First, I do thank you for your service! Does it snow in Jerusalem? My understanding of the Torah places the Passover festival at the beginning of March. The law requiring the whole of Israel to congregate in Jerusalem three times a year is problematic from my point of view in more ways than one, but...does it snow in Jerusalem in March? Thank you again for your availability.
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Yes, it snows in Jerusalem. However, it does not snow each year, and when it does it is generally a light and fleeting snowfall. The sun melts it during the day in nearly every case.
There have been years when there has been a heavier snow that has closed things down for a number of days, but the Jerusalem Municipality (and the residents!) has learned to prepare well for such an event.
Passover is supposed to be in the Spring according to the Torah. Since the Jewish calendar is lunar-based and therefore 354.5 days long, the calendar has been established to include a leap year (an extra month before Passover) seven times in every 19-year cycle. In this manner, in theory, Passover will be celebrated in a month that is neither too cold or too warm.
I don't ever recall there being snow during Passover, or hearing about this, but I agree it could be somewhat difficult to ascend to Jerusalem on snowy paths! If it would not be possible, the people would be exempt from the mitzvah of bringing a Passover sacrifice and ascending to the Temple, until the weather permitted. By the way, in the time of the Temple, the decision of whether to add an extra month to the year, and therefore to determine when Passover would be that year, was made by the members of the highest Jewish religious court - the Sanhedrin - and they would use their wisdom make sure that Passover would be celebrated in good weather.