Merger of two equal pure plumes ($Q_c=2Q,M_c=2M,F_c=2F,{\Gamma }_c=\sqrt{2}$)

If two equal pure plumes merge while keeping their common velocity and reduced gravity, conservation of cross-sectional area gives radius $\sqrt{2}$ times the individual radius. All three integral fluxes double. The plume balance parameter increases by $2^3/2^{5/2}=\sqrt{2}$, so the immediate combined plume is a lazy plume. Similarity attraction may restore pure balance farther downstream, but instantaneous pure balance does not follow from adding two pure plumes.

For a merged pair held at the original velocity and reduced gravity, pure plume balance requires the original individual radius, not the larger area-preserving radius. The required pure area is half the combined area. Shrinking to that area at unchanged velocity and reduced gravity would halve volume, momentum and buoyancy fluxes, contradicting conservation of the merged fluxes. A conservative merger must change its properties or begin away from pure balance.