## 3 Results and Discussion
### Analysis of Water Vapor Flux Divergence and Water Vapor Flux  
Water vapor flux and water vapor flux divergence are the most direct types of evidence for predicting precipitation [43]. Therefore, this article also considers water vapor flux and water vapor flux divergence as feedback factors for changes in low-level atmospheric environmental characteristics. Among them, water vapor flux was introduced to quantify the direction and magnitude of water vapor transport and where it was concentrated. Watervapor flux divergence describes the amount of water vapor dissipated per unit time and per unit volume. Figure 7: Surface temperature map for Tonga, April 2019–April 2022. The black line depicts the volcano’s outline before it erupted on 15 January 2022. P1, P2, P3, and P4 are the feature points. In this paper, the water vapor flux, water vapor flux dispersion, and the anomaly level of water vapor flux dispersion and water vapor flux from 14 to 17 January 2022, were analyzed, where the climate state was the data from 14 to 17 January 1981 to 2010. As shown in Figure 9, a positive divergence of water vapor flux indicates that water vapor is diverging, while a negative divergence indicates that water vapor is converging. On 14 January (Figure 9a), there was a weak water vapor convergence zone over the Tonga volcano, and on 15 January (Figure 9b), this weak water vapor convergence zone strengthened due to increased water vapor transport from high latitudes. On the 16 th (Figure 9c), the Tonga volcano had a weak water vapor divergence zone over 850 hpa, and the northeast air flow inhibited the water vapor transport. On the 17 January (Figure 9d), the water vapor divergence increased, and the easterly airflow increased, inhibiting water vapor transport over the ocean at high latitudes. As shown in Figure 10, there was an anomalous area of water vapor irradiation over Tonga volcano on 14 January (Figure 10a), and the westerly airflow brought high-latitude water vapor transport, and this anomalous area of irradiation over Tonga volcano was strengthened on 15 January (Figure 10b). Water vapor transport was also strengthened at 850 hPa. On 16 January (Figure 10c), there was a weak anomalous area of water vapor irradiation over Tonga volcano, and the easterly airflow suppressed water vapor transport. On 17 January (Figure 10d), the anomalous area of water vapor irradiation was strengthened, and the easterly airflow was also strengthened. A low-pressure center is also known as a cyclone [44]. Figure 9 shows that the direction of water vapor transport largely coincides with the direction of atmospheric circulation [45; 46]. A large amount of water vapor is accumulated in the southern cyclone of Tonga, where most of the water vapor comes from the diversion of the low-pressure center in eastern Tonga. The intensity of water vapor transport in the southern cyclone reached the highest value on the second day after the eruption of the HTHH volcano (Figure 9c). On 15 January, the southern cyclone of Tonga moved toward New Zealand under the influence of the volcanic eruption (Figure 9b). At the same time, a low-pressure center in eastern Tonga began to build up and move towards Tonga, transporting large amounts of water vapor to Tonga. As shown in Figure 10, a westerly wind formed under the influence of the low-pressure center in southern and eastern Tonga. This westerly wind transported a large amount of water vapor to southern and eastern Tonga. The water vapor  
Figure 8: Time series curve of temperature at four characteristic points in Tonga, April 2019–April 2022. The dotted line indicates missing data for the month. transport intensity of the southern cyclone in Tonga reached the highest value on the 16 th (Figure 10c), consistent with the PWV results. The anomalous water vapor of the southern cyclone in Tonga on the 16 th was most likely from the water vapor released by the HTHH volcanic eruption [47].