Here you go, pdq...

"The Himyar and Berber proselytes are overshadowed by the Khazars, who ruled from the fourth century, along the Volga, and in the eastern Ukraine, the Crimean peninsula and modern Georgia. The Silk Road and the Don and Volga gave the kingdom a rich and flourishing trade, and means for a powerful army. While the "spoken Khazar language consisted of Hunnic-Bulgarian dialects with others from the Turkic family," "[t]here is no doubt…that the Khazars' sacred tongue and written communication was Hebrew." The Khazars converted over time between the mid-eighth and mid-ninth centuries, and for the same reason "that accounted for Himyar's conversion…[t]he desire to remain independent in the face of mighty, grasping empires…Had the Khazars adopted Islam…they would have become subjects of the caliph. Had they remained pagan, they would have been marked for annihilation by the Muslims…Christianity, of course, would have subjected them to the Eastern Empire." Conversion began with the elite and included the larger population over time. Khazar Judaism was substantially rabbinical, though Kairate Judaism, a Protestant-like sect which regards the Hebrew Bible as solely authoritative, may have flourished also."

http://theglobalrealm.com/2010/02/06/the-invention-of-the-jewish-people/

By the way, the Khazars are not the only non-Jewish population that converted en mass to Judaism. There are others did as well, and they are discussed in that article. So many of the other Jews who are migrating to Israel also do not have Middle Eastern origins.