Note that Adar II is the "real" Adar, the one in which Purim is celebrated, the one in which yahrzeits for Adar are observed, the one in which a 13-year-old born in Adar becomes a Bar Mitzvah. In the fourth century, Hillel II established a fixed calendar based on mathematical and astronomical calculations.This calendar, still in use, standardized the length of months and the addition of months over the course of a 19 year cycle, so that the lunar calendar realigns with the solar years.These three phenomena are independent of each other, so there is no direct correlation between them.
The civil calendar used by most of the world has abandoned any correlation between the moon cycles and the month, arbitrarily setting the length of months to 28, 30 or 31 days.
A few years ago, I was in a synagogue, and I overheard one man ask another, "When is Chanukkah this year?
" The other man smiled slyly and replied, "Same as always: the 25th of Kislev." This humorous comment makes an important point: the date of Jewish holidays does not change from year to year.
The months drift around the seasons on such a calendar: on a 12-month lunar calendar, the month of Nissan, which is supposed to occur in the Spring, would occur 11 days earlier in the season each year, eventually occurring in the Winter, the Fall, the Summer, and then the Spring again.
On a 13-month lunar calendar, the same thing would happen in the other direction, and faster.