 Weak hyperbolicity of the unsteady top-hat plume

For plume area $A$ and reduced gravity $g^{\prime }$, the principal matrix in variables $(A,w,g^{\prime })$ is $\left(\begin{array}{ccc}w& A& 0\\ 0& w& 0\\ 0& 0& w\end{array}\right)$. At $A>0$ it has a triple real speed $w$ but only two independent eigenvectors. Thus the model is weakly hyperbolic, not strongly or strictly hyperbolic. Real speeds alone do not provide a complete set of characteristic fields or a uniform high-frequency estimate. Shape factors representing nonuniform radial profiles can alter this degeneracy.