From AMS Glossary
The state of a fluid that is characterized by both barotropy and piezotropy, that is, pressure and density surfaces coincide.
This condition guarantees that the fluid will remain barotropic at all future times. Example: an ideal homentropic fluid (i.e., a fluid with homogeneous potential temperature) with uniform composition.
Gill, A. E. 1982. Atmosphere–Ocean Dynamics. Academic Press, . p. 228.