## 3 Results and Discussion
### Analysis of Atmospheric Circulation  
In addition, atmospheric circulation serves as a direct factor affecting water vapor flux and water vapor flux divergence. This article also studied the characteristics of regional circulation changes during volcanic eruptions. Specifically, from Figure 5a, it can be seen that there was low pressure near 30\({}^{\circ}\) S on the south side of the 500 hPa Tonga volcano on 14 January. High pressure on the north side, corresponding to the low-level 850 hPa Tonga volcano, also had low pressure near 30\({}^{\circ}\) S on the south side (Figure 5b), while the high-pressure field was south of 30\({}^{\circ}\) S as well as on the north side of Tonga volcano, which was located on the north side of the cyclone with a prevailing northwest wind. The low pressure at 500 hPa on the south side of the volcano moved southward on 15 January (Figure 5c), and the volcano was covered by high pressure with solid northwesterly winds; the low pressure at 850 hPa on the south side of the volcano also moved southward (Figure 5d), while weak low pressure appeared on its northeast side. On the 16 th (Figure 5e), the  
Figure 3: Distribution of water vapor anomalies from 1 January 2022 to 24 January 2022; blue is the area of PWV anomaly decrease, and red is the area of PWV anomaly increase. Black triangle is Tonga’s position. 500 hpa volcano was located on the west side of the high pressure, and easterly winds prevailed over it. Figure 4: Water vapor differences at regional stations in Tonga for January and February 2022; red is positive, green is negative and colorless locations indicate missing data for that day. The red box is the HTHH volcano eruption time. The red box indicates the time period from 13 January to 16 January, when the Tonga volcano erupted. In contrast, the corresponding 850 hpa volcano was located on the south side of the low pressure (Figure 5f), and easterly winds prevailed, bringing warm advection. On the 17 th (Figure 5g), the 500 hpa volcano was still located on the west side of the high pressure, and easterly winds prevailed over it. In contrast, the corresponding 850 hpa volcano was  
Figure 5: The 850 hPa (**b**,**d**,**f**,**h**) and 500 hPa (**a**,**c**,**e**,**g**) temperature fields (color shaded areas, unit: \({}^{\circ}\)C) and wind fields (arrows, unit: m\(\cdot\)s\({}^{-1}\)) for January 2022. All red boxes indicate low pressure centers south of Tonga, and all green boxes indicate low pressure centers east of Tonga. The red and green boxes indicate changes in the two low pressure centers. located on the south side of the low pressure (Figure 5h), and easterly winds brought warm advection. Using the circulation data from 1981 to 2010 as the climate state, anomaly levels of 850 hPa and 500 hPa geopotential height field, temperature field, and wind field were obtained from 14 to 17 January 2022 (Figure 6). Figure 6: The 850 hPa (**b**,**d**,**f**,**h**) and 500 hPa (**a**,**c**,**e**,**g**) temperature anomaly fields (color shaded areas, unit: \({}^{\circ}\)C) and wind anomaly fields (arrows, unit: m\(\cdot\)s\({}^{-1}\)) for January 2022. Climate state is from 1981 to January 2010 data. The red box indicates the low pressure center on the south side of Tonga. The green box indicates the low pressure center on the east side of Tonga. From Figure 6, we can see a low-pressure anomaly on the southwest side of 500 hpa Tonga volcano on 14 January (Figure 6a) and a low-pressure anomaly on the corresponding lower 850 hpa (Figure 6b), with westerly winds prevailing over the volcano, bringing cold advection from the south. The wind field converged on the volcano's east side, and the convergence of cold and warm advection was conducive to precipitation.