The equations that describe a steady ideal-fluid motion have a first integral that expresses a functional relationship between the potential vorticity P, the density ϱ, and the Bernoulli function B: P = F(ϱ,B).
Welander, Pierre. 1971. "Some exact solutions to the equations describing an ideal-fluid thermocline." Journal of Marine Research 29, (2). https://elischolar.library.yale.edu/journal_of_marine_research/1202