The process in which the anatomical structures of the dorsal aorta are generated and organized. The dorsal aorta is a blood vessel in a single-pass circulatory system that carries oxygenated blood from the gills to the rest of the body. In a single-pass circulatory system blood passes once through the heart to supply the body once. [ ZFA:0000014 GOC:bf http://purl.obolibrary.org/obo/UBERON_0005805 GOC:dgh http://en.wikipedia.org/wiki/Aorta ]