We report air temperature and humidity changes during the two solar eclipses of 26 December 2019, and of 21 June 2020, respectively, in the cities of Al-Hofuf and Riyadh in Saudi Arabia. During the December eclipse the Sun rose already eclipsed (91.53% of the area covered) while the June eclipse, although also annular in other places of the Arabian Peninsula, was just partial at Riyadh (area covered 72.80%). This difference apparently affected the observed response on the recorded variables of temperature, relative humidity (RH) and vapor pressure (VP) in the two events. Change in these variables went unnoticed for the first eclipse since it was within the natural variability of the day; yet for the other, they showed clearly some trend alterations, which we analyze and discuss. A decrease in temperature of 3.2 {deg}C was detected in Riyadh; however, RH and VP showed an oscillation that we explain in the light of a similar effect reported in other eclipses. We found a time lag of about 15 min measured from the eclipse central phase in this city. We made an inspection of related fluctuations and dynamics from the computed rates of the temporal variation of temperature and RH. Trying to identify the influence of solar eclipses in similar environments we have made a broad inter-comparison with other observations of these variables in the Near East, northern Africa and in the United States. We compare our results with results obtained by other authors working with the December eclipse but in the United Arab Emirates and Oman, which showed dissimilar results. These inter-comparisons show how effectively the lower atmosphere can respond to a solar eclipse within a desert environment and others similar. As a preamble, a historical revision of temperature and humidity in the context of eclipse meteorology is also included.
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