A Relationship between the Bernoulli Function and Potential Vorticity

Abstract

In 2-dimensional steady adiabatic unforced systems, both the Bernoulli function B and potential vorticity Q are functions of the stream function ψ and Q is the derivative of B with respect to ψ, i.e., Q = dB/dψ. In this note, this formula is extended to 3-dimensional steady adiabatic systems. Even if there exists dissipation, essentially the same formula holds as far as the inner product between the dissipative force and the velocity is everywhere negative. An application is presented to the problem of lee vorticity.