The authors study different ways to partition a quasi-geostrophic flow.

When relative vorticity is zero, the streamfunction decomposes into a term for the solution with homogeneous boundary conditions and a term due to the boundary conditions. These two term both go to infinity for large scale while their sum remains finite (Section 2b; see also Lau and Holopainen 1984). This reminds me of the Ekman flow and the geostrophic flow that both go to infinity toward the equator, canceling each other keeping their sum finite.

In their Section 3, they decompose numerically an idealized example with a realistic stratification. Because the example is not in thermal wind balance, the vorticity is zero although the example was constructed with zero vorticity in mind. They write that the fraction of PV that is converted from the stretching term to the relative vorticity term depends primarily on the horizontal scale of the wave.