This paper studies the vulnerability of the worldwide air transportation network (WATN) during a global catastrophe such as COVID-19. Considering the WATN as a weighted network, many airport connections could be completely or partially disrupted during such extreme events. However, it is found that existing weighted metrics cannot reflect the impact of connection capacity reduction on network connectivity. Therein, this work proposes a novel network efficiency metric termed as layered weighted network efficiency (LWNE) metric to measure the connectivity of the air transportation networks (ATNs) and study their vulnerability in response to different levels of disruptions, including airport level, country level, and global level. The most critical airport connections and their impact on network connectivity are identified. It is found that the critical connections are mostly between so-called bridge airports but not core airports in the WATN. By examining the impact of partial link disruptions, it is found that some connections mainly serve local travel demand and are very robust to partial disruptions, while the others connecting global hubs are sensitive to partial disruptions. Further, the WATN is robust to the individual disconnection of most countries; however, it is vulnerable to the simultaneous disconnection of countries that serve international transfers. Interestingly, the WATN is insensitive to the disconnection between any two countries, even those with sizeable domestic ATNs. Concerning global disconnections, as long as all the international connections hold 10% of their original flights, the WATN can still expect 40% of its pre-disruption performance. This paper deepens the understanding of ATNs under extreme events and provides a method for studying transportation networks' vulnerability facing global disruptions.