulations and WFE calculations based on discrete sampling and ray segmentation are performed to quantitatively evaluate the transient behaviors of the NC-related WFE. It clearly exhibits apparent oscillatory characteristics, which are coupled by main oscillation with low frequencies and minor oscillation with high frequencies. Moreover, the generation mechanisms of two types of oscillations are studied. The conspicuous oscillation frequencies of the main oscillation caused by heated telescope mirrors with varying dimensions are primarily lower than 1 Hz, suggesting that active optics may be adopted to correct the main oscillation of NC-related WFE while the adaptive optics may correct the minor oscillation. Furthermore, a mathematical relationship between WFE, temperature rise, and mirror diameter is derived, revealing a significant correlation between WFE and mirror diameter. Our work suggests the transient NC-related WFE should be considered as one of the critical supplements to the mirror seeing evaluation.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Ground-based solar telescope has been recognized as one of the main observation instruments to investigate solar activities. During ground-based solar telescope’s operation, nearly 5∼10% of the solar radiation incident on the telescope mirror would be absorbed by its coating, leading to an increase in the temperature in the mirror. In fact, it would raise the temperature of the mirror up to nearly 8 degrees higher than the ambient temperature [1,2]. In the absence of wind or in the case when the wind speed is low, buoyancy-driven natural convection (NC) will appear above the hot mirror of ground-based solar telescope. The dynamic inhomogeneous temperature fields of NCs, would degrade the image quality of ground-based solar telescope. In long-term observations of ground-based solar telescope, it has been found that, when there is no wind or when the wind speed is minimal, the image quality of the ground-based solar telescope is very poor, and the capability to correct image quality of the adaptive optics system would be significantly reduced. It is generally believed that the image quality degradation of ground-based solar telescope caused by NCs is greater than the forced convection, although experiments are rare [3]. The reason may be that natural convection injects more heat into the air domain of the optical path than forced convection. Conventionally, the methods of evaluating image quality degradation caused by airflow disturbances above the heated telescope mirror and unheated telescope mirror are primarily based on the Kolmogorov turbulence model and the frozen turbulence hypothesis [4–17]. However, as driven by strong buoyancy, the large-scale air motions and temperature variations change anisotropically, which is divergent from homogeneous isotropic Kolmogorov turbulence. Therefore, seeing parameters (seeing FWHM, Fried parameter, etc.) calculated based on Kolmogorov turbulence model are not accurate enough to evaluate the effect of the NC on telescope image quality. The uneven temperature field of NCs leads to the actual wavefront of the ground-based solar telescope deviating from its real case. Time-averaged WFE has been employed to evaluate the image quality degradation caused by hot airflow above the heat-stop of solar telescope at a wind speed of 1.5 m/s [18]. However, the transient information and structural evolution of the hot airflow on image quality would be eliminated by using time-averaged calculation method. Till now, the transient behaviors of the NC-related WFE have been rarely investigated. In the absence of understanding to the transient behaviors and frequency characteristics of the NC-related WFE, the potential of using adaptive optics and active optics to improve the NC-related WFE of solar telescope mirror is still not clear. It calls for the quantitative evaluation of transient behaviors of NC-related WFE in the terms of improving telescope image quality. In addition, similar to solar telescopes heated by solar radiation, various reflectors of high-power laser systems are also heated by laser light, which not only causes thermal deformation of the reflector surface but also causes NC disturbances [19–21]. However, there have been rare research reports on laser beam quality degradation caused by NC.
In this work, given that NC's anisotropic temperature field changes at slow timescale, a method to quantitatively evaluate the transient behaviors and frequency characteristics of the NC-related WFE above heated mirror is elucidated, which adopts transient computational fluid dynamics (CFD) simulations and WFE calculation based on discrete sampling and ray segmentation. As revealed by the calculation results, the WFE exhibits obvious oscillatory behaviors, which are coupled by low-frequency main oscillations and minor high-frequency oscillations. Moreover, possible generation mechanisms of two typical oscillations are presented.
2. Method description, modeling and validation
The proposed quantitative evaluation method for the transient behaviors and frequency characteristics of NC-related WFE is shown in Fig. 1.
Fig. 1. Schematic diagram of quantitative evaluation method for the NC-related WFE transient behaviors.
Firstly, large eddy simulation (LES) was employed for transient CFD simulation of NC, by which the time evolution of the temperature fields was determined. Secondly, a functional equation was adopted to build up the link between the refractive index of the air and its temperature. Afterwards, an array of light rays was obtained via discrete sampling of the light beam incident on the mirror, and each ray was meshed into a considerable number of small segments. Then the optical path length (OPL), the function of meshing length and the refractive index of the air, were calculated, by which the WFE could be obtained subsequently. The transient behaviors of the WFE were investigated by calculating the WFEs at different evolution times. Finally, the optical path difference (OPD) between the respective light ray and the average OPL of the incident beam was calculated. The root-mean-square (RMS) OPDs of all rays was calculated, defined as the RMS WFEs. The RMS WFEs at different evolution times were further calculated so as to analyze the transient behaviors.
2.1 Transient CFD simulation of NC
Transient CFD simulation was performed using the Ansys Fluent software. The overall settings for the simulation can be found in Fig. 2(a). The green cylinder represents a telescope mirror with a diameter of 30 cm and a thickness of 1 cm. The air domain in the CFD simulation was set as a rectangular hexahedron with the length, width, and height of 430 cm, 430 cm, and 601 cm, respectively, to ensure the development of the NC above the mirror would not be affected by the computational boundary conditions.
Fig. 2. (a) Computational domain and settings. (b) Inflation layer around the mirror. (c) Enlarged view of the boxed region in (b).
An inflation layer was built to accurately assess the boundary effects. (see, Fig. 2(b)). The height of the first layer was 0.05 cm, and the height of the inflation layer was set to gradually increase layer by layer. The sufficient mesh density and the extra inflation layer ensured that the CFD simulation results could be mesh-independent. The primary material property of the air domain was selected as the incompressible-ideal-air at 20℃ in the normal atmosphere. The mirror surface was set to a constant temperature boundary of 28℃, which was in accordance with the previous research results [1,2]. The outer boundary of the air domain was set as the open boundary at zero wind speed. The analysis type of CFD calculation was set with a transient state to obtain the structural evolution and time-resolved information of the NC. Table 1 lists the models and key settings of CFD simulation.
Table 1. Model and key settings
2.2 Air refractive index conversion
Refractive index of the air is a function closely related to the air temperature, pressure, and humidity. Notably, as the contribution of the air humidity on its refractive index is less than 1% [22], it is thus not taken into the consideration in this study. In the wavelength range from 0.3 µm to 1.69 µm, the relationship between the air refraction index and temperature, pressure, and the wavelength could be expressed in Eq. (1) [23].
The applicable temperature range is -40℃∼ 100℃, and the pressure ranges from 80 to 120 kPa. P, T, $\lambda $ denote the local atmospheric pressure (unit: Pa), the air temperature (unit: K), the wavelength (unit: μm), respectively. The refractive index of air could be easily calculated according to the temperature field of NC.
2.3 Transient behaviors of the WFE
The NC’s uneven temperature field was used to calculate the WFE at different evolution time, and then the transient behaviors of the WFE could be obtained. The incident light for the telescope mirror was set as a square light beam with a wavefront of 30 × 30 cm2, which was perpendicularly incident on the mirror surface (Fig. 3(a)). The length of the square light beam was expressed as H.
Fig. 3. Schematic diagram of square light beam incident on the mirror surface perpendicularly (a). Schematic diagram of discrete sampling of the square light beam (b). Schematic diagram of ray segmentation and temperature values for central area rays (c).
The square light beam was sampled at an equal interval of 1 cm, which would give 961 (31×31) rays, expressed as $\textrm{L}{\textrm{R}_\textrm{i}}$(i = 1, 2, 3, …, 961) (Fig. 3(b)). The individual ray was separated into N segments of different lengths, expressed by $\textrm{LR}{\textrm{S}_{\textrm{ij}}}$(i = 1, 2, 3, …, 961; j = 1, 2, 3, …, N). The lengths of segments were expressed as ${L_j}$ (j = 1,2,3,…, N) (Fig. 3(c)). The lengths of ${L_j}$ (j = 1, 2, 3, …, 100) were 1 mm, and the others were 1 cm. Given the tiny air temperature difference of each segment, the temperature at the segment’s endpoint was regarded as its average temperature, which was expressed as ${T_{ij}}$(i = 1, 2, 3, …, 961; j = 1, 2, 3, …, N). Meanwhile, the air refractive index at each segment, denoted as ${n_{ij}}$, which could be obtained by bringing ${T_{ij}}$ into Eq. (1). The OPL of the respective ray, denoted as $OP{L_i}$(i = 1,2,3,…961), was written as follows:
The wavefront (denoted as ${W_r}$) of square light beam was obtained by arranging the OPL of the respective ray according to its position. The value of WFE, denoted as ${W_e}$, was written as:
where ${W_i}$ was the wavefront of the square light beam above the mirror at the air temperature of 20°C.The time-dependent WFE caused by the NC’s uneven temperature fields could be obtained using the above method. Fourier transform was employed to obtain the frequency characteristics for the WFE. The quantitative evaluation of transient behaviors and frequency characteristics of the WFE could be used as a complement to the traditional mirror seeing evaluation to provide more insights on the effect of NC on image quality. It was also beneficial to analyze the capability of active optics and adaptive optics to correct WFEs caused by NC.
2.4 Transient behaviors of RMS WFE
The RMS WFE was adopted to evaluate the image quality degradation caused by NC. The total number of all incident rays on the mirror was denoted as M. The calculation of the RMS WFE was presented as follows. The OPL of the respective ray incident on the mirror was denoted as $OP{L_k}$(k = 1, 2, 3, …, $\textrm{M}$), and average OPL, denoted as AOPL, was expressed as:
The difference between OPL of the respective ray and $AOPL$ was expressed as $\Delta O{P_k}$(k = 1,2,3,…,M):
The RMS WFE (${W_{rms}}$) was expressed as:
The average value, oscillation amplitude, and frequency characteristics of the RMS WFE could be calculated in accordance with the time-resolved RMS WFE caused by the NC.
2.5 Experimental validations
In order to validate our evaluation method, we have measured the transient air temperature and OPL caused by the NC above the mirror and compared them to the numerical calculation results. Here, we should point it out that to measure the entire temperature field of the NC is challenging. Therefore, the air temperature at two separate locations above the mirror surface was measured without loss of generality, using the omg-usb-temp-5203 temperature collector and the coco-002 temperature sensor with a diameter of 50 µm. These devices could give the temperature measurement frequency and accuracy of 2 Hz and ±0.2℃, respectively. Figure 4(a) presents the schematic diagram depicting the air temperature measurement setup. The mirror made of red copper with a diameter of 30 cm and a thickness of 1 cm was heated for use to obtain the actual temperature of the NC. Mirror and square heat-insulating plate with a hole in the middle were placed on the stable temperature platform. Besides the thermal sensors mentioned above to measure the air temperature at two heights above the mirror, another two temperature sensors were employed to monitor the mirror temperature and surrounding air temperature.
Fig. 4. Schematic diagrams of temperature measurement experiment (a) and OPL measurement experiment (b).
The structural evolution of the NC would lead to rapid changes in WFEs. Due to the limitation of the low response speed, wavefront detection instruments commonly used to measure the surface shape of optical elements (e.g., Zygo interferometer and 4D interferometer) still was not able capture the rapidly changing WFE caused by NC. The changes in WFE essentially leads to the change in the OPL of light incidence on the mirror. Without loss of generality, the measured results of the central ray’s OPL were compared with the calculated results. Figure 4(b) shows the schematic diagram of the OPL measurement experiment. Laser interferometer IDS3010 was employed to measure the OPL, which was equipped with a laser probe and a reflector to emit and reflect laser ray, respectively. The IDS3010 had an operating wavelength of 1530 nm, a measurement accuracy of 2 nm, and a measurement period of 80 ms. The laser probe was placed above the mirror's center. The distance between the laser probe and the mirror was denoted as H.
3. Results and discussion
3.1 Transient CFD simulation of NC
Figure 5 depicts temporal temperature field snapshot of NC at a designed mirror temperature, ambient temperature and wind speed of 28℃, 20℃ and 0 m/s, respectively.
A thermal boundary layer was first generated on the mirror surface. Very shortly after, the periphery of the thermal boundary layer would break in about 2∼3 s. The stability loss of the thermal boundary layer from the periphery toward the central region precisely parallels the observed stability loss process of thermal boundary layer induced by a heated copper plate [24]. The heat was injected into the upper air through the convection heat transfer. After about 12 s, a typical NC of mushroom shape (also known as a natural convective plume) with a coronal cap was formed. The structure of the NC is similar to that of buoyancy plume generated by a heat source [25]. In principle, the typical evolution process of the NC could be assigned into an initial stage and a main stage, as shown in Fig. 6.
The initial stage was rooted in the thermal boundary layer, where the air received heat from the underneath mirror. The heated air would be driven to go upwards by the strong buoyancy and continued to entrain the surrounding air, by which the section of the initial stage tended to shrink subsequently. With the continuous entrainment of the surrounding air, the temperature difference between the NC and the ambient air decreased gradually, so the main stage with a ‘coronal cap’ was steadily formed. As the coronal cap floated upwards and got far away from the mirror, and the temperature rise of the coronal cap would fade away slowly, during which a new coronal cap was formed under the old one. The life cycle of a coronal cap in the main stage was almost same, with an interval of about 3.2 s, as presented in the boxed areas of A/B/C/D/E in Fig. 5. The exfoliated coronal cap would gradually break into smaller pieces and finally merged into the surrounding air, reaching a maximum height of nearly 1 m above the mirror. The arrows in Fig. 5 indicate the complete evolution process of a coronal cap. The presence of the periodic coronal caps would lead to significant periodic changes in the temperature field of the NC, whereas its velocity field would also exhibit an oscillatory behavior, which is similar to the periodic oscillatory motion of the axisymmetric thermal plume obtained by Direct Numerical Simulation [26]. The periodic oscillatory motion of plume is known as the “puffing” phenomenon. Puffing has been observed experimentally [27–31] and studied through numerical simulations previously [32–35]. However, the mechanism of the puffing phenomenon has not been fully recognized. Instantaneous velocity vectors in Fig. 7 demonstrate the air motion of NC above the hot telescope mirror. The center of NC is the ascent airflow, and the outer side is the vortex ring.
Fig. 7. Snapshot of the instantaneous velocity vectors of NC: 12 s (left), 16 s (middle), and 20 s (right).
Velocity vectors in Fig. 7 have shown a clear scenario that the vortex ring entrained the surrounding air into the center, and the NC has different vortex ring intensity at different evolution period of time. The CFD simulation results for the vortex ring structure are similar to the previous measurement results [24] and numerical simulation results [36] about the vortex ring. Moreover, the initial stage of the NC undergoes periodic expansion and shrinkage, as demonstrated in Fig. 5, which is a consequence of the radial inward flow induced by the vortex ring during entrainment [26]. The CFD simulation in this work demonstrated that the large-scale air motions and air temperature variation of the NC obviously did not conform to the isotropic Kolmogorov turbulence, whereby traditional mirror seeing parameters based on the Kolmogorov turbulence model cannot fully evaluate the effect of the NC on telescope image quality. The NC generated above the mirror with a temperature rise of 8 degrees in the air at 20℃ was termed “NC8” for convenience.
3.2 Transient behaviors of WFEs
Figure 8(a)–(d) present the 3D WFEs caused by NC8 at 0.8, 4.8, 8.8 and 12.8 sec, respectively.
Fig. 8. WFEs caused by the NC8 at (a) 0.8 second, (b) 4.8 sec, (c) 8.8 sec, (d) 12.8 sec, respectively.
As depicted in Fig. 5 and Fig. 8, a uniform thermal boundary layer emerged on the mirror surface at 0.8 sec, and the OPD at the plate bottom is about -29.7 nm. The depth of the ‘pan’-shaped WFE was getting deeper at 4.8 s. As the NC evolves, the shapes of the WFE become more complicated, resulting in a larger OPD, with a maximum OPD of 169.4 nm at 8.8 s. NC evolved into a typical structure with a coronal cap at 12.8 s, and the WFE gradually evolved into an inverted cone. The OPD at the bottom of the inverted cone is about -380.2 nm at 12.8 sec. According to the simulation results, the air temperature in NC above hot telescope mirror was higher than surrounding air. Given by Eq. (1), the wavefront of the incident beam after passing through NC was smaller than ideal wavefront in the air at 20℃. Thus, the WFE would have a negative value. A closer examination of the transient behaviors of the WFE caused by NC8 was performed based on the tracing of the calculated OPD of the central ray, as shown in Fig. 9(a).
Fig. 9. (a) Time-dependent OPD of the central ray of NC8, and (b) the corresponding Fourier spectra of the inset of (a). WFEs caused by NC8 at second oscillation: (c) 14.16 sec, (d) 16.00 sec, (e) 17.28 sec, respectively.
The OPD represents the OPL difference between the central ray passing through the NC in contrast to passing through the static state air at 20℃. The air temperature of the NC was higher than 20℃, so the OPD was a negative value. At the condition of NC8, an oscillatory characteristic for the OPD could be clearly identified, with the OPD promptly decreasing from zero to - 513 nm at 11.36 sec. It rebounded to – 380 nm at 12.8 sec, and then further dropped to – 680 nm, thus forming the first oscillation with a peak-to-valley height (PVH) of 300 nm. After the second oscillation from 14.16 sec to 17.28 sec with a weakened PVH of 207 nm, it entered a plateau phase with periodical stable oscillatory behaviors. In the whole duration time of 60 sec, there were 15 significant large-amplitude dominant oscillations, which are called the main oscillation in this work. The inset of Fig. 9(a) depicts the zero-averaged OPD evolution with time in boxed region. The ideal OPL of central ray at 20℃ air is constant, so the zero averaged central ray’s OPD in the inset represents the oscillations of the calculated OPL around the zero-mean. The oscillatory behavior of the OPD of the central ray represents the oscillatory behavior of the WFE caused by NC8. Figure 9(c)–(e) depict the front views of the WFEs at 14.16, 16.00, and 17.28 sec, respectively, as highlighted by the orange circles in Fig. 9(a). It clearly shows that the two oscillatory behaviors of OPD and WFE refer to one-to-one correspondence. In general, after a few circles of oscillations, the WFEs caused by NC8 in the plateau phase would quickly stabilize, ending up an apparent oscillatory behavior with a maximal PVH of nearly 100 nm. Figure 9(b) illustrates the corresponding Fourier spectrum obtained from the OPD time history in the inset. The dominant frequency of oscillatory behavior of WFE was about 0.32 Hz, which is equal to the generation frequency of the coronal cap. The above results clearly verified that the oscillatory behavior of the WFE is dominated by the puffing phenomenon. The Fourier spectrum of Fig. 9(b) suggests that 4 individual frequencies below 1 Hz with conspicuous amplitude can be identified, while high-frequency oscillations above 1 Hz are significantly weaker. High-frequency oscillations are coupled with the main oscillations, as marked in the four boxed regions of the insets of Fig. 9(a).
As discussed before, solar telescope mirrors will be heated by solar radiation, and many NCs will appear above many solar telescope mirrors. The maximum OPD of total WFE caused by all NCs refers to the sum of the OPD caused by each NC. The deformation (several microns) of the deformable mirror for adaptive optics will be largely consumed by these NCs, especially in a windless environment, thus reducing the capability of adaptive optics to eliminate the effect of atmospheric turbulence on image quality. According to the calculated results shown in Fig. 9, the conspicuous oscillation frequencies of the main oscillation were concentrated in the calibration frequency range (0.01∼1 Hz) of the active optics. Thus, active optics holds the potential to correct the main oscillations caused by the NC-related WFE.
3.3 Transient behavior of RMS WFE
The transient behavior of RMS WFE can be adopted to evaluate the overall effect of the NC on the image quality degradation. Figure 10(a) depicts the time history of the RMS WFE caused by NC8.
At the presence of NC, the RMS WFE increased rapidly. This RMS WFE of NC8 reached the maximum of 196 nm at 25.28 sec and then entered a plateau phase with an oscillatory behavior. The PVH of the oscillatory behavior was primarily in the range of 8∼10 nm. The average RMS WFE of NC8 during the plateau phase was determined to be 193.44 nm. With the same method, we calculated the average RMS WFE as the temperature increases and mirror diameters, as presented in Table 2. The formula in regard to the average RMS WFE (${R_{MS}}$), temperature rise ($\Delta T$), and mirror diameter ($L$) is obtained by a nonlinear fitting method, which is expressed as follows:
Table 2. Average RMS WFE for varying temperature rises and mirror diameters
The $a$, $b$, and $c$ values in Table 3 are the point estimates for the coefficients of the Eq. (7), and the interval range is the interval estimate of the 95% confidence interval. RMSE is the root mean square value of the error between the 12 average RMS WFEs and the 12 fitted values obtained by bringing the point estimate into the Eq. (7). R-square is the coefficient of determination.
Table 3. Coefficients of nonlinear fitting
The coefficients from the nonlinear fitting indicates that the average RMS WFE is dependent on both the increase in mirror temperature and the mirror diameter. Particularly, the average RMS WFE is proportional to ∼$\Delta {T^{0.861}}$ and ∼${L^{0.288}}$.
The inset of Fig. 10(a) shows the zero averaged RMS variations over time as highlighted in the boxed region. As shown in Fig. 10(b), the dominant frequency was 0.32 Hz, corresponding to the generation frequency of the coronal cap. The main frequencies were below 1 Hz, similar to those of central ray’s OPD (see, Fig. 9). Using the same method, we calculated the Fourier spectrum of RMS’ plateau phase for a 90 cm diameter mirror as the temperature rises, as displayed in Fig. 11.
The results reveal that the main frequencies of RMS WFE, caused by NC generation above a larger mirror, also fall below 1 Hz. In principle, with larger mirror and less temperature rise, the generation frequency of the coronal cap of mirror’s NC would be reduced. It further confirms that active optics is able to correct the NC-related WFE of large telescope. In fact, the error budget system based on RMS WFE of all optical elements has been extensively used in the optical system design of astronomical telescope. Significantly, our calculation results have indicated the RMS WFE caused by NC is not negligible. RMS WFE caused by NC effect may be incorporated into the error budget system to optimize the error allocation in the machining and assembly process of the optical elements for the solar telescope.
3.4 Experimental validations
We first compared the CFD simulated and measured results for the air temperature change at the same location in the NC8. Figure 12(a) presents the time history of the CFD simulated air temperature of NC8 at 5/6 cm above the mirror center.
Fig. 12. (a) Air temperature varying with time at the 5/6 cm above the mirror center, (b) the corresponding Fourier spectrum of the boxed region in (a). (c) is the monitored temporal air temperature at 5/6 cm above the mirror center, and the Fourier spectra are illustrated in (b).
With the appearance of the NC8, the air temperatures increased rapidly from 7 s and entered a plateau phase at nearly 18 s. The air temperatures were nearly 3 to 4 ℃ higher than the ambient temperature. The air temperature would increase as the distance getting closer to the mirror. The air temperatures exhibited apparent oscillatory behaviors, and its PVH values primarily ranged from 0.2 to 1 ℃. Figure 12(b) illustrates the Fourier spectrum of the temperature change over time in the boxed region of Fig. 12(a). The frequency with the maximum relative amplitude was also 0.32 Hz, equal to the generation frequency of the coronal cap of NC8, suggesting that the oscillatory behavior of air temperature was also derived from the puffing. The oscillation frequencies with conspicuous amplitudes were lower than 1 Hz, similar to the Fourier spectra of the central ray’ OPD in Fig. 9(b). Furthermore, the relative amplitude above 1 Hz is not as significant.
Figure 12 (c) shows the measured transient air temperatures and the corresponding Fourier spectra presented in Fig. 12(d). The measured air temperature at the two locations are nearly 2 to 4 ℃ higher than the ambient temperature. Similarly, the air temperatures are also found to exhibit the oscillatory behaviors. The Fourier spectra are more complicated as compared to the simulated results, and no apparent prominent frequency could be distinguished, which is not completely consistent with the calculated results. However, oscillations with larger amplitudes predominantly occur in the region with frequencies lower than 0.4 Hz, similar to the simulated results as shown in the inset of Fig. 12(b). The oscillatory behaviors of air temperature caused by the puffing can be used to evaluate the presence of natural convection above the mirror. Figure 13 presents the CFD simulated temporal air temperature at 1/5/10 cm above the center of a 30 cm diameter mirror at temperature rises of 0.5 ℃ and 1°C, respectively.
Fig. 13. (a) Air temperature varying with time at the 1/5/10 cm above a 30 cm diameter mirror with a temperature rise of 0.5 ℃ (a) and 1℃ (b).
For a temperature rise of 1°C, the air temperatures exhibit apparent oscillatory behaviors, and its PVH values primarily range from 0.1 to 0.3 ℃. When the temperature rise is reduced to below 0.5°C, no oscillatory behaviors of air temperatures in the plateau phase are observed anymore. The absence of oscillatory behavior in air temperature indicates that the hot airflow above the mirror surface is no longer dominated by natural convection. This suggests the impact of natural convection on the image quality degradation becomes negligible under such conditions.
Temperature measurements at different heights have shown that the air temperature at 110 cm above the center of the mirror would be no longer affected by the NC8. Therefore, the distance between the laser probe and the mirror was set to be 110 cm. To show the oscillatory behaviors of OPL more clearly, the measured data of OPL were zero-averaged, as shown in Fig. 14(a).
Fig. 14. Zero averaging of the measured OPL time history of NC8 (a) and the corresponding Fourier transform spectrum (b). (c) Enlarged view of the boxed region in (b).
The measured results show that the PVH values of the main oscillations primarily ranged from 50 to 150 nm, and a number of minor oscillations are coupled with the main oscillations. Some typical minor oscillations are shown in the boxed regions of Fig. 14(a). The PVH values of the minor oscillation are mainly concentrated in tens of nanometers. Compared with the calculation results, the Fourier spectrum of measured OPL is much more complicated. However, the measured results were similar to the calculated results in terms of the outline of frequency distribution and the low frequency components of the main oscillation. The dominant oscillations mainly fall in the low-frequency region below 1 Hz, which is consistent with the calculated results. The frequency of the minor oscillation is much higher than the maximum calibration frequency (1 Hz) of the active optics. No predominant frequency component of the minor oscillation is found in the Fourier spectrum in Fig. 14(b). The oscillation amplitude of the minor oscillation is relatively small, adaptive optics can be used to improve the image quality degradation caused by the minor oscillation of WFE. The numbers of minor oscillations of measured results were much larger than the calculated results. The reason for the reduction of minor oscillations in the calculation results is due to the fact that LES has made moderate simplifications in the calculations for small-scale air motions.
As discussed before, the main oscillation derives from the large-scale air motion caused by the puffing. We believe that the minor oscillation may be derived from the self-turbulization. As confirmed in Fig. 5, the exfoliated structure of coronal cap is broken into small fragments step by step, which would continuously evolve, irregularly breaks into tiny pieces and then merges into the surrounding air. The self-turbulization is the transition from the NC’s main stage to the mature air turbulence caused by the inherent instability of NC, which is originated by the mixing of warm air in NC and the surrounding air, the enhancement of the diffusion effect, as well as the disappearance of the buoyancy driving effect. The temperature oscillation amplitude of these small fragments is lower than that of NC’s puffing, and these small fragments would show different motion directions and temperature gradients. Therefore, the oscillation amplitude of the OPL and WFE caused by small fragments, i.e., minor oscillation, will be smaller than the main oscillation caused by puffing. The air motion and temperature variation of small fragments are fairly isotropic, satisfying the homogeneous isotropic Kolmogorov turbulence model to some extent. Therefore, the effect of small fragments on image quality can be evaluated by mirror seeing parameters. However, considering that large-scale convective air motion is dominant in NC above the hot telescope mirror, it is inaccurate to evaluate the image quality degradation caused by NC only by mirror seeing parameters.
4. Conclusions
In this work, we have proposed a method to quantitatively calculate the transient behaviors and frequency characteristics of the NC-related WFE. The transient CFD simulation based on LES is adopted to obtain the temperature fields of the NC at different evolution time, and then the WFE and its RMS value are calculated by discrete sampling and ray segmentation. Our results demonstrated that the large-scale air motions and temperature variation of the NC above telescope mirror obviously do not conform to the isotropic Kolmogorov turbulence model. Astronomical seeing parameters based on the Kolmogorov turbulence model cannot fully evaluate the effect of NC on telescope image quality. The method proposed in this study is a supplement to the mirror seeing evaluation. When the temperature of a 30 cm mirror is 8 degrees higher than the ambient temperature in a windless environment, the shape of NC-related WFE evolves from a “pan” to an inverted cone, and then enters a plateau phase with oscillatory behaviors. The oscillatory behavior of NC-related WFE consists of the main oscillation caused by puffing and the minor oscillation with high-frequencies caused by the self-turbulization. The conspicuous oscillation frequencies of the main oscillation caused by heated mirrors at varied diameters are mainly below 1 Hz, and its oscillation amplitude can exceed 100 nanometers. Active optics can be used to correct the main oscillations, while the minor oscillations can be corrected by adaptive optics. The RMS WFE caused by NC is not negligible, and it probably should be integrated into error budget system of solar telescope to optimize the error allocation of machining errors and assembly errors of the optical elements. It is important to note that the RMS WFE is not solely related to temperature rise, but also to mirror diameter. A mathematical relationship has been derived to describe the correlation between RMS WFE, temperature rise, and mirror diameter, which can be used to quantitatively evaluate image quality degradation caused by heated telescope mirrors, potentially. The method proposed in this study can quantitatively evaluate the effect of opto-mechanical systems of ground-based solar telescope and high-power laser system on image (laser beam) quality and optimize the telescope/high-power laser system’s design of the opto-mechanical systems and thermal control systems.
Funding
National Natural Science Foundation of China (11673064, 11773069, U1931124).
Disclosures
The authors declare that there are no conflicts of interest related to this article.
Data availability
Data underlying the results presented in this paper may be obtained from the authors upon reasonable request.
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