In a county organized under a charter adopted under Act No. 293 of the Public Acts of 1966, being sections 45.501 to 45.521 of the Michigan Compiled Laws, whose charter prescribes an elected county executive, and which county has a population of more than 2,000,000 at the time the charter is adopted, the powers and duties of the drain commissioner, under this act, shall be performed by a person or persons designated in accordance with the county's charter.