1) please advise when tears come naturally during morning prayer or Minha or during reciting Tehhilim, is it forbidden on Shabbos? 2) what are the tears nature or source? should a person control it?

ב"ה Shalom From your short description as to the cause of your tears, it seems that they stemmed from an emotion of feeling close to Hashem and not due to sadness. Such crying is permitted on Shabbat as Rav Akiva did when he read Shir Hashirim on Shabbat . (שו"ע רפח, ב, ומשנ"ב ד) All the best and Chag same'ach