Integrals of the vorticity equation. Part II: Special two-dimensional flows

In Part 1. a general integral of the 2D vorticity equation was obtained. This is a formal solution for the vorticity of a moving tube of air in a 2D unsteady stratified shear flow with friction. This formula is specialized here to various types of 2D flow. For steady inviscid flow, the integral reduces to an integral found by Moncrieff and Green if the flow is Boussinesq and to one obtained by Lilly if the flow is isentropic. For steady isentropic frictionless motion of clear air, several quantities that are invariant along streamlines are found. These invariants provide another way to obtain Lilly's integral from the general integral.