On generalized vorticity-conservation laws

We present here a new type of generalization of the conservation law of circulation, in two dimensions, and of the helicity-conservation law in three dimensions assuming zero potential vorticity. The conserved quantities keep their usual expressions ∫∫ k · rot v d 2 r , ∫∫∫ v · rot v d 3 r , respectively, as in the barotropic case. These generalizations are based upon the general formulation of conservation laws in terms of potentials (here the Clebsch potentials). The conserved currents that we derive are expressible in terms of the ordinary physical variables ( r , t , v , P , ρ) only, as they should.