Growing mode of power-law plume similarity ($\det (\lambda I-J)={\lambda }^3+(q+2m+f){\lambda }^2+(3qm/2+qf+2mf)\lambda +qmf$)

The linearized relative-flux matrix is $J=[(-q,q/2,0);(m,-2m,m);(f,0,-f)]$. Its characteristic polynomial has constant term $qmf<0$ for physical positive-buoyancy similarity, so at least one real eigenvalue is positive. The growing mode behaves as $z^{\lambda }$ and makes the similarity trajectory unstable with increasing height, rather than with physical time.